Robust Variable Selection: BMA, LASSO, and WALS

Downloads

Each dataset is available as a labeled Stata .dta and its source file.

⇩ Download all data (ZIP)stata_codebook.do

Run stata_codebook.do in Stata once to attach long-form per-variable notes to the .dta files.

Load directly in code

Every file loads straight from GitHub (raw URLs). Swap the file name to load any dataset.

Stata

* Stata 14+ : `use` reads an https URL directly
global BASE "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_bma_lasso_wals/data/"
use "${BASE}synthetic-co2-cross-section.dta", clear
describe
notes

Python

!pip install -q pyreadstat
import pandas as pd
BASE = "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_bma_lasso_wals/data/"
df = pd.read_stata(BASE + "synthetic-co2-cross-section.dta")

# load every dataset at once
files = ["synthetic-co2-cross-section"]
data = {f: pd.read_stata(BASE + f + ".dta") for f in files}

# pyreadstat (richest metadata) reads LOCAL files -> download first
import pyreadstat, urllib.request
urllib.request.urlretrieve(BASE + "synthetic-co2-cross-section.dta", "synthetic-co2-cross-section.dta")
df, meta = pyreadstat.read_dta("synthetic-co2-cross-section.dta")

Copy and paste this snippet in Google Colab app. https://colab.research.google.com/notebooks/empty.ipynb

R

# R : haven::read_dta auto-downloads an https URL
library(haven)
BASE <- "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/content/post/r_bma_lasso_wals/data/"
df <- read_dta(paste0(BASE, "synthetic-co2-cross-section.dta"))

Overview & sources

Companion data for a hands-on R tutorial comparing three principled responses to the variable selection problem — Bayesian Model Averaging (BMA), the LASSO, and Weighted Average Least Squares (WALS) — on a fully synthetic cross-section of 120 fictional countries. Twelve candidate regressors compete to explain log CO₂ emissions: 7 have true nonzero effects and 5 are pure noise deliberately correlated with GDP and the true predictors, creating realistic multicollinearity. Because the data-generating process is known, the data carries its own “answer key” against which each method is graded. The convergence of mechanically distinct methods on the same variables — four are flagged by all three (triple-robust) — illustrates methodological triangulation. The entire DGP is open and reproducible (set.seed(2017)).

Data sources

Cite this data

Please cite this dataset as follows.

APA

Mendez, C. (2026). Three Methods for Robust Variable Selection: BMA, LASSO, and WALS [Data set]. https://carlos-mendez.org/post/r_bma_lasso_wals/

Hoeting, J.A., Madigan, D., Raftery, A.E., & Volinsky, C.T. (1999). Bayesian Model Averaging: A Tutorial. Statistical Science, 14(4), 382–417. Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society, Series B, 58(1), 267–288. Magnus, J.R., Powell, O., & Prufer, P. (2010). A Comparison of Two Model Averaging Techniques with an Application to Growth Empirics. Journal of Econometrics, 154(2), 139–153.

BibTeX

@misc{mendez2026rbmalassowals,
  author       = {Mendez, Carlos},
  title        = {Three Methods for Robust Variable Selection: BMA, LASSO, and WALS},
  year         = {2026},
  howpublished = {\url{https://carlos-mendez.org/post/r_bma_lasso_wals/}},
  note         = {Data set}
}

@article{hoeting1999bma,
  author  = {Hoeting, Jennifer A. and Madigan, David and Raftery, Adrian E. and Volinsky, Chris T.},
  title   = {Bayesian Model Averaging: A Tutorial},
  journal = {Statistical Science},
  volume  = {14}, number = {4}, pages = {382--417}, year = {1999}
}
@article{tibshirani1996lasso,
  author  = {Tibshirani, Robert},
  title   = {Regression Shrinkage and Selection via the Lasso},
  journal = {Journal of the Royal Statistical Society, Series B},
  volume  = {58}, number = {1}, pages = {267--288}, year = {1996}
}
@article{magnus2010wals,
  author  = {Magnus, Jan R. and Powell, Owen and Prufer, Patricia},
  title   = {A Comparison of Two Model Averaging Techniques with an Application to Growth Empirics},
  journal = {Journal of Econometrics},
  volume  = {154}, number = {2}, pages = {139--153}, year = {2010}
}

Variable explorer search & filter all 14 variables

Type to filter by name or label, or use the chips to filter by type. Each row shows a mini distribution. Click a header to sort.

Cross-file variable index

Which file each variable appears in (● = present).

Construction & formulas

The target model regresses log CO₂ on the 12 candidate regressors:

• General model: log_co2 = β₀ + Σⱼ βⱼ xⱼ + ε, j = 1…12.
• BMA — posterior inclusion probability (PIP): PIPⱼ = Σ_{k: j ∈ Mₖ} P(Mₖ | y) — total posterior mass on models containing variable j; PIP ≥ 0.80 is the robustness threshold (Raftery 1995). Posterior mean E[βⱼ | y] = Σₖ β̂ⱼ,ₖ · P(Mₖ | y).
• LASSO — L1 penalty: β̂ = argmin (1/2n)·‖y − Xβ‖² + λ‖β‖₁; λ chosen by 10-fold CV (lambda.min / lambda.1se). Post-LASSO refits OLS on the selected variables for unbiased magnitudes.
• WALS — Laplace prior: split y = X₁β₁ + X₂β₂ + ε into focus regressors X₁ (here the intercept) and auxiliary regressors X₂ (the 12 candidates); orthogonalize X₂ and average each coefficient independently under p(γⱼ) ∝ exp(−|γⱼ|/τ), the same prior that underlies the LASSO penalty. Robustness flagged at |t| ≥ 2.

Synthetic data-generating process (set.seed(2017), n = 120). GDP drives the system: log_gdp ~ N(8.5, 1.5²). True predictors and noise are built as linear functions of log_gdp plus Gaussian noise, so the noise variables are correlated with GDP yet have zero true effect. The outcome is log_co2 = 2.0 + 1.200·log_gdp + 0.008·industry + 0.012·fossil_fuel + 0.010·urban_pop + 0.004·democracy + 0.500·trade_network + 0.005·agriculture + N(0, 0.3²); the five noise variables (log_trade, fdi, corruption, log_tourism, log_credit) enter the outcome with coefficient exactly 0.

The datasets

Switch datasets with the tabs. Each shows the full variable dictionary plus a sortable statistics table with mini distributions and data coverage.

Expand all datasets expand to search (Ctrl/⌘+F) or print across all datasets

Known limitations & caveats

• Synthetic data. There is no real data behind this tutorial. The 120 “countries” and all 13 numeric variables are simulated; results are internally consistent with the calibrated DGP but are not empirical evidence about real CO₂ emissions or their determinants.
• Built-in answer key. The true coefficients are known by design (7 nonzero, 5 exactly zero). The point of the dataset is to grade each method against this truth; that luxury does not exist with real data, which is exactly why the post recommends using all three methods.
• Deliberate multicollinearity. The noise variables are constructed to be correlated with GDP and the true predictors, so a naive OLS can show spurious significance. This is intentional, to make variable selection genuinely hard.
• Power, not bias, limits detection. All methods reach perfect specificity, but small true effects (agriculture, β = 0.005; and the borderline urban_pop / democracy for BMA) are hard to detect at n = 120 — a sample-size limitation, not a flaw in any method.
• Cross-section only. No time dimension and no panel structure; the methods here address model uncertainty, not endogeneity, nonlinearity, or heteroskedasticity, which real applications must handle separately.