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An introduction to an R package that brings Bayesian inference, panel data econometrics, and rigorous validation to one of political economy’s most enduring empirical debates.
Why This Package Exists
Here is a question that has occupied economists for over two centuries: when you pay for something, does the price you pay bear any systematic relationship to the labor required to make it?
Adam Smith thought so. David Ricardo refined the idea. Karl Marx built an entire theory of exploitation on it. And since the mid-twentieth century, empirical researchers have been trying to measure the strength of this correspondence with real-world data.
The challenge has always been methodological. The datasets are panel data — prices observed across many economic sectors over many time periods — and they demand techniques that respect both the cross-sectional structure (different industries behave differently) and the temporal dimension (relationships can shift over time). A simple scatterplot of values against prices, however illustrative, will not settle the question.
valueprhr is an R package built to close this methodological gap. It provides a complete, reproducible pipeline: from raw price matrices to model estimation, from Bayesian inference to out-of-sample validation, from structural break detection to side-by-side model comparison. It was designed for political economy, but as we will see, its toolkit applies to any panel data problem where you need to assess the correspondence between two variables across entities and time.
The Core Idea (in Plain Language)
In the classical and Marxian tradition, the value of a commodity is determined by the total labor time — direct and indirect — required to produce it. If a table requires 10 hours of socially necessary labor and a chair requires 5, the table’s value is twice the chair’s.
This gives rise to what economists call direct prices (denoted pd): prices that are strictly proportional to the labor embodied in each commodity. They represent what prices would be if they perfectly mirrored labor content.
But capitalism does not work that way. Capital flows between sectors seeking the highest return, and competition tends to equalize the rate of profit across industries. The prices that emerge from this process are called prices of production (denoted pπ). They redistribute surplus value: sectors with higher organic composition of capital (more machinery relative to labor) tend to have prices of production above their direct prices, and vice versa.
The central empirical question is: despite this redistribution, how closely do direct prices and prices of production correspond?
The standard test is a log-linear regression:
ln(pπit) = α + β · ln(pdit) + uit
where i indexes sectors and t indexes time periods.
Three hypotheses are at stake:
- β ≈ 1: a one-percent increase in direct prices is associated with roughly a one-percent increase in production prices (proportionality).
- R² ≈ 1: direct prices explain the vast majority of the variation in production prices.
- Stability: the relationship holds consistently across time periods.
If all three hold, the labor theory of value has strong empirical support. valueprhr gives you the tools to test each one rigorously.
What’s Inside the Package
valueprhr organizes its functionality into six modules. Here is what each does and why it matters.
1. Data Preparation
Real-world data rarely arrives in the format econometric methods require. The package accepts two data frames in wide format (rows = years, columns = sectors) — one for direct prices, one for production prices — and converts them into the long-format panel structure that econometric models expect.
library(valueprhr)# Wide format: Year | Agriculture | Manufacturing | Mining | ...direct <- read.csv("direct_prices.csv")production <- read.csv("production_prices.csv")# Convert to long panel: Year, Sector, direct, production, log_direct, log_productionpanel <- prepare_panel_data(direct, production, log_transform = TRUE)head(panel)#> Year Sector direct production log_direct log_production#> 1 1960 Agriculture 45.2 48.1 3.81 3.87#> 2 1961 Agriculture 46.0 49.0 3.83 3.89#> ...
The function prepare_log_matrices() does the same job but returns matrix format, which is what the multivariate methods (PLS, CCA) need.
2. Panel Data Models
This is where the core econometrics happens. The package implements two complementary specifications:
Two-Way Fixed Effects (FE) controls for both sector-specific and time-specific unobserved heterogeneity:
Yit = αi + γt + β · Xit + εit
In plain terms: every sector has its own baseline (some sectors are systematically more expensive), every year has its own macroeconomic conditions (inflation, crises), and the model isolates the within variation to estimate the core relationship.
fe <- fit_twoway_fe(panel, robust_se = TRUE, cluster_type = "group")print(fe)#> Two-Way Fixed Effects Model#> ============================#> Observations: 1200 | Sectors: 20 | Years: 60#> R-squared: 0.9876 | Adjusted R-squared: 0.9870#>#> log_direct coefficient:#> Estimate = 0.9754, SE = 0.0123, t = 79.30, p = 0.0000
The cluster_type = "group" option computes cluster-robust standard errors at the sector level, which accounts for serial correlation within each sector’s time series.
Mundlak Correlated Random Effects (CRE) takes a different route. Instead of dummy variables for every sector, it decomposes the predictor into a within-sector component (how Xit deviates from sector i‘s average) and a between-sector component (the sector average itself):
Yit = α + βW · (Xit − X̄i) + βB · X̄i + ui + εit
In data science language: this is a way to control for group-level confounders without the computational cost of N dummy variables. If βW = βB, the within and between effects are the same, and a simpler Random Effects model suffices. If they differ, the relationship between values and prices operates differently within a sector over time than across sectors.
# Add Mundlak termspanel_cre <- create_mundlak_data(panel, x_var = "log_direct")# Fit the modelcre <- fit_mundlak_cre(panel_cre, include_time_fe = TRUE)print(cre)#> Mundlak Correlated Random Effects Model#> =========================================#> Within-sector effect (beta_W): 0.9680#> Between-sector effect (beta_B): 0.9912#>#> Mundlak test H0: beta_W = beta_B#> F-stat = 2.14, p-value = 0.1438#> -> Fail to reject H0: RE/CRE specification is consistent
The function test_mundlak_specification() formalizes this check. A low p-value means you should stick with Fixed Effects; a high p-value means the simpler model is adequate.
The package also includes a Panel Granger Causality test (the Dumitrescu-Hurlin procedure), which tests whether past values of direct prices help predict current production prices — and vice versa.
panel_granger_test(panel, lags = c(1, 2))#> direction lag W_stat Z_stat p_value significant#> 1 direct -> production 1 8.432 3.126 0.0018 TRUE#> 2 direct -> production 2 6.215 2.441 0.0146 TRUE#> 3 production -> direct 1 5.890 2.103 0.0354 TRUE#> 4 production -> direct 2 4.012 1.332 0.1828 FALSE
3. Bayesian Models
Classical (frequentist) estimation gives you a single point estimate for β. Bayesian methods give you a full probability distribution over possible values, incorporating your prior beliefs and updating them with the data.
In econometric language: instead of β̂ = 0.975 ± 0.012, you get a posterior distribution showing that β lies between 0.95 and 1.00 with 95% probability.
The package offers two Bayesian approaches:
Sector-by-Sector Bayesian GLM fits an independent Bayesian linear model for each sector, using weakly informative priors (the rstanarm package handles the MCMC sampling via Stan). Each sector gets its own slope and intercept, along with Leave-One-Out Cross-Validation (LOO-CV) scores.
bayes <- fit_bayesian_glm_sectors( direct, production, chains = 4, iter = 4000)print(bayes$summary_table)#> Sector beta_mean beta_sd beta_lower beta_upper elpd looic n_obs#> 1 Agriculture 0.982 0.025 0.933 1.031 -42.3 84.6 60#> 2 Manufacturing 0.971 0.031 0.910 1.031 -38.7 77.4 60#> 3 Mining 0.958 0.042 0.876 1.041 -45.1 90.2 60#> ...
In data science language: LOO-CV is a principled way to assess out-of-sample predictive performance without holding out data. The LOOIC (LOO Information Criterion) is the Bayesian analogue of AIC — lower is better.
Bayesian Hierarchical Model goes further by pooling information across sectors. Instead of treating each sector in isolation, it assumes that sector-specific slopes are drawn from a common population distribution:
βi ~ N(μβ, σβ2)
Sectors with less data “borrow strength” from the population mean. This is especially valuable when some sectors have short time series.
hier <- fit_bayesian_hierarchical(panel, include_time = TRUE)print(hier)#> Bayesian Hierarchical Model#> ============================#> Observations: 1200 | Sectors: 20#>#> LOO-CV:#> ELPD = -312.45#> LOOIC = 624.90#>#> Population-level effects:#> parameter mean sd 2.5% 97.5%#> 1 (Intercept) 0.1423 0.0892 -0.032 0.317#> 2 log_direct 0.9734 0.0145 0.945 1.002#> 3 Time_scaled 0.0031 0.0018 -0.0004 0.007
4. Multivariate Analysis
When the number of sectors (N) is large relative to the number of time periods (T), standard regression becomes unstable. This is the “small T, large N” problem common in panel data. The package offers three multivariate techniques to handle it:
Partial Least Squares (PLS) extracts latent components that explain covariance between direct prices and production prices. It handles multicollinearity gracefully and is widely used in chemometrics, genomics, and now in value-price analysis.
matrices <- prepare_log_matrices(direct, production)pls <- fit_pls_multivariate( matrices$X_clean, matrices$Y_clean, max_components = 8)print(pls)#> Partial Least Squares (PLS) Regression#> =======================================#> Optimal components: 3#>#> R-squared by component:#> n_components R2_train R2_cv#> 1 1 0.942 0.938#> 2 2 0.971 0.965#> 3 3 0.984 0.980
Canonical Correlation Analysis (CCA) finds linear combinations of direct prices and production prices that are maximally correlated. In econometric language: CCA extracts the “shared economic signal” — the common factor driving both sets of prices.
cca <- run_sparse_cca(matrices$X_clean, matrices$Y_clean, n_components = 3)print(cca)#> Canonical Correlation Analysis#> ===============================#> Components: 3#>#> Canonical correlations:#> CC1: r = 0.9987 (Var X: 92.3%, Var Y: 91.8%)#> CC2: r = 0.9841 (Var X: 5.1%, Var Y: 5.4%)#> CC3: r = 0.9523 (Var X: 1.8%, Var Y: 1.9%)
The first canonical correlation above 0.99 indicates an extremely tight structural link between the two price systems.
Panel VAR captures dynamic feedback: do lagged values of direct prices predict current production prices, and vice versa?
pvar <- fit_panel_var(panel, lags = 2, transformation = "fd")
5. Cross-Validation
Standard k-fold cross-validation violates temporal ordering. If you train on 1960–1990 and test on 1985–1990, future information leaks into the training set. The package implements two time-aware approaches:
Rolling Window CV trains on t₀ … tW, tests on tW+1 … tW+H, then rolls the window forward.
cv <- rolling_window_cv( panel, window_sizes = c(20, 30), step_size = 2, test_horizon = 3)print(cv$summary)
Leave-One-Sector-Out (LOSO) trains on all sectors except one and predicts the held-out sector. This tests cross-sectional generalization: does the value-price relationship estimated from other sectors hold for agriculture? For mining? For finance?
loso <- leave_one_sector_out(panel)print(loso$summary)#> metric mean sd#> 1 RMSE 0.04521 0.01832#> 2 MAE 0.03587 0.01456#> 3 R_squared 0.96120 0.02340
An average R² above 0.96 in LOSO-CV means the relationship generalizes robustly across sectors.
6. Structural Break Tests
Has the value-price relationship been stable over time? Or did it shift at some point — due to globalization, a methodological change in data construction, a technological revolution, or a regime shift in profit rate equalization?
The package aggregates the panel to a time series and applies a battery of tests:
breaks <- test_structural_breaks(panel, break_date = 1990)print(breaks)#> Structural Break Tests#> ========================#> Time-series observations: 60#>#> Chow Test:#> Break date: 1990#> F-stat = 1.8420, p = 0.1687#>#> supF / Bai-Perron Test:#> supF = 5.2130, p = 0.0842#> Breaks detected: 0
A non-significant result is actually good news here: it means the value-price correspondence has been structurally stable across the entire sample period.
The Full Pipeline in One Command
If you want to run everything at once — data preparation, FE and CRE models, cross-validation, structural break tests, and model comparison — the package offers a single entry point:
results <- run_full_analysis( direct, production, run_bayesian = FALSE, # Set TRUE if you have rstanarm installed run_cv = TRUE, run_breaks = TRUE, verbose = TRUE)# Access everythingprint(results$comparison)print(results$cv_summary)cat(format_break_results(results$breaks))
You can then export the comparison table and CV results to CSV:
export_results_csv( results$comparison, results$cv_summary, output_dir = "results/")
Beyond Political Economy: General Panel Data Applications
Although valueprhr was built for the specific question of value-price correspondence, its methods are general-purpose panel data tools. Any research problem involving the relationship between two variables observed across entities and time can benefit from the package:
- Health economics: Does out-of-pocket spending track underlying treatment costs across regions over time?
- Environmental economics: Do carbon prices reflect the embodied emissions of goods across industries?
- Education: Do standardized test scores correspond to instructional expenditure across school districts over decades?
- Finance: Do book values predict market valuations across sectors?
- Any two-variable panel regression where you need fixed effects, Mundlak decomposition, robust standard errors, time-aware cross-validation, or structural break detection.
The key requirement is that your data has a panel structure (entities × time) and that the distributional assumptions of the models are reasonable for your context. The methods — two-way FE, Mundlak CRE, Bayesian hierarchical models, PLS, CCA, rolling-window CV, structural break tests — are econometric staples that transcend any particular application domain.
Installation and Dependencies
The package requires R ≥ 4.1.0. Core functionality depends only on base R and the Metrics package. Extended features (Bayesian models, panel data infrastructure, structural break tests) are handled through soft dependencies that are loaded on demand:
# Install from GitHubinstall.packages("devtools")devtools::install_github("isadorenabi/valueprhr")# Optional: install all suggested packages at oncesuggested <- c( "rstanarm", "loo", "plm", "lme4", "pls", "vars", "panelvar", "strucchange", "lmtest", "sandwich", "dplyr", "tidyr", "tibble")install.packages(suggested[!sapply(suggested, requireNamespace, quietly = TRUE)])
Note for Bayesian models:
rstanarmrequires a working C++ toolchain — Rtools on Windows, Xcode Command Line Tools on macOS, orbuild-essentialon Linux.
What Makes This Package Methodologically Different
Three features distinguish valueprhr from a hand-rolled analysis:
- Time-aware validation. Most applied work reports in-sample R² as evidence of fit. valueprhr pairs every model with rolling-window and leave-one-sector-out cross-validation, giving you out-of-sample performance that is honest about temporal dependence and cross-sectional generalization.
- The Mundlak decomposition. By splitting effects into within-sector and between-sector components, the package lets you test whether the value-price relationship operates at the sector level (structural), at the temporal level (cyclical), or both. This is a nuance that most empirical studies in this literature overlook.
- Bayesian hierarchical pooling. Sectors with short time series are a common headache. The hierarchical model lets small sectors borrow statistical strength from the population, producing more stable estimates than independent sector-by-sector regressions.
A Note on the Underlying Data
The wiki documentation mentions that market price indices used in this framework are constructed by temporally disaggregating the aggregate Consumer Price Index using the Input-Output matrix as a structural indicator, relying on closed-form Bayesian solutions from the BayesianDisaggregation library. This is a methodological detail worth understanding: the sectoral prices are not raw market quotes but statistically consistent decompositions of the macroeconomic aggregate. This ensures that the estimated sectoral price movements add up to the observed CPI, a property that many ad hoc sectoral price datasets lack.
Citation
If you use valueprhr in your research:
@software{gomezjulian2025valueprhr, author = {Gómez Julián, José Mauricio}, title = {valueprhr: Value-Price Analysis with Bayesian and Panel Data Methods}, year = {2025}, url = {https://github.com/isadorenabi/valueprhr}, note = {R package version 0.1.0}}
Author: José Mauricio Gómez Julián — ORCID — isadore.nabi@pm.me
License: MIT
Repository: github.com/IsadoreNabi/valueprhr
The labor theory of value is either one of the most important ideas in the history of economics or one of the most contested. Either way, it deserves better tools than a spreadsheet and a prayer. valueprhr brings the full machinery of modern econometrics to the question — and lets the data speak for itself.