“Is that to say we are against Free Trade? No, we are for Free Trade, because by Free Trade all economical laws, with their most astounding contradictions, will act upon a larger scale, upon the territory of the whole earth; and because from the uniting of all these contradictions in a single group, where they will stand face to face, will result the struggle which will itself eventuate in the emancipation of the proletariat.”
Karl Heinrich Marx · Marx-Engels Collected Works, Vol. VI, p. 290
When calculus is first studied, the integral usually appears as a procedure for accumulating quantities over an interval. Later, the same idea is extended: integration is no longer carried out only along an axis, but also over curves, regions of the plane, volumes, and surfaces. Line integrals, multiple integrals, and surface integrals are precisely some of the forms taken by this generalization.
In general terms, line integrals allow us to accumulate a quantity along a curve. Multiple integrals allow us to do so over regions of two or more dimensions. Surface integrals, in turn, allow us to integrate over a surface located in space. At the level of university calculus, this distinction is enough to understand why each kind of integral requires different techniques while keeping in view that all of them belong to the same mathematical idea: infinitesimally summing a magnitude over a given domain.
The main theorems associated with these integrals are also related. For line integrals there is the Fundamental Theorem for Line Integrals; when studying multiple integrals one encounters Fubini’s Theorem, the Pappus–Guldinus Theorems, and Green’s Theorem; and in the study of vector fields on surfaces one encounters Stokes’ Theorem and Gauss’ Theorem, also known as the Divergence Theorem.
The fundamental difference does not lie in the idea of integration itself, but in the geometric object over which accumulation is carried out.
Green, Stokes, and Gauss are not literally the same theorem in an elementary calculus course, but they express a common structure: they relate an integral carried out in the interior of a region to another integral carried out over its boundary. In a more general treatment, this unity is formalized through the generalized Stokes theorem; however, for an undergraduate course it is enough to understand them as different manifestations of the same principle.
Line Integrals
A line integral is an integral whose domain of integration is a curve. Instead of moving through the points of an interval on the real axis, one moves through the points of a path that may lie in the plane or in space. To perform the calculation, the usual procedure is to parameterize the curve.
To parameterize means to describe the coordinates of every point on the curve through an auxiliary variable, usually \(t\). In the case of a straight line, the parameterization may take a simple form such as \(x=c+kt\). For a general curve, however, it is more appropriate to think in terms of expressions such as \(x=x(t)\), \(y=y(t)\), and, when necessary, \(z=z(t)\). The procedure therefore does not require a single constant direction vector; that feature belongs to the particular case of a straight line.
FIGURE 1 · A scalar line integral accumulates the values of a function along a curve. Original source indicated in the article: Greg School.
There are two especially important cases. In a line integral of a scalar function, a magnitude is accumulated along the curve. If a wire has variable density, for example, a line integral can be used to obtain its total mass. In a line integral of a vector field, by contrast, one is usually interested in the effect of the field along the displacement: the classical example is the work done by a force along a path.
Scalar line integral: ∫C f ds
Vector line integral: ∫C F · dr
FIGURE 2 · Different kinds of line integrals: integration with respect to arc length and integration of vector fields along a path. Original source indicated in the article: YouTube.
When the curve is closed, the symbol \(\oint\) is often used. The circle drawn over the integral sign indicates precisely that the path returns to its starting point. This detail is particularly important when studying circulation, because the integral is taken around the entire closed boundary.
The Fundamental Theorem for Line Integrals
The Fundamental Theorem for Line Integrals plays a role analogous to the Fundamental Theorem of Calculus in one variable. If a vector field is conservative and can be written as the gradient of a potential function, then the integral between two points depends only on the initial point and the final point, not on the particular path followed from one to the other.
If F = ∇φ, then
∫C F · dr = φ(B) − φ(A)
This is one of the first indications of an idea that will appear repeatedly: certain complicated integrals can be transformed into simpler expressions when the geometric structure of the problem allows the appropriate theorem to be applied.
Multiple Integrals
Multiple integrals generalize the process of integration to domains involving more than one variable. A double integral is carried out over a two-dimensional region and a triple integral over a three-dimensional region. Their geometric interpretation depends on what is being integrated.
For example, if the constant function \(1\) is integrated over a region of the plane, the double integral gives the area of that region. If a nonnegative function \(f(x,y)\) is integrated over a region \(R\), the integral may be interpreted as the volume lying below the surface \(z=f(x,y)\) and above \(R\). Likewise, a triple integral of \(1\) over a region of space gives its volume. One should therefore not mechanically identify “double integral” with “volume” or “triple integral” with “hypervolume”: the meaning depends on the integrand and on the domain.
FIGURE 3 · Interpretation of a double integral as the accumulation of infinitesimal sections over a region. Original source indicated in the article: AlgebraHD.
Fubini’s Theorem
Fubini’s Theorem is fundamental because, under the usual conditions studied in calculus, it allows a multiple integral to be computed by means of iterated integrals. In practical terms, this means integrating first with respect to one variable and then with respect to another. In triple integrals there are as many as six possible orders of integration, and choosing a suitable order can simplify the calculation considerably.
∬R f(x,y) dA
may be computed, depending on the region, as
∫ [ ∫ f(x,y) dy ] dx
or as
∫ [ ∫ f(x,y) dx ] dy
Conceptually, Fubini’s principle is not limited to triple integrals. It is a general result concerning integration on product spaces. In an undergraduate calculus course, however, it is most often applied to double and triple integrals, which are also the cases that can be represented geometrically most easily.
FIGURE 4 · A triple integral can be set up using different orders of integration. The geometry of the region determines which order is most convenient. Original source indicated in the article: SlidePlayer.
The Pappus–Guldinus Theorems
The two Pappus–Guldinus theorems relate centroids to objects generated by revolution. Intuitively, both say that when a figure rotates around an axis, a geometric magnitude generated by that rotation can be calculated by multiplying the original magnitude by the distance traveled by its centroid.
The first theorem relates the length of a plane curve to the area of the surface of revolution generated by that curve. The second theorem relates the area of a plane region to the volume of the solid of revolution generated when it is rotated around an axis that does not intersect the region. This is the relevant formulation of the two classical theorems in calculus.
Green’s Theorem
Green’s Theorem establishes a relation between a double integral over a plane region and a line integral along the closed curve forming its boundary. If \(C\) is the positively oriented boundary of a region \(R\), then, under the usual hypotheses:
∮C P dx + Q dy
=
∬R (∂Q/∂x − ∂P/∂y) dA
The conceptual importance of the theorem is greater than the formula itself. Green’s Theorem allows an accumulation performed around a boundary to be replaced by one performed throughout the interior, or vice versa. It therefore forms a natural bridge between line integrals and double integrals.
Surface Integrals
A surface integral is an integral carried out over a surface located in space. Here it is useful to distinguish two cases, in a way analogous to what occurred with line integrals.
In a scalar surface integral, a scalar function is accumulated over the surface. If a curved sheet has variable surface density, for example, an integral of this kind can provide its mass. By contrast, when a vector field is integrated through the dot product with the normal vector, one computes the flux of the field through the surface.
Scalar surface integral:
∬S f dS
Flux of a vector field:
∬S F · n dS
FIGURE 5 · Surface integral of a scalar function. The surface may be described by a parameterization, and the area element must be adjusted to its geometry. Original source indicated in the article: SlidePlayer.
If a surface can be written as \(z=g(x,y)\), it may be parameterized by \((x,y,g(x,y))\). The differential surface element is not simply \(dx\,dy\), because it must account for the local inclination of the surface. For the graph \(z=g(x,y)\), the factor that appears is:
dS = √(1 + (∂g/∂x)² + (∂g/∂y)²) dA
This factor corrects the area projected onto the \(xy\)-plane so that it becomes the actual area on the inclined surface. If the surface is described in another way—for example, by \(x=g(y,z)\)—the corresponding parameterization is used.
Gauss’ Theorem or the Divergence Theorem
When \(S\) is a closed surface enclosing a volume \(V\), Gauss’ Theorem relates the total flux of a vector field through the surface to the divergence of the field throughout the interior volume. If \(\mathbf F=(P,Q,R)\), its divergence is:
div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
and the theorem states:
∯S F · n dS
=
∭V div(F) dV
The symbol \(\oiint\)—or, typographically, a double integral sign with a circle—indicates that the surface is closed. Conceptually, Gauss’ Theorem says that the net flux leaving the boundary of a volume is determined by the total contribution of the “sources” and “sinks” of the field within that volume.
Green, Stokes, and Gauss: One Underlying Idea
At this point we can return to the initial observation and formulate it more precisely. Green’s, Stokes’, and Gauss’ theorems should not be treated as isolated results. All three connect what happens in a region with what happens on its boundary, although each one does so in a different geometric situation.
Green works in the plane: it relates a double integral over a region to a line integral along its boundary. Stokes works with an oriented surface in space: it relates the circulation of a field around the curve forming its boundary to an integral of the curl over the surface. Gauss works with a volume: it relates the flux through the closed surface enclosing it to the divergence throughout the volume.
Conceptual Scheme
Green: plane region ↔ boundary curve.
Stokes: surface ↔ boundary curve.
Gauss: volume ↔ boundary surface.
This resemblance is not accidental. In more advanced courses, the generalized Stokes theorem allows these results to be gathered within a single mathematical structure. Even without differential forms, however, the central intuition can already be understood at undergraduate level: under suitable conditions, an integral over the interior can be transformed into an integral over the boundary, and vice versa.
In this sense, the progression from line integrals to multiple integrals and surface integrals is not an arbitrary collection of techniques. It is a gradual extension of the same idea of accumulation and, at the same time, an introduction to one of the deepest relations in vector calculus: the relation between a region and its boundary.
Comparative Summary
Type of integral
Domain
Typical interpretations
Related theorems
Line integral
Curve
Mass of a wire, work, circulation
Fundamental Theorem for Line Integrals; Green
Double integral
Plane region
Area, volume under a surface, mass of a lamina
Fubini; Green
Triple integral
Region in space
Volume, mass, volumetric accumulation
Fubini; Gauss
Surface integral
Surface in space
Weighted area, surface mass, flux
Stokes; Gauss
The difference among these integrals can therefore be summarized simply. A line integral accumulates along a path; a multiple integral accumulates over a region; and a surface integral accumulates over a surface. What changes is the geometry of the domain and, with it, the differential element that must be used. The fundamental idea of integration, however, remains the same.
To integrate is to accumulate. What changes from one integral to another is where the accumulation takes place and what magnitude is being accumulated.
A guided reading of Toward a Dialectical-Materialist Interpretation of Topology, by José Mauricio Gómez Julián
Guided Reading · History of Mathematics · Philosophy of Science
· · ·
Imagine a coffee mug made of rubber. We can stretch it, compress it,
twist it, and gradually deform it into something resembling a doughnut.
To ordinary geometry, the mug and the doughnut are very different
objects: they have different curvatures, lengths, and proportions.
Topology, however, asks a deeper question: did their essential structure
really change, or did we merely alter their metric appearance?
That question — what may vary without an object ceasing to belong to the
same structural class — lies at the conceptual heart of topology. But
José Mauricio Gómez Julián’s essay seeks to go one step further. Rather than merely
presenting mathematical definitions, it reconstructs the history of the
discipline in order to ask what philosophical meaning lies in studying
precisely those properties that remain through certain
transformations.
The journey moves through Leibniz, Euler, Cantor, Dedekind, Poincaré,
Peano, Brouwer, and Hausdorff; through the bridges of Königsberg, the
paradoxes of dimension, set theory, continuity, and homeomorphisms.
Eventually, these threads are brought together in a proposal: to read
topological structure from a dialectical-materialist perspective, as a
mathematical way of thinking about the relationship among
transformation, invariance, structure, and qualitative change.
01 · The Fundamental Problem
From Measuring Objects to Studying Relations
For centuries, thinking geometrically meant above all thinking in terms
of magnitudes: lengths, areas, angles, distances, and proportions.
Topology introduces a change in perspective. What matters is no longer
exclusively how much something measures, but also
how its parts are related.
The essay finds a decisive antecedent in Gottfried Wilhelm Leibniz. In
the seventeenth century, Leibniz imagined a geometria situs, a
“geometry of position”: a discipline in which the relative arrangement
of elements would take priority over their magnitude. The intuition was
remarkably modern. Two configurations might differ in their metric
dimensions and yet share something deeper in their organization.
Topology asks less about how much an object measures than about which
relations survive when its shape changes.
This is also the first useful key for readers coming from economics or
the social sciences. An economic network may change enormously in the
volume of its transactions without necessarily changing its basic
pattern of connections; an institution may grow or shrink while
preserving certain internal relations; a political structure may
undergo quantitative modifications without yet experiencing a
qualitative transformation in its organization.
This does not automatically turn such questions into problems of
mathematical topology. It does, however, help us grasp the intuition
that interests the essay: distinguishing between changes of magnitude
or appearance and changes of structure.
02 · Königsberg, 1736
Euler and the Birth of a New Way of Seeing
One of the foundational episodes in this history takes place in the
Prussian city of Königsberg. The city was divided by the Pregel River
and connected by seven bridges. The problem was easy to state: was it
possible to take a continuous walk crossing every bridge exactly once?
Leonhard Euler realized that the distances, the sizes of the islands,
and the lengths of the bridges were irrelevant. Each landmass could be
replaced by a point, and each bridge by a connection between points.
The physical problem was thus transformed into an abstract structure.
The Mathematical Idea
What we would now call a graph preserves only the
information relevant to the problem: which vertices are connected by
which edges. The Königsberg problem is a problem of an
Eulerian traversal: it asks whether every edge can
be traversed exactly once.
Euler showed that this was impossible. All the relevant vertices had
odd degree, whereas a traversal using each edge exactly once can have
only zero or two vertices of odd degree.
Yet for the historical argument of the essay, the decisive point is not
merely the solution. It is the method of abstraction. Euler
deliberately removed information about magnitude in order to preserve
a structure of relations. A real city, with water, bridges, and
distances, became a mathematical object whose organization could be
studied independently of scale.
03 · A Productive Crisis
Cantor and the Strange Problem of Dimension
The next major leap appears in the nineteenth century with Georg Cantor
and set theory. Cantor discovered that the points of a line segment and
the points of a square can be placed in one-to-one correspondence:
both sets have the same cardinality.
This result was profoundly counterintuitive. A segment appears
one-dimensional and a square two-dimensional. How could they contain,
in a precise sense, the “same number” of points?
An Essential Distinction
Having the same cardinality does not mean having the
same topological dimension. That was precisely the problem: counting
points is not enough to capture what intuitively distinguishes a line
from a surface.
In philosophical terms, the contradiction between geometric intuition
and set-theoretic result forced mathematics to reformulate the
question. If dimension could not simply be reduced to the number of
coordinates or to the cardinality of points, a deeper structural
property had to be discovered.
The essay places particular emphasis on episodes of this kind:
contradictions do not appear merely as unpleasant accidents in
science, but as engines of conceptual development. A notion that once
seemed self-evident — “dimension” — becomes problematic, and by
becoming problematic it forces the construction of a deeper theory.
1676 · Leibniz
Imagines a geometry based on position rather than magnitude.
1736 · Euler
Reduces the Königsberg problem to a structure of vertices and
connections.
1877 · Cantor
Correspondence between sets of different apparent dimensions
destabilizes the old geometric intuition.
Late 19th Century
Dedekind, Peano, and others force mathematics to distinguish among
cardinality, continuity, and dimension.
Early 20th Century
Poincaré, Brouwer, and Hausdorff consolidate the problems that will
shape modern topology.
04 · When a Definition Also Asks About the World
Poincaré: Continuity, Cuts, and Meaning
Henri Poincaré occupies a special place in the story because his
questions about dimension did not arise solely from technical
difficulties. He was also interested in understanding why we experience
space as three-dimensional, what relationship exists between
mathematical geometries and physical space, and where our geometric
intuitions come from.
His idea was to think about dimension through cuts. In
intuitive terms, the dimension of a continuum could be investigated by
asking what kind of object must be removed in order to divide it. A
line can be disconnected by removing a point; separating a surface
generally requires something of higher dimension.
The essay grants this idea particular philosophical significance. A
dimension no longer appears merely as a coordinate drawn along an axis,
but begins to be related to the way in which the parts of a space are
connected.
A structure is defined not only by its components, but by the system of
relations that makes those components into a whole.
Poincaré did not thereby provide the final mathematical word on
dimension. His proposal encountered difficulties and would eventually
be replaced by more robust formulations. Yet for the historical reading
developed in the article, that is precisely the point: a formulation
may be mathematically superseded while still preserving a fertile
philosophical intuition.
05 · The Consolidation of the Discipline
Brouwer, Hausdorff, and Modern Topology
L. E. J. Brouwer brought the problem of dimension to a new level of
rigor. Among his fundamental contributions was the invariance of
dimension: Euclidean spaces of different dimensions cannot be
equivalent through a homeomorphism. A line and a plane do not become
structurally identical no matter how ingenious the correspondence
between their points may be.
The result is important because it separates two ideas that Cantor had
forced mathematicians to confront: two sets may have the same
cardinality and yet possess different topological structures.
Felix Hausdorff, in turn, contributed to transforming topology and set
theory into increasingly abstract and systematic disciplines. By the
beginning of the twentieth century, mathematical “space” no longer had
to be imagined as a physical room filled with geometric points. Its
elements could be functions, sequences, or other abstract objects.
This generalization is decisive. Topology ceases to be merely a strange
geometry of deformable surfaces. It becomes a language for speaking
about continuity, neighborhoods, convergence, connectedness, and
structure across enormously broad classes of mathematical objects.
06 · The Mathematical Core
What Is a Topology, Really?
We can now state the idea precisely. Let X be a set. A
topology on X is a collection τ of subsets of X —
called open sets — satisfying certain rules.
∅, X ∈ τ
arbitrary unions of members of τ belong to τ
finite intersections of members of τ belong to τ
The pair (X, τ) is called a
topological space. What matters is that τ determines
what it means to be “near,” what continuity means, and how the space is
organized without requiring any numerical notion of distance.
A metric may tell us that two points are 3.7 units apart. A topology
can study relations of proximity and continuity even when no distance
function exists at all.
Concept
Intuition
Metric
Allows distances between points to be quantified.
Topology
Describes a structure of neighborhoods, continuity, and more
general spatial relations.
Homeomorphism
A continuous bijection with continuous inverse between two
topological spaces.
Topological invariant
A property that remains unchanged under homeomorphisms.
The Famous “Rubber-Sheet Geometry”
From here comes the classical metaphor. We may stretch, compress, or
twist an object as long as we do not cut it or glue together parts that
were previously separate. A circle can be deformed into an ellipse
without leaving its topological class.
A sphere can be deformed into an ellipsoid. Creating a hole in the
sphere in order to transform it into a torus, however, requires a
topologically radical modification: we are no longer merely changing
distances and curvatures, but the structure of the object itself.
The Central Point
In topology, “preserving structure” does not mean preserving visual
appearance or distances. It means preserving those relations encoded
by the topological structure. Mathematically, the relevant notion of
equivalence is the homeomorphism.
07 · From Formalism to Meaning
The Dialectical-Materialist Reading
Up to this point, we have topology. The specifically philosophical move
of the essay begins when it asks what this kind of mathematics tells us
about the relationship among structure, transformation, and
permanence.
The author’s proposal begins by distinguishing between changes that
affect certain properties of a system without destroying its
fundamental structure and changes that do alter that structure. The
distinction immediately recalls a central category of dialectics: not
every quantitative modification yet constitutes a qualitative change.
A topological object may be stretched, twisted, or deformed within
certain limits while retaining its invariants. But when it is torn, when
a new connection is created, or when an essential connection is
eliminated, a different class of structure appears.
The essay interprets this difference through the dialectical relation
between form and essence. Form may
vary considerably while certain internal relations remain stable; when
transformations reach the very organization constitutive of the system,
change ceases to be merely formal and becomes qualitative.
Invariance does not mean immobility: something may change profoundly in
appearance while preserving, through those transformations, a
determinate structure.
This is perhaps the most interesting conceptual bridge proposed by the
text. “Remaining” and “changing” cease to be mutually exclusive
absolutes. A system can change precisely because it possesses a
structure within which certain changes are possible. That same
structure also determines which transformations would cease to count as
internal modifications and instead become a rupture.
Homeomorphism and Structure
The homeomorphism therefore acquires special philosophical importance
for the author. Technically, two spaces are homeomorphic when there
exists between them a continuous bijection whose inverse is also
continuous. Philosophically, the article interprets this as a
formalization of the idea that externally different configurations may
share the same structural organization.
This is not because a mug and a torus are “the same thing” in every
possible sense, but because a particular level of abstraction permits
them to be treated as equivalent with respect to the properties studied
at that level.
This connects with another important epistemological thesis of the
essay: every science abstracts. Physics, chemistry, biology, economics,
and mathematics isolate particular relations in order to investigate
them. To abstract does not necessarily mean to deny the rest of
reality; it means provisionally selecting which relations will be
treated as essential for a particular problem.
08 · An Excursion Beyond Mathematics
From Abstract Space to DNA
To show that topological language is not confined to geometric
exercises, the essay turns to a particularly suggestive example: the
structure of DNA.
DNA molecules can form coiled, knotted, and interlinked structures.
During real biological processes, enzymes known as
topoisomerases can temporarily cut a strand, allow
changes in the molecule’s entanglement, and then reconnect it. Knot
theory and other topological tools are useful precisely for describing
aspects of these configurations.
Here, the old metaphor of “cutting and gluing” ceases to be merely a
pedagogical image. The connectivity of a molecular structure can
undergo physically real modifications.
Why the Example Matters
The article uses DNA as an epistemological illustration: changing a
quantity — length, twist, distance — is not the same thing as changing
the constitutive relations of a structure. When a connection is
broken and recomposed, the kind of transformation is qualitatively
different.
From the dialectical perspective developed by the author, this case
illustrates a more general idea: systems possess relatively stable
properties, but that stability exists within processes of
transformation. Some transformations may accumulate or reach a point at
which a qualitatively different organization emerges.
09 · The Thesis in Perspective
What the Essay Proposes — and What It Does Not
It is useful to distinguish carefully between two levels. The first is
strictly mathematical: topological spaces, continuity,
homeomorphisms, invariants, and dimension have formal definitions and
results that do not depend on accepting a Marxist philosophy.
The second level is interpretive. The article argues that the history
and conceptual structure of topology can be understood particularly
fruitfully through dialectical-materialist categories: structure,
relation, transformation, invariance, essence, form, and qualitative
change.
In other words, the argument is not that a theorem of topology can be
derived from Marx. Nor does it claim that a homeomorphism and a
dialectical contradiction are literally the same concept. The project
is to seek a structural correspondence: to show that
certain relations formally discovered by mathematics may acquire
epistemological meaning when placed within a more general conception of
change and structure.
Seen in this way, the historical reconstruction is not decorative.
Cantor challenges an inherited intuition about dimension; Peano shows
that continuity can produce phenomena that intuition did not
anticipate; Poincaré attempts to redefine the problem; Brouwer
introduces new proofs and new abstractions; Hausdorff helps systematize
the language. The modern concept emerges through conflicts,
reformulations, and successive theoretical developments.
That historical movement is precisely what makes the article’s
dialectical reading attractive: a scientific theory does not appear
finished from the outset. Its categories develop through concrete
contradictions that force earlier concepts to be revised, some of their
elements preserved, and others abandoned.
The Question That Remains
Perhaps the most powerful intuition a non-specialist reader can take
away is this: knowing something does not consist solely in measuring
its visible properties. We may also ask which relations make that
thing the structure it is, which modifications it can undergo without
ceasing to preserve that structure, and what kind of transformation
would have to occur for a different structure to emerge.
Topology provides an extraordinarily precise mathematical language
for one version of that question. Gómez Julián’s essay proposes that
dialectical materialism, in turn, provides a way of interrogating its
philosophical meaning.
· · ·
The journey that begins with bridges, points, and lines thus ends with a
much broader question. What does it mean to say that something remains
“the same” while changing? Which transformations are accidental with
respect to a structure, and which alter what constitutes it? How can
continuity be distinguished from rupture?
These are mathematical questions when we speak of topological spaces.
But they are also questions that reappear, in different forms, when we
study physical, biological, economic, or social systems. The
philosophical wager of the article is precisely that this recurrence
should not be treated as a merely verbal coincidence: it deserves to be
investigated as a correspondence among forms of structure,
transformation, and invariance.
To follow the complete historical development, mathematical
definitions, and philosophical argument in their original formulation:
ON THE POLITICAL ECONOMY OF THE SYRIAN CONFLICT (2017)
JOSÉ MAURICIO GÓMEZ JULIÁN
I · SectionIntroduction
The key to understanding the present is understanding the past, just as the key to looking toward the future is understanding the present: “Men make their own history, but they do not make it as they please, under circumstances chosen by themselves, but under circumstances directly encountered, given and transmitted from the past.” (Marx, The Eighteenth Brumaire of Louis Bonaparte, 1975).
The Syrian political conflict is a conflict among the interests of nation-states and, consequently, it cannot be understood without first understanding what a State is, because otherwise it will also be impossible to understand which sectors of society carry their interests from the national sphere into the international sphere through the State itself.
In general terms, this concept appears implicitly as far back as the philosophers of ancient Greece, although it is in Machiavelli’s work that the word “State” appears for the first time, where he says: “All states, all dominions that have held and hold power over men have been and are either republics or principalities.” (Machiavelli, 2006, p. 13). Subsequently, throughout history, different political theorists would define it in different ways. Here, two orthodox definitions will be introduced and then contrasted with a heterodox definition. The two orthodox definitions belong to the last classical treatise writer (Hegel) and the last modern treatise writer (Weber).
Hegel writes on the matter: “The State, precisely, as universal and objective freedom, in the free autonomy of the individual will; the State, which as a real and organic spirit, a) of a people, b) through the relations of the specific national spirits, c) realizes and manifests itself in Universal History as the universal spirit of the world. The Law of the State is supreme.” (Hegel, 1968, p. 33).
Weber, for his part, writes: “For sociology the reality ‘state’ is not necessarily composed of its juridical elements; or, more precisely, it does not derive from them. In any case there is no collective personality in action. When one speaks of the ‘state,’ the ‘nation’ (…) one refers only to the development, in a certain form, of the social action of a number of individuals, whether real or constructed as possible (…)” (Weber, 2002, p. 12). Having clarified this, he states: “The State, like the political associations that preceded it, is a relation of domination of men over men based on the means of legitimate coercion (that is, regarded as legitimate). Thus, for it to subsist it is necessary that the dominated submit to the authority of those who dominate in each case. When and why they do so can be understood only when the internal motives of justification and the external means on which domination rests are known.” (Weber, 2002, p. 1057).
Machiavelli’s definition explicitly says little or nothing about the political motives underlying the formation of the State; implicitly, however, it suggests the State’s repressive role or, in his words, “the dominions that have held and hold power over men.” In Hegel’s case, a romantic vision of the State is presented, which is hardly surprising, since for this author the State was the will of God. Historical evidence, of course, shows that the State has little or nothing to do with the free autonomy of individual freedom and therefore cannot be universal and objective freedom either, since individual freedom is the sine qua non condition of universal or general freedom. It is precisely from this definition that the naïve view arises that the State is all of us. If this were so, there would be no way in which the State could be the organic spirit of a people, although it does possess a real existence as a social fact[1] through the relations among specific national spirits. Ultimately, if the State were all of us, there would be no need to create institutions that protect the rights of individuals against the State itself.
Weber’s definition sheds light on the necessary condition for the State to subsist (the submission of the dominated to the dominant) and the sufficient condition (possessing means of coercion regarded as legitimate). The necessary condition of subsistence is, in turn, the circumstance that makes its existence possible, while the sufficient condition is the most important circumstance guaranteeing its perpetuation. Yet the most important question in terms of political philosophy is precisely the one Weber refuses to answer: why does one sector of society accept submission to another sector? How does a State concentrate the monopoly of physical violence?
The preceding questions could be answered from Rousseau’s point of view. He wrote: “I suppose men to have reached the point at which the obstacles to their preservation in the state of nature exceed the forces that each individual can employ to maintain himself in that state. Then this primitive condition can no longer subsist, and the human race would perish unless it changed its manner of being. Now, since men cannot generate new forces, but only unite and direct those that already exist, they have no other means of preserving themselves than to form by aggregation a sum of forces capable of overcoming resistance, to bring them into play toward a single end, and to make them act together and in conformity. This sum of forces can arise only from the cooperation of many; but since the strength and freedom of each man are the principal instruments of his preservation, how can he commit them without harming himself and neglecting the obligations he owes to himself? This difficulty, reduced to my subject, may be stated in the following terms: ‘To find a form of association which will defend and protect with the common force the person and goods of each associate, and by which each, uniting with all, nevertheless obeys only himself and remains as free as before.’ Such is the fundamental problem to which the Social Contract provides the solution. The clauses of this contract are so determined by the nature of the act that the slightest modification would render them useless and without effect; so that, although they may never have been formally stated, they are everywhere the same and have everywhere been tacitly recognized and admitted, until, when the social pact is violated, each person recovers his original rights and regains his natural freedom, losing the conventional freedom for which he had renounced the former.” (Rousseau, 2007, pp. 45–46).
Rousseau’s view, although broadly correct, contains a certain romanticism and leaves questions unanswered. It is romantic to think that a social institution as complex and historically rooted as the State would disappear at the slightest violation of the contract established among the sectors of society involved in that contract. Nor does Rousseau explain the historical pattern that determines how the clauses of such a contract are formulated at each historical moment in the transition from one form of State to another. All sciences seek, in their object of study (for the social sciences, that object is society), to establish general patterns of behavior that allow them to predict future changes in that object, because the ultimate purpose of the social sciences is not merely to explain a social fact, but to predict the behavior of the fact analyzed over time[2]; in that sense, explaining it is only an inevitably necessary step toward achieving that end.
Once the above is established, the following question arises inexorably: is there a general pattern across the different societies that have existed in human history which explains why the clauses of the different social contracts that emerged in the transition from one social formation to another were established in a particular way? The answer is yes. Rousseau already leaves implicit a fundamental point: necessity as the foundation of the birth of the social contract or, what amounts to the same thing, a clear position on the fundamental problem of philosophy—that material causes determine and subordinate the realm of ideas.
Rousseau is not alone in taking this position. As can be seen, Weber also attributes the subsistence of the State (what has here been called the “sufficient condition”) to possession of the means of legitimate coercion, that is, the monopoly of physical violence. Weber’s first cause or necessary condition is the submission of the dominated to the dominant, which is itself explained by internal motives of justification (which will evidently have an explanation rooted in the historical conditions of the moment) and the external means (which are objective) on which domination rests.
Durkheim also proceeds along these lines when he states: “Thus, we must consider social phenomena in themselves, independently of the subjects who form a representation of them; they must be studied from the outside, as external things, for it is as such that they present themselves to us (…) A thing is recognized principally by the fact that it cannot be modified by a simple decree of the will. This does not mean that it resists every modification, but rather that to produce a change it is not enough merely to desire it; an effort, more or less arduous, is also required because of the resistance it opposes to our action and which, moreover, cannot always be overcome. We have already seen that social facts possess this property. Far from being a product of our will, they determine it from outside; they are like molds into which we are compelled to cast our actions.” (Durkheim, 2009, pp. 83–84). Machiavelli states that “Principalities are either hereditary (…) or new.” (Machiavelli, 2006, p. 13), and throughout his work he explains that new principalities are generally acquired by arms and virtue or by the arms and fortune of others. Comte, for his part, writes: “But this provisional instinct, without which science would then have lacked suitable nourishment, must ultimately become habitually subordinated to a just systematic appreciation, as soon as the full maturity of the positive state has made it possible always to apprehend sufficiently the true essential relations of each part with the whole, so as constantly to offer a broad horizon to the most eminent investigations while nevertheless avoiding all puerile speculation.” (Comte, 2017, p. 23). Finally, Marx, an author who will be examined in greater depth below, likewise takes a position along the same lines.
As can be seen, one of the two great theorists of Political Law[3], one of the greatest political theorists in human history, and the four founding fathers of Sociology are clear, despite their theoretical differences, on the point under discussion[4].
Having established the above, we can proceed to the next stage of the analysis. Humanity has witnessed four different social formations: a) primitive community, b) slavery, c) feudalism, and d) capitalism.
If the Social Sciences are indeed sciences and, moreover, the emergence of the different social contracts corresponding to each of the social formations set out above obeys objective factors, the first task is to determine which objective factor is the most important of all. This poses no great intellectual challenge, because before anything else the human species seeks, and has always sought, to obtain the means of subsistence necessary for survival. In this sense, the production of those means of subsistence stands as humanity’s most important material or objective factor, although, contrary to what economicism assumes, it is not the only one.
“In the production of their life, men enter into definite relations that are necessary and independent of their will, relations of production which correspond to a definite stage of development of the material productive forces. The totality of these relations of production constitutes the economic structure of society, the real basis on which a legal and political superstructure rises and to which definite forms of social consciousness correspond.” (Marx, A Contribution to the Critique of Political Economy, 1989, pp. 7–8).
It is therefore the relations of production—those relations which members of a society establish in response to necessity at a particular historical level of technological development—that condition social life and around which a particular culture arises, reinforcing them and allowing them to remain stable over time. This in turn means that a particular social contract is constructed around the relations that members of society establish in order to secure the production of their means of subsistence. These social relations of production will also express the degree of development of the productive forces of the period, which in turn will shape the way in which the clauses of the social contract are designed. Now, do members of society establish these relations on equal terms? The answer is emphatically no. Once human beings became sedentary, the enslavement of members of one gens by another and the emergence of the social division of labor gave rise to the slave mode of production. Equality was thereafter erased from the fate of human society, and the social relations of production established in every historical-social formation became a clear expression of the relation between dominant and dominated: slaveholders and slaves, feudal lords and serfs, capitalists and proletarians. The common denominator of these power relations among social classes, expressed in the legal relations of each period (which are no more than the expression of relations of production on the juridical plane), is that one social class possesses the material means for producing wealth—the means of production—while another social class possesses only its labor power with which to secure its subsistence.
This distribution of the means of production does not, of course, obey the whims of history or random events whose explanations are isolated in character. Rather, in general terms[5], it results from one sector of society finding itself, through historical evolution, in a privileged position that it uses to accentuate its differences from the rest of society, accumulating wealth from that position and achieving a monopoly over the means of exercising physical violence. In Europe, for example, the capitalist class has its origins in the Middle Ages among small merchants, a status accessible only to free men rather than serfs. By buying in the countryside at a lower price than they sold in the city, they began the accumulation of wealth that in its contemporary form we know as capital. Yet the primitive accumulation of capital was not limited to this: independent producers were also expropriated from their lands by force of gunpowder and blood, and decrees were even issued by different kings permitting the opening of the tombs of saints and other religious figures of Catholicism in order to extract any piece of gold—including dental prostheses—which could then be minted into coin.
Once the foregoing has been established, the State can therefore be defined as the political organization of the economically dominant class whose ultimate purpose is to maintain the existing order and crush the resistance of the other classes through possession of the monopoly of physical violence. It is therefore unsurprising that the process of State formation consisted in setting apart a special public power with its army, police, prisons, and coercive institutions of various kinds.
In this sense, speaking of democracy, defined as a “Form of political regime based on the proclamation of the principles of the power of the people, the freedom and equality of citizens. Democracy presupposes recognition of the principle of the subordination of the minority to the majority, the electability of the principal organs of State power, and the existence of political rights and freedoms.” (Oníkov & Shishlin, 1980, p. 140), lacks real meaning.
Contemporary evidence of the foregoing is provided by the 2001 Nobel laureate in Economics, Joseph Stiglitz, who writes: “Finance is not the only area in which foreign investment has been an ambiguous blessing. In some cases, new investors persuaded governments (often with bribes) to grant them special privileges, such as tariff protection. In many cases the U.S., French, or other advanced industrial-country governments exerted pressure, reinforcing the notion in developing countries that it was perfectly appropriate for authorities to intervene in the private sector and presumably collect from it. In some cases, the role of the State seemed relatively innocuous (although not necessarily incorruptible). When U.S. Secretary of Commerce Ron Brown traveled abroad, he was accompanied by American businessmen seeking contacts with those emerging markets and entry into them. Presumably, the chances of obtaining a seat on the plane increased if one made significant campaign contributions.” (Stiglitz, 2003, p. 100).
· · ·
II · SectionHistorical Background of Syria
Syria is a country whose population is concentrated in the northwest of its territory, which is due to the fact that only 0.06% of its surface area has water. According to (Datosmacro, 2017), Syria has a population of 18,502,413 inhabitants and a total area of 185,180 km.2, which is equivalent to a population density of approximately 100 inhabitants per square kilometer.
According to (OPEC, 2017), Syria has 2.5 billion barrels of oil in reserve, ranking 31st in the world; however, these are only proven oil reserves[6]. According to (CIA, 2017), Syria also ranks 43rd worldwide in terms of proven natural-gas reserves, with 240,700,000,000 m.3
This significant quantity of natural resources, while not representing proportions of global reserves as important as those of other countries—for example Iran (the country with the 5th-largest proven oil reserves and the 2nd-largest proven natural-gas reserves) or Iraq (the country with the 4th-largest proven oil reserves and the 11th-largest proven natural-gas reserves)—is sufficient reason for different nations to have a geopolitical interest in Syria. There is, however, an even more important reason: Syria borders Iraq, and Iraq in turn borders Iran, as can be seen below:
Figure 1
Source: (RT, Economía, 2017)
As can be seen, the gas pipeline runs from the Iranian coast across the Iran–Iraq and Iraq–Syria borders until it reaches the Syrian coast. Its final purpose is to supply European consumers, as well as consumers in the countries through which the pipeline passes. Syria is therefore geopolitically fundamental: whoever controls that country controls the pipeline’s final outlet and, with it, the supply of natural gas to Europeans; if, in addition, Iraq and Iran are controlled, one has in hand one of the most profitable businesses a company could imagine.
The matter does not end there. To move the oil extracted from any of these countries, one must also go around the entire Arabian Peninsula or around the entire African continent.
Figure 2
Source: (EIA, 2017)
Since Euclid, it has been known that the shortest distance between two points is a straight line. Construction of the Trans-Arabian Pipeline began in 1947 and was administered primarily by the U.S. company Bechtel[7]. The pipeline was originally intended to terminate in Haifa, which at the time lay within the British Mandate for Palestine; because of the establishment of the State of Israel, however, an alternative route through Syria was established, and Lebanon was selected for one of the export terminals at Sidon. As (Little, 2003, pp. 12–13) notes, the Syrian government initially opposed the plan but ratified construction of the pipeline in 1949 after a military coup overthrew the democratic government there, and oil transport through the pipeline finally began in 1950. This coup against the government of the day—which had been chosen through free elections—was promoted by the CIA, and authoritative voices such as (Massad, 2017)[8], (Little, 2003)[9], and even declassified CIA records confirm it. Although that pipeline is no longer strategically important today, it serves as a historical reference point for understanding who has continued to destabilize Syria and why, a matter that will be examined in greater depth below.
Although the Trans-Arabian Pipeline is no longer strategically important in geopolitical terms, Syria certainly remains so, because going around the Arabian Peninsula and crossing the Suez Canal significantly increases transport costs and also the time required to realize oil as a commodity.
Figure 3
Source: (Puzzle de la Historia, 2017)
As can be seen, going around the Arabian Peninsula in order to reach the Suez Canal (where fees are paid) involves traveling 11,600 kilometers, while avoiding the Suez Canal requires going around Africa, a distance of 19,800 kilometers. To give the reader a sense of scale, traveling through the Suez Canal would amount to 22.3 times the distance between Paso Canoas and Peñas Blancas, while avoiding it would mean traveling 38.05 times that same distance[10].
Would it not be simpler to trade the oil if it could be taken out in a straight line through Syria? The answer is yes. Doing so, however, would require political control of Syria. This makes the country a fundamental geopolitical point, because whoever controls Syria will control not only those reserves of natural resources—oil and natural gas—but also the most efficient commercial route for placing them in Western markets.
This highly strategic commercial position—in terms of any commodity, not only oil and natural gas—has made Syria desirable to various empires throughout world history, from the Persians and Greeks, through the Romans, and ultimately the Turks.
Let us reconstruct some of that history in order to see the political evolution of the territory now known as Syria and thereby understand both the reasons for its relatively recent emergence as a nation-state and the climate of instability that has arisen within it and in its surroundings.
The Roman Empire underwent an administrative division initiated with the tetrarchy[11] of Emperor Diocletian (284–305) and consolidated by Emperor Theodosius I (379–395), who divided it between his two sons: Arcadius became emperor of the East and Honorius emperor of the West. What remained of the Roman Empire after Odoacer (King of the Heruli) deposed the last Western Roman emperor (Romulus Augustulus) in 476 was conquered by Islam, becoming the Ottoman Empire[12] on May 29, 1453. The Ottoman Empire would then endure until the First World War (WWI).
After WWI, small countries appeared on the map for the first time, among them Syria on September 1, 1918. Like every historical event, this occurrence was not arbitrary. As with the emergence of Iraq, Jordan, and others, it resulted from the fact that the French and British could not defeat the Ottoman Empire alone, making it necessary not only to ally with one another but also to secure the support of civilians residing within the empire. To win their favor, they promised them “Greater Palestine,” that is, a single nation for all Arabs.
In secret, however, France and the United Kingdom concluded the Sykes–Picot Agreement[13], which consisted of distributing the geopolitical control they would exercise over the territory occupied by the Ottoman Empire in the event that they won the war. Napoleon is said somewhere to have remarked that, if one wished to succeed in politics, the key was to promise everything and fulfill nothing; that is precisely what the Europeans did, and the promise of a single nation for all Arabs was never fulfilled. Instead, as part of a distribution of Ottoman territory among the European powers of the time, they created many relatively small nation-states.
Figure 4
Source: (Encyclopædia Britannica, 2017)
It was in this context, after the victory of the European nations, that Syria appeared and came under French control, a state of affairs that lasted until the end of the Second World War (WWII).
As reported in (U.S. Department of the Army, 2017), shortly after the end of WWII, during 1944 the Syrian government assumed the functions of fourteen administrative departments that had been under direct French control since 1920. These included departments responsible for matters such as customs, social affairs, excise taxes, supervision of concessionary companies, and oversight of tribes. France retained control of social, cultural, and educational services, as well as the Levant Special Forces[14], which were used for security purposes. Despite French opposition, the Soviet Union in July and the United States in September 1944 granted Syria and Lebanon unconditional recognition as sovereign states; British recognition followed a year later. These Allied nations pressured France to evacuate Syria.
The new Syrian government demanded the immediate and unconditional transfer of the Levant special troops to Syrian control or their dissolution, and threatened to form a national army unless that step was taken. France, however, made the withdrawal of the troops conditional on Syria signing a treaty that would assign France a privileged position in the country.
In January 1945, the Syrian government announced the formation of a national army, and in February it declared war on the Axis powers (principally Germany, Italy, and Japan). In March, the country became a founding member of the United Nations (UN), an indication of its sovereign status, and in April it affirmed its commitment to the idea of Arab unity by signing the Arab League pact.
Before introducing Israel’s role in Syria’s historical and current situation, it is necessary to discuss the ideological-religious conflicts in the region.
Shortly after the end of WWII, the Arab Socialist Ba’ath Party was founded, hereafter referred to as BA’ATH in reference to the ideology of this pan-Arab party, which combined the old aspirations for a single Palestinian nation with a secular ideology and socialist foundations. It came to power in 1963 in Syria and Iraq; the Syrian and Iraqi factions of the BA’ATH clashed in 1966 and thereafter acted independently. This produced an ideological fragmentation in the region, dividing it essentially into two blocs: Syria’s allies and Iraq’s allies.
Shortly before the BA’ATH came to power, Syria united with Egypt (with which it did not even share a border) to form the United Arab Republic between 1958 and 1961. The alliance did not last long, but it endured long enough for Egypt to persuade Syria to nationalize its natural resources. The failure of this union was not due to random or arbitrary factors either. In 1961 there was a coup d’état in Syria sponsored by the radical Syrian army, after which the country and its population endured a tyrannical government for two years; another coup followed in 1963, the one in which the BA’ATH took power in both Syria and Iraq in that same year.
In this 1963 coup, Háfez al-Assad appeared for the first time as a political figure and became head of the BA’ATH State; the same occurred in Iraq with Saddam Hussein, also a member of the BA’ATH Party. With the Cold War, which began in 1947, Syria aligned itself with the Soviet Union, while a recently created nation-state aligned itself with NATO. Israel must now be introduced into the international setting.
Between 1936 and 1939, a revolt took place in the Palestinian territories controlled by the United Kingdom—those formed after the defeat of the Ottoman Empire—whose cause was protest against Jewish immigration, since Arabs and Jews possessed ideological-religious differences that were irreconcilable within the framework of Middle Eastern culture. The revolt arose because of the United Kingdom’s consent to Jewish immigration into Palestinian territories[15] and its refusal to hold democratic elections to decide the future of the Jews. Since Palestinians had historically occupied those territories for centuries, they possessed an overwhelming demographic majority that would have resolved the elections in their favor.
Thus, after the French and British withdrew from what had formerly been Ottoman territory, and amid the ideological-religious conflicts between Palestinians and Jews in the area, the United Nations General Assembly met on November 29, 1947, following the report of the Peel Commission (which evaluated the causes of the conflict), and approved a plan dividing Palestine into two States. Curiously, the plan awarded 53.6% of the territory to the Jews and 48.7% to the Arabs, while determining that Jerusalem—which represented an area equivalent to 2.7%—would be a corpus separatum administered by the United Nations. The fact that, for political reasons (Israel would be a Western ally in the Middle East), the Jews were given lands that had been occupied by Palestinians for centuries, and that despite Palestinians making up 67% of the population compared with 33% represented by Jews, the latter were awarded a larger share of the territory, led the countries representing Palestine to intervene militarily in the territory assigned to the Jews two weeks after the UN resolution was adopted. Jerusalem subsequently ceased to be a corpus separatum and became the capital of the State of Israel, with the complicity and silence of its Western allies. According to Israeli professor David Grossman, this would later mean that “since 1967, those who set the agenda and benefit from the budgets (in Israel) are the right and the settlers, a small devout group that has kidnapped an entire nation and confiscated our future.” (Grossman, 2010).
The foregoing caused Israel’s neighbors not to recognize it as a legitimate nation-state and, once the Europeans had departed, to attack Israel instead.
Since then, through force and the complicity of the West, and since the artificial creation of the State of Israel, geographic changes have taken place in the proportion of territory occupied by Palestinians, as shown below.
Figure 5
Source: (Miller, 2017)
This problem extends beyond geography; it concerns the distribution of resources per capita and, with it, quality of life. We will now briefly examine the variation in population density from before 1948 to 2014.
Given the geographic change undergone by Palestinian territory, and using the total-population figures from (Datosmacro, 2017) beginning in 1970, we can calculate changes in Palestine’s population density. According to (Simpson, 2017), before 1948 Palestinian territory covered 26,626 km.2 and had an estimated population of 1,061,270 inhabitants, equivalent to 39.86 inhabitants per square kilometer. If Palestinian territory became 48% of the original in 1947 with the UN resolution, then the territory would have amounted to 12,780.48 km.2; assuming that the population did not vary significantly during that period (and even if it did, there are insufficient data to perform the calculation), the resulting population density would have been 83.03. In 1967 Palestinian territory was only 22% of the original, equivalent to 5,857.72 km.2; given that by 1970 the Palestinian population had risen to 1,124,293, this would imply a population density of 191.93 inhabitants per square kilometer. Finally, in 2014 Palestinian territory was only 12% of the original, equivalent to 3,195.12 km.2, and with a population of 4,422,143 reported by the source cited above, the resulting density would be 1,384.03 inhabitants per square kilometer. This would imply an approximate increase in population density of 3,372.23%. It is not difficult to imagine the economic difficulties that this increase in demographic pressure has implied for the Palestinian people in terms of implementing social policies, income-distribution policies, and other related government actions, even without counting the destruction of wealth caused by the wars with Israel. Conversely, it is also not difficult to imagine the economic advantages this has afforded Israel, whose population, as seen above, was the minority occupying those territories, without yet considering the substantial financing provided by the United States to Israel in various political and social areas, including the war against the Palestinian people and against countries that do not align themselves with its interests, as detailed by (Sparrow, 2017). The United States also used economic leverage in the international arena: it was the country that contributed the most to UNESCO (22%) and withdrew that support from the institution after Palestine was admitted, as reported by (El Mundo, 2017).
What have the industrialized powers done in the face of such a violation of the civil rights of the Palestinian population? Absolutely nothing. They have simply made requests that have no coercive force, leaving Israel free to comply with them or not; naturally, those requests have not been obeyed, while the blood of hundreds of victims has been shed. According to (TeleSUR, 2017), between April and May 1948—the first year of the State of Israel—the Israelis took several Arab cities under the UN partition plan, but on April 10 the first massacre by Jewish militiamen against 100 Palestinian peasants in the village of Deir Yassin was recorded, forcing thousands of Arab civilians to flee the region. In 1967, during the Six-Day War, which began on June 5, Israel militarily occupied the West Bank and Gaza Strip, Syria’s Golan Heights, and the Sinai Peninsula. During the military offensive, Mirage 3 aircraft of the Israeli army destroyed more than 400 Arab aircraft. Those battles reportedly left 776 Israelis dead, 2,563 wounded, and 15 prisoners. On the Arab side, there were at least 24,000 dead, 45,000 wounded, and 6,000 prisoners. Between 1975 and 1981, the number of deaths officially recorded by Palestinian and UN organizations was 170, although reports from human-rights organizations speak of more than 3,000 additional deaths not officially recorded.
In 1982, the death toll rose sharply, with 11,275 deaths recorded in the Lebanon War, where the Zionist army under then Minister of Defense Ariel Sharon launched “Operation Peace for Galilee” on June 6, with the objective of destroying the Palestine Liberation Organization (PLO), which was concentrated in Beirut after being expelled from Jordan in 1970. In the Sabra and Shatila massacre, 675 Israeli soldiers, around 9,800 Syrian soldiers, and Palestinian militiamen died. Between 2000 and 2010, an estimated 7,342 Palestinians, mostly civilians, died as a result of bombings and land and sea attacks by the Israeli army. Finally, during Israel’s prolonged occupation of Palestine, more than 1,500 civilians died and 1,215 Palestinians were displaced because of the demolition of homes by Israeli authorities. In the Gaza Strip conflict, 86% of the dead were Palestinians.
Ultimately, “Does the so-called international community exist? Is it anything more than a club of merchants, bankers, and warriors? Is it anything more than the stage name the United States adopts when it puts on theater?” (Galeano, 2017).
Háfez al-Assad’s arrival in power did not alter the conflicts already present in the region, specifically between Syria and Israel. Later, with the Cold War placing the two countries in different ideological blocs, tensions intensified and Syria entered into an open war against Lebanon.
Syria’s complications were not yet over, and the ideological frictions within the region were exploited by its allies. Before proceeding to describe that social fact, however, the genesis of the political-ideological factions among the Palestinian people will be introduced.
Contrary to what is commonly assumed, Arabs are not necessarily Muslims, although most of them are. The following table presents this segmentation.
Table 1
Source: Author’s own elaboration.
The proportion between Sunnis and Shiites is shown below.
Figure 6
Source: (RT, Sunitas y chiitas: ¿Qué es lo que los separa?, 2017)
It is estimated that Sunnis account for roughly 80–90% and Shiites for 10–20%. In Syria, because Háfez al-Assad’s government belonged to the Shiite minority while governing a Sunni majority, strong opposition movements began to emerge and were suppressed by gunpowder and blood. This internal conflict in Syria was exploited by Sunni Islamists, who took up arms against Háfez al-Assad; once again, the revolt was put down at the cost of many lives. Háfez al-Assad died in 2000 and was succeeded by Bashar al-Assad.
As (Rafizadeh, 2017) notes, once Bashar al-Assad came to power he proposed neoliberal-style economic openings with the goal of joining the World Trade Organization (WTO). These policies even led the United States to lift its opposition to Syria entering the organization, and with 153 members in favor, the WTO granted Syria observer status[16]. However, as the cited author observes, this economic reformism was not accompanied by income-redistribution policies (as is characteristic of the neoliberal model) and generated social unrest in the country. This forced the Syrian president to return to his father’s economic policies.
Several years later, the “Arab Spring”[17] reached Syria. There, “dozens of people demonstrated in Dar’a in the south of the country demanding freedom, political reforms and action against corruption, peaceful acts controlled by the Syrian police without violent incidents. The march against the government was launched on Internet social networks and supported by thousands of people who backed a Syrian revolution against President Bashar al-Assad until freedom and justice were achieved, but mobilization in the streets was limited. In response, Al-Assad announced economic reforms, amnesty for prisoners, and greater Internet access. The situation of instability continued throughout 2011, attracting international attention and raising the possibility that the UN Security Council might impose sanctions similar to those applied to Libya.” (EcuRed, 2017).
Some argue that the “Arab Spring” was a social movement sponsored by the United States to destabilize governments in the region that were not aligned with its interests, as investigative journalist Lee Stranahan put it: “When Barack Obama took office as president in 2008 with Hillary Clinton as Secretary of State, one of his first foreign-policy steps was his speech in Cairo, Egypt, where he stated, in the presence of members of the Muslim Brotherhood, that ‘a new era had begun in the U.S. attitude toward Islam’ (…) What followed was the wave of Arab Spring uprisings throughout the region, which, despite initially being regarded as ‘bold movements of people desperate for freedom,’ were in fact ‘a way for Islamists to dethrone the more secular military dictators of the region (…) The U.S. openly called for the overthrow of Al Assad despite the fact that it was clear that the outcome of this overthrow would be the seizure of control by Islamist groups (…) Syria proved resistant’ to Obama and Clinton’s regime-change efforts, even though the U.S. supplied weapons to the so-called ‘rebels’ that ultimately ended up in the hands of Islamist groups.” (Stranahan, 2017).
Likewise, according to U.S. attorney John F. Kennedy Jr.: “Our war against Bashar al Assad did not begin with the peaceful civil protests of the Arab Spring in 2011, but in 2000, when Qatar offered to build a $10 billion gas pipeline through Saudi Arabia, Jordan, Syria, and Turkey.” (Kennedy, 2017).
Julian Assange, founder of WikiLeaks, takes a similar line, stating: “The Syrian chapter goes back to 2006, when in a very important cable Ambassador Roebuck, based in Damascus, responds to a discussion about a plan to overthrow the Assad Government in Syria (…) When the United States needs to do something, it brings together the different arms of U.S. power, which include the military, Intelligence, finance, commerce, and its information power.” (Assange, 2015). Assange also states that “Within those military strategies, the journalist highlights the role of apparently civilian institutions, such as the National Endowment for Democracy (NED) and the United States Agency for International Development (USAID) (…) Assange also mentions the alignment of the International Monetary Fund (IMF) with Washington’s foreign policy when it comes to weakening or destabilizing countries through their finances.” (HISPANTV, 2017).
One fact does not, of course, exclude the other. From the standpoint of Western conceptions of political philosophy, many countries in the region have unfinished business regarding the political freedoms they grant their citizens. After the democratic opening, however, Bashar revived his neoliberal policies by promoting privatizations in order to smooth relations with the United States and its allies, while making no concession to the opposition movement in Syria. Thus, despite his economic-liberalization policies, as (RPP Noticias, 2017) reports, the United States and the other aligned countries called for the Syrian president’s resignation, and Obama announced economic sanctions against Syria, including a prohibition on trade with the Arab country. In the words of the U.S. Department of State: “The United States government has repeatedly called on President Bashar al-Assad to step down and has led the efforts of the international community to work toward a negotiated political solution to the conflict.” (U.S. Department of State, 2017).
In its attempts to overthrow Bashar al-Assad, the United States financed the creation of the Islamic State, according to Julian Assange: “Thus, for example, the disastrous intervention—absolutely disastrous—in Libya, the destruction of Gaddafi’s Government, which led to the occupation by Islamic State of large segments of that country, the flows of weapons going to Syria, guided by Hillary Clinton toward jihadists inside Syria, including Islamic State, are there in those emails. There are more than 1,700 emails from the Hillary Clinton collection that we have released on Libya alone.” (RT, 2017).
In addition, the Obama administration allowed Islamic State to grow in the expectation that this would help force the overthrow of the Syrian president, as John Kerry stated at the time: “We knew that it (Daesh) was growing, we were watching, we saw Daesh strengthening and we thought that threatened Al-Assad (…) We thought that we could probably get Al-Assad to come and negotiate, but instead of negotiations we found that Al-Assad got Putin to support him (…) The Russians have intervened because they do not want to see a Daesh government […] Unfortunately, the Russians have changed the situation. They have changed the equation.” (RT, 2017).
Figure 7
Source: (Heritage for Peace, 2017)
Ultimately, the reader should remember that 241 years ago the United States, France, and England were the United States, France, and England and governed themselves; 100 years ago, by contrast, Syria did not exist on the map, and 72 years ago it did not govern itself. Moreover, unlike the Western countries just mentioned, Syria never decided its own borders: they were decided for it on the basis of geopolitical convenience rather than criteria grounded in its economic, demographic, cultural, and social needs. The country therefore contains profound design errors—errors in its geographical architecture. It is difficult to imagine implementing social projects in a country such as Syria, where water exists only in a small part of the territory and where a desert of 200,000 square miles lies between the population and its sources of natural resources (as can be seen in Figure 7), according to (Encyclopædia Britannica, 2017). It is also a country that has continually been destabilized and even controlled by the principal powers of the West, and where a profound ideological-religious contradiction, combined with foreign interventionism, makes it extraordinarily difficult to achieve the minimum social and political cohesion required to move forward as a nation-state. It is therefore not surprising that the BA’ATH repeatedly uses repression to remain in power, or that this produces social eruptions that reach the level of civil war, as has occurred in various Western countries during the past century when they faced socioeconomic problems of far smaller magnitude than Syria’s.
· · ·
III · SectionWars as a Mechanism for Offsetting the Fall in the Average Rate of Profit
“War, then, is not a relation of man to man, but of State to State, in which individuals are enemies only accidentally, not as men nor even as citizens, but as soldiers: not as members of their country, but as its defenders. Finally, a State can have as its enemy only another State, and not men, since between things of different natures no true relation can be established.” (Rousseau, 2007, p. 42).
As Rousseau argues, wars set States against one another. In the first section of this research, however, it was shown that States respond to the interests of the dominant class at each specific stage of historical-social development. When two nation-states confront one another, what lies behind the conflict is a confrontation between the interests of the dominant classes within each of them. The constant interventionism of the United States in Syria is not the product of chance or coincidence; rather, it obeys inexorable economic laws belonging to the capitalist mode of production, specifically the law of the tendency of the average rate of profit to fall.
“(…) as the historical-natural process we call human society unfolds, Capitalism develops the productive forces to an unprecedented level and thereby generates an increase in the proportion of accumulated surplus value allocated to the acquisition of constant capital at the expense of the proportion allocated to variable capital (because technological innovation reduces necessary labor time and increases surplus labor time). Proportionally, however, the reduction in necessary labor time will be smaller than the increase in total capital or, what amounts to the same thing, increases in the rate of surplus value will be smaller than increases in total capital, and thus a downward tendency in the average rate of profit will emerge, which Marx called the law of the tendency of the rate of profit to fall.” (Gómez, 2017, pp. 51–52)
In weighted-average terms, the foregoing means simply that the technological innovations undertaken by capitalist firms will not proportionally offset the increase in total investment and, consequently, the average rate of profit will tend downward. There are, however, counteracting causes that turn the law into a law of tendency. In Marx’s words: “(…) we see that instead of the difficulty with which economists have hitherto struggled, namely explaining the fall in the rate of profit, the inverse difficulty arises: explaining why this fall is not greater or more rapid. This is undoubtedly due to the play of influences that counteract and neutralize the effects of this general law, giving it merely the character of a tendency, which is why we present the fall in the general rate of profit here simply as a tendency to fall.” (Marx, Capital, Volume III, 2010, p. 232).
Marx enumerated the most general causes that make the fall in profit a tendential law: an increase in the degree of exploitation of labor (the intensity and/or duration of the working day), a reduction of wages below their value (below the value of labor power), the cheapening of the elements that make up constant capital (fixed capital and intermediate consumption), relative overpopulation (the role played by the mass of unemployed workers in wage determination), foreign trade, and the increase in share capital (which generates dividends).
Despite these counteracting causes, however, the fall in the rate of profit asserts itself tendentially, as shown below.
Figure 8
Source: Author’s own elaboration based on (Marquetti & Foley, 2017).
Although capitalism has been the first socioeconomic formation (SEF) to exhibit cyclical crises and to make use of elements foreign to the essence of its economic philosophy, it has also shown an enormous capacity to reinvent itself and recover. In this sense, it has increasingly and more clearly employed mechanisms that were previously less prominent—for example, wars.
Figure 9
Source: (Allianz Global Investors, 2017)
Figure 9 shows a Kondratiev long wave, named after the economist Nikolai Kondratiev, who argued that the capitalist economic cycle as a whole follows periodic fluctuations in economic activity that can be represented as waves. Each wave has different stages. The starting point represents the beginning of economic activity, which expands until it reaches a peak; after that, markets become saturated as a result of overproduction (production greater than what the market, given a particular level of purchasing power, can absorb), causing capitalist economies to slow until they reach what we now know as financial crises.
In Kondratiev’s words: “It is generally recognized today that the dynamics of economic life under the capitalist social order are not simple and linear in character, but rather complex and cyclical. Nevertheless, science has clarified very little about the nature and types of cyclical movements, which resemble oscillations. When we speak of cycles in economics, we generally refer to cycles of activity lasting seven to eleven years. But it is evident that these movements are not the final type of economic cycle. In reality, the dynamics of economic life are more complicated. In addition to the cycles already mentioned, which we shall agree to call ‘intermediate,’ it has recently been shown that the existence of still shorter oscillations, lasting approximately three and a half years, is probable. But this is not all. There are real grounds for supposing the existence of large oscillations of approximately fifty years in the capitalist economy, a fact that makes the problem of economic dynamics even more complex.” (Kondrátiev, 2017)
Kondratiev is referring here to the three principal cycles of economic activity: Kitchin cycles (short duration), Juglar cycles (intermediate duration), and Kondratiev cycles (long duration). This section will focus on the latter, the long cycles.
It is true that Kondratiev’s analysis is rough and relatively mechanical because it lacks a prior theoretical formulation. This leaves him unable to establish precise duration intervals for the long waves or explain the causes of those intervals; nor does it allow him to examine deeply either the reasons that give rise to crises or those that permit capitalist economies to recover, much less the causes of the point-to-point transition within the wave. Nevertheless, it does provide a general idea of the behavior of economic activity and makes it possible to identify certain patterns of behavior that can be connected with what has been developed throughout this research.
As can be seen in Figure 9, each Kondratiev cycle has corresponded to some technological revolution. The expansionary phase begins once the innovation is introduced and, as a snowball effect develops—that is, as the innovation spreads beyond the innovating firm to imitating firms—economic activity flourishes. Once all sectors of the economy, or at least the most important ones, have adopted the innovation and produced what their production-possibility frontier permits, the economy stagnates. After some time, stagnation gives way to slowdown and finally to economic recession. It can also be seen how a new technological revolution begins the recovery and opens the curtain on a new phase, or wave, of the economic cycle. Statistically speaking, these long waves have been found to last between forty and sixty years, tending more toward the upper than the lower limit.
What Kondratiev proves unable to explain, however, is why economic crises become deeper as the economic cycle advances. Nor does he explain why it becomes more difficult to emerge from them, or the consequences this has for the average rate of profit, capital accumulation, and competition. In general terms, this can be explained by the fact that as capitalist economies develop—and with them technological development—it becomes increasingly difficult to carry out revolutions in production techniques. Among other factors, competition forces firms constantly to introduce small innovations; as society reaches a given degree of development of its productive forces, these become increasingly costly (as explained above) and their impact on the average rate of profit becomes progressively smaller. At the same time, every SEF has shown insurmountable limits to its capacity for technological innovation. For primitive communities, for example, it would have been impossible to construct the carriages and armor produced by the Roman Empire under the slave mode of production, just as it would have been impossible for the Romans to conceive the construction of mills powered by water or wind, and for those societies in turn to discover nuclear energy. What does this mean? Simply that as historical-social development advances, each mode of production gradually exhausts its capacity for technological innovation, which also means that technological revolutions become increasingly less close at hand.
The foregoing has two fundamental consequences. The first is an intensification of the downward tendency of the average rate of profit; the second is the centralization of capital, that is, the deepening of inefficient and weakly competitive market structures such as monopolies, monopsonies, oligopolies, oligopsonies, and others. This in turn has important effects on capital accumulation. The most important is that market structures lacking efficiency and competitiveness move away from the average rate of profit because their profit margins rise as a result of their market power[18], while firms with an average level of technology simultaneously see their rate of profit fall and firms with below-average technology are penalized by competition through bankruptcy and are absorbed by other firms, generally those with greater market power. This economic phenomenon occurs because technological innovations, for the reasons already discussed, become increasingly costly.
The matter does not end there, because market structures to which the average rate of profit does not apply and which possess significant market power—for example, monopolies—are free to slow technological innovation because they do not face competition[19]. In this way they move still farther from the downward tendency of the average rate of profit, since their profit margins are above the average. It is not uncommon to observe, for example, that monopoly profit rates tend to rise during financial crises while the average rate of profit plummets.
It might seem that these kinds of market structures, which promote inefficiency and lack of competitiveness, are an economic alchemy for capitalist firms. Since Adam Smith, however, economic science has understood the inverse relationship between wages and profits. Consequently, the measures by which firms distance themselves from the fall in the average rate of profit can only be possible at the cost of implementing the factors mentioned earlier by Marx. There is various evidence of this. One recent example occurred in France, an industrialized economy whose large companies were nevertheless forced to introduce greater flexibility into the labor market. Actions of this type by those operating within such market structures generate social discontent and, on many occasions, large generalized protests, as reported by (Yárnoz, 2017), one of which ended with 60 French people detained and 40 injured in a single day.
There is, of course, a palliative measure for this problem which, while not representing a definitive solution, contributes to slowing the natural course of economic laws: wars.
Figure 10
Source: (Kondratieff Winter, 2017)
As can be seen, wars have played a major role in the recovery of capitalist economies when they have gone through crises of overproduction. This is clearly not a new phenomenon: it can be observed at least as far back as 1812, when capitalism had existed for only 23 years if its political establishment in France is dated to the storming of the Bastille in 1789. What explains this? The answer is quite simple. Part of the unemployed labor force leaves the economy, which raises wages (given the resulting shifts in the labor-supply curve, holding labor demand constant); purchasing power therefore rises and production is stimulated again. In addition, many unemployed workers die in war (so firms do not have to pay severance compensation, for example), while the Government assumes responsibility for subsidizing their families (with taxes paid by those same families). The “benefits” of war for the economy do not end there. In many countries war destroys land and industry, reducing the stock of available fixed capital. Holding other factors constant, if demand for fixed capital remains unchanged while its supply declines, its market price rises. Machinery that had been undergoing depreciation as a result of use and new technologies therefore experiences a slowing of that process, which in turn slows the need for innovation. Finally, war itself promotes economic activity by activating industries closely linked to weapons manufacturing and even industries that benefit significantly from arms production. These factors lead to higher stock-market valuations for companies wholly or partly connected to this industry, because greater economic activity increases their profitability.
Industries most closely linked to arms production are those that belong directly to the weapons industry. “These are the first 10 companies on the 2011 ranking (figures in parentheses correspond to the 2010 ranking):
1 (1). Lockheed Martin (United States). Missile systems, electronics, and aerospace. Sales of $36.270 billion in 2011. Net profits: $2.655 billion. 123,000 employees (132,000).
2 (3). Boeing (United States). Aircraft, electronics, missiles, aerospace. Sales of $31.830 billion. Net profits of $4.018 billion. 171,700 employees (160,500).
3 (2). BAE Systems (United Kingdom). Aircraft, artillery, missiles, military vehicles, ships. Sales of $29.150 billion. Net profits of $2.349 billion. 93,500 employees (98,200).
4 (5). General Dynamics (United States). Artillery, electronics. Sales of $23.760 billion. Net profits of $2.526 billion; 95,100 employees (90,000).
5 (6). Raytheon (United States). Missiles, electronics. Sales of $22.470 billion. Net profits of $1.896 billion. 71,000 employees (72,400).
6 (4). Northrop Grumman (United States). Aircraft, electronics, missiles, warships. Sales of $21.390 billion. Net profits of $2.118 billion. 72,500 employees (117,100).
7 (7). EADS (EU). Aircraft, electronics, missiles. Sales of $16.390 billion. Net profits of $1.442 billion. 133,120 employees (121,690).
8 (8). Finmeccanica (Italy). Aircraft, artillery vehicles, missiles. Sales of $14.560 billion. Net profits of $902 million. 70,470 employees (75,200).
9 (9). L-3 Communications (United States). Electronics. Sales of $12.520 billion. Net profits of $956 million. 61,000 employees (63,000). 10 (10). United Technologies (United States). Aircraft, electronics, engines. Sales of $11.640 billion. Net profits of $5.347 billion. 199,900 employees (208,220).”
Source: (Moreno, 2017)
As can be seen, of the nine companies mentioned, six are American and the other three belong to countries allied with the United States. Even public officials can benefit from war, as has been the case with Donald Trump. According to (Boggioni, 2017), Trump owns shares in Raytheon, the company that manufactures Tomahawk missiles, and therefore benefited from having initiated the attack on Syria, because Raytheon’s shares appreciated, as can be seen below.
Figure 11
Source: (Raytheon, 2017)
Nor is the impact of wars on industries only partially connected with warfare negligible, such as steel producers, pharmaceutical companies, oil producers, among others.
In modern times, oil is fundamental to all productive processes, while natural gas also plays an important role in combustion, to the point that many regard it as the fuel of the future. If to the foregoing we add the savings in transport and commercialization costs represented by the possibility of moving these resources out in a straight line (through Syria), the significant quantity of resources the country possesses, its geopolitically strategic position on the map, and the law of the tendency of the average rate of profit to fall, it becomes entirely natural to understand the reasons why the United States conducts its current foreign policies toward Syria.
Puzzle de la Historia. (April 14, 2017). Construcción y Nacionalización del Canal de Suez. Retrieved from Puzzle de la Historia: http://www.puzzledelahistoria.com/?cat=2144
U.S. Department of the Army. (April 14, 2017). Syria. Retrieved from Federal Research Division of the Library of Congress: http://countrystudies.us/syria/10.htm
Weber, M. (2002). Economía y Sociedad. Madrid: Fondo de Cultura Económica.
[1] This refers to Durkheim’s definition of a social fact, which consists of “Every way of acting, whether fixed or not, capable of exercising an external constraint upon the individual; or again, every way of acting which is general throughout a given society while at the same time existing in its own right, independently of its individual manifestations.” (Durkheim, 2009, p. 68).
[2] In general, all sciences have as their ultimate aim the prediction of changes over time in their object of study.
[3] Rousseau is referred to here; the other is Thomas Hobbes.
[4] Of course, this is not a reference to a mechanistic approach in which the objective world determines the world of ideas. There is no doubt that the world of ideas has influence over and a capacity to transform the objective world, as Durkheim expressed. What is argued here is that, in the last instance, the material conditions of existence determine the world of ideas.
[5] For further discussion, one may consult Engels’s The Origin of the Family, Private Property and the State, as well as Chapter XXIV of Volume I of Marx’s Capital, entitled “The So-Called Primitive Accumulation,” for the case of Europe, or Rafael Menjívar Larín’s Acumulación Originaria y Desarrollo del Capitalismo en El Salvador for the general case of Latin America.
[6] Proven oil reserves are those that can be extracted profitably with current technology, taking into account the price of oil in the year being analyzed. These estimates are based on geological and engineering analyses, so the quantities established enjoy a high degree of confidence.
[7] The largest construction and civil-engineering company in the United States.
[8] Arab intellectual who serves as Professor of Modern Politics and History at Columbia University.
[9] Professor of History at Clark University.
[10] According to Google Maps, the distance between the two borders is 520.4 kilometers.
[11] A form of government in which power is shared by four persons who call themselves tetrarchs.
[12] Turks who conquered what is now known as Syria and its surroundings, founding that empire.
[13] Its name derives from the fact that it was secretly negotiated by François Georges-Picot (a French diplomat and lawyer) and Mark Sykes (a lawyer by profession and member of the English nobility).
[14] Military units formed during the period of the French Mandate in Syria, in addition to the Army of the Levant. These were French armed forces that occupied part of Western Asia between WWI and WWII and were composed of locally recruited personnel, who ultimately became the basis of the national armies of Syria and Lebanon.
[15] Due in some measure to the persecution they suffered from the so-called “Axis of Evil,” composed principally of Germany, Italy, and Japan.
[16] Countries that are in the process of negotiating their accession to the WTO.
[17] “The name by which the media identified the series of popular and political demonstrations that took place in the Arab region mainly from the beginning of 2011 and led to the fall of the dictatorships of Ben Ali in Tunisia and Hosni Mubarak in Egypt, the intensification of violence in Yemen, and the civil war in Libya. Some international analysts have found points of similarity between the Arab Spring and anti-crisis demonstrations in Europe, Asia, and the United States.” (EcuRed, 2017).
[18] When speaking of the existence of an average rate of profit, reference is made to a weighted average of sectoral rates of profit. The weighting factor is simply each firm’s share of the total capital existing in an economy—that is, the proportion that its investment represents of the total investment available to a society at a given moment in time. The existence of this average rate does not presuppose a static economy; on the contrary, it arises from the very economic dynamic by which capitalist firms rapidly migrate from one productive branch to another over time in search of the highest profit margins for their investments. It is precisely this process that produces an approximate equalization of sectoral profit rates and thereby makes the existence of the average rate of profit possible.
[19] In the case of market structures such as oligopolies, the issue can vary. Although most of the time they collude and behave as a monopoly, on some occasions they compete, and their analysis then becomes more similar to that of competitive market structures than to that of noncompetitive ones.
1.1. Argonne researchers posit way to locally circumvent Second Law of Thermodynamics For more than a century and a half of physics, the Second Law of Thermodynamics, which states that entropy always increases, has been as close to inviolable as any law we know. In this universe, chaos reigns supreme. But researchers with the U.S. Department of Energy’s (DOE’s) Argonne National Laboratory announced recently that they may have discovered a little loophole in this famous maxim. Their research, published in Nature Scientific Reports, lays out a possible avenue to a situation where the Second Law is violated on the microscopic level.
The Second Law is underpinned by what is called the H-theorem, which says that if you open a door between two rooms, one hot and one cold, they will eventually settle into lukewarm equilibrium; the hot room will never end up hotter.
But even in the twentieth century, as our knowledge of quantum mechanics advanced, we didn’t fully understand the fundamental physical origins of the H-theorem.
“What we did was formulate how these beautiful abstract mathematical theories could be connected to our crude reality.”
Recent advancements in a field called quantum information theory offered a mathematical construction in which entropy increases.
“What we did was formulate how these beautiful abstract mathematical theories could be connected to our crude reality,” said Valerii Vinokur, an Argonne Distinguished Fellow and corresponding author on the study.
The scientists took quantum information theory, which is based on abstract mathematical systems, and applied it to condensed matter physics, a well-explored field with many known laws and experiments.
“This allowed us to formulate the quantum H-theorem as it related to things that could be physically observed,” said Ivan Sadovskyy, a joint appointee with Argonne’s Materials Science Division and the Computation Institute and another author on the paper. “It establishes a connection between well-documented quantum physics processes and the theoretical quantum channels that make up quantum information theory.”
The work predicts certain conditions under which the H-theorem might be violated and entropy — in the short term — might actually decrease.
As far back as 1867, physicist James Clerk Maxwell described a hypothetical way to violate the Second Law: if a small theoretical being sat at the door between the hot and cold rooms and only let through particles traveling at a certain speed. This theoretical imp is called “Maxwell’s demon.”
“Although the violation is only on the local scale, the implications are far-reaching,” Vinokur said. “This provides us a platform for the practical realization of a quantum Maxwell’s demon, which could make possible a local quantum perpetual motion machine.”
For example, he said, the principle could be designed into a “refrigerator” which could be cooled remotely — that is, the energy expended to cool it could take place anywhere.
The authors are planning to work closely with a team of experimentalists to design a proof-of-concept system, they said.
The study, “H-theorem in quantum physics,” was published September 12 in Nature Scientific Reports. Other authors on the study were G.B. Lesovik of Russia’s L.D. Landau Institute for Theoretical Physics and Switzerland’s Theoretische Physik; A.V. Lebedev of Theoretische Physik; and M.V. Suslov of the Moscow Institute of Physics and Technology.
The study was supported by the U.S. Department of Energy’s Office of Science, the Swiss National Foundation, the Pauli Center for Theoretical Studies at ETH Zurich and the Russian Foundation for Basic Research.
Argonne National Laboratory seeks solutions to pressing national problems in science and technology. The nation’s first national laboratory, Argonne conducts leading-edge basic and applied scientific research in virtually every scientific discipline. Argonne researchers work closely with researchers from hundreds of companies, universities, and federal, state and municipal agencies to help them solve their specific problems, advance America’s scientific leadership and prepare the nation for a better future. With employees from more than 60 nations, Argonne is managed by UChicago Argonne, LLC for the U.S. Department of Energy’s Office of Science.
The U.S. Department of Energy’s Office of Science is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of the most pressing challenges of our time. For more information, visit the Office of Science website.
2. Additional Information, Part II
The second law of thermodynamics states that entropy cannot decrease. In classical physics, this follows from Boltzmann’s H-theorem, which applies to solutions of his kinetic transport equation. There is no quantum analogue of this equation that would allow a quantum H-theorem to be proved. Physicists at Argonne National Laboratory have published in Scientific Reports a quantum version of the H-theorem for systems whose evolution is governed by a unital quantum channel (channels are used in quantum information theory).
There are quantum channels that are not unital. For this reason, these physicists, as well as many media outlets reporting on the finding, suggest that local violations of the second law of thermodynamics may be observable (mediated by quantum Maxwell’s demons). However, the article does not present any concrete example; moreover, the two examples of non-unital channels that it does present comply with the second law. Future studies will have to determine whether such local violations are possible in real physical systems, as well as identify a real example of a quantum Maxwell’s demon.”
The article is G. B. Lesovik, A. V. Lebedev, …, V. M. Vinokur, «H-theorem in quantum physics,» Scientific Reports 6: 32815 (12 Sep 2016), doi: 10.1038/srep32815, arXiv:1407.4437 [quant-ph].
Ludwig Boltzmann’s H-theorem (1872, 1896) is the statistical foundation of the second law of thermodynamics. This theorem states that if is the spatiotemporal distribution of the density of the molecules of an ideal gas at time , at position , and with velocity , then the entropy given by cannot decrease; that is, . To prove this, Boltzmann’s equation for the function f is used under the assumption that the particles colliding in the gas have velocities independent of their positions (the so-called molecular chaos hypothesis). Quantum mechanics, however, relates positions and velocities through Heisenberg’s uncertainty relations.
In 1929, John von Neumann proposed a quantum explanation for the origin of entropy growth. Entropy is defined in terms of the density matrix of quantum mechanics as . It can be shown that this function does not decrease, even without resorting to a kinetic transport equation, by using the measurement procedure of quantum mechanics. Since then, many physicists have attempted to obtain a quantum version of Boltzmann’s H-theorem (a good overview can be found in the book by Jochen Gemmer, Mathias Michel, and Günter Mahler, Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems, Lecture Notes in Physics, Springer, 2009).
The new article draws on quantum information theory (QIT) and proposes describing the quantum dynamics of a system by means of a quantum channel (QC). In information theory, a (data) channel is the theoretical model of a transmission medium through which information-carrying signals travel from a sender to a receiver. Any quantum system can be interpreted as a (quantum) channel through which information flows. Using quantum channels makes it possible to obtain a quantum formulation of the H-theorem: entropy does not decrease if the evolution of the system is described by a unital quantum channel. The proof presented in the new article applies only to unital channels and makes no claim about non-unital channels.
A quantum channel is a positive, trace-preserving map of the density matrix, denoted ; it is said to be unital if . In general, a quantum channel need not be unital; see, for example, G. G. Amosov, “Estimating the output entropy of a tensor product of two quantum channels,” Theoretical and Mathematical Physics 182: 397–406 (2015), doi: 10.1007/s11232-015-0270-6. The new quantum H-theorem concerns unital channels, but it reaches no conclusion about non-unital ones. The authors of this work suggest that, in a quantum system whose evolution is described by a non-unital quantum channel, entropy might decrease locally and the second law of thermodynamics might therefore fail to hold.
In addition to the mathematical proof of the quantum H-theorem, the article provides three concrete examples of quantum systems modeled by quantum channels. The three figures in this post correspond to each of them. One is unital, while the other two are non-unital. However, in both non-unital systems the second law is still satisfied. As you have read, the article does not present any example of a non-unital channel in which the law is violated, even locally. The authors suggest that such violations may be possible, but they do not provide a demonstration of this claim. This detail appears to have been overlooked by some media outlets reporting on the story.
In short, this is a very interesting article, but one whose interpretation requires great care. Casually claiming that local violations of the second law of thermodynamics may exist in quantum systems is not the same as presenting a concrete example of such a violation. Quantum physics is subtle, and quantum information theory even more so. Claims of this kind should therefore be treated with caution.
Probability · Statistical Theory · Unsupervised Learning
From Bayes Estimators to Gaussian Mixtures
A guided reading of José Mauricio Gómez Julián’s 2020 study on
Bayesian decision theory, cluster analysis, density estimation,
the EM algorithm, and model-based clustering in R.
Research conducted in 2020José Mauricio Gómez JuliánApprox. 15-minute read
Statistics often becomes difficult not because any single idea is
impossible to understand, but because several ideas must suddenly be
held together at once. A probability distribution leads to an
estimator; an estimator leads to a loss function; a loss function
leads to an optimization problem; an optimization problem leads to an
algorithm; and the algorithm finally produces something that looks
deceptively simple on the screen: a handful of clusters. Gómez
Julián’s 2020 study, On Bayes Estimators, Cluster Analysis and
Gaussian Mixtures, is essentially an attempt to reconstruct that
entire chain of reasoning before asking R to perform the calculation.
Its destination is model-based cluster analysis with finite
Gaussian mixture models. But the paper deliberately takes the
long road. Before arriving at Gaussian mixtures and the
Mclust() routine, it moves through
posterior probability, Bayesian estimators, loss functions, mean
squared error, information criteria, data mining, machine learning,
supervised and unsupervised learning, density estimation,
k-means, the expectation-maximization algorithm, categorical
and Dirichlet distributions, optimization, parameters and
hyperparameters. The point is not simply to list definitions. It is to
show why these concepts meet inside the same statistical machine.
Reading note · what this article will do
The original 2020 study is much broader than a conventional software
tutorial. This guided reading therefore concentrates on its main
statistical architecture: how Bayesian reasoning,
cluster analysis, Gaussian mixture
models, the EM algorithm, and
BIC-based model selection fit together, and what
happens when that framework is applied in R.
The immediate motivation is educational research. Gómez Julián begins
from a doctoral study by Villegas Barahona concerned with academic
performance, latent dimensions, directly observed student variables,
CUR matrix decomposition, and the construction of a statistical model
capable of supporting academic and administrative decision-making.
That earlier project provides the practical problem; the 2020 study
asks what statistical theory one must understand in order to follow
the machinery being used.
This matters because a statistical package can make a difficult
procedure look trivial. A researcher can type a command, obtain a
classification, inspect a graph, and move on. Yet the command silently
presupposes answers to difficult questions. What is being estimated?
What does it mean for observations to belong to different groups?
What happens when those memberships are not observed? How should the
number and geometry of the groups be chosen? What quantity is the
algorithm maximizing? And how much uncertainty remains after a point
has been assigned to a cluster?
These are not merely programming questions. They are questions about
probability, inference, geometry and decision-making.
The paper’s distinctive strategy is therefore foundational: instead
of treating Gaussian mixture models as a black box, it reconstructs
the conceptual staircase leading to them.
A cluster on a computer screen is the final visible result of a much
longer argument about probability, hidden structure, estimation and
optimization.
Conceptual summary of Gómez Julián’s 2020 framework
There is another reason this approach is useful outside statistics.
Political scientists, economists, sociologists and public-policy
researchers frequently work with populations that are heterogeneous.
Countries, households, firms, voters or students may appear in one
dataset while actually belonging to several statistically distinct
subpopulations. If those subpopulations are not directly labelled,
the analytical task is no longer simply to estimate an average. It is
to infer the hidden structure that may have generated the observations.
That is where cluster analysis and
Gaussian mixture models eventually enter. But Gómez
Julián begins one layer deeper: with Bayes and the logic of updating
knowledge when new evidence arrives.
· · ·
02 · Bayesian foundations
Bayes as a rule for learning from evidence
At its simplest, Bayes’ theorem tells us how a probability should
change when we acquire relevant information. Suppose we have a
hypothesis \(H\) and observe some data \(D\). Bayesian updating
connects four quantities:
\( P(H\mid D)=\dfrac{P(D\mid H)\,P(H)}{P(D)} \)
posterior = likelihood × prior / marginal probability of the data
The prior, \(P(H)\), represents the state
of information before the new evidence is incorporated. The
likelihood, \(P(D\mid H)\), tells us how
compatible the observed evidence is with the hypothesis. The
denominator \(P(D)\) normalizes the calculation. The result,
\(P(H\mid D)\), is the posterior: the
probability conditional on having observed the new evidence.
For readers coming from economics, there is a useful analogy. Imagine
beginning with a set of beliefs about the likely position of an
economy, then receiving new information about employment, inflation
or production. The point of Bayesian updating is not that the old
information disappears. Rather, prior information and new evidence
are combined according to a precise probabilistic rule. The posterior
then becomes the informational starting point for whatever decision
comes next.
The key intuition
Bayes’ theorem is not yet a clustering algorithm. Its importance
here is more fundamental: Gaussian mixture models repeatedly
ask conditional-probability questions. Given an observed
data point, how probable is it that the point came from component
1, component 2, component 3, and so on? Once group membership is
hidden rather than directly observed, posterior probabilities
become a natural language for reasoning about that uncertainty.
This is one of the conceptual bridges that makes the paper coherent.
What begins as an abstract discussion of conditional probability will
later reappear in a very concrete form: each observation can carry a
probability of membership in each possible Gaussian
component. That is already a major difference between a Gaussian
mixture model and the familiar hard assignment produced by ordinary
k-means.
03 · Statistical decision-making
From posterior probability to a Bayes estimator
Updating probabilities is only part of the story. Eventually, an
analyst has to do something with the posterior
distribution. A parameter must be estimated, a prediction must be
produced, a model must be selected, or an observation must be
assigned—perhaps provisionally—to a group.
This is why Gómez Julián’s 2020 study moves from Bayes’ theorem into
decision theory. Once several possible estimates or
actions are available, the statistical problem can be expressed as a
question of consequences: if the unknown quantity is really
\(\theta\), what is the cost of reporting some estimate
\(\hat{\theta}\)?
That cost is represented by a
loss function. The exact form of the
function depends on what kinds of errors matter in the problem under
study. One particularly important case, and the one emphasized in
the paper, is squared-error loss:
\( L(\theta,\hat{\theta})=(\hat{\theta}-\theta)^2 \)
a larger distance between the estimate and the unknown parameter
produces a disproportionately larger loss
Squaring the error has two immediate consequences. First, positive
and negative deviations no longer cancel each other. Second, large
errors are penalized more heavily than small ones. The associated
expected loss is therefore closely connected with the familiar
mean squared error.
\( \mathrm{MSE}(\hat{\theta}) =
E_{\theta}\!\left[(\hat{\theta}-\theta)^2\right] \)
mean squared error as an expected measure of estimation error
Bayesian decision theory adds one decisive ingredient: rather than
evaluating loss while treating the parameter as an unknown fixed
object outside the probability calculation, the posterior
distribution is used to average the possible consequences of a
decision. The relevant quantity becomes the
posterior expected loss.
\( \rho(a\mid x)
= \int_{\Theta} L(\theta,a)\,
p(\theta\mid x)\,d\theta \)
posterior expected loss: consequences averaged over current
uncertainty about the parameter
A Bayesian decision rule selects the action that minimizes this
quantity. Under squared-error loss, something especially elegant
happens: the optimal estimate is the
posterior mean.
\( \hat{\theta}(x)
= E(\theta\mid x)
= \int_{\Theta}
\theta\,p(\theta\mid x)\,d\theta \)
Bayes estimator under quadratic loss
The intuition is straightforward. The posterior distribution
describes what values of \(\theta\) remain plausible after observing
the data. If squared distance is what we wish to minimize, then the
posterior mean is the point that minimizes the average squared
distance to all those possible values.
A useful distinction
Bayesian updating tells us how uncertainty changes
after observing evidence. Bayesian decision theory
tells us how to convert that updated uncertainty into an action.
The first produces a posterior distribution; the second combines
that posterior with a loss function.
This distinction becomes important later. A Gaussian mixture model
does not merely compute probabilities. It uses probabilities as part
of an iterative estimation problem in which unknown component
memberships and unknown component parameters have to be inferred
together.
The study also discusses the broader statistical idea of
Bayes risk: the expected loss associated with a
decision rule when uncertainty about the parameter is itself
represented probabilistically. Within the decision-theoretic
framework adopted in the paper, a Bayes estimator is the estimator
chosen because it minimizes the relevant expected loss.
Probability describes uncertainty; a loss function gives that
uncertainty consequences.
The bridge from inference to decision theory
· · ·
04 · Unsupervised learning
Why clustering is different from classification
The paper then changes scale. It moves from the estimation of an
unknown parameter toward a broader machine-learning question:
how can structure be discovered in a dataset when the
observations do not already come with known class labels?
This is the defining setting of
unsupervised learning. In supervised
learning, the training data contain an outcome or label that the
algorithm is asked to reproduce or predict. A model may learn, for
example, whether a loan applicant defaulted, which party a respondent
voted for, or what numerical value a dependent variable took.
In unsupervised learning there is no such answer key. The algorithm
receives observations and their measured characteristics, but not a
pre-existing declaration that observation 17 belongs to type A while
observation 18 belongs to type B. The structure itself must be
inferred from patterns in the data.
Classification versus clustering
In classification, classes are known during
training and the model learns how to assign new observations to
them. In clustering, the groups are not given in
advance. The method attempts to discover a useful grouping from
similarities, differences and distributional structure within the
observed data.
Gómez Julián places cluster analysis at the center of
this unsupervised-learning problem. In its most general form,
clustering means partitioning observations according to shared
characteristics so that observations within a cluster are relatively
similar and observations belonging to different clusters are
relatively dissimilar.
That formulation sounds simple, but it conceals one of the deepest
difficulties in clustering: there is not always one uniquely
obvious way to divide a dataset. The same cloud of points
may plausibly be described as containing two broad groups, several
narrower subgroups, or a hierarchy in which larger groups contain
smaller ones.
The study uses this ambiguity to distinguish two broad families.
Partitional clustering divides observations into
non-overlapping groups at a selected level. Hierarchical
clustering, by contrast, organizes groups within groups,
producing a nested structure that can often be represented as a
tree.
This is more than a technical distinction. It reminds us that a
cluster is not simply an object waiting in the data to be
photographed. A clustering procedure embodies a definition of what
similarity means, how distance is measured, what geometry is
permitted, and at what scale differences are considered important.
To ask how many groups are in a dataset is already to ask what
counts as a group.
Why clustering is fundamentally a modelling problem
This is particularly important for economists and political
scientists. Suppose countries are represented by unemployment,
literacy, poverty and public education expenditure. A clustering
algorithm may detect statistically distinct configurations of those
variables. But the resulting groups should not automatically be
treated as substantive political or economic “types.” Statistical
grouping is evidence about structure; interpretation still requires
theory and knowledge of the phenomenon being studied.
Before clustering, the paper also emphasizes the importance of
data preprocessing. Outliers, different measurement
scales, irrelevant variables and missing values can alter the
apparent geometry of the dataset. Normalization may therefore matter
when distance is central to the algorithm, while variable reduction
can be useful when irrelevant dimensions obscure rather than clarify
structure.
· · ·
05 · A first clustering model
What k-means actually assumes
To understand why Gaussian mixtures are useful, Gómez Julián first
introduces one of the best-known clustering algorithms:
k-means.
The basic idea is geometric. Choose a number of groups,
\(K\). Associate each group with a center, or
centroid. Then assign every observation
to the group whose centroid is closest. The centroids are updated
from the observations assigned to them, and the process is repeated
until the configuration stabilizes.
\( \displaystyle
\min_{C_1,\ldots,C_K}
\sum_{k=1}^{K}
\sum_{x_i\in C_k}
\lVert x_i-\mu_k\rVert^2
\)
the familiar k-means objective: minimize within-cluster squared
distance from each observation to its centroid
Even readers who have never implemented the algorithm can visualize
its logic. Imagine placing \(K\) pins on a map. Each observation is
sent to the closest pin. The pins are then moved to the centers of
the observations assigned to them, and the assignment is repeated.
Eventually the pins and memberships stop changing substantially.
This procedure is powerful and computationally convenient, but its
simplicity imposes a geometric structure. The distance-to-centroid
logic works most naturally when clusters are compact and roughly
spherical—or circular when visualized in two dimensions.
Real datasets need not cooperate. A cluster may be long and narrow,
tilted diagonally through the feature space, tightly concentrated in
one direction and widely dispersed in another. Two groups can also
overlap. Once these possibilities appear, distance to a single center
may no longer describe the structure adequately.
There is another limitation that becomes central to the paper.
Ordinary k-means makes what is known as a
hard assignment. An observation is placed in cluster
1 or cluster 2 or cluster 3. The algorithm does not naturally say:
“there is a 72% probability that this observation belongs to cluster
1 and a 28% probability that it belongs to cluster 2.”
Two limitations to remember
The transition from k-means to Gaussian mixtures in the
study is motivated by two especially important ideas:
cluster geometry and
uncertain membership. Gaussian mixtures can model
clusters with covariance structure and can assign probabilistic,
rather than purely deterministic, membership.
This prepares the central conceptual turn of the study. Instead of
thinking of a cluster merely as a collection of points around a
centroid, we can think of it as a
probability distribution.
· · ·
06 · Latent structure
The central idea: a population can be a mixture
Suppose we observe the distribution of some variable across an entire
population. At first glance we see only one dataset. But what if that
population is actually composed of several subpopulations generated
by different statistical processes?
This is the fundamental intuition behind a
mixture model. The overall probability
distribution is represented as a weighted combination of several
component distributions.
\( \displaystyle
f(x_i;\Psi)
=
\sum_{k=1}^{G}
\pi_k f_k(x_i;\theta_k)
\)
finite mixture model · each component has its own parameters and
contributes according to its mixture weight
Here \(G\) is the number of components.
\(f_k(x_i;\theta_k)\) is the density of component \(k\), determined by
its own parameters \(\theta_k\). The quantity \(\pi_k\) is the
component’s mixture weight, satisfying
\(\pi_k>0\) and
\(\sum_{k=1}^{G}\pi_k=1\).
The weights are important. If 70 percent of the population appears
to have been generated by one component and 30 percent by another,
the two component densities should not contribute equally to the
overall population density. The mixture weights encode their relative
prevalence.
But the deepest feature of the model is something we do
not directly observe: the
component identity of each observation. The dataset
contains \(x_i\), but it does not ordinarily contain an additional
column supplied by nature saying:
“this point was generated by Gaussian component 3.”
Component membership is therefore a
latent variable. It is hidden structure
inferred from the observed data.
Observed and latent quantities
The observations \(x_1,\ldots,x_n\) are visible. The component
labels that generated them are not. A finite mixture model therefore
links observable data to
unobservable group membership, while estimating
the parameters and relative weight of each component.
This is why the paper treats mixture models as naturally connected
to hierarchical and latent-variable modelling. There is one level at
which an observation belongs to some unobserved component, and
another level at which the observed value is generated according to
the probability distribution associated with that component.
If the component distributions are Gaussian, the model becomes a
Gaussian mixture model, or GMM:
\( \displaystyle
f(x)
=
\sum_{k=1}^{G}
\pi_k\,
\mathcal{N}(x\mid\mu_k,\Sigma_k)
\)
Gaussian mixture model · each latent group is represented by a
Normal density with its own mean and covariance structure
In one dimension, each component has a mean and a variance. In
several dimensions, the mean becomes a vector
\(\mu_k\), while dispersion and dependence among variables are
represented by the covariance matrix \(\Sigma_k\).
The covariance matrix is precisely what gives Gaussian mixture
models their geometric flexibility. It allows one cluster to be
narrow, another broad, another elongated, and another oriented along
a diagonal direction in multivariate space.
The paper therefore presents Gaussian mixtures as a probabilistic
generalization of the more rigid centroid-based intuition associated
with k-means. Instead of asking only which center is closest,
the model asks a richer question:
given the estimated distributions, how probable is it that
this observation came from each component?
\( \displaystyle
P(Z_i=k\mid x_i)
=
\frac{
\pi_k\,
\mathcal{N}(x_i\mid\mu_k,\Sigma_k)
}{
\sum_{j=1}^{G}
\pi_j\,
\mathcal{N}(x_i\mid\mu_j,\Sigma_j)
}
\)
posterior probability that observation i belongs to component k
Now the earlier discussion of Bayes becomes visibly relevant. The
model begins with component weights and component densities and,
conditional on an observed point, calculates updated probabilities
of component membership.
An observation near the center of one component and far from all
others may receive an assignment probability close to one. An
observation lying in an overlapping region may receive substantial
probability under two or more components. This is
soft classification: the uncertainty surrounding
membership is retained instead of being immediately discarded.
A Gaussian mixture does not merely divide the data. It proposes a
probabilistic account of how several hidden subpopulations could
have generated the observed population.
The core modelling idea of the 2020 study
But we have now reached an apparent circularity. To estimate the mean,
covariance and weight of each component, we would like to know which
observations belong to which component. Yet determining which
observations belong to which component is precisely what requires
knowing those means, covariances and weights.
Solving that circular problem is the task of one of the most important
algorithms in latent-variable statistics:
expectation-maximization.
07 · Hidden information
EM: learning when group membership is unknown
The difficulty facing a Gaussian mixture model can now be stated
precisely. We observe the data points, but we do not observe the
component from which each point was generated. If those memberships
were known, estimating the parameters of each Gaussian component
would be relatively straightforward. But the memberships themselves
depend on parameters that are still unknown.
Gómez Julián’s 2020 study approaches this problem through the
classical expectation-maximization algorithm, or
EM, developed by Dempster, Laird and Rubin. The
broader setting is estimation from incomplete data:
there is information that would make the estimation problem easier,
but that information is not directly observed.
In mixture modelling, the missing piece is especially intuitive.
Imagine that each row of the dataset secretly carries an additional
variable saying which Gaussian component generated it. If that hidden
variable were visible, we would have what can be thought of as the
complete data. In reality, only the measured
variables are observed.
The missing-data interpretation
In a Gaussian mixture model, the data point itself is observed,
but its generating component is latent. EM treats this hidden
information as the missing part of an otherwise more convenient
statistical problem.
The ingenious feature of EM is that it does not demand that this
missing information somehow become directly observable. Instead, it
alternates between two calculations. Each calculation makes the
other possible.
The E-step: estimate the hidden memberships
Begin with some current values for the component parameters: the
means, covariance matrices and mixture weights. Given those values,
calculate how probable it is that each observation belongs to each
component.
These probabilities are often called
responsibilities. Component \(k\) takes
responsibility for observation \(i\) in proportion to how plausible
that observation is under the component’s Gaussian density and how
prevalent that component is in the mixture.
\( \displaystyle
\gamma_{ik}
=
P(Z_i=k\mid x_i,\Psi)
=
\frac{
\pi_k\,
\mathcal{N}(x_i\mid\mu_k,\Sigma_k)
}{
\sum_{j=1}^{G}
\pi_j\,
\mathcal{N}(x_i\mid\mu_j,\Sigma_j)
}
\)
E-step intuition · estimate the probability that each observation
belongs to each Gaussian component
Notice what has happened. The hard, unknown statement “observation
\(i\) belongs to cluster \(k\)” has been replaced with a set of
probabilities. An observation may be overwhelmingly associated with
one component, or it may sit in an overlapping region and divide its
probability between several components.
The M-step: update the model
Once those expected memberships have been calculated, the algorithm
turns the problem around. It now treats the probabilistic memberships
produced by the E-step as information for re-estimating the
parameters.
The means, covariance matrices and mixture weights are updated so
that the likelihood of the observed data increases under the newly
estimated mixture.
One EM cycle
E-step: using the current model parameters,
estimate the hidden component memberships.
M-step: using those estimated memberships,
re-estimate the model parameters by maximizing the relevant
likelihood criterion.
Then the algorithm goes back to the E-step. The new parameters imply
new membership probabilities; those new probabilities imply new
parameter estimates; and the process continues iteratively.
\( \Psi^{(0)}
\rightarrow
\text{E-step}
\rightarrow
\text{M-step}
\rightarrow
\Psi^{(1)}
\rightarrow
\text{E-step}
\rightarrow
\text{M-step}
\rightarrow \cdots \)
expectation and maximization alternate until the fitted solution
stabilizes
The process stops when the parameter estimates—or equivalently the
likelihood improvements—change so little that the algorithm is
considered to have reached convergence.
This makes EM a particularly elegant response to the apparent
circularity encountered at the end of the previous section. We needed
cluster membership to estimate the distributions, but we needed the
distributions to estimate cluster membership. EM solves the problem
by alternating between the two conditional tasks.
Estimate what is hidden using the current model; then improve the
model using what you have just estimated.
The iterative logic of expectation-maximization
There is an important qualification. EM is an optimization algorithm,
not a magical guarantee that every possible starting point will lead
to the globally best solution. Mixture-model likelihoods can contain
multiple local optima. Initialization and model specification can
therefore matter. What EM guarantees at the operational level is an
iterative procedure for improving the likelihood until a stationary
solution is reached.
· · ·
08 · Statistical geometry
Why Gaussian mixtures can see ellipses
The next step in Gómez Julián’s argument is geometric. In one
dimension a Gaussian distribution is described by a mean and a
variance. Move into two or more dimensions, however, and variance is
no longer sufficient. The relationships among variables must also be
represented.
This is the role of the covariance matrix,
\(\Sigma_k\). For component \(k\), the mean vector \(\mu_k\)
determines its center, while \(\Sigma_k\) determines how the
probability mass spreads through multivariate space.
\( X\mid Z=k
\sim
\mathcal{N}(\mu_k,\Sigma_k) \)
each Gaussian component possesses its own center and covariance
geometry
In two dimensions, contours of equal Gaussian density form
ellipses. This provides an intuitive way to read
covariance. A nearly circular ellipse indicates similar dispersion
in different directions. An elongated ellipse indicates much greater
variation along one direction than another. A tilted ellipse signals
covariance between the variables.
This is precisely where Gaussian mixture clustering becomes more
flexible than the elementary geometric picture supplied by
k-means. A centroid alone tells us where a cluster is
centered. A covariance matrix also tells us its
volume, shape and orientation.
Think geometrically
Two clusters may have centers that are equally far apart while
still being statistically very different. One may be compact and
almost circular; another may be broad and strongly elongated.
Gaussian mixture models can represent this difference because the
covariance matrix is part of the model.
The mclust framework studied in
the paper exploits this fact systematically. Instead of fitting only
one possible covariance structure, it considers a family of Gaussian
models obtained by placing different restrictions on the
volume, shape and
orientation of the component ellipsoids.
Gómez Julián discusses the 14 multivariate Gaussian models available
in the version of mclust studied
in the 2020 research. Their compact names—such as
EEE,
VEV,
VVI or
EEV—encode restrictions on those
geometric properties.
Example
Geometric idea
EEE
Equal volume, equal shape and equal orientation across
components
VEV
Variable volume, equal shape and variable orientation
VVI
Diagonal covariance structure with variable volume and shape
EEV
Equal volume and shape, with orientation allowed to vary
These codes are not decorative software jargon. They describe
competing statistical hypotheses about the geometry of the hidden
groups. Should all clusters have the same spread? May one be larger
than another? Must their ellipses point in the same direction? Is a
diagonal covariance matrix enough, or does the data require rotated
ellipsoids?
Seen this way, model-based clustering is doing more than deciding
where to draw boundaries. It is comparing alternative
generative descriptions of the data.
In model-based clustering, the shape of a cluster is not an
afterthought. It is part of the hypothesis being estimated.
Covariance as statistical geometry
· · ·
09 · Model selection
BIC and the problem of choosing a model
Gaussian mixtures create a new problem precisely because they are
flexible. We may fit different numbers of components, and for each
number of components we may consider different covariance
structures. Which model should be preferred?
Maximized likelihood alone is not enough. Adding parameters usually
gives a model more freedom to accommodate the observed data, so raw
fit can improve simply because the model has become more complicated.
If complexity is never penalized, the procedure is pushed toward
increasingly elaborate specifications.
This motivates the
Bayesian Information Criterion, introduced by
Gideon Schwarz and discussed at length in the 2020 study. In one
common notation,
\( \mathrm{BIC}
=
-2\log \hat{L}
+
k\log n \)
conventional minimization form · fit is balanced against a penalty
that increases with model complexity
Here \(\hat{L}\) is the maximized likelihood, \(k\) is the number of
estimated parameters and \(n\) is the sample size. The first term
rewards fit; the second penalizes additional parameters.
An equivalent sign convention is often written so that
larger values are preferred:
\( \displaystyle
\log \hat{L}
–
\frac{k}{2}\log n
\)
Schwarz’s maximization form · the same fit-versus-complexity logic
expressed with the opposite orientation
A practical warning about signs
Readers sometimes see “choose the smallest BIC” in textbooks and
then encounter mclust output
where the preferred model has the largest BIC
value. This is a matter of convention. The criterion can be written
with opposite signs. What matters is using the convention adopted
by the software or source consistently.
In mclust, BIC therefore becomes
the mechanism for comparing combinations of
component number and
covariance parametrization. The software can fit a
collection of candidate Gaussian mixture models and compare them
rather than forcing the researcher to stipulate one geometry in
advance.
Conceptually, this is a competition among explanations. A
one-component model says that a single Gaussian population is
sufficient. A two-component model says that two latent
subpopulations provide a better account after accounting for the
additional parameters. A five-component model makes an even more
elaborate claim. BIC asks whether the gain in likelihood is large
enough to justify that extra complexity.
What BIC is doing in this paper
BIC acts as a bridge between estimation and
model selection. EM estimates the parameters of a
candidate Gaussian mixture. BIC helps decide which candidate
structure—among different numbers and geometries of components—is
comparatively preferable.
This distinction is essential. EM does not, by itself, answer every
modelling question. Given a specified mixture structure, it provides
a way to estimate its parameters. Model selection operates at another
level: it compares alternative structures.
The result is a layered procedure. First define candidate probability
models. Then estimate them. Then compare them. Finally inspect the
resulting classification and ask whether the statistical structure is
substantively meaningful.
\( \text{candidate models}
\rightarrow
\text{EM estimation}
\rightarrow
\text{BIC comparison}
\rightarrow
\text{selected clustering structure} \)
the model-based clustering workflow developed toward the applied
section of the study
We are now ready for the final step of the 2020 investigation:
seeing what this machinery actually produces in R. Gómez Julián
closes the substantive analysis with two types of application. The
first uses the canonical Iris dataset; the second moves into
social and economic data from the World Bank, combining indicators
of education expenditure, literacy, unemployment and poverty.
10 · Applied examples in R
From Iris flowers to World Bank indicators
After more than one hundred pages of theoretical preparation, Gómez
Julián’s 2020 study finally lets the statistical machinery run. This
last substantive section is useful precisely because the preceding
discussion changes the meaning of what would otherwise look like a
few lines of R code. By this point, a call to
Mclust() is no longer merely a
software command. It invokes finite Gaussian mixtures, latent
membership, maximum-likelihood estimation through EM, alternative
covariance geometries and BIC-based model comparison.
The paper provides two kinds of illustration. First comes the
canonical Iris dataset distributed with R. Then the analysis
moves to a dataset assembled from World Bank
indicators, bringing the method into a setting much closer
to economics, political science and public-policy research.
How to read the output
When Mclust() reports a model
such as VEV,
EEE or
VVI, it is describing the
covariance structure selected for the Gaussian components. When it
reports a number of components, it is describing the number of
mixture components preferred by the model-selection procedure
among the candidates fitted.
The Iris example
The first application uses the four familiar quantitative variables
in the Iris dataset. Gómez Julián runs:
mod1 <- Mclust(iris[,1:4])
summary(mod1)
Gaussian model-based clustering of the four measured Iris variables
The reported solution is a
VEV Gaussian finite mixture with
two components. The 150 observations are partitioned
into clusters containing 50 and
100 observations, respectively. The reported
log-likelihood is \(-215.726\), while the output gives a BIC of
\(-561.7285\) and an ICL of \(-561.7289\).
Dataset
Selected structure
Clustering
Iris
VEV · 2 Gaussian components
50 / 100 observations
The accompanying plots make visible the two layers of the procedure.
One panel compares BIC values across candidate covariance models and
different numbers of components. Another displays the resulting
classification across pairs of the measured variables. The graph is
therefore not merely showing clusters after the fact: it also gives
the reader a view of the model-selection problem that produced them.
This example is deliberately straightforward. Its role is to show
that the theoretical discussion of mixture densities, EM,
covariance structure and BIC can be condensed operationally into a
remarkably short piece of R code.
· · ·
A social-science example using World Bank data
The second application is more directly connected to the concerns of
economists and policy researchers. Gómez Julián constructs an
example using World Bank data and four variables for 2018:
Variables used in the 2020 application
Public expenditure on education as a percentage
of GDP; unemployment as a percentage of the total
labour force; poverty incidence according to the
national poverty line; and the adult literacy rate
for persons aged 15 and above.
The R workflow imports the separate datasets, selects the 2018
observations, joins them by country, removes rows for which the
required combination contains missing values, and then applies
Mclust() to the resulting
numerical variables.
When all four indicators are considered together, only
13 complete observations remain in the dataset used
by the code. The reported model is
EEE with
nine components: ellipsoidal Gaussian clusters with
equal volume, equal shape and equal orientation.
\( n=13,\qquad G=9,\qquad
\text{model}=\mathrm{EEE} \)
four-variable World Bank example reported in the study
The output reports a log-likelihood of \(-77.62892\), 54 degrees of
freedom, BIC \(-293.7651\) and ICL \(-293.7727\). The component
counts are extremely small: the nine clusters contain respectively
1, 2, 2, 1, 1, 1, 2, 2 and 1 observations.
Gómez Julián then repeats the model-based clustering exercise using
smaller combinations of variables. This is particularly revealing
because the available sample size changes sharply depending on which
World Bank indicators must be simultaneously observed.
Variables
n
Model
Components
BIC
Education expenditure + literacy
39
XXI
1
−475.3468
Education expenditure + unemployment
71
VVI
2
−679.1997
Education expenditure + poverty
17
EEV
5
−193.6785
The contrast is striking. For public education expenditure and adult
literacy, the fitted solution contains only
one component among 39 complete observations. For
education expenditure and unemployment, the selected model is
VVI with
two components, containing 49 and 22 observations.
For education expenditure and poverty, the result is an
EEV specification with
five components, whose sizes are 3, 3, 4, 4 and 3.
These examples demonstrate something that can easily disappear when
one speaks abstractly about “the number of clusters.” The number of
components is not an intrinsic number attached forever to a set of
countries. It depends on the variables being modelled, the available
observations, the candidate covariance structures and the statistical
criterion used to compare those models.
An important inferential boundary
The output reported in this section is
cluster analysis. It describes statistical
structure found by the fitted Gaussian mixture models. By itself,
such an exercise does not establish that education expenditure
causes unemployment, literacy or poverty, nor does it estimate the
magnitude of a causal effect. Those would be different inferential
questions requiring a different research design.
This distinction is especially valuable for policy analysis. A
cluster can reveal that some countries occupy similar regions of a
multivariate statistical space. That may motivate substantive
investigation. It does not, by itself, explain historically or
causally why those countries occupy that region.
The four-variable result deserves similar care. Nine components from
only thirteen complete observations is exactly the kind of output
that should be read together with the sample size and model
complexity rather than reduced to the phrase “nine types of
countries.” The study reports the statistical fit; substantive
interpretation requires returning from the model to the empirical
object being studied.
· · ·
11 · The larger lesson
What the 2020 study is really teaching
The most important feature of Gómez Julián’s investigation may be its
refusal to begin with the software. The paper could have been a short
tutorial showing how to call
Mclust(), inspect BIC and plot a
classification. Instead, it constructs a long conceptual route from
probability and estimation to the final clustering output.
That route matters because the elements are genuinely connected.
Bayesian reasoning introduces conditional
probability and updating. Decision theory explains
how probability distributions can be connected to estimators and
loss. Unsupervised learning introduces the problem
of discovering structure without known labels.
Cluster analysis gives that problem a statistical
form. Mixture models reinterpret an apparently
homogeneous population as the superposition of latent
subpopulations. Gaussian mixtures give those
subpopulations flexible probabilistic geometry.
EM estimates models whose membership information is
hidden. And BIC provides a way to compare competing
specifications.
\( \text{Bayes}
\rightarrow
\text{estimation}
\rightarrow
\text{latent variables}
\rightarrow
\text{mixtures}
\rightarrow
\text{EM}
\rightarrow
\text{BIC}
\rightarrow
\text{clustering} \)
a compressed map of the conceptual route reconstructed in the
2020 study
One can also read the paper as an argument for understanding
statistical methods structurally. A model is not simply an equation.
It includes assumptions about what is observable, what is latent,
which probability family describes the data, how parameters are
estimated, what geometries are permitted and how competing
specifications are compared.
Gaussian mixture models make that point unusually visible. The
observable cloud of data is only the surface. Beneath it lies a
proposed generative structure: component distributions, latent
memberships, mixture weights, means and covariance matrices. The
analyst does not observe this machinery directly. It is inferred.
The visible dataset is the starting point. The statistical model is
a hypothesis about the hidden structure capable of producing it.
A central intuition running through the study
This is also why the difference between
hard and soft classification is so
significant. Saying that a country, person or flower belongs to
“cluster 2” suppresses information. A Gaussian mixture can instead
preserve the fact that an observation may lie near the frontier
between several plausible components. Probability makes ambiguity
measurable.
Likewise, covariance transforms clustering from the simple idea of
distance from a center into a richer account of statistical
geometry. Groups may differ not only in location but also in
dispersion, shape and orientation. And BIC reminds us that greater
flexibility comes at a cost: a model must earn its additional
complexity through improved fit.
For economists, econometricians and political scientists, perhaps the
most transferable lesson is therefore methodological. If a
population may contain qualitatively different statistical regimes,
forcing every observation into a single homogeneous distribution can
conceal structure. Mixture models offer one formal way of asking
whether the aggregate pattern may instead be generated by several
latent components.
But the converse warning is equally important. Discovering a
statistically preferred partition does not relieve the researcher of
the obligation to understand the real phenomenon. A component is a
component of a statistical model. Whether it corresponds to a
meaningful social class, institutional regime, developmental
configuration, biological population or merely a feature of the
available sample must be established with substantive knowledge and
further evidence.
In one sentence
Gómez Julián’s 2020 investigation is a theoretical guided tour of
the ideas required to understand how
Gaussian finite mixture models can discover latent
structure in unlabeled data, how
EM estimates that structure, and how
model-selection criteria such as BIC help decide which
probabilistic representation to retain.
· · ·
12 · Closing perspective
Statistics before software
There is a useful reversal at the heart of this study. Modern
statistical computing encourages us to begin with a function and
discover afterward what it does. Gómez Julián’s 2020 text proceeds
in the opposite direction: first reconstruct the mathematical and
statistical concepts, then approach the function.
That choice makes the paper unusually broad. Posterior probability,
Bayes estimators, loss functions, BIC, data mining, machine learning,
clustering, density estimation, vector quantization,
k-means, EM, categorical and Dirichlet distributions,
optimization, parameters, hyperparameters and Gaussian finite
mixtures all appear because the final procedure stands at the
intersection of those ideas.
For a technically trained reader, the value of this route is that it
exposes the architecture hidden beneath a familiar R command. For a
reader from political science, economics or another applied field,
it offers something equally useful: an intuitive path into a method
that otherwise arrives wrapped in matrix algebra and probability
notation.
And the practical lesson is simple. When an algorithm reports that
the data contain one group, two groups or five, the interesting
question is not merely what did the software return? It is:
what statistical model made that answer possible, what
assumptions gave the groups their shape, and what kind of statement
about reality is the result actually capable of supporting?
Good statistical practice begins where the automatic output ends:
with an attempt to understand what the model has actually measured.
Final reflection on Gómez Julián’s 2020 study
José Mauricio Gómez Julián · 2020
On Bayes Estimators, Cluster Analysis and Gaussian Mixtures:
A General Theoretical Analysis of densityMclust in R
and Statistical Theory
When Counts Refuse to Behave: Understanding the Negative Binomial II as a Measurement Instrument
A guided reading of José Mauricio Gómez Julián’s 2020 essay Some Reflections on the Negative Binomial Distribution II as a Measurement Instrument—tracing the argument from geometric series and probability mixtures to overdispersion, latent heterogeneity, and simulation in R.
Reading note. This essay explains the paper on its own terms while keeping the mathematics technically precise. Where a qualification is needed—especially in the passage from an exponential mixing distribution to the general Negative Binomial II—it is marked explicitly rather than silently altering the paper’s argument.
The question behind the paper
Count data appear everywhere. An economist counts firm failures, strikes, defaults, patents, accidents, or entries into a market. A political scientist counts protests, cabinet changes, violent events, legislative vetoes, or international disputes. A biologist counts surviving organisms, mutations, infections, or offspring. The elementary model for many such problems is the Poisson distribution. But real counts often fluctuate more than a Poisson model allows. Their variance is larger than their mean: the data are overdispersed.
The Negative Binomial II—usually abbreviated NB2—is one of the central statistical instruments for precisely that situation. Yet Gómez Julián’s paper is not content to present the NB2 probability mass function, list its moments, and move on. Its organizing question is broader: what kind of object is a probability distribution, where does it come from, what other distributions does it contain or presuppose, and what is gained when we understand its construction rather than merely its final formula?
The paper therefore has two explicit axes. The first, and more important one, studies NB2 as the outcome of a wider theoretical structure involving hierarchical models and probability mixtures. The second studies NB2 as an individual distribution—its form, interpretation, moments, and practical use, including hand calculations and R. The intended unity between these axes is philosophical as well as mathematical: probability distributions are treated as instruments for measuring natural and social phenomena under uncertainty, and the paper argues that their history, formal structure, scientific interpretation, and application should not be torn apart.
The distribution is easier to understand when we see not only the finished formula, but also the process that produces it.
· · ·
Why begin with the geometric series?
The paper begins surprisingly far away from count-data regression: with the geometric series, its historical roots, and its relation to the binomial theorem. This is deliberate. Gómez Julián wants the reader to see the Negative Binomial II as part of a mathematical genealogy rather than as a formula that appeared fully formed.
The route is roughly this: the geometric series provides a simple infinite expansion; differentiation exposes a recurring combinatorial pattern; that pattern is used to motivate the binomial expansion; and replacing the ordinary exponent with a negative one leads to the negative binomial series. The point is not merely algebraic. The paper repeatedly emphasizes the movement from simpler structures to more general ones and from one family of mathematical objects to another.
1 + x + x2 + x3 + ··· = 1 / (1 − x), |x| < 1
↓
binomial expansion
↓
negative binomial series
The paper’s algebraic genealogy in compressed form
For a nontechnical reader, the important idea is simple: a probability distribution can be understood through the transformations and relationships that generate it. The paper later reinforces this visually with a large network diagram of probability distributions, reproduced from ProbOnto, in which distributions are connected by transformations, limiting relations, and special cases. NB2 is therefore presented as one node in a densely connected mathematical ecology, not as an isolated technique.
Probability as a measurement problem
Before building the NB2, the paper stops to ask what “probability” means. This is not a decorative philosophical detour. Gómez Julián’s position is that formal probability calculus and philosophical interpretation cannot be completely divorced, because statistical conclusions depend on what we think probabilities are measuring and on how the scientific problem is conceptualized.
The paper adopts an explicitly objective and dialectical-materialist orientation. Randomness is treated primarily as an epistemological condition: events appear random because their causes are unknown, too numerous, or too complicated to represent completely. At the same time, the Kolmogorov axioms provide the formal mathematical framework that makes probability calculations coherent. The paper also draws on objective Bayesianism to argue that degrees of belief should be constrained by evidence, scientific theory, and the probability calculus rather than reduced to arbitrary personal opinion.
Why this matters statistically
A model is not chosen only because its formula is convenient. The scientific description of the process determines which random variables, conditional relationships, latent quantities, and parameterizations are meaningful. That principle becomes concrete once the paper turns to hierarchical models.
A family, not an isolated formula
The paper next introduces the distributions needed for its construction: Bernoulli, Binomial, Poisson, and Exponential. Each plays a distinct role.
Distribution
Plain-language role
Role in the paper
Bernoulli
One trial with two possible outcomes.
The elementary unit from which repeated success/failure experiments are built.
Binomial
Counts successes in a fixed number of Bernoulli trials.
The first level of the hierarchical construction.
Poisson
Counts occurrences when events arrive with a given mean rate.
Makes the number of opportunities or events itself random.
Exponential
Models waiting time or positive continuous variation associated with a Poisson process.
Introduces variation in the Poisson rate across observational units.
This sequence already contains the paper’s methodological intuition. A complex phenomenon can be decomposed into simpler probabilistic stages, each corresponding to a different part of the scientific story. Instead of forcing all uncertainty into a single flat formula, a hierarchical model lets uncertainty enter at more than one level.
Hierarchies, mixtures, and latent variables
A hierarchical model specifies variables and parameters in stages. A mixture model appears when a parameter in one probability distribution is itself treated as a random quantity governed by another distribution. The parameter that seemed fixed at the lower level becomes variable at the higher level.
That is the key conceptual move in the paper. It is also why Gómez Julián brings in the language of latent variables: some of the forces producing observed variation may not be directly observed, but their consequences can still be represented probabilistically. In applied work, this is familiar. Two factories, municipalities, firms, hospitals, or individuals may face different underlying event rates even if we initially write one common Poisson equation for all of them.
The paper links this mathematical construction to the Hegelian distinction between Being-in-itself and Being-for-itself. Stripped of the philosophical vocabulary, its statistical meaning is fairly intuitive. Studying NB2 “in itself” means studying the wider process and network of relationships from which it emerges. Studying NB2 “for itself” means taking the marginalized distribution as a distinct object and examining its own formula, parameters, moments, and applications.
The first mixture: Binomial inside Poisson
The first major construction uses a biological example. Imagine an insect that lays many eggs. Conditional on a mother having laid Y eggs, each egg survives independently with probability p. The number of surviving eggs, X, is therefore Binomial. But instead of fixing the number of eggs Y, the paper lets it vary according to a Poisson distribution with mean λ.
X | Y ~ Binomial(Y, p)
Y ~ Poisson(λ)
First hierarchical model
Marginalizing means asking for the distribution of X after summing out the intermediate variable Y. Algebraically, we combine all the possible values of Y, weighted by how probable each one is. The result is elegant:
X ~ Poisson(λp)
After marginalizing over Y
In modern probability language, this is a version of Poisson thinning. If the total number of opportunities is Poisson and each opportunity independently survives with probability p, then the surviving count is also Poisson, with its mean reduced from λ to λp. The paper also derives the same conclusion through iterated expectations:
E[X] = E{E[X | Y]} = E[pY] = pλ
For the paper, this is more than a computational trick. It shows how a hierarchical process that appears to contain two random layers can be “compressed” into a simpler marginal law without erasing the scientific interpretation that motivated the hierarchy.
When the Poisson rate itself varies
The next step introduces another level of heterogeneity. Suppose there are many insect mothers, and the Poisson mean is not the same for every mother. The paper now treats the rate itself as random:
X | Y ~ Binomial(Y, p)
Y | Λ ~ Poisson(Λ)
Λ ~ Exponential(β)
Three-stage hierarchy in the paper
The statistical intuition is important. Variation does not occur only in the observed count; it can also occur in the underlying rate that generates the count. Once the rate differs across observational units, the final distribution becomes more dispersed than a single-rate Poisson model. This is precisely the kind of latent heterogeneity that makes the negative binomial family useful in economics, epidemiology, demography, political science, and many other count-data settings.
The paper shows that the mean of the full hierarchy can be obtained by repeatedly conditioning and averaging, arriving at E[X] = pβ under its parameterization. It then integrates out the random Poisson rate and identifies the resulting expression with the negative binomial form.
Technical qualification added for accuracy
There is an important distinction here. An Exponential distribution is a Gamma distribution with shape parameter equal to 1. Therefore, a Poisson–Exponential mixture produces the geometric distribution, which is a special case of the negative binomial with r = 1. The general NB2 with arbitrary dispersion parameter r arises from a Poisson–Gamma mixture. Thus, the paper’s core intuition—random heterogeneity in the Poisson rate generates negative-binomial-type overdispersion—is correct, but the fully general NB2 requires the Gamma mixing family rather than the Exponential distribution alone.
This qualification actually sharpens the paper’s broader message. The geometric distribution, the exponential distribution, the gamma family, Poisson processes, and the negative binomial are not unrelated objects. They sit inside a network of special cases and mixture relationships. The more general Gamma mixing distribution preserves the same hierarchical logic while extending the model beyond the r = 1 case.
· · ·
The NB2 “for itself”: what the finished distribution tells us
Once the hierarchy has been “compressed,” the paper turns to the Negative Binomial II as an object in its own right. One common parameterization writes the probability of observing x failures before the r-th success as:
Here, p is the success probability and r is the target number of successes. The distribution answers a reversed version of the familiar Binomial question. The Binomial asks: how many successes occur in a fixed number of trials? The negative binomial asks: how many failures occur before a fixed number of successes is reached?
The paper’s most important statistical property appears in its first two moments. With q = 1 − p:
μ = E[X] = rq / p
Var(X) = rq / p2
= μ + μ2/r
That final equality is the bridge to modern count-data econometrics. The Poisson distribution imposes Var(X) = μ. NB2 allows:
Var(X) = μ + αμ2, where α = 1/r
In other words, variance can grow faster than the mean. The smaller r is—or, equivalently, the larger the heterogeneity parameter α is—the more dispersion the model permits. As heterogeneity vanishes, NB2 approaches the Poisson benchmark. This is why the paper presents NB2 as a more flexible instrument for count data when the Poisson equality between mean and variance is too restrictive.
Feature
Poisson
Negative Binomial II
Mean
μ
μ
Variance
μ
μ + αμ²
Extra heterogeneity
Not separately modeled
Captured through α (or r)
Typical use
Equidispersed counts
Overdispersed counts
For an econometrician, this variance function is often the fastest route to understanding NB2. For a broader reader, an intuitive translation is enough: NB2 expects the world to be more uneven than a simple Poisson process. Some units have persistently higher event rates than others; unobserved conditions vary; clusters form; the same mean can coexist with much wider dispersion.
The maintenance example: counting failures before the fourth alarm
The paper gives a concrete industrial interpretation. Imagine fixed capital—a machine—producing parts. A part is either satisfactory or defective. A monitoring system treats a defective part as the event of interest because it signals the need for maintenance. Assume independent Bernoulli trials and a constant defect probability.
Gómez Julián sets r = 4, p = 0.005, and asks for the probability of observing exactly x = 100 failures before the fourth success, using the negative binomial mass function. Substitution gives:
P(X = 100 | r = 4, p = 0.005)
≈ 0.000067
The number is tiny—about 6.7 chances in 100,000. The statistical interpretation is not that “100 failures happen and then four successes happen” as two separate blocks. Rather, among an ongoing sequence of independent trials, exactly 100 non-events occur before the fourth event of interest is reached.
The paper checks the result manually, with a Texas Instruments calculator, and with R. It then plots the corresponding distribution, illustrating how a small success probability pushes substantial probability mass toward relatively large counts before the required number of successes is accumulated.
What the R simulation is trying to show
The final applied part of the main text shifts from evaluating a probability to generating pseudo-random data. The paper constructs a custom card-drawing experiment: repeatedly sample from a 52-card deck until a specified rank appears, record how many draws were required, and repeat the experiment many times. It then compares the histogram produced by that “from first principles” counting procedure with a histogram generated by R’s built-in rnbinom function.
The figures for 50, 100, 150, and 200 repetitions show the same qualitative pattern: a strongly right-skewed count distribution with many small values and a long tail. The pedagogical purpose is clear. Software is not magic. A built-in random generator is implementing a probabilistic structure that can also be approximated through an explicit sequence of elementary trials.
A convention to watch in R
R’s rnbinom convention counts the number of failures before a specified number of successes. A hand-built routine that counts the total number of draws including the successful draw differs by one when size = 1. For an exact one-to-one comparison, the manual routine and the software call should use the same counting convention. This does not erase the pedagogical point of the simulation, but it matters for exact numerical equivalence.
The broader lesson is useful far beyond R. Simulation can reveal what a distribution means operationally: not merely how its formula looks, but what repeated mechanism would generate data with that shape.
NB1, Bayes, and the annexes: why the paper keeps widening the frame
The annexes extend the same relational approach. The paper distinguishes NB2 from NB1, emphasizing that different negative-binomial parameterizations answer slightly different counting questions. In the NB1 presentation used there, the random variable is the total number of Bernoulli trials needed to reach r events of the chosen type; in the NB2 presentation, the random variable is restricted to the number of failures before those r successes.
The paper also returns to conditional probability, total probability, inverse probability, and objective Bayesianism. This may seem far removed from overdispersed count data, but it serves the same philosophical thesis: statistical formulas must be understood through the relationships they encode. Conditional probability is not merely a ratio; it represents a structured dependence between events. Bayesian updating is not merely algebra; it connects prior knowledge, evidence, and posterior assessment.
Finally, the paper gives a more general NB2 expression in terms of the Gamma function and reports estimators based on the first two sample moments. This again links the abstract distribution to empirical work: the population parameters acquire meaning only because sample information provides a route to estimation.
What should we take away?
Gómez Julián’s paper is best read as an extended argument against treating the Negative Binomial II as a black-box formula. Its distinctive contribution is not a new NB2 estimator or a new regression algorithm. It is an attempt to reconstruct the distribution through several layers at once: historical, algebraic, probabilistic, computational, applied, and philosophical.
For the nontechnical reader, the central statistical lesson can be stated in one sentence: when counts vary more than a single-rate Poisson model permits, the extra variability can often be understood as heterogeneity in the underlying event rate, and the negative binomial family provides a natural way to represent it.
For the econometrician, the key signature is the NB2 variance function, Var(Y) = μ + αμ². For the mathematician, the paper is an invitation to follow the transformations linking geometric series, binomial expansions, gamma functions, conditional distributions, and marginalization. For the philosopher of science, its central claim is methodological: the formal instrument, the scientific object, the history of the instrument, and the interpretation of uncertainty should be studied in relation rather than isolation.
And for the applied researcher, perhaps the most useful question is the simplest one: what process would have to be operating for this distribution to be a sensible measurement instrument? Once that question is asked, NB2 stops being just a convenient correction for overdispersion. It becomes a hypothesis about how heterogeneity enters the data-generating process.
A distribution is most informative when its probability law and its generating story tell the same scientific story.
In one compact map
Bernoulli gives the elementary success/failure trial;
Binomial aggregates such trials at a fixed size;
Poisson makes the number or rate of occurrences stochastic;
Gamma heterogeneity lets that Poisson rate vary across units;
marginalizing the latent rate yields the Negative Binomial II, whose variance can exceed its mean.
This explainer follows the architecture and substantive aims of Gómez Julián’s paper while separating the paper’s own philosophical framing from the technical qualifications added here for mathematical precision.
When a Regression Looks Excellent — and Still May Be Wrong
What money, nominal GDP, gold, and the U.S. economy from 1990 to 2011 can teach us about the difficult art of specifying an econometric model.
A GUIDED READING OF GÓMEZ JULIÁN (2020)
RAMSEY RESET · ARCH · OLS · TIME SERIES · MODEL SPECIFICATION
· · ·
A regression can look spectacular and still be asking the wrong question. It can explain almost all of the observed variation in a variable, produce highly significant coefficients, draw a fitted line that hugs the data remarkably closely—and nevertheless contain a structural flaw. That tension is the real subject of Gómez Julián’s study of hypothesis testing and the optimal specification of a regression model. The empirical setting is monetary: the paper examines the U.S. money supply between 1990 and 2011 in relation to nominal GDP and the international price of gold. But the deeper lesson concerns econometric reasoning itself. How do we move from a model that merely fits to one whose specification we have serious statistical reasons to trust?
That distinction matters far beyond monetary economics. Political scientists estimate the effects of institutions on growth, epidemiologists study relationships between exposures and outcomes, sociologists model inequality, and economists routinely estimate relations among prices, production, investment, employment, money, and financial variables. In every case, a regression equation compresses a complex process into a manageable mathematical form. The danger is obvious: a convenient equation can conceal dynamics that the equation itself does not represent.
The central methodological idea
The paper treats hypothesis tests not as decorative statistics added after a regression has already been estimated, but as instruments for interrogating the model itself. A high coefficient of determination is not the end of the investigation. It is the beginning of a sequence of questions: Are the residuals independent? Is their variance stable? Are the parameters stable? Is the assumed functional form adequate? Are the time series stationary? Could an apparently excellent fit be spurious?
This guided reading
The economic hypothesis behind the regression
Why the United States, and why 1990–2011?
The deceptively impressive first OLS result
Why residual diagnostics change the story
Ramsey RESET and the problem of functional form
ARCH, volatility, and dependence hidden in squared errors
Unit roots, spurious regression, and cointegration
What the investigation ultimately establishes—and what remains open
1. The economic model under examination
The empirical exercise does not begin from a purely data-driven search for correlations. It begins from an economic proposition inherited from an earlier 2016 study. In the formulation used here, the quantity of money in circulation is treated as a function of the prices of commodities in the economy and of the international price of gold, under simplifying conditions concerning the velocity of circulation and the relative stability of the variables during the period under examination.
For the U.S. application, commodity prices are represented empirically by nominal GDP—called Current GDP in the dataset—while the second explanatory variable is the international gold price. Money in circulation, denoted by M, becomes the dependent variable.
Conceptual relation
M = f(Current GDP, Gold)
M → quantity of money in circulation
Current GDP → nominal output, used in the paper as an empirical expression of commodity prices
Gold → international price of gold
This direction of explanation is important. The paper is not merely asking whether money, output, and gold happen to move together. Its theoretical framework assigns a specific economic meaning to their relationship. The statistical model is therefore supposed to represent a prior conception of how the monetary process works.
That distinction between statistical relationship and economic meaning later becomes one of the paper’s recurring themes. A regression can identify mathematical dependence among measured variables, but the interpretation of that dependence does not emerge automatically from the regression coefficients. It is supplied by the theoretical framework within which the statistical model has been constructed.
A regression can tell us how variables move together under a particular specification. It cannot, by itself, tell us what those movements mean.
This is especially important for readers outside economics. Imagine that two political indicators move almost perfectly together over twenty years. The regression does not know whether one causes the other, whether both respond to a third process, whether their common trend creates a misleading correlation, or whether the relation changes after an institutional break. Those are questions about the structure of the phenomenon—and therefore about model specification.
2. Why study the United States from 1990 to 2011?
The paper deliberately uses the same broad historical interval as the earlier study while moving the empirical setting from El Salvador to the United States. The choice is not presented as a neutral convenience. Gómez Julián argues that the social and economic structure of the United States makes the period particularly useful because it contains episodes that could expose weaknesses in a simple regression specification.
The interval includes the long expansion associated with the post-Cold-War era, the rise and collapse of the dot-com boom, the financial crisis that erupted in 2008, and the first years of its aftermath. In particular, the observations for 2009, 2010, and 2011 are treated as analytically valuable because they incorporate the macroeconomic disturbance following the financial crisis.
Why those last three observations matter
A model may behave quite well during relatively tranquil periods and reveal its weaknesses only when the system is disturbed. Including 2009–2011 therefore acts, in effect, as a small stress test: does the relationship estimated from the preceding years survive when the economy moves through an unusually turbulent episode?
The paper repeatedly insists that economic time series are records of historical processes rather than interchangeable observations floating in abstract statistical space. The year 2009 is not simply “observation number 20.” It follows 2008. The history that produced it matters. This idea becomes crucial once autocorrelation, structural stability, unit roots, and cointegration enter the discussion.
This historical perspective also explains why the paper resists treating every statistical problem as something that can simply be removed through data cleaning. If an economic series contains persistence, structural change, or stochastic trends, those properties may reflect features of the economic process itself. A model should therefore try to understand them before mechanically eliminating them.
3. The first regression looks almost too good
The starting empirical model is a conventional multiple linear regression estimated by ordinary least squares. In simplified notation,
Initial OLS specification
Mt = β0 + β1GDPt + β2Goldt + εt
For a reader encountering econometrics for the first time, the idea is straightforward. The regression chooses the values of the coefficients β that make the predicted money supply as close as possible—in the least-squares sense—to the observed money supply. The residual ε for each year is the part that the equation fails to explain.
And at first sight, the results are extremely impressive.
Term / statistic
Estimate
Reported p-value
Intercept
−2.130 × 10¹²
7.67 × 10⁻⁶
Current GDP
0.8392
6.29 × 10⁻¹⁴
Gold
2.044 × 10⁹
8.10 × 10⁻⁵
R²
0.9834
—
Adjusted R²
0.9817
—
Overall F statistic
563.3
< 2.2 × 10⁻¹⁶
An R² of roughly 0.983 means that, within this fitted linear specification, approximately 98.3 percent of the observed variation in the dependent variable is accounted for by the regressors. Both slope coefficients also appear extremely significant under the conventional t tests. If one judged the regression only by these headline numbers, the temptation would be to declare victory.
98 percent explained variation sounds like the end of the story. In this paper, it is where the interesting story begins.
Why? Because statistical significance is conditional. A t statistic has the interpretation we normally give it only if the probabilistic structure assumed by the model is sufficiently appropriate. An R² can be enormous because two variables genuinely share an economically meaningful relation—but it can also become enormous when several non-stationary time series drift through time in broadly similar directions.
Likewise, a visually excellent fit does not prove that the functional form is correct. A straight line may approximate a curved relationship surprisingly well over a short sample. Nor does it establish that the unexplained component behaves as required by the classical regression framework.
The residuals therefore become witnesses. If they contain systematic temporal structure, changing variance, nonlinear information, or persistent patterns, then the model has left something important behind.
4. What does “well specified” actually mean?
This is the conceptual bridge between the economic question and the statistical machinery that occupies the rest of the study. A regression model is not adequately specified merely because its fitted values resemble the observations. Its mathematical form must also be compatible with important features of the data-generating process.
01Functional form
Is the assumed linear relationship sufficient, or do squared, cubic, logarithmic, or otherwise nonlinear terms contain systematic information that the original model misses?
02Error structure
Are regression errors approximately independent and homoscedastic, or does the past help predict their magnitude or variance?
03Temporal structure
Are the variables stationary, or do persistent stochastic trends make conventional regression inference potentially misleading?
The paper therefore deploys a broad diagnostic battery: the Ramsey RESET specification test, ARCH tests, White’s heteroskedasticity test, variance inflation factors for collinearity, residual autocorrelation tests, residual normality tests, CUSUM and CUSUM-of-squares procedures for parameter stability, and—later in the investigation—unit-root and cointegration techniques.
The philosophical point behind that battery is worth emphasizing. No single diagnostic test is treated as an oracle. Different tests examine different potential failures, and they can even produce apparently conflicting indications. Model evaluation therefore becomes cumulative: one result raises a suspicion, another narrows the possibilities, a modification removes one defect but perhaps exposes another, and the investigation gradually moves toward a more adequate representation.
A useful way to think about hypothesis testing
Instead of asking only, “Is this coefficient statistically significant?”, the paper encourages a broader sequence of questions: What assumption is being tested? What would rejection imply about the model? If a defect is detected, what new specification should be considered? And after modifying the model, what new diagnostics become necessary?
This transforms hypothesis testing from a ritual of stars and p-values into a method of model criticism. In that sense, the tests are not merely judging individual coefficients. They are helping the researcher decide whether the mathematical representation itself should survive in its present form.
5. The first crack: the errors remember the past
The paper’s subsequent diagnostics show why the impressive R² cannot be taken at face value. In the empirical discussion, a significant problem of residual autocorrelation emerges. In plain language, the unexplained error associated with one year is not behaving as though it were disconnected from errors in surrounding years.
That matters because ordinary least squares is at its cleanest when the error process does not carry precisely this kind of systematic temporal dependence. If residuals are autocorrelated, the equation is effectively saying: “I have explained the systematic part of the process,” while the residuals answer: “No—you have left some systematic time structure inside me.”
Intuition
Corr(εt, εt−1) ≠ 0
If today’s unexplained component contains information about yesterday’s unexplained component, the errors are not behaving as independent noise.
In a genuinely historical system this is hardly an exotic possibility. Production in one year depends partly on previous production; investment decisions inherit previous expectations and balance sheets; monetary magnitudes accumulate institutional and financial history. Time links observations together.
The crucial econometric question is therefore not whether such dependence can exist—it obviously can—but whether the chosen model has represented it adequately. If it has not, temporal dependence migrates into the residuals.
The study responds by introducing an additional term constructed from changes in nominal GDP. This modification improves the residual autocorrelation diagnostics substantially. Yet the author explicitly acknowledges an important limitation: the choice is empirically motivated rather than derived from an independent theoretical justification, and the paper does not claim to have searched exhaustively over the optimal number of lags, compared every plausible dynamic specification, or estimated the full family of VAR-type alternatives.
An important limitation acknowledged by the study
The modified specification should therefore be understood as a provisional diagnostic solution, not as the final uniquely correct model. Its purpose is methodological: to show how one statistical test can reveal a problem, motivate a modification, and force the researcher to test the new model again.
And that is exactly what happens. Fixing one problem does not end the investigation. Once the residual autocorrelation appears to improve, other diagnostics—especially the behavior of squared errors and the Ramsey specification tests—raise a more difficult possibility:
What if the problem is not merely that the regression forgot part of the past, but that the relationship itself has been written in the wrong mathematical form?
That is where the paper’s two central diagnostic tools enter the story: the ARCH test, which asks whether the variance of the errors carries temporal structure, and the Ramsey RESET test, which asks whether nonlinear combinations contain explanatory information that the original regression has failed to capture.
Those tests are the subject of the next stage.
End of Stage I · Next: RESET, ARCH, CUSUM and the anatomy of misspecification
· · ·
6. A model can pass one test and fail another
Once the residuals become the object of investigation, an important principle emerges: there is no single thing called “the regression assumptions.” Different diagnostic tests ask different questions about different aspects of the model. Passing one of them therefore does not imply passing the others.
This distinction becomes unusually clear in the paper’s U.S. regression. The residuals look reasonably well behaved in several respects. The normality test does not provide sufficient evidence to reject Gaussian residuals. The variance inflation factors are modest, so severe multicollinearity does not appear to be driving the estimates. White’s test likewise provides no strong evidence of ordinary heteroskedasticity.
And yet a residual-autocorrelation test detects temporal dependence.
Diagnostic
Null hypothesis
Reported result
Reading at 5%
White
No heteroskedasticity
p ≈ 0.232
Do not reject
Breusch–Godfrey LM, order 1
No residual autocorrelation
p ≈ 0.023
Reject
ARCH, order 1
No ARCH effect
p ≈ 0.991
Do not reject
Residual normality
Residuals are normally distributed
p ≈ 0.105
Do not reject
There is nothing logically contradictory about that combination. The errors may have an approximately constant variance and an approximately normal marginal distribution while still being serially dependent. In other words, the size and shape of the error distribution can look acceptable even though errors occurring close together in time are related.
One residual, several questions
White asks whether the variance changes systematically. Breusch–Godfrey asks whether errors are serially correlated. ARCH asks whether the conditional variance itself displays temporal dependence. A normality test asks about the shape of the residual distribution. These are related questions, but they are not interchangeable.
This is one reason diagnostic work cannot be reduced to a checklist in which every test is interpreted as another version of “good model” versus “bad model.” Each test illuminates a different part of the statistical structure.
7. Ramsey RESET: asking whether the equation has the wrong shape
Residual autocorrelation raises the possibility that something systematic has escaped the regression. But it does not by itself tell us exactly what. One possibility is temporal dynamics. Another is an omitted variable. A third is that the assumed mathematical form of the relationship is itself inadequate.
The Ramsey Regression Specification Error Test—RESET—is designed as a broad diagnostic for precisely this sort of misspecification.
The intuition is easier than the name suggests. Begin with a fitted regression:
Original model
Mt =
β0 +
β1GDPt +
β2Goldt +
εt
From that equation we obtain fitted values, M̂. RESET then asks whether nonlinear transformations—such as M̂² or M̂³, or related nonlinear transformations of the regressors—contain explanatory information that the original equation failed to capture.
RESET intuition
Mt =
β0 +
β1GDPt +
β2Goldt +
γ1M̂t2 +
γ2M̂t3 +
ut
NULL HYPOTHESIS → the additional nonlinear terms do not improve the model in a statistically significant way.
REJECTION → the original specification has left systematic nonlinear information unexplained.
This does not mean that RESET discovers the uniquely correct alternative equation. A rejection does not tell us, by itself, “use a quadratic model” or “add precisely this missing variable.” RESET is better understood as a smoke detector: it can tell us that something about the specification deserves attention without necessarily identifying the source of the smoke.
The results are revealing—but not uniform
The paper runs several versions of RESET in R. This matters because the conclusion depends somewhat on how the additional nonlinear information is constructed.
RESET variant in R
Power
p-value
At α = 0.05
Fitted values
2
0.06463
Do not reject
Regressors
2
0.1216
Do not reject
Principal-component form
2
0.04188
Reject
Fitted values
3
0.01402
Reject
Regressors
3
0.1177
Do not reject
Principal-component form
3
0.05739
Do not reject
That pattern deserves a careful interpretation. There is evidence of specification trouble, but not a unanimous verdict across every implementation. Two versions reject adequate specification at the conventional 5 percent threshold, while several others do not. Some of the non-rejections are nevertheless close to conventional thresholds.
The Gretl analysis tells a similar story. A specification test using squares and cubes jointly reports a p-value of roughly 0.037, providing evidence against the null of adequate specification. A logarithmic nonlinearity test also rejects linearity at approximately the 5 percent level, while other nonlinear formulations do not.
The important result is therefore not “every possible RESET test rejects.” It is subtler: the apparently excellent linear regression is vulnerable to nonlinear specification diagnostics.
This distinction is fundamental. If several plausible transformations reveal predictive structure that the original regression leaves unexplained, the investigator no longer has good reason to regard the initial linear form as obviously sufficient.
And notice how far we have moved from the seductive R² of 0.983. The regression still fits the observed money series extremely closely. Nothing has happened to erase that fit. What has changed is our interpretation of what that fit proves.
8. ARCH: when the errors are calm today because they were calm yesterday
The second test emphasized by the paper is the ARCH test associated with Robert Engle’s work on autoregressive conditional heteroskedasticity.
Its central insight is that ordinary autocorrelation and volatility dependence are different phenomena. Suppose the signs of successive errors are difficult to predict: a positive error may be followed by a negative one, then another positive one. Standard autocorrelation could therefore be weak.
But suppose large errors tend to arrive near other large errors, while small errors tend to cluster around small errors. The signs may appear irregular while the magnitudes clearly remember the past.
The key transformation
εt → εt2
Squaring removes the sign and emphasizes magnitude.
If ε² at time t is systematically related to earlier ε² values, the variance contains temporal structure.
This is the core of the ARCH idea. The paper introduces it partly through the language of heavy-tailed distributions, but the technically decisive point is more specific: an ARCH test is a test for dynamics in conditional variance, not simply a generic test for heavy tails. Heavy tails and volatility clustering can be related empirically, but they are not the same property.
In the initial 1990–2011 regression, the reported first-order ARCH p-value is approximately 0.991. At the conventional 5 percent level, there is therefore no evidence for rejecting the null hypothesis of no first-order ARCH effect.
This produces an instructive combination:
ASerial dependence
The ordinary residuals show evidence of first-order autocorrelation.
BConstant variance
White’s test does not provide evidence of general heteroskedasticity.
CNo initial ARCH
The squared residuals do not show evidence of first-order ARCH dependence.
So the temporal problem detected at this stage concerns the residuals themselves rather than a clear clustering of their squared magnitudes. That diagnostic distinction becomes important when the model is modified.
9. CUSUM versus CUSUM of squares: two views of stability
Another question is whether the regression coefficients remain stable throughout the sample. If the relationship among money, nominal GDP, and gold changes substantially over time, a single set of coefficients estimated for 1990–2011 may be concealing several different regimes.
The paper approaches this through CUSUM and CUSUM of squares. Both procedures examine cumulative residual behavior, but they do not have identical statistical properties or identical sensitivity to different forms of instability.
And here the results appear to disagree.
The apparent conflict
The ordinary CUSUM analysis indicates parameter instability beginning around the transition from 2006 to 2007. The CUSUM-of-squares graph, by contrast, remains within its confidence boundaries and therefore points toward parameter stability over the sample.
The paper does not simply average these conclusions. It turns to methodological literature comparing the tests through Monte Carlo simulations and gives greater analytical weight to CUSUM of squares, which the cited research describes as more reliable under several forms of misspecification and more resistant in settings involving non-predetermined regressors, cointegration, or weak serial correlation.
This is another useful lesson in statistical practice. Two tests can disagree without either calculation being mechanically “wrong.” Tests have different power against different alternatives. They react differently to the location, form, and magnitude of structural change. Their assumptions also matter.
Statistical evidence is not produced by counting how many tests say “yes” and how many say “no.” One must ask what each test can actually detect.
Still, the ordinary CUSUM result remains economically suggestive. The timing is difficult to ignore: instability appears near the years immediately preceding the 2008 crisis. The paper treats this not as proof that the crisis has been statistically detected in advance, but as an observation deserving further investigation.
That caution is important. A diagnostic graph can reveal an unusual pattern. Explaining the historical cause of that pattern requires more than the graph itself.
10. Repairing the model creates a new model—and therefore new problems
Having identified residual autocorrelation, the paper experiments with an additional GDP-related term. This is where the logic of hypothesis testing becomes especially visible: diagnosis leads to respecification, and respecification immediately requires another round of diagnosis.
A notation that requires care
The paper describes the added variable in places as involving the second differences of nominal GDP, while also identifying it as Current GDP (t−2) and naming the Gretl variable CurrentGDPt2. These are not mathematically identical operations: a two-period lag, GDPt−2, is different from a second difference, Δ²GDPt. For that reason, the safest reading is to preserve the paper’s reported variable and results while recognizing that its verbal description and notation are not fully aligned.
The author is explicit about another limitation: there is no independent theoretical argument offered for choosing this particular modification. It is adopted because the empirical results improve relative to an alternative tried in the study. Nor does the investigation claim to have performed an exhaustive search over optimal lag structures, Monte Carlo lag selection, VAR models, or every competing dynamic specification.
That restraint matters. The added term is a provisional econometric repair, not a declaration that the final structural model has been discovered.
The modified regression nevertheless produces another strikingly high fit:
Statistic
Modified model
Interpretive point
R²
≈ 0.9904
Fit improves still further
Adjusted R²
≈ 0.9888
Remains extremely high
Durbin–Watson
≈ 0.879
Still far below the benchmark value near 2
White test
p ≈ 0.054
Borderline, but not rejected at 5%
The Breusch–Godfrey sequence becomes especially revealing. For lag orders two through six, the reported p-values rise above 0.05. Yet the first-order test remains significant, with a p-value of roughly 0.020.
A careful reading therefore requires a qualification. The paper subsequently describes the modification as solving the autocorrelation problem across the examined lag structure, but the displayed first-order LM result still rejects the null of no autocorrelation at 5 percent. The safest conclusion from the numerical output is consequently that the modification substantially changes and weakens the detected autocorrelation pattern, but does not unambiguously eliminate first-order residual autocorrelation.
Why this matters methodologically
A researcher should privilege the reported statistical output over the desire for a tidy narrative. If one diagnostic remains problematic, the model remains diagnostically open—even when most of the other numbers improve.
Something else happens as well. In the modified model, RESET becomes much more forceful. Gretl reports p-values of approximately 0.0036 for a squares-only specification test and 0.0014 for tests involving cubes and the joint squares-and-cubes formulation.
In other words, adding temporal structure may alleviate one problem while making another easier to see. The improved fit does not rescue the linear specification. On the contrary, the nonlinear diagnostics now give stronger evidence that the equation is still leaving systematic structure unexplained.
The ARCH diagnostics also become more interesting. The first-order ARCH test in the modified specification reports a p-value of approximately 0.069. That is not significant at 5 percent, but it is below 10 percent. The higher-order tests reported for orders two and three are much less suggestive of an ARCH effect.
This is why the paper regards the altered ARCH behavior, together with the persistently low Durbin–Watson statistic, as a warning rather than as a clean resolution of the model’s problems.
11. Hypothesis testing as an iterative method of criticism
At this point the investigation has reached a result more important than any individual p-value.
The original model looked exceptionally strong by conventional headline measures. Diagnostics then revealed residual dependence. A modified specification changed that dependence but did not make every warning disappear. RESET continued—and in some versions became even more emphatic—in suggesting that the linear form was incomplete.
The workflow therefore looks less like a one-shot estimation problem and more like a sequence:
The diagnostic cycle
Theory
→ Specification
→ Estimation
→ Diagnostics
→ Criticism
→ Respecification
→ New Diagnostics
This is the methodological heart of the paper. Hypothesis tests are valuable not because a p-value mechanically declares what reality is, but because carefully chosen tests can reveal where a particular mathematical representation ceases to behave as it should.
And the next problem is more fundamental still.
Everything discussed so far presumes that conventional regression inference is meaningful for these time series. But GDP, the money supply, and gold prices evolve historically and often trend strongly through time. If those series are non-stationary, then an extraordinarily high R² and spectacular t statistics may arise even when a conventional regression is statistically deceptive.
Before asking whether the coefficients are impressive, one must ask whether the time series permit those coefficients to mean what we think they mean.
That takes the investigation from regression diagnostics into one of the deepest problems in applied time-series econometrics: unit roots, spurious regression, and cointegration.
And there the apparently simple monetary regression becomes a much more interesting object.
End of Stage II · Next: Unit Roots, Spurious Regression and Cointegration
· · ·
12. The deeper problem: what if the series themselves do not stand still?
The specification diagnostics have already complicated the apparently excellent OLS regression. But the paper now introduces a problem that reaches deeper than residual autocorrelation or an omitted nonlinear term: non-stationarity.
To understand why this matters, consider what a conventional regression quietly hopes for. If we observe a variable over time, we would like its probabilistic behavior to be sufficiently stable that observations from different years remain meaningfully comparable. In the weak form of stationarity emphasized in the paper, the mean and variance remain constant through time, while the covariance between two observations depends on the distance separating them rather than on the calendar date at which they occur.
Weak stationarity — intuition
E(Yt) = μ
·
Var(Yt) = σ²
·
Cov(Yt,Yt−k) = γ(k)
The statistical environment does not systematically change merely because the calendar advances.
Economic series frequently make this assumption difficult. Nominal GDP grows over decades. Monetary aggregates expand. Asset prices can wander for long periods. A shock today can alter not merely the next observation but the level around which future observations evolve.
This is the setting in which the concept of a unit root becomes important.
Take the simplest autoregressive process:
AR(1) representation
Yt = ρYt−1 + ut
If |ρ| is comfortably below one, shocks tend to fade. But when ρ = 1, the equation becomes
The unit-root case
Yt = Yt−1 + ut
ΔYt = ut
A shock enters the level of the series permanently rather than being mechanically pulled back toward a fixed mean.
In the lag-operator notation discussed in the paper, this is
Lag-polynomial form
(1 − L)Yt = ut
Setting the lag polynomial equal to zero gives 1 − L = 0, hence L = 1: the “unit root.”
The practical consequence is crucial. A unit-root process carries a stochastic trend. Its path depends cumulatively on past shocks. A disturbance is not merely temporary noise around an unchanged trajectory; it becomes part of the history from which the next observation begins.
In a stationary world, shocks fade around a stable statistical center. In a unit-root world, shocks become part of the path.
This is one reason the paper devotes considerable attention to the historical character of economic time series. Its broader argument is that persistence should not automatically be treated as dirty data to be removed before “the real” analysis begins. Persistent movement may itself contain information about the underlying economic process.
13. A macroeconomic dispute becomes a lesson in time-series econometrics
At this point the paper makes a long historical detour through the debate surrounding the 2008–2009 recession, especially the disagreement involving Gregory Mankiw, Paul Krugman, Brad DeLong, and others over whether output losses associated with recessions should be expected to disappear through unusually rapid subsequent growth.
The technical issue beneath the polemics is highly relevant to the paper. Imagine that output collapses during a recession. There are two very different statistical stories one might tell.
ATrend-stationary story
The recession pushes output temporarily below an underlying path. Once the shock disappears, unusually strong growth can return output toward that old trajectory.
BUnit-root story
The recession changes the level from which the economy subsequently evolves. Growth may resume, but part of the lost output is not automatically recovered.
CEmpirical question
Deciding between these stories is not semantic. It changes both forecasting and the interpretation of macroeconomic shocks.
The paper sides strongly with the view that economists should not presume automatic return to an old trend. It invokes both the unit-root literature and later evidence that, on average across countries, output following crises may remain permanently below its previous trend path.
For the monetary regression, however, the importance of this debate is methodological rather than merely historical. If macroeconomic series can inherit shocks permanently, then fitting one trending level series against another requires considerable caution.
And that leads to a famous statistical trap.
14. The beautiful regression that may mean nothing
Suppose two completely unrelated variables both rise persistently with time. A regression of one on the other can produce a high R², apparently significant t statistics, and an elegant fitted line—even though there is no meaningful relationship connecting them.
This is the classic problem of spurious regression.
Why non-stationarity is dangerous
When unrelated non-stationary variables share persistent trends, ordinary regression machinery may mistake common trending behavior for substantive explanatory power. The usual t and F inference can then become unreliable.
This possibility suddenly makes the original result—R² ≈ 0.983—look rather different. Earlier it appeared to be the model’s greatest achievement. Now its sheer magnitude, combined with a relatively low Durbin–Watson statistic, becomes part of the reason for investigating whether the fit might be deceptive.
The paper invokes the traditional rule of thumb associated with this problem: when an estimated time-series regression displays a very high R² together with a much lower Durbin–Watson statistic, suspicion of spurious regression is warranted. The rule is diagnostic rather than a proof, so the analysis cannot stop there.
A high R² is evidence of fit. It is not, by itself, evidence that the variables share a genuine long-run relation.
This is perhaps the most important reversal in the investigation. The same statistic can look reassuring at the beginning of an analysis and suspicious later, once we know more about the temporal properties of the data.
15. Dickey–Fuller: testing whether shocks die or accumulate
To investigate the problem, the paper turns to the Dickey–Fuller family of unit-root tests, especially the Augmented Dickey–Fuller test (ADF) and the ADF–GLS / DF–GLS variant.
A convenient way to see the logic is to rewrite the autoregressive process in changes:
ADF logic — simplified
ΔYt
=
α + βt + γYt−1
+
Σ δiΔYt−i
+
εt
NULL HYPOTHESIS → unit root.
If γ is consistent with zero, the previous level does not generate the type of mean reversion required to reject the unit-root hypothesis.
The lagged differences are included to absorb additional serial dependence.
The test can be estimated under different deterministic specifications: without a constant or deterministic trend, with a constant, or with both constant and trend. That choice is not cosmetic. A series that looks non-stationary under one specification may behave differently once a deterministic trend is explicitly modeled.
The paper also emphasizes lag selection and employs information criteria—including variants of AIC and BIC—because too few lags can leave residual autocorrelation behind, while too many can consume substantial degrees of freedom in an already tiny sample.
A crucial statistical clarification
In standard hypothesis-testing language, a p-value is not the probability that the null hypothesis is true, nor the probability that rejecting it would constitute a Type I error. It is the probability, assuming the null hypothesis and the test model are true, of obtaining a test statistic at least as extreme as the one observed. Throughout this guided reading, p-values are interpreted in that conventional frequentist sense.
What happens to the variables?
The resulting picture is not clean enough to permit a one-line declaration that every series behaves identically.
For the money supply, the paper reports unit-root evidence in levels under the specifications it examines. The behavior of nominal GDP and the GDP-related term differs across the tests and specifications, while the gold-price series also changes classification depending on whether a constant and deterministic trend are included.
The important point for the argument is therefore not that “all four variables possess exactly the same order of integration.” The important point is that non-stationarity is sufficiently pervasive to make the validity of a levels regression a serious econometric question.
The paper then examines first differences.
Differencing
ΔYt = Yt − Yt−1
Instead of modeling the level—say, the stock of money—we model its change from one period to the next.
If a series is I(1), differencing it once should ordinarily produce an I(0), or stationary, process. But the paper’s results are again mixed. Some first-difference specifications become stationary, while others continue to exhibit unit-root behavior depending on the deterministic terms included.
Series discussed
First-difference result reported by the paper
Nominal GDP
Results depend on the specification; the paper finds stationary behavior in relevant variants but not uniformly in every formulation.
Money supply
Unit-root behavior without deterministic terms; stationarity appears when constant and/or trend specifications are considered.
Gold price
Unit root in the specifications without trend; stationarity is obtained when drift and deterministic trend are incorporated.
This prevents differencing from serving as a simple universal cure in the paper’s exercise. The investigation consequently returns to the levels relation and asks a more interesting question.
16. Two non-stationary series can still belong together
Non-stationarity does not automatically imply that a regression is spurious.
Suppose two variables wander over time, each carrying a stochastic trend. If they wander independently, regressing one on the other may indeed create a meaningless statistical relationship. But suppose their long-run movements are linked so that a particular linear combination of them is stationary.
Then the two series are said to be cointegrated.
Cointegration — simplest intuition
Yt ~ I(1),
Xt ~ I(1)
but
Yt − βXt ~ I(0)
Each series can wander individually while their long-run discrepancy remains bounded.
A useful image is two hikers walking along an irregular mountain path while connected by a rope. Neither remains in a fixed location; both may wander far from where they began. But they cannot drift arbitrarily far apart. Their individual coordinates are non-stationary, while the distance imposed by the rope remains constrained.
Cointegration says that variables may wander without wandering independently.
This distinction rescues an important class of long-run economic relationships from the spurious-regression problem. A levels regression among non-stationary variables can be meaningful if those variables share a genuine cointegrating relation.
The paper therefore moves from asking about the variables individually to asking about their joint stochastic structure.
17. Engle–Granger: look at what remains after the long-run relation
The first route explored is the Engle–Granger logic. Estimate a candidate long-run relation and obtain its residuals:
Candidate long-run equation
Mt
=
β0
+
β1GDPt
+
β2Goldt
+
ut
Test whether ût is stationary.
The intuition is powerful. If money, GDP, and gold merely share unrelated trends, their regression error can itself wander indefinitely. If instead their combination represents a stable long-run relationship, deviations from that relationship should not drift without bound.
The question behind Engle–Granger
After fitting the proposed long-run relationship, do the residuals behave like another non-stationary process, or do they return toward a statistically stable range?
The paper reports an awkward result. Some of the individual series do not satisfy the unit-root conditions in the same way, and the residual-based evidence does not yield the completely clean configuration the author was seeking. A reported p-value of approximately 0.099 becomes part of this inconclusive picture at the 5 percent significance level.
Rather than treating that ambiguity as the end of the exercise, the paper moves to a multivariate method: the Johansen approach.
Technical note on terminology
The paper describes one “cointegrating regression” as a regression that explicitly includes time and calls time the variable that cointegrates the series. That is not the standard definition of Engle–Granger cointegration. In conventional econometrics, cointegration concerns whether a linear combination of integrated variables—typically examined through the residuals of their levels relationship—is stationary. A deterministic time trend may be included in a specification, but time itself is not thereby “the cointegrating variable.” The empirical sequence reported in the paper is retained here, while this distinction is made explicit to avoid conflating the two concepts.
18. Johansen: stop examining one equation at a time
Engle–Granger is naturally built around a single candidate cointegrating relation. Johansen’s method approaches the problem as a system. Instead of choosing one variable as the sole dependent variable and relegating the others to the right-hand side, the variables are embedded in a vector autoregressive structure.
The paper applies Johansen’s procedure to a four-variable system containing the money supply, current GDP, the additional GDP-related term, and gold.
The central object can be represented through a vector error-correction form:
Johansen framework — schematic form
ΔYt
=
ΠYt−1
+
Σ ΓiΔYt−i
+
εt
The rank of the matrix Π contains the essential information.
r = 0No cointegrating relation
There is no stationary long-run linear combination detected among the integrated variables.
0 < r < nCointegration
One or more independent long-run relations constrain the joint movement of the variables.
r = nFull rank
In the standard framework, the system does not have the structure of a collection of I(1) variables requiring cointegration analysis in the usual sense.
Johansen supplies two familiar testing strategies: the trace statistic and the maximum-eigenvalue statistic. They approach the rank question slightly differently, but both ask how many independent long-run relations are supported by the system.
For the purpose of this paper’s argument, the most important null hypothesis is
The decisive null
H0: r = 0
No cointegrating relationship exists.
The paper tests more than one deterministic specification
Johansen results are reported under alternative assumptions about deterministic components: a system treated essentially as a pure random walk, another incorporating drift, and another allowing drift together with a deterministic trend.
Across these cases, the paper emphasizes the same finding: the null hypothesis of zero cointegrating relations is rejected.
Johansen setting discussed
Key hypothesis
Paper’s interpretation
No constant / pure random-walk setting
r = 0
Rejected
Specification with drift
r = 0
Rejected
Drift + deterministic trend
r = 0
Rejected
The displayed output for the first case is particularly emphatic: the trace statistic for r = 0 is reported with a p-value effectively equal to zero. The other deterministic specifications likewise reject the zero-rank hypothesis.
The paper therefore takes the Johansen results as evidence that the strong comovement observed in the regression is not merely an accidental correlation among unrelated trending series.
The variables may be non-stationary in isolation, yet the system can still contain a statistically detectable long-run relation.
19. So was the original regression spurious?
The paper’s final answer to this particular question is no.
That answer is not obtained from the original R², from the significance of the GDP and gold coefficients, or from the visually close correspondence between predicted and observed money. All of those were available near the beginning of the investigation.
The conclusion arrives only after a much longer chain:
From apparent fit to long-run validation
High R²
→ residual diagnostics
→ specification concerns
→ unit-root tests
→ suspicion of spurious regression
→ first differences
→ residual cointegration analysis
→ Johansen system tests
On the paper’s interpretation of the Johansen evidence, rejecting r = 0 means that at least one long-run relation exists within the system. This removes the specific fear that the original high fit is nothing more than the classic nonsense regression created by unrelated stochastic trends.
But—and this is essential—“not spurious” does not mean “correctly specified.”
The two questions are logically distinct.
QUESTION 1Is the regression spurious?
Johansen evidence is used by the paper to answer: apparently not; a cointegrating relation is present.
QUESTION 2Is the specification optimal?
No. RESET and the residual diagnostics have already provided reasons to doubt the simple linear formulation.
QUESTION 3Is the final dynamic model known?
No. The paper explicitly leaves VAR, SVAR, VEC and related dynamic approaches as avenues for subsequent work.
This is the point at which the investigation becomes much more interesting than a standard “significant coefficient” exercise. The monetary variables may possess a genuine long-run statistical relationship while the particular linear equation initially chosen to represent that relationship remains incomplete.
The central distinction
Cointegration can defend a long-run relationship against the accusation of spurious regression without defending the exact regression specification used to represent that relationship.
A theoretically meaningful relationship can be real and yet nonlinear. It can be real and yet dynamic. It can be real and yet require several equations rather than one. It can survive the test for spuriousness while still failing a specification test.
That distinction prepares the final question of the paper—and of this guided reading:
What, after all these tests, has actually been established about money, prices, gold, and the role of hypothesis testing—and what would go beyond the evidence?
That is where the final stage begins.
End of Stage III · Next: What the Study Establishes, Its Limits and the Final Synthesis
· · ·
20. After all those tests, what has actually survived?
We can now return to the deceptively simple regression with which the investigation began. Money in circulation was modeled as a function of nominal GDP and the international price of gold. Ordinary least squares produced an extraordinarily high R², highly significant coefficients, and fitted values that visually followed the observed monetary series very closely.
Had the investigation stopped there, the conclusion would have been easy: the model fits remarkably well.
But it did not stop there.
Residual diagnostics raised questions about serial dependence. RESET tests showed that nonlinear information could matter. CUSUM and CUSUM-of-squares did not tell exactly the same story about parameter stability. A modification involving the GDP-related term changed some diagnostic results while generating new questions. Unit-root tests then exposed a deeper danger: the variables themselves contained enough non-stationary behavior to make a conventional levels regression potentially misleading.
Finally, cointegration analysis provided the paper with a way out of the spurious-regression problem. The Johansen tests rejected the hypothesis of zero cointegrating relations under the deterministic specifications examined. The author therefore concludes that the regression is not merely the artificial product of unrelated trending time series.
The result in one sentence
The paper finds evidence that the variables share a genuine long-run statistical relationship, while simultaneously finding that the particular linear regression used to represent that relationship is not yet an optimally specified model.
That combination is not a contradiction. It is arguably the most important substantive lesson of the entire exercise.
21. A relationship can survive even when a model does not
It is useful to separate three propositions that are easily collapsed into one.
01The variables are associated
The estimated regressions display very strong statistical comovement among money, nominal GDP, and gold.
02The association is not merely spurious
The Johansen results are interpreted as evidence of cointegration and therefore against the hypothesis that the levels relationship is simply a nonsense regression among unrelated trends.
03The linear equation is optimal
This proposition does not survive. Specification diagnostics leave substantial reason to search for a better mathematical representation.
This distinction is fundamental in applied econometrics. Empirical criticism of a particular equation does not necessarily refute the underlying relationship that motivated it. Conversely, evidence that a relationship exists does not confer immunity on the equation chosen to describe it.
A failed specification is not automatically a failed theory; and a successful fit is not automatically a successful specification.
That is why the paper’s methodological emphasis on hypothesis testing is so important. The tests progressively separate questions that a single regression output tends to blur together.
22. What the study does not establish
The safest way to understand the contribution is also to specify its limits.
First, the paper does not discover a final, uniquely correct dynamic equation for the U.S. monetary process. The additional GDP-related variable is explicitly presented as an empirically motivated modification rather than as the consequence of a completed theoretical derivation. The author also acknowledges that the analysis does not exhaustively determine the optimal lag structure through simulation or systematically compare all possible dynamic alternatives.
Second, the existence of cointegration does not by itself settle every substantive economic question. It indicates that the variables possess a long-run statistical structure that constrains their joint evolution. The economic meaning and direction assigned to that structure come from the theoretical framework in which the regression was constructed.
The paper itself makes precisely this broader methodological distinction earlier in the analysis: regression models represent relationships among variables, while the meaning and direction of those relationships must be interpreted through the relevant theoretical framework.
Cointegration is important—but it is not everything
Rejecting the hypothesis of zero cointegrating relations helps answer the specific statistical question of whether the apparent long-run comovement can be dismissed as a spurious regression. It does not automatically identify the best functional form, the complete dynamic structure, or every causal mechanism operating among the variables.
Third, the empirical sample is small. The principal period contains only twenty-two annual observations, from 1990 through 2011. That matters particularly once additional regressors, lag structures, nonlinear terms, and multivariate time-series procedures enter the analysis. Every extra parameter consumes scarce information.
The study should therefore be read as an econometric investigation of specification and diagnosis—not as the final possible model of the monetary structure of the United States.
23. Why 2009–2011 turned out to matter
One of the paper’s more interesting empirical exercises is the comparison between specifications containing the first years after the 2008 crisis and versions ending in 2008.
A naïve expectation might be that adding three extraordinarily turbulent years would simply make the regression worse. But the diagnostics do not yield such a simple result.
For example, the paper reports a Durbin–Watson statistic of approximately 0.91 for 1990–2011 compared with roughly 0.51 for 1990–2008. Neither value is particularly reassuring in isolation, but the longer sample moves closer to the conventional benchmark near two.
The CUSUM comparison is also notable. The paper identifies five observations associated with instability in the 1990–2011 version, versus ten in the shorter 1990–2008 exercise, while CUSUM of squares indicates stability in both cases.
And after the provisional GDP-related modification, the longer period again performs better in some autocorrelation diagnostics: the paper reports disappearance of higher-order autocorrelation earlier in the 1990–2011 sample than in the shortened sample.
A useful empirical lesson
Adding crisis observations does not mechanically “contaminate” an econometric model. Sometimes precisely those observations reveal the structure of the process more clearly—or alter the diagnostics in ways that force the investigator to rethink what the apparently tranquil years were telling them.
This reinforces the paper’s historical orientation. A crisis is not merely an inconvenient outlier standing outside the system. For an economic analysis, crisis observations can be part of the phenomenon one is trying to understand.
24. Why the next natural step is a system rather than another isolated equation
By the end of the investigation, several clues point in the same direction.
The variables possess substantial temporal persistence. Residual autocorrelation appears in important specifications. Nonlinearity remains a concern. The possibility of endogenous interaction is raised. Cointegration indicates that the variables may be tied together over the long run. And Johansen’s framework itself naturally treats them as members of a multivariate dynamic system.
For that reason, the paper points toward the VAR family and related approaches—VAR, structural VAR, near-VAR, and vector error-correction models—as natural candidates for further investigation.
From a single equation to a dynamic system
One equation:
Mt = f(GDPt, Goldt)
↓
Dynamic system:
Yt =
A1Yt−1 +
A2Yt−2 +
· · · +
εt
Rather than making one variable purely dependent and the others purely explanatory, a vector system can represent feedback through time among several variables.
This is particularly relevant because the original theoretical framework concerns a social process whose variables need not be independent in any simple contemporaneous sense. Money, production, prices, financial conditions, and monetary institutions evolve together through time.
If the variables are cointegrated, a vector error-correction model becomes especially natural because it can combine short-run changes with the long-run relation identified by cointegration.
What the paper leaves open
These alternatives are proposed as future directions rather than estimated as the definitive replacement model in the present study. The paper explicitly leaves VAR, SVAR, VEC and related specifications for subsequent investigation.
This matters because intellectual restraint is part of good model building. Discovering that the current model is incomplete does not imply that one has already discovered its successor.
25. The real protagonist of the paper is not money—it is hypothesis testing
At first glance, a reader could reasonably classify the study as a paper about money, nominal GDP, gold, and the United States between 1990 and 2011. Those variables provide the empirical material, and the economic framework determines what the regression is intended to mean.
But structurally, the study is about something broader: how an econometric model is criticized.
The sequence matters.
The complete logic of the investigation
Economic theory
↓
Initial regression specification
↓
OLS estimation
↓
Residual and stability diagnostics
↓
Detection of autocorrelation and specification problems
↓
Provisional respecification
↓
New diagnostics
↓
Unit-root analysis
↓
Suspicion of spurious regression
↓
Cointegration analysis
↓
Rejection of zero cointegration
↓
A relationship that survives—but a model still awaiting improvement
Seen this way, hypothesis testing is not a final bureaucratic stage appended to an otherwise completed model. It is one of the mechanisms by which the model evolves.
A null hypothesis establishes a provisional benchmark. Data are used to confront that benchmark. Rejection—or failure to reject—changes what questions are sensible to ask next. Another specification is then examined. New assumptions become testable. New weaknesses appear.
The procedure is therefore iterative rather than ceremonial.
The value of a statistical test lies not merely in deciding whether a p-value crosses 0.05, but in forcing the model to expose what it has failed to represent.
This is also why apparently contradictory diagnostics need not be embarrassing. White can fail to detect heteroskedasticity while Breusch–Godfrey detects autocorrelation. CUSUM can flag instability while CUSUM of squares remains inside its boundaries. A modification can improve one diagnostic and worsen another. Unit-root evidence can initially threaten a levels regression while cointegration later provides evidence that the long-run relation is not spurious.
That is not statistical chaos. It is what happens when different tests are aimed at different properties of a complicated object.
26. What non-specialists should take away
For a reader without formal econometric training, the deepest lesson can be stated with almost no mathematics.
Suppose someone presents a regression claiming that variable X explains variable Y. The graph looks excellent. The R² is 0.98. The coefficients have several stars beside them.
None of those facts is meaningless. But none of them completes the argument.
You would still want to know whether the errors contain patterns, whether the relationship changes through time, whether the assumed straight line is hiding curvature, whether the variables are simply trending together, and whether the apparent relationship survives methods designed specifically for non-stationary time series.
This paper is, in effect, an extended demonstration of that principle.
FITDoes the equation follow the data?
R² and fitted values help answer this question.
DIAGNOSISDoes the model behave statistically as assumed?
Residual, heteroskedasticity, autocorrelation, stability, and specification tests address this layer.
STRUCTURECan the temporal relationship itself be trusted?
Unit-root and cointegration analysis address the deeper time-series problem.
Only after all three levels have been considered does the meaning of an impressive regression become substantially clearer.
27. What economists and econometricians should take away
For a technical reader, the paper is most interesting when understood as a worked example of specification uncertainty.
The progression from a static OLS equation toward questions of lag structure, serial correlation, conditional variance, functional form, structural stability, integration, and cointegration illustrates an important fact about empirical work: model choice is rarely separable from model criticism.
The paper does not deliver the last word on that process. Indeed, its own endpoint makes clear that further specification work is required.
That incompleteness is not incidental to the argument. It is part of the argument.
The strongest methodological conclusion
An “optimal” specification cannot be identified simply by maximizing fit. The researcher must progressively examine whether the statistical assumptions, temporal structure, functional form, and multivariate relationships implied by the model are compatible with the observed data.
This also explains why no single information criterion, goodness-of-fit statistic, residual test, or cointegration statistic can substitute for the entire modeling process. Each operates within a defined question. Model adequacy emerges from the architecture of those questions taken together.
28. The larger epistemological lesson
There is a broader idea running beneath the econometrics.
A statistical model is an abstraction. Its usefulness depends on what it preserves from the process it represents and what it leaves outside. The residual is therefore not merely mathematical garbage produced after estimation. It is the quantitative trace of everything the fitted equation did not capture.
Sometimes that remainder behaves approximately like noise. Sometimes it remembers yesterday. Sometimes its variance clusters. Sometimes it reveals a nonlinear structure. Sometimes the entire apparent relationship is generated by common stochastic trends. And sometimes variables that wander individually nevertheless reveal a stable long-run connection when studied jointly.
Hypothesis tests give the investigator systematic ways to distinguish among those possibilities.
The better question is not “Does my regression fit?” but “What evidence would reveal that the mathematical form I chose is inadequate?”
That change of question alters the role of statistics. Instead of being used only to confirm an equation, statistical inference becomes a method for attempting to falsify inadequate representations and forcing the researcher toward better ones.
The resulting model may therefore become more complicated as the investigation progresses. But that complexity is not necessarily a defect. If the process itself is dynamic, historically dependent, nonlinear, and jointly determined, a simpler equation can achieve elegance only by omitting part of what needs to be explained.
29. Final assessment: an excellent fit, a real relation, an unfinished model
We can now describe the paper’s journey in three phrases.
IAn excellent fit
The initial OLS regression explains an exceptionally large proportion of the observed variation in the money series.
IIA statistically defensible long-run relation
Johansen cointegration results lead the paper to reject the interpretation that the relationship is merely spurious.
IIIAn unfinished specification
Nonlinearity and dynamic structure remain unresolved, so the linear regression cannot be treated as the final model.
The last point is crucial because the temptation in empirical work is often to search for a specification that finally produces reassuring statistics. This investigation points in another direction. Every apparently satisfactory result can generate another legitimate question.
A high R² leads to residual analysis. Residual dependence leads to dynamic modification. Modification leads to another specification test. Trending variables lead to unit-root analysis. Unit roots lead to the possibility of spurious regression. That possibility leads to cointegration. Cointegration, finally, saves the existence of a long-run relation without saving the original equation from further criticism.
The paper therefore ends somewhere more intellectually interesting than “the model works” or “the model fails.”
The relationship survives the investigation. The specification does not emerge untouched from it.
And that is precisely why hypothesis testing matters.
Its deepest role is not to decorate a regression table with asterisks. It is to organize doubt: to convert suspicions about a model into questions that can be confronted with data, to reveal which assumptions remain defensible, and to indicate where the next round of modeling must begin.
In the study examined here, that process starts with three variables—money, nominal GDP, and gold—and ends with a much broader lesson. Good econometrics is not the art of finding an equation that looks convincing. It is the discipline of discovering how far an equation can be trusted before the evidence forces us to build a better one.
· · ·
In one paragraph
Gómez Julián’s investigation begins with an OLS model of the U.S. money supply as a function of nominal GDP and the international price of gold for 1990–2011. Despite an exceptionally high R² and highly significant coefficients, diagnostic testing reveals residual and specification problems that motivate further investigation. Ramsey RESET raises doubts about linearity; autocorrelation and stability tests reveal temporal complications; unit-root analysis raises the danger of spurious regression; and first-difference results do not provide a completely simple resolution. The analysis therefore moves to cointegration. After an inconclusive intermediate stage, Johansen tests reject zero cointegration under the specifications examined, leading the paper to conclude that the regression is not spurious. Yet the same investigation also concludes that the model remains statistically improvable, particularly in its specification, and points toward richer dynamic approaches such as VAR, SVAR and VEC models. The methodological lesson is consequently larger than the monetary application: hypothesis testing is most useful when treated as an iterative instrument for criticizing, revising and progressively improving a statistical model rather than as a final ritual for attaching significance stars to coefficients.
Unit Roots, Profitability, and Marx’s Falling Rate of Profit
A guided reading of José Mauricio Gómez Julián’s 2020 essay on non-stationarity,
permanent economic shocks, investment, and the long-run dynamics of
capitalist profitability.
GÓMEZ JULIÁN · 2020 · EXPLAINER · ≈ 15 MIN READ
1. The question beneath the statistics
A recession ends. Factories reopen, employment begins to rise and gross
domestic product starts growing again. But has the economy actually
returned to the path it was following before the crisis? Or has the
crisis changed the path itself?
That deceptively simple question sits at the heart of Gómez Julián’s
essay On the Law of the Tendential Fall in the Average Rate of
Profit: Unit Roots and Non-Stationarity of Time Series. The paper
brings together a technical problem in time-series econometrics and a
much older problem in political economy: Karl Marx’s claim that the
average rate of profit is subject to a long-run tendency to fall.
The connection may initially seem strange. A unit root
belongs to the vocabulary of stochastic processes, forecasting and
econometrics. Marx’s falling rate of profit belongs to theories of
accumulation, technological change, crisis and class relations. Gómez Julián’s
argument is that they meet at one fundamental idea:
economic history may leave permanent traces.
If shocks alter the future path of an economy rather than merely
disturbing it temporarily, history is not noise around the model.
History becomes part of the model.
The paper therefore treats non-stationarity as more than an annoying
statistical property that must be removed before running a regression.
It interprets it as evidence against a picture of capitalism in which
every disturbance is followed by an automatic return to an unchanged
long-run equilibrium.
The route from that proposition to Marx, however, has several stages.
Understanding them separately is the easiest way to see both what is
powerful about the argument and where econometric caution is required.
· · ·
2. Stationarity: does the economy forget?
In everyday language, something stationary remains roughly where it
is. In time-series analysis the meaning is more precise. A weakly
stationary process has a constant mean and variance through time, and
the covariance between two observations depends on how far apart they
are—not on the historical date at which they occur.
The intuitive issue is memory. Imagine an economic
variable fluctuating around a stable center. A recession pushes it
downward, but forces inside the system progressively pull it back.
Given enough time, the effect of the original shock fades. The process
“forgets.”
Now imagine another process. A negative shock pushes the variable
downward and the next period begins from that lower level. Later shocks
are added to everything that happened before. The effects accumulate.
There is no built-in statistical mechanism guaranteeing a return to
the old path. This second world is the natural habitat of the
unit root.
Process
What happens after a shock?
Long-run intuition
Stationary
The shock gradually dies out.
The series tends to return toward a stable distribution.
Trend-stationary
The shock dies out around a deterministic trend.
The long-run path moves predictably with time.
Difference-stationary
The shock changes the level of the series permanently.
The level follows a stochastic trend; differences may be stationary.
This distinction matters enormously for economics. If output is
trend-stationary, a recession is essentially a temporary displacement
from a pre-existing trajectory. If output contains a unit root, the
recession can change the trajectory from which the future proceeds.
Growth may resume without the lost output ever being recovered.
A useful distinction
Recovery of the growth rate is not the same thing as
recovery of the level. An economy can start growing at
3 percent again while remaining permanently poorer than it would
have been had the crisis never occurred.
3. What a unit root actually means
The paper introduces the idea through the simplest autoregressive
model. Let the current value of a variable depend on its previous value
plus a new disturbance:
Yt = ρYt−1 + ut
ρ is the autoregressive coefficient; ut is the new shock.
When the absolute value of ρ is below one, the influence
of an old shock becomes progressively smaller. The process is
mean-reverting under the usual conditions. But when
ρ = 1, we obtain:
Yt − Yt−1 = ut
or, equivalently, ΔYt = ut
This is the canonical random walk. Today’s level contains yesterday’s
level in full. Every shock is therefore incorporated into the future
path. The variance of the level grows with the horizon, and conventional
statistical inference applied mechanically to such levels can become
misleading.
This is why unit roots are associated with the famous problem of
spurious regression. Two unrelated trending series can
produce an impressive coefficient, a high R-squared and apparently
significant test statistics simply because both contain persistent
stochastic trends.
The econometric response is not to declare all relationships between
non-stationary variables meaningless. Rather, researchers ask whether
the variables are cointegrated—whether some stable
long-run combination of them exists—or otherwise transform and model
the series in a way consistent with their integration properties.
Technical translation
A unit root does not mean that a variable literally moves at random
in every economically relevant sense. It means that, within the
statistical representation, shocks to the level are not forced to
disappear. The economic interpretation of those shocks still
requires theory.
· · ·
4. The Mankiw–Krugman dispute: will lost output come back?
Gómez Julián devotes a substantial part of the paper to a remarkably public
argument that unfolded after the 2008 financial crisis. The Obama
administration’s Council of Economic Advisers expected a strong
rebound. The underlying intuition was familiar: unusually weak growth
during a recession should eventually be followed by unusually strong
growth as the economy returns toward normal.
Gregory Mankiw objected. Drawing on earlier work with John Campbell, he
argued that aggregate output behaves much more persistently than the
conventional picture of temporary deviations from a stable trend would
suggest. If a fall in output can have a permanent component, then one
cannot simply assume that everything lost during a recession will be
recovered through subsequent above-normal growth.
Conditional recovery versus unconditional forecasting
Mankiw’s point was subtler than “recoveries never happen.” If one knew
with certainty that a recession had ended, then strong post-recession
growth might indeed be likely. But a forecast made in real time does
not possess that information. There remains some probability that the
recession will continue, that another contraction will follow, or that
part of the loss reflects a permanent change rather than temporary
under-utilization.
Arnold Kling supplied an especially intuitive version of the contrast.
Suppose output is low because frightened households temporarily postpone
purchases. Once fear disappears, spending can rebound: that resembles a
trend-stationary story. But suppose resources have been committed to the
wrong houses, technologies or forms of human capital. Those resources
cannot simply be unspent. The economy may grow again, but some of the
loss remains in history.
Krugman and DeLong: look at unemployment
Paul Krugman and Brad DeLong approached the problem from another angle.
High unemployment and low capacity utilization indicate unused
resources. If unemployment is far above normal, they argued, one should
expect it to fall; combined with Okun’s law, that creates a case for
unusually rapid output growth during recovery.
DeLong supported the argument with a relationship between unemployment
and subsequent GDP growth. Mankiw’s response was econometric rather
than rhetorical: he suspected that the apparent relationship was being
driven disproportionately by observations surrounding the exceptionally
strong recovery after the 1981–82 recession.
The subsequent calculations discussed in Gómez Julián’s paper were revealing.
For the full sample, the regression produced an adjusted R-squared of
about 11 percent and a t-statistic of 3.5. Removing eight quarters
associated with the Reagan-era rebound reduced the adjusted R-squared
to about 5 percent and the t-statistic to 2.1. The relation did not
literally vanish, but its apparent strength became substantially more
dependent on a particular historical episode.
A later study by David Cushman, also reviewed in the paper, went further.
Cushman asked what a conventional econometric forecaster working in
2009 might have concluded. His results supported Mankiw’s skepticism
about the stronger rebound projections and judged DeLong’s dynamic
Okun-law specification to perform poorly relative to alternative
forecasts.
Why this episode matters for the paper
Gómez Julián treats this debate as empirical evidence against automatically
assuming mean reversion in macroeconomic aggregates. The central
issue is not whether recessions are followed by recoveries. They
usually are. The issue is whether a recovery necessarily restores
the counterfactual path that existed before the recession.
5. From persistent shocks to profitability
At this point the paper changes scale. The argument is no longer merely
about forecasting GDP after a recession. Gómez Julián asks what kind of
economic mechanism could make history matter so persistently.
Olivier Blanchard provides an important bridge. Writing about financial
crises, Blanchard observed that, across countries, output often does
not return to its old trend path after a crisis.
Instead, the economy can remain permanently below it. That is a
profound distinction: the rate of growth can normalize while the level
of productive activity remains permanently reduced.
Gómez Julián then connects this phenomenon to Marxian political economy.
Marx’s law of the tendential fall in the rate of profit is not a claim
that profitability must decline mechanically every year. It is a claim
about a long-run force generated by capitalist accumulation, operating
through a system that also contains counteracting influences and
recurrent cyclical recoveries.
In broad Marxian terms, profit must be evaluated relative to the capital
advanced to obtain it. Technical development raises productivity, but
capitalist competition also encourages firms to substitute machinery,
infrastructure and other forms of accumulated capital for living labor.
The paper emphasizes the resulting relationship between the
organic composition of capital, profitability and the
development of productive forces.
The crucial insight is that a cyclical rebound and a secular tendency
are perfectly compatible. A profit rate can fall, recover sharply,
experience another boom and still exhibit a lower long-run trajectory
across successive historical cycles.
Figures 4–7 in the paper
Gómez Julián reproduces estimates assembled by Michael Roberts showing a
declining long-run rate of profit for major G20 economies, a rising
organic composition of capital alongside falling profitability, and a
longer historical series in which repeated recoveries occur inside a
broader downward movement. The figures are used as corroborating
evidence for the Marxian tendency, not as new estimates produced by
Gómez Julián himself.
This is the point at which non-stationarity acquires its political-
economic meaning in the paper. If crises leave lasting scars, and if
the variable organizing accumulation itself evolves historically, then
“returning to normal” cannot simply mean returning to an eternal
statistical center. What counts as normal after one historical cycle
may already differ from what counted as normal before it.
6. Do profits lead investment?
The argument still requires another link. Even if profitability follows
an important long-run trajectory, why should it organize the broader
movement of output and employment? Gómez Julián’s answer is
investment.
Investment expands productive capacity, creates demand for machinery
and construction, reorganizes labor and shapes future production. If
changes in profitability systematically precede changes in investment,
then profitability becomes a plausible transmission mechanism between
Marx’s theory of accumulation and the macroeconomic path observed in
time-series data.
Kothari, Lewellen and Warner
Using U.S. corporate data, they find that profits and stock returns
predict changes in investment up to roughly a year and a half ahead
and absorb much of the predictive content attributed to variables
such as interest rates, volatility, credit spreads and Tobin’s q.
Michael Roberts
The paper uses Roberts’s international estimates of profitability
to argue that the long-run decline is visible not merely in a
single recession but across a much broader historical sequence of
booms, crises and partial recoveries.
José A. Tapia Granados
Using 251 quarters of U.S. data, Tapia compares competing endogenous
theories of the business cycle and reports evidence more consistent
with profits leading investment than with investment independently
determining subsequent profits.
Tapia’s result is particularly important for Gómez Julián because it reverses a
familiar Keynesian or Minskyan narrative in which autonomous investment
“calls the tune.” In the interpretation favored by the paper, expected
profitability is what gives capitalists the incentive to accumulate.
Investment therefore reacts to profit conditions, and movements in
profitability propagate into the wider economy.
Figures 8–11
The tables reproduced from Tapia show the behavior of profits,
investment and wages around U.S. expansions and recessions; regressions
in which lagged profits help explain investment; and Granger-causality
tests in which profits contain substantial predictive information for
subsequent private fixed investment.
The paper’s final causal picture can therefore be reconstructed as a
sequence:
The proposed chain
profitability → investment → production and employment →
crisis/recovery path
If profitability is historically conditioned, and investment depends
strongly on profitability, then successive periods of accumulation do
not begin from a clean slate. Each begins with a capital stock, a profit
environment and a productive structure inherited from the preceding
period.
· · ·
7. What the evidence can—and cannot—establish
This is also where precision becomes especially important. The paper
brings together several empirical facts that can be mutually
reinforcing, but they are not logically interchangeable. A careful
reading should keep four distinctions in view.
Four econometric guardrails
A unit root does not imply a downward trend.
It implies persistence: shocks to the level need not disappear.
A unit-root process can wander upward, downward or in both
directions. The sign of a long-run profitability tendency must
come from additional theory and evidence.
A falling series need not contain a unit root.
A variable may decline around a deterministic trend while its
deviations from that trend remain stationary. “Falling” and
“non-stationary” answer different statistical questions.
Economic plausibility does not by itself eliminate
spurious regression. Temporal precedence and a strong
substantive mechanism are important for causal interpretation,
but regressions among persistent series still require
appropriate treatment of unit roots, cointegration and dynamic
specification.
Granger causality is predictive, not automatically
structural. If profits Granger-cause investment,
past profits improve forecasts of investment conditional on
the model. That is meaningful evidence about temporal ordering,
but additional assumptions are required to establish the full
causal mechanism.
These qualifications do not destroy Gómez Julián’s central argument. They make
its strongest defensible form clearer.
Non-stationarity contributes evidence for persistence and
historical dependence. Blanchard contributes evidence that
major crises can leave output permanently below its old trajectory.
The profitability literature cited by Gómez Julián contributes a separate claim:
the average rate of profit displays a long-run declining tendency.
Kothari, Lewellen and Warner and Tapia contribute another link:
profitability contains important information about subsequent
investment.
The Marxian conclusion emerges from the combination of these
propositions, not from the unit root alone.
Non-stationarity supplies the memory. Profitability supplies the
direction proposed by the theory. Investment supplies the transmission
mechanism.
This distinction is essential because it transforms a potentially
overextended statistical claim into a much richer research program.
Instead of asking whether one test can “prove Marx,” the relevant
questions become: How persistent are shocks to output and profitability?
Are breaks temporary or permanent? How should the profit rate itself be
measured? Does profitability lead accumulation across countries and
historical regimes? And does the long-run trajectory survive alternative
specifications?
Those are empirical questions—and therefore questions on which Marxian,
Keynesian and mainstream econometric approaches can genuinely confront
one another using evidence.
8. Why this matters beyond econometrics
It would be easy to treat the unit-root controversy as a specialist
dispute about the properties of an autoregressive coefficient. Gómez Julián’s
paper insists that much more is at stake.
Consider the expression the economy will return to normal. It
sounds descriptive, almost innocent. Statistically, however, it embeds
a hypothesis. It assumes that there exists some stable reference path
toward which the economy tends to return after a disturbance.
If the process is strongly path-dependent, that assumption may fail.
A banking crisis can destroy firms, interrupt careers, cancel
investment projects and alter the capital stock. A prolonged recession
can change the composition of production. Investment not undertaken
today means productive capacity that does not exist tomorrow. The
future therefore reflects not only current conditions but also the
sequence by which those conditions were reached.
For political scientists, this has an immediate implication. Economic
crises cannot always be understood as temporary deviations after which
politics resumes on an unchanged material foundation. If output,
employment, investment and profitability carry historical scars, then
crises can alter the terrain on which subsequent distributional
conflicts and policy decisions occur.
For economists, the message is equally important. Choosing between a
stationary and non-stationary representation is never merely a cosmetic
preprocessing decision. It changes what the model says an economic
shock is. In one representation, the shock is a temporary
displacement. In another, it becomes part of the state from which all
subsequent development proceeds.
And for readers of Marx, the paper offers an unusual bridge between
nineteenth-century political economy and modern time-series reasoning.
Marx’s theory is historical in structure: accumulation changes the
conditions under which the next round of accumulation occurs. Gómez Julián’s
use of non-stationarity gives that historical intuition a statistical
analogue. The economy need not circle eternally around an unchanged
center because the process itself can transform the point from which
the next movement begins.
9. The argument in one view
Stripped of its polemical edges, the paper can be summarized as a
five-part argument.
Macroeconomic time series can be highly persistent.
Treating every recession as a temporary deviation from an invariant
trend is therefore an empirical assumption, not a statistical law.
Some major shocks leave permanent output losses.
The post-crisis economy may grow again without recovering the level
it would otherwise have reached.
Profitability is central to capitalist investment.
The evidence reviewed in the paper indicates that profits possess
substantial predictive power for subsequent investment.
The average rate of profit exhibits a historical tendency.
The Marxian studies cited by Gómez Julián interpret long-run profitability
data as showing a secular downward movement interrupted by cyclical
recoveries and counter-movements.
The economy therefore has memory.
If profitability conditions accumulation and crises alter the
subsequent path, capitalist development should be modeled as a
historical process in which past states help create future ones.
That final proposition is the most intellectually interesting part of
Gómez Julián’s essay. The significance of non-stationarity is not that a
statistical test can settle a two-century debate in political economy.
It is that the statistical language of permanent shocks, stochastic
trends and path dependence is difficult to reconcile with a simplistic
image of economic history as temporary noise around an eternally
self-restoring equilibrium.
Marx’s falling rate of profit and the unit-root literature are not the
same theory, and one does not mechanically prove the other. But they
meet around a common challenge: what if the economic system
carries its own history forward?
Once that possibility is admitted, a crisis is not merely something
that happens to an otherwise unchanged economy. It becomes one
of the events through which the economy itself is historically made.
· · ·
Conclusion: taking history seriously
The paper begins with an econometric distinction but ends with a claim
about the nature of political economy. Stationarity describes a world
capable, under appropriate conditions, of forgetting. Unit-root
behavior describes a world in which disturbances can survive inside
the future.
Gómez Julián argues that the second image is more compatible with the historical
character of capitalist development and with Marx’s analysis of
accumulation. The evidence on permanent output losses challenges easy
assumptions of automatic restoration; the evidence on profits and
investment gives profitability a mechanism through which it can shape
the real economy; and the cited long-run profit-rate estimates provide
the directional component required for the Marxian argument.
The technically careful conclusion is therefore stronger when stated
modestly. Unit roots do not demonstrate the falling rate of profit.
What they undermine is the presumption that economic disturbances must
disappear without changing the long-run path. Once persistence is
combined with evidence about the historical movement of profitability
and its relationship to accumulation, the paper’s central thesis comes
into view.
The deepest question is not whether an economy eventually grows again.
It is whether, after history has happened, the old path still exists to
be returned to.
Principal works discussed in the paper
Campbell, J. Y. & Mankiw, N. G. (1987).
Are Output Fluctuations Transitory?
Blanchard, O. (2009).
Sustaining a Global Recovery.
Cushman, D. O. (2013).
Paul Krugman Denies Having Concurred With an Administration Forecast: A Note.
Kothari, S., Lewellen, J. & Warner, J. (2017).
The Behavior of Aggregate Corporate Investment.
Roberts, M. (2020).
A World Rate of Profit: A New Approach and
More on a World Rate of Profit.
Tapia Granados, J. A. (2013).
Does Investment Call the Tune? Empirical Evidence and Endogenous Theories of the Business Cycle.
A general theory of the cycles of the dynamics of the capitalist system in the long run — now available in English
Every few years the same story is told twice. First, that the economy has entered a new era in which the old rules no longer apply. Then, some months later, that what happened was an accident: a shock, a bubble, a virus, a war. Both tellings share a premise so quiet that it is rarely examined — that the rise and the fall are separate events, and that a good theory of the good years need not be a theory of the bad ones.
The thesis released today argues the opposite, and then goes to some length to measure it. The boom and the crisis are not two phenomena but two moments of one: the crisis of overproduction is the mechanism by which capitalism restores the conditions of an accumulation that its own success had eroded. Devaluation clears the field; new methods of production are introduced under duress; profitability recovers on the ruins. The recovery is not the negation of the crisis. It is its product.
That claim is old. What is new here is the attempt to make it decidable — to state it in a form that quarterly data on the United States economy between 1992 and 2024 could have contradicted, and then to check whether they do.
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Three questions, and why the order matters
The investigation is organised around one general objective — to analyse the long-run cyclical behaviour of U.S. capitalism in the light of the dominant economic theories — and three specific ones, asked strictly in this order:
Which theory explains and predicts best? Not which is most elegant, or most widely taught, but which survives being pointed at the data.
Which factors generate the cycle? Economic and extra-economic alike — the thesis refuses in advance to treat wars and monetary policy as noise sitting outside a clean economic mechanism.
By which rules do those factors interact? A list of causes is not a theory. The theory is in the grammar that binds them.
The order is not decorative. A great deal of applied economics answers the third question with machinery borrowed from a theory it never subjected to the first. Here the selection of the framework is itself a result, defended before it is used.
Five families of an old argument
Before measuring anything, the thesis maps the terrain. Economic thought on the cycle is sorted into five groups: the pre-Kondratieff non-heterodox schools; the Kondratieff school; the post-Kondratieff marginalist and neoclassical schools; the heterodox schools; and the historiographic vision of long waves, which reads the cycle through the archives rather than through the equations.
With that map in hand, three long-running disputes are adjudicated rather than summarised. Does the crisis originate in overproduction or in underconsumption?Is a sustained expansion of credit a symptom of recovery, or of the exhaustion of the conditions that made recovery possible?Is there really an inverse relation between inflation and unemployment, or is the appearance of one an artefact of the precariousness of the labour market? Each is answered, and each answer carries consequences later, when the model is specified.
A framework that states its own conditions of failure
A substantial part of the theoretical apparatus is devoted to a materialist characterization of the dialectical method: its fundamental categories, a Marxist ontology built from a metalogical gnoseology, and an explicit treatment of verification, falsification and decidability. The purpose is unglamorous and indispensable — to fix, in advance, which propositions of the theory are empirically decidable and which are interpretive. Without that boundary, no amount of subsequent statistics can tell you what has been tested.
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Ten dials, seven of them internal
The empirical core is a Bayesian generalized linear model of the growth of U.S. real output, estimated with Hamiltonian Monte Carlo and cross-validated against machine-learning and deep-learning competitors. It retains thirteen coefficients across ten factors. Seven are economic:
Factor
What it registers
Net Average Rate of Profit (ARoP)
The central variable of the accumulation process, and the one whose long-run tendency the theory predicts.
Elasticity of the gross rate of surplus value to the average organic composition of capital
How the exploitation of labour power responds when the technical structure of capital changes.
Non-residential fixed investment
The pace of accumulation in the productive sector; the hinge between boom and crisis.
Inventory-to-sales ratio
The gap between producing value and realising it on the market.
S&P 500
Financialization, entering through a natural cubic spline with three degrees of freedom.
Non-financial private sector credit
The credit system as the accelerator and the brake, splined with two degrees of freedom.
Capitalist R&D spending
The innovative impulse; the second largest coefficient in the model.
And three are extra-economic: military spending (splined with three degrees of freedom), the federal surplus or deficit, and the effective federal funds rate. Their presence is not a concession to realism. It follows from the argument that an imperial economy counteracts the tendency of its own profit rate to fall by means that are not internal to its national accounts.
The Average Rate of Profit carries the fourth largest coefficient of the thirteen — behind only the intercept, R&D spending, and one basis function of the splined S&P 500. The conclusion the author draws from its behaviour is worth quoting in substance: what is favourable to the global process of capital accumulation is not thereby favourable to the dynamics of aggregate growth. The two are not the same quantity, and treating them as one is precisely the confusion the cycle punishes.
Note, too, what the splines are doing. Three of the ten factors would not sit still in a straight line. That is not a technical footnote: it is the first quantitative sign that the interaction of these factors involves thresholds and turning points rather than a stable proportionality.
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Not random. Chaotic.
“Unpredictable” and “random” are not synonyms, and the difference decides what kind of science economics can be. A random system has no internal structure to find. A chaotic one is rigidly determined and still unpredictable at long horizons, because arbitrarily small differences in initial conditions grow exponentially apart.
Three measurements place the U.S. economy in the second category. The Lyapunov exponent is positive (approximately $0.0515$): small perturbations amplify rather than dissipate. The correlation dimension is not an integer ($3.32798$): the attractor reconstructed by Takens’ theorem has a fractal structure, patterns repeating across scales of time and magnitude — which is what “cyclical, but not periodic” means when it is stated precisely. And recurrence quantification finds high determinism alongside variability in laminarity and in the maximum diagonal line length: underlying deterministic structures that themselves evolve.
Read together, these say something a forecaster should find sobering and a theorist should find encouraging. The long-horizon forecast is not merely hard; it is structurally bounded. But the structure that bounds it is real, stable and measurable — which is exactly what a theory of the cycle needs to have something to explain.
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The shape of time
The most unusual instrument in the thesis is topological. The idea is to stop asking how big the numbers are and start asking which observations can see which. Convert the series into a directed visibility graph — a link from one quarter to another when the second is visible from the first over the intervening data — and study the order structure that results.
Two topologies are built on it, and they disagree in an informative way.
The coarser Alexandrov topology, built on temporal reachability, turns out to be connected. At the level of its order structure the economy is globally a single piece: every observation is bound to every other by chains of temporal visibility. There is no quarter that stands apart.
The finer Nada topology is locally fragmented — six components under the natural visibility graph, thirty-six under the horizontal one. Zoom in, and the fabric shows seams: structural discontinuities at the level of closed neighbourhoods.
Global unity and local rupture at once. That duality is not a contradiction to be resolved; it is the object being described. And a third measurement gives the whole thing a direction: the bitopological analysis yields $D = +4$, meaning that expansions generate more temporal visibility than contractions. The cycle is not symmetric in time. Growth accumulates gradually and in view; collapse happens abruptly and blind. Run the film backwards and it is recognisably the wrong film.
⚠️ Why you must not “clean” the crises
There is a habit in applied work of treating extreme values as contamination and smoothing them away by discontinuous imputation. Here that habit is shown to be a category error with a measurable price. The extreme fluctuations of the 2020 crisis belong to a connected block even under the finer topology; severing them is a topological rupture, not a cleaning operation. The thesis reports the consequence directly: models fitted after such imputation performed worse, because one was using predictors suited to one phenomenon — real output growth — to predict a qualitatively different one: real output growth after the crisis had been removed from it. The crises are not noise around the cycle. They are the cycle.
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The grammar of the cycle
The third question receives a seven-part answer. The factors interact through feedback (the rate of profit shapes investment, investment shapes the organic composition of capital, which feeds back into the rate of profit); time lags (R&D and fixed investment pay out on a delay, and the delay is itself cycle-generating); non-linearity (thresholds and regime changes, which is why three factors needed splines); deterministic chaos; sectoral interdependence between the department producing means of production and the one producing means of consumption; topological structure, global connectedness with local fragmentation; and the influence of the global context, which is how military spending and the S&P 500 enter a nominally domestic account.
The unifying claim is that each phase of the cycle contains the seed of its own negation. New methods of production introduced during the crisis lay the foundations of the next boom; the overaccumulation of the boom prepares the ground for the next crisis. Innovation initially arrests the fall of the profit rate and ultimately deepens it — through the way the degree of exploitation of labour power responds, over time, to the very methods introduced to raise it.
What a cycle is for
The thesis closes on a question most treatments never pose. If the cycle is a mechanism, what does it accomplish? Two answers, at different depths. Its intermediate practical end is to restart the process of capital accumulation once instability has reached a critical level — this the mechanism achieves, repeatedly, at a cost borne unevenly. Its definitive practical end is to lay the material and spiritual conditions for a reorganization of the fundamental productive structure of society, one capable of a stability beyond what the capitalist mode of production can reach within its own limits.
What this establishes, and what it does not
The evidence supports the claim that classical Marxist economic theory possesses the greatest explanatory and predictive capacity for long-run cycles among the theories examined here, on this economy, over this period. It is a comparative result on the United States between 1992 and 2024, quarterly — not a universal proof, and not a forecast. The thesis is explicit about the cost of its own data: the Average Rate of Profit and the average rate of surplus value were available only annually through 2020, and completing the series to 2024 required temporal disaggregation and prediction, which puts a wider band of uncertainty around the most recent quarters. The philosophical, historical, conceptual and statistical scope of each result is distinguished in the text, and results unfavourable to the hypotheses are reported alongside the favourable ones.
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About this edition
This is the English edition of a thesis originally written in Spanish and submitted to the Universidad Latina de Costa Rica for the degree of Licentiate in Economics. It is interdisciplinary by construction, drawing on Marxist political economy, dialectical and historical materialism, the history and historiography of economic thought, the philosophy and methodology of science, econometrics, Bayesian statistics, the theory of complex systems and topology.
The edition carries a Note on the Translation that fixes the rendering of the terms whose Spanish usage is technical and not interchangeable with their nearest English cognates — gnoseology, sublation, long wave, solvent demand, technique — and records the editions from which quotations are taken, including the two distinct English and Spanish editions of the Soviet philosophical dictionary, which are cited under different transliterations because they are different books with different pagination.
Costa Rica’s Democratic Myth: How Indigenous Dispossession Built the Country’s Agrarian Wealth
Reading Notes · Political Economy · Latin American History
Costa Rica’s Democratic Myth: How Indigenous Dispossession Built the Country’s Agrarian Wealth
A plain-language reading of a groundbreaking study that dismantles the “White Legend” — and asks why a “democratic” state systematically violates the very rights it claims to protect.
Based on: Gómez Julián, J. M. (2026). Revista de Historia, N.º 93, pp. 1–45.
Costa Rica likes to think of itself as the exception. A small, peaceful republic nestled between two oceans, crowned with universal healthcare, no army since 1948, and a democratic tradition so stable that political scientists have called it the “Switzerland of Central America.” This flattering self-image — what scholars have long called the “White Legend” (Leyenda Blanca) — is one of the most durable stories in Latin American politics. It is also, according to a meticulous new study by Costa Rican economist José Mauricio Gómez Julián, profoundly misleading.
Published in the Revista de Historia (University of Costa Rica, No. 93, 2026), the article sets out to do something ambitious: not merely to describe the well-documented inequality in Costa Rican land ownership — earlier scholars had already done that — but to explain why a formally democratic state produces, protects, and reproduces extreme agrarian concentration, generation after generation, and why the systematic violation of indigenous territorial rights is not an unfortunate glitch in the system but one of its operating mechanisms.
The answer, the author argues, lies in understanding land dispossession not as a historical accident but as a structural feature of capital accumulation — a process that began with Spanish colonialism, was formalized by the liberal republic, was turbocharged by coffee and banana exports, and continues today under the twin guises of “conservation” and legal impunity.
0.7258
Gini coefficient for land concentration in Costa Rica’s southern region (Coto Brus). For context: 0 means perfect equality, 1 means one person owns everything. Sweden’s is around 0.53.
The Numbers Behind the Myth
Before we get to history, let’s sit with some uncomfortable statistics — the kind that rarely appear in Costa Rica’s tourism brochures.
The Gini coefficient — the standard measure of inequality, where 0 is perfect equality and 1 is total concentration — tells a stark story. In Costa Rica’s southern region, land concentration reaches 0.7258. In Guanacaste’s cattle ranches, it climbs as high as 0.9. Nationally, the figure hovers around 0.796. For perspective: the top 1% of farms holds more than 37% of all agricultural land, while the bottom 39% of farms collectively own just 0.2%.
And the people who work the land? Between two-thirds and three-quarters of Costa Rica’s agricultural workforce is landless — a proportion higher than anywhere else in Central America. In some regions like Coto Brus, more than 90% of farms lack legal titles.
These figures are not from a radical pamphlet. They come from rigorous academic work by scholars like Mitchell Seligson, Marc Edelman, Charles Brockett, Jeffery Paige, and James Dunkerley — researchers who compared census data with actual property records and found that the state’s own statistics systematically underestimate the degree of concentration. Gómez Julián builds on their empirical foundation but adds a layer they largely did not: a categorical political-economic explanation of why this pattern persists within a formally democratic state.
“Nearly three-quarters of the Costa Rican peasantry today are landless” — a fact that contradicts the national story of a country of small, equal farmers.
A Timeline of Dispossession: From Encomienda to Enclave
The article reconstructs the history of indigenous land loss in Costa Rica across five major phases. Each one reveals a different mechanism, but all serve the same structural function: clearing land and resources for capital accumulation.
1. Colonial Foundations (pre-1821)
Spanish conquest introduced three key instruments of dispossession: encomiendas (assigning indigenous labor to settlers), reducciones (forced resettlement into concentrated villages), and composiciones fraudulentas (legal maneuvers to convert indigenous land into colonial property). Roman law justified the entire operation by declaring indigenous lands to be property of the Spanish Crown. Though later laws like the Leyes Nuevas of 1542 formally attempted to protect indigenous peoples, the legislation on “vacant lands” (baldíos) declared as empty the very territories where indigenous communities lived, making them available for non-indigenous appropriation. By the end of the colonial period, a fragile but real legal recognition of “indigenous property” still existed through reducciones and “indigenous neighborhoods.” This legal floor is precisely what the post-colonial liberal state would systematically dismantle.
2. The Liberal Republic Strikes (1821–1880)
Independence did not bring recognition — it accelerated dispossession. In 1841, under the dictatorship of Braulio Carrillo, the Código General formally abolished the communal indigenous land regime as a protected legal category. Then, in 1882, Presidential Decree No. XIV declared vast stretches of northern territory — including the homeland of the Maleku people (historically called “Guatuso”) — to be baldíos: vacant state land open to appropriation by non-indigenous settlers and ranchers. The Maleku suffered what the article describes as one of the most intense proportional territorial and demographic reductions in Costa Rican indigenous history.
1841
Year the Código General, under dictator Braulio Carrillo, formally abolished communal indigenous land tenure — legalizing de facto what colonial expansion had already done.
3. Coffee, the Beneficio, and Silent Proletarianization (1830–1930)
The coffee boom created Costa Rica’s agrarian structure — and its contradictions. Unlike Guatemala or El Salvador, Costa Rica did not develop enormous haciendas. The largest farm was 604 hectares; the average was 41. By 1963, 58.5% of cultivated land was in units smaller than 35 hectares. This is the image the “White Legend” celebrates: a nation of smallholders.
But the fine print tells a different story. Over 70% of the rural population was landless — a higher proportion than anywhere else in Central America. Power concentrated not through land-grabbing alone but through the beneficio (coffee-processing mill): in 1850, just sixteen beneficios controlled 85% of coffee exports, and access to the beneficio determined a small producer’s ability to finance and sell their crop. Credit systems tied small producers to merchant-landowners in relationships of dependency. Land prices inflated roughly twenty-fold in thirty years, incentivizing small producers to sell their parcels and then sell their labor.
Critically, the Huetar people of Orosí resisted the coffee frontier around 1860 — an episode largely absent from standard histories of the coffee cycle. The “virtual absence of indigenous population” in the Central Valley during the great coffee expansion, which Dunkerley cited as one factor explaining Costa Rica’s distinct political development, was not a pre-existing geographic condition. It was, the article argues, the result of colonial epidemics (severe die-offs between 1576 and 1581), the legal dissolution of communal landholding in the 19th century, and physical displacement in the face of the advancing coffee frontier.
4. Bananas, the Atlantic Enclave, and the Displacement of the Bribri and Cabécar (1870–1940)
On Costa Rica’s Caribbean side, a parallel story unfolded — this one driven by foreign capital. The Soto-Keith Contract of 1883 granted Minor Keith 800,000 acres of state land, a 99-year railway lease, and sweeping tax exemptions. By 1890, his operation was the largest banana producer on Earth. The standard narrative presents this as development on “vacant land.”
It was not. The Bribri and Cabécar peoples who inhabited the Talamanca valley were forced to move inland, deeper into their ancestral territory, because the banana concession included a significant portion of indigenous lands along the Sixaola river basin. The article calls this one of the key regional mediations of the dispossession process: the banana enclave was not merely a deal between a state and a transnational — it was a concrete mechanism of indigenous dispossession, legally articulated through the fiction of the baldío and materially executed through corporate control of Caribbean space.
The United Fruit Company (UFCO) would later reduce its Atlantic plantations, face a major strike in 1934, and eventually shift production to the Pacific after the sigatoka disease devastated crops in 1938. But the damage was done: the “absence of indigenous population” in banana-region analyses reproduces, with different words, the same ideological device that operates in the coffee narrative — it naturalizes absence by hiding the dispossession that produced it.
5. Limited Reform, Titulación, and the Persistence of Inequality (1957–1990)
Costa Rica did attempt reform. International Convention 107 of the ILO (approved domestically in 1959) recognized indigenous and tribal peoples’ rights. Law 2825 of 1961 repealed the baldíos framework and declared such lands “state property” grounded in the social function of ownership. The Indigenous Law of 1977 (Law 6172) reserved portions of territory for indigenous peoples and granted full legal capacity to their communities through Asociaciones de Desarrollo Integral (ADIs).
Yet the contradictions were glaring. The 1961 law allowed expropriation of idle land but required cash payment at market value — a provision that severely limited implementation. By 1970, only 1,272 families had been settled on roughly 35,400 hectares. The titling program backed by USAID reached a partial result of about 20,462 titles (some 339,761 hectares) by September 1979, at a rate of 425 titles per month against a theoretical capacity of 900–1,000.
The results were telling. After receiving a title, the proportion of beneficiaries with access to credit rose from 18% to 31.7% — but the Gini coefficient for titled land was about 0.708, and when all land possessed by beneficiaries (titled plus other holdings) was included, it rose to nearly 0.781, almost identical to the national figure. Titling did not correct structural inequality — it potentially consolidated it. As the article puts it: these programs reached only a fraction of landless families while maintaining the fundamental structures of concentration.
“Titling does not correct structural inequality and may consolidate it.”
Conservation as Dispossession: The Modern Face of an Old Story
One of the most striking arguments in the article is that the Costa Rican state itself — not just private ranchers or transnational corporations — has become a primary agent of indigenous dispossession, and it has done so wearing the respectable clothes of environmentalism.
In 1982, Costa Rica created the Parque Internacional La Amistad (PILA), which was recognized as a UNESCO Biosphere Reserve that same year and as a World Heritage Site in 1983. What the celebratory narrative omits is that the park appropriated, as state-managed protected area, significant extensions of ancestral territory claimed by the Bribri and Cabécar peoples — without prior, free, and informed consultation. Its buffer zone includes the indigenous territories of Ujarrás, Salitre, and Cabagra. Inside the Biosphere Reserve, sixteen indigenous territories of five different peoples are situated. The state thus re-emerged, in this phase, as the principal agent of dispossession, this time through the legal figure of conservation.
A similar dynamic operates through the Pago por Servicios Ambientales (PSA) — Costa Rica’s flagship Payment for Environmental Services program, administered by FONAFIFO. As documented by researcher Vargas Mena, the PSA has operated inside officially recognized indigenous territories. The forest, the water, and even the air (in the form of the carbon market) are commodified as “environmental services” whose monetary yield flows out of indigenous territory or is captured through institutional mechanisms that subordinate community decisions to the timelines and requirements of external agencies. This is not, the article argues, a correction of agrarian capitalism — it is a new phase of it.
Add to this the contemporary presence of pineapple and palm oil multinationals in the southern zone, whose expansion encroaches on Bróráni, Bribri, and Cabécar territories. Their externalities — intensive water use, agrochemical contamination, pressure on adjacent lands — operate as mechanisms of environmental and economic dispossession on indigenous communities.
16
Indigenous territories of five different peoples situated inside the PILA Biosphere Reserve — appropriated without prior, free, and informed consultation.
Recovering What Was Lost — and Dying for It (2011–Present)
Faced with the persistent failure of the state to enforce legally recognized rights, indigenous movements launched what they call recuperaciones de facto: community-led recoveries of territories formally inscribed under the ADIs but materially usurped by non-indigenous occupants. Beginning in 2011 in places like Salitre, Cabagra, and Térraba, these recoveries triggered the organized, violent reaction of ranchers determined to suppress them.
The consequences have been lethal. On March 18, 2019, Bribri leader Sergio Rojas Ortiz was assassinated. On February 24, 2020, Brórán leader Jethy Rivera was killed. Both murders occurred despite the existence of precautionary measures from the Inter-American Commission on Human Rights (Measure 321-12, issued in 2015). The impunity has been near-total: the Rojas case was archived in January 2024 with no detained suspects. In the Rivera case, the material perpetrator was initially convicted in February 2023, but the appellate court in Cartago reversed that conviction in July 2023, and in a second trial in August–September 2024, the accused was acquitted — despite his confession — under the principle of in dubio pro reo.
The UN Special Rapporteur on the Rights of Indigenous Peoples documented that these attacks remain unpunished, with Costa Rican authorities failing to investigate even when perpetrators are known to the community. The cycle of dispossession–resistance–repression, the article argues, reveals that the de facto recoveries are not isolated incidents but the empirical manifestation of an ongoing class struggle over the means of production, mediated by a persistent colonial condition.
“The contemporary accumulation by dispossession requires the same extra-economic violence that Marx identified in primitive accumulation.”
A Broader Pattern: The State Always Chooses Sides
The indigenous story does not exist in isolation. The article places it within a broader pattern of how the Costa Rican state has responded to rural land conflicts throughout its modern history — and the answer is consistent: it favors capital.
The 1983 crisis is emblematic. When United Brands announced it was leaving, the state bought its land at $35,000 per hectare — a price that demonstrated the state’s weakness vis-à-vis transnationals. Standard Fruit demanded $200 million in credits and fired 700 workers. In the first eighteen months of the Monge administration alone, 185 rural conflicts were registered. In July 1983, 3,500 hectares of United Brands land were occupied by union militants and campesino federation members — and they were evicted by 600 police officers.
Land occupations surged across administrations: 70 under Trejos, 500 during the Figueres (second) and Oduber governments, 120 in Carazo’s first year alone. The “land question” was never limited to Guanacaste, where cattle-ranching concentration had always been extreme and conflictive — it was a national structural feature.
Rebuilding Political Categories from the Ground Up
Here is where the article makes its most ambitious theoretical move. Rather than importing Marxist categories as ready-made tools, Gómez Julián claims to reconstruct them inductively from the historical evidence itself. The sequence is deliberate: facts first, categories second, return to facts with new interpretive power third.
The State, in class societies, is the social organization of the political power of the economically dominant class. Law is the system of enforceable norms established by state power — it is the legal organization of that political power. Government is the administration of state institutions, and crucially, the economically dominant class is not necessarily the governing class: those who hold power are not always those who administer it, but the governing class is factually subordinate to the dominant class.
Democracy, in its most general sense, is the government of the majority over the minority. But this concept was coined in slave-owning Athens of the 5th century BC, where only adult male citizens could vote, excluding women, slaves, and foreigners. The democracy of adult male citizens was a dictatorship for everyone else — the overwhelming majority. This dialectical relationship extends to all class societies: every democracy is a dictatorship for those systematically excluded from the majority’s will, and every dictatorship preserves an internal “democracy” for the class that imposes its hegemony. The relationship is dialectical, not identical.
The Rule of Law (Estado de Derecho) is the idea that the state’s power is regulated and guaranteed by law — meaning the state creates law and then voluntarily submits to its own creation. But the article argues this is a sophism: states do not create law to submit to it; they create it as a regulatory framework founded on the interests of the politically and ideologically dominant class. As evidence, the author points to Article 105 of Costa Rica’s own Constitution, which prohibits referendums on budgetary, tax, fiscal, monetary, credit, pension, security, and administrative matters — that is, on virtually every fundamental issue of national social life. The validity of any referendum result is ultimately subject to the decision of the Legislative Assembly. The state, in other words, has structurally exempted itself from popular will on every matter that counts.
And corruption? It is not an aberration. Defined as the practice in which a public official formally deviates from the functions for which they were entrusted with power, corruption “formally” (not factually) departs from stated commitments. Why is the deviation formal rather than factual? Because, within this framework, corruption is a necessary mechanism for the interests of the dominant class to impose themselves over the rest of society while electoral democracy reigns — one of several mechanisms that enable factual dictatorship through the alienation of the dominated classes.
Three Processes, Two Levels of Abstraction
The theoretical heart of the article lies in the way it reconstructs the relation among primitive accumulation, accumulation by dispossession, and what it calls acumulación entrelazada. In this post, I translate the latter deliberately as intertwined accumulation. The choice matters because it allows us to distinguish the concrete process reconstructed in the Costa Rican case from the broader concept of entangled accumulation developed by Guilherme Leite Gonçalves and Sérgio Costa.
The distinction is not merely terminological. At the level of objective historical processes, primitive accumulation and accumulation by dispossession can possess relative autonomy: they are distinguishable mechanisms with their own immediate forms, temporal locations, and material effects. Relative autonomy does not mean isolation. It means that one can identify each process in reality without denying that, under concrete historical conditions, they coexist, penetrate one another, and alter the conditions under which the other operates.
Primitive Accumulation
Marx’s original problem: the violent historical separation of producers from their means of production and the construction of the property relations required for capitalist production. In the Costa Rican sequence, this includes colonial dispossession, Carrillo’s 1841 abolition of communal tenure, the 1882 decree that treated Maleku territory as vacant land, and the Soto-Keith concession of 1883. These are not simply old events remembered by a later capitalism; they establish legal and material conditions that subsequent accumulation inherits.
Accumulation by Dispossession
David Harvey’s formulation identifies the continuing and contemporary use of expropriatory practices: privatization of land and common goods, suppression of collective rights, commodification of previously non-commodified spheres, colonial or neo-colonial appropriation of assets, and the use of credit, law, and state power to transfer resources. In the Costa Rican case, this logic appears in contemporary territorial occupation, conservationist appropriation, environmental-service markets, and other mechanisms that convert collectively preserved resources into fields of accumulation.
Intertwined Accumulation
As reconstructed here from Gómez Julián’s Costa Rican case, intertwined accumulation names the concrete relation of mutual conditioning and reinforcement among relatively distinguishable accumulation processes. It is the point at which an earlier expropriation creates the legal, spatial, or social conditions for a later dispossession; the later dispossession reproduces or deepens the effects of the earlier one; and both become connected to new fractions of capital, state institutions, and wider circuits of accumulation. In this sense, intertwined accumulation is not a synonym for either primitive accumulation or accumulation by dispossession, nor does it abolish their analytical distinction.
Relative Autonomy Does Not Mean Separation
This way of reading the historical record avoids a false choice. We do not need to say either that primitive accumulation is a closed episode belonging only to the birth of capitalism, or that every later dispossession is simply the same phenomenon under a new name. A concrete process may be identified as primitive accumulation when it restructures property relations by separating communities from the conditions of their reproduction. A contemporary process may be identified as accumulation by dispossession when already existing rights, assets, commons, or public resources are transferred into circuits of capital. And a third analytical problem appears when the effects of these processes become causally linked.
That third problem is what intertwined accumulation captures here. The relevant relation is not merely chronological succession. It is a material interaction: one process changes the conditions of possibility of another. Earlier legal destruction of communal tenure can make later private occupation easier; later occupation can normalize and reproduce the property regime created by the earlier rupture; state regulation can connect both to transnational commodity chains, tourism, conservation finance, or carbon markets. The processes remain distinguishable, but their effects are no longer additive. They become synergistic.
The arrow does not mean a rigid historical sequence. It represents a causal relation that may operate across different periods, regions, and institutional forms.
The Costa Rican mapping makes this visible. Carrillo’s abolition of communal tenure created a legal environment within which the Huetar coffee-frontier dispossession could advance. Decree XIV of 1882 converted Maleku territory into legally available space and enabled its appropriation by ranchers and settlers. The Soto-Keith concession connected territorial dispossession in Talamanca to transnational banana capital. PILA and the PSA later inserted ancestral territory into conservationist and environmental-value circuits, linking local territorial conflict to tourism, state administration, and global carbon capital. The post-2011 de facto recoveries reveal the contemporary collision between the collective indigenous regime and the individual capitalist regime produced through this long history.
Entangled Accumulation: The Broader Category
This is where Gonçalves and Costa’s concept of entangled accumulation becomes especially useful. Their 2019 formulation is deliberately broader. It seeks a comprehensive category for the interdependencies that have structured capitalist expansion across history: exploitation and expropriation; wage and slave or otherwise unfree labor; state power and illegal violence; capitalist and non-capitalist economies; different geographical scales; different historical periods; and intersecting social hierarchies such as class, race, ethnicity, and gender.
In that broader framework, primitive accumulation, accumulation by dispossession, financial expropriation, superexploitation, and other forms do not follow a rigid chronology. They may coexist in the same period and space, disappear and re-emerge, and become connected through law, politics, corruption, culture, discourse, and the changing boundary between state and market. Entangled accumulation therefore operates at a higher level of abstraction: it is a general category for the historical interpenetration of multiple mechanisms and dimensions of capitalist accumulation.
The two concepts can therefore be related without collapsing them into one another. Intertwined accumulation, as used in this post to interpret the Costa Rican article, is the more concrete category: it identifies a specific process of causal articulation and synergy among distinguishable modes of accumulation inside a determinate historical formation. Entangled accumulation, in Gonçalves and Costa’s sense, is the more general category under which such concrete articulations can be theorized together with many other forms of interdependence.
Relationship between the concepts
Eintertwined ⊂ Eentangled
This set-theoretic expression is an interpretive shorthand used in this post, not a formula stated verbatim by either paper. It means that the specific Costa Rican process can be treated as one concrete determination within the wider family of interdependencies captured by entangled accumulation.
This distinction also clarifies why the Costa Rican article can speak of the convergence of primitive accumulation, accumulation by dispossession, and intertwined accumulation without reducing all three to one undifferentiated process. The first two refer to mechanisms that can be identified with relative autonomy; the third refers to their concrete articulation when they coexist and reinforce one another. At the same time, that entire configuration can be located within Gonçalves and Costa’s more encompassing theory of entangled accumulation. What looks like a terminological inconsistency at a purely abstract level thus becomes, at the level of historical reality, a useful distinction between mechanisms, their interaction, and the general category capable of containing both.
The key distinction is between a mechanism and a relation among mechanisms: primitive accumulation and accumulation by dispossession can be identified separately, while intertwined accumulation names the historically concrete synergy through which they condition and reproduce one another.
A Timeline of Dispossession: Key Milestones
1542–1880
Laws on baldíos, early decrees (e.g., Guatuso). Legal-administrative dispossession: Crown → State → Private hands.
1830–1890
Coffee boom and beneficio system. Concentration via processing and credit. Huetar resistance in Orosí (c. 1860). Dissolution of communal land regime.
The article introduces one more theoretical tool that deserves attention: the Marxist concept of ground rent — both natural and differential. Contemporary indigenous struggles, the author argues, cannot be explained solely by the seizure of land as a means of production. They also revolve around the natural resources those lands contain: water, forests, biodiversity, scenic beauty, carbon sequestration capacity. The yield from these resources can be converted into rent that is appropriable by various fractions of capital — agroindustrial, extractivist, touristic, conservationist (private or state-run).
The PSA inside indigenous territories, the nature tourism surrounding national parks created on ancestral lands, the monetization of environmental services in the carbon market, and adjacent agroindustry are all concrete devices through which capital captures or redistributes rents whose material base is precisely the territories that indigenous peoples have preserved through their historical struggle. Indigenous territorial defense is therefore simultaneously defense of communal use-value against capitalist exchange-value, and a material obstacle to the full realization of differential rent by capital.
What This Means — and Why It Matters Beyond Costa Rica
The article’s conclusions are blunt. The “White Legend” of Costa Rican democratic exceptionalism does not survive contact with the evidence. A country where nearly three-quarters of the peasantry is landless, where the Gini coefficient for land ownership approaches 0.8 nationally and reaches 0.9 in cattle regions, where over 90% of farms in some zones lack legal titles, and where indigenous leaders are assassinated with impunity for reclaiming territories that the state itself has legally recognized as theirs — that country is not a democratic anomaly in an unequal region. It is a case study in how formal democracy and extreme agrarian concentration coexist, and how the latter is structurally enabled by the former.
The argument is not that Costa Rica is uniquely evil. It is that the myth of its exceptionalism serves a function: it legitimizes the very regime of accumulation that makes extreme inequality possible. The “White Legend” is not a miscalculation or a public-relations excess. It is, in the article’s framework, an ideological device that is functional to the system it obscures. As long as Costa Rica believes it is democratic and equal, the structural mechanisms that produce dispossession operate without scrutiny.
And the final, perhaps most unsettling implication: the article concludes that non-indigenous civil society participation is essential to defend rights that are “fundamentally irreconcilable with the needs of accumulation.” The rights of indigenous peoples over their territories, in other words, cannot be secured within the existing accumulation regime without a political rupture that extends beyond indigenous communities alone. This is not a call for charity or solidarity in the conventional sense. It is a structural observation: the system that dispossesses indigenous peoples is the same system that produces landlessness among non-indigenous peasants, and their interests converge at the point where the logic of accumulation is challenged.
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A Note on the Source
This post is primarily a reading of José Mauricio Gómez Julián, “Acumulación por despojo y el mito del excepcionalismo democrático costarricense: el despojo indígena como rasgo estructural de la concentración agraria,” Revista de Historia, No. 93 (January–June 2026), pp. 1–45, published by the Universidad Nacional de Costa Rica. The article is available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International license. The author is an independent researcher specializing in political economy and statistical methodology, with a degree in Economics from the Universidad Latina de Costa Rica.
The conceptual clarification developed above also engages Guilherme Leite Gonçalves and Sérgio Costa, “From primitive accumulation to entangled accumulation: Decentring Marxist Theory of capitalist expansion,” European Journal of Social Theory (2019), DOI 10.1177/1368431018825064. To keep the analytical levels distinct, this post deliberately reserves entangled accumulation for Gonçalves and Costa’s broad category and uses intertwined accumulation for the more concrete process reconstructed from the Costa Rican case. That terminological distinction is an interpretive choice of this post rather than a claim that the two papers themselves explicitly establish the same hierarchy in those exact words.
The article is explicitly situated within Marxist political economy and historical materialism. It does not claim neutrality: it argues that the theoretical framework it employs has superior explanatory power for the phenomena it documents, and it reconstructs its core political categories (State, Law, Democracy, Rule of Law, Corruption) inductively from the historical evidence rather than importing them as axioms. Readers who disagree with the theoretical framework may still find the empirical documentation — drawn from census data, property records, legislation, UN reports, Inter-American Commission records, and direct indigenous testimony — valuable in its own right.
The data referenced throughout — Gini coefficients, titling statistics, acreage figures, occupation percentages — are drawn from the sources the article cites (Seligson, Edelman, Brockett, Dunkerley, Paige, Guevara and Chacón, Vargas Mena, Zúñiga et al., and others). No figures have been invented or extrapolated beyond what the article itself presents.
“The history of the communities indigenous from the conquest to the present reveals not an exceptional democratic inclusion but a systematic dispossession as a structural feature of capital accumulation.”