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Is Using Shapley Value Good for Covariate Selection in Panel Models?

This paper proposes and validates a fixed-effects Shapley-value method for covariate selection in panel models, demonstrating through simulations and empirical applications that it offers a transparent, computationally efficient, and parsimonious alternative to existing approaches while accurately recovering true active sets and improving model performance.

Original authors: Htwe Ko, Pathairat Pastpipatkul

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Htwe Ko, Pathairat Pastpipatkul

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of economics, researchers often face a puzzle that looks simple but is deeply tricky: how to figure out which factors truly drive a specific outcome, like the growth of a nation's income. Imagine a scientist looking at a massive spreadsheet containing data from dozens of countries over many years. They have a list of potential causes—things like education spending, foreign investment, inflation rates, and population growth. The problem is that there are too many candidates to test them all at once without creating confusion, and economic theory rarely points to a single, perfect list of ingredients. For decades, statisticians have tried to solve this by using various mathematical shortcuts to filter out the noise, but these methods often leave researchers guessing about which variables are actually doing the heavy lifting and which are just along for the ride. The core challenge is not just finding a model that fits the data well, but understanding exactly how much credit each individual factor deserves for that fit, especially when those factors are tangled together in complex ways.

A team of researchers at Chiang Mai University has proposed a fresh way to tackle this problem, borrowing a concept from game theory to bring clarity to the chaos of economic data. Instead of treating the selection of variables as a statistical guessing game, they framed it as a cooperative game where every potential factor is a player. In this game, the goal is to measure how much each player contributes to the team's success, which in this case is the ability of the model to explain changes in national income. The researchers used a mathematical tool called the Shapley value, a method originally designed to fairly divide winnings among players in a cooperative game. By applying this logic to economic data, they could calculate the exact average contribution of every single variable, accounting for every possible combination of factors. This approach treats the fixed effects of the data—the unique, unchanging traits of each country and each year—as a constant background, allowing the researchers to focus purely on how the changing variables interact to explain the outcome.

The researchers tested their new method, which they call Fixed Effects Shapley-Value covariate selection, using a series of computer simulations. They created thousands of fake economic datasets where they knew the true answer beforehand: they knew exactly which variables were real drivers and which were just random noise. In these tests, the method proved remarkably effective. As the amount of data grew, the method became almost perfect at identifying the true drivers and ignoring the noise. It successfully separated the signal from the static, recovering the correct set of variables in nearly every instance when the sample size was large enough. The simulations showed that while some older methods would keep including irrelevant variables just because they happened to look good in a specific combination, this new approach consistently assigned zero or negative value to the useless factors, effectively filtering them out.

To see if this worked in the real world, the team applied their method to actual economic data from forty-five countries, split into groups of twenty developed nations and twenty-five developing nations, covering the years from 2000 to 2023. They started with a full model containing ten different economic indicators, such as government debt, trade openness, and exchange rates. When they ran the Shapley value analysis, the results revealed a clear hierarchy of importance that differed depending on the type of country. For the developed nations, trade openness and foreign direct investment emerged as the most powerful drivers of growth, while for developing nations, foreign investment and population growth took the lead. The method also highlighted variables that, despite appearing significant in traditional statistical tests, actually contributed very little to the overall explanation of growth. In some cases, it even assigned negative value to certain factors, indicating that including them actually made the model worse by adding unnecessary complexity without adding real explanatory power.

The final models produced by this method were much simpler and more focused than the original, all-encompassing versions. In the developed country group, the method selected seven key variables; in the developing group, it narrowed the list down to just three; and for the combined group, it chose four. These streamlined models performed just as well as the full models in explaining the data, but with a much cleaner structure. When the researchers compared their results to other advanced statistical techniques, they found a strong agreement on the most important variables, but the new method offered a distinct advantage: it provided a single, definitive list of the best variables rather than a confusing set of probabilities. This means that a researcher can now look at the data and say with confidence exactly which factors are driving growth, without having to guess which variables to keep or discard. The study concludes that this game-theoretic approach offers a transparent, fair, and computationally efficient way to solve one of economics' oldest problems, turning a messy list of possibilities into a clear, evidence-based story about what actually matters.

Technical Summary: Fixed-Effects Shapley-Value Covariate Selection in Panel Models

Problem Statement
The paper addresses the persistent challenge of covariate selection in panel data models, particularly in macroeconomic contexts where researchers face a "menu" of plausible determinants that often exceeds the number of observations or reliable estimation capacity. Traditional approaches, such as stepwise regression, Lasso-type penalization, and information-criterion searches over all $2^p$ subsets, face limitations: they become computationally inflexible as the number of covariates ( $p$ ) exceeds roughly twenty, their conclusions are sensitive to the chosen criterion, and they fail to provide an exact decomposition of a model's explanatory power into individual regressor contributions. Furthermore, existing methods often conflate selection with shrinkage or lack a transparent mechanism to attribute the "value" of the model to specific variables while accounting for fixed effects.

Methodology: Fixed-Effects Shapley-Value (FESV)
The authors propose the Fixed-Effects Shapley-Value (FESV) method, which frames covariate selection as a cooperative game-theoretic problem.

Game Formulation:
- Players: Each candidate covariate is treated as a player in a transferable-utility game.
- Non-Players: Unit ( $\alpha_i$ ) and time ( $\gamma_t$ ) fixed effects are treated as structural components that are always present; they are partialled out via the within-transformation before coalition evaluation and are never selected.
- Characteristic Function ( $v$ ): The value of a coalition (subset of covariates $S$ ) is defined as the adjusted $R^2$ of the within-estimator regression using only variables in $S$ . This function is net of fixed effects and penalized for degrees of freedom.
- Alternative Functions: The study also considers unadjusted $R^2$ and negative BIC as characteristic functions to test robustness.
Shapley Value Calculation:
- The Shapley value ( $\phi_i$ ) for each covariate is computed as the weighted average of its marginal contribution across all possible coalitions ( $2^p$ subsets).
- Properties: The method leverages the Shapley value's axiomatic properties: efficiency (sum of values equals the total explained variation), symmetry, additivity, and the null-player property (irrelevant variables receive zero or negative value).
- Computation: The exact Shapley vector is calculated by enumerating all $2^p$ coalitions, fitting the within-regression for each, and applying combinatorial weights.
Selection Rules:
- Rule A (Positive Sign): Selects all covariates with a positive Shapley value ( $\phi_i > 0$ ). This serves as a diagnostic but tends to over-select in finite samples due to sampling noise.
- Rule B (Penalized Nested): Orders covariates by decreasing Shapley value and selects the top- $k$ coalition that maximizes a penalized objective: $J(k) = v(M_k) - \gamma k$ , where $\gamma$ is a user-defined penalty (e.g., 0.01). This rule effectively controls false positives by charging a cost for adding variables that do not sufficiently improve the adjusted fit.

Key Contributions

Exact Decomposition: Unlike stepwise or penalized methods, FESV provides an exact, non-residual decomposition of the full model's adjusted $R^2$ into per-covariate contributions.
Separation of Selection and Estimation: The method performs selection without shrinkage. Once the optimal subset is identified via the Shapley game, standard OLS on the within-transformed data is used for final estimation, preserving coefficient interpretability.
Handling Fixed Effects: The approach explicitly treats fixed effects as non-players, ensuring selection is robust to the specification of unit and time effects.
Theoretical Consistency: The paper establishes that under standard regularity assumptions, the estimated Shapley values converge to the population values, and Rule B consistently recovers the true active set as $N \times T \to \infty$ .

Empirical and Simulation Results

Monte Carlo Simulations:
- Simulations across five sample sizes ( $N \times T$ ranging from 50 to 4000) with 6 covariates (3 active, 3 noise) demonstrate that Rule B recovers the true active set with near-perfect precision and recall (F1 score $\approx$ 1.0) at moderate sample sizes ( $N \times T \geq 450$ ).
- Rule A maintains high recall but suffers from a persistent false-positive rate (precision $\approx$ 0.79) because sampling noise occasionally assigns small positive values to noise covariates.
- The simulations confirm that the penalized characteristic function (adjusted $R^2$ ) assigns negative values to noise covariates in expectation, enabling effective selection.
Empirical Application (Growth Regressions):
- Applied to panel data of 45 countries (20 developed, 25 developing) from 2000–2023 to explain GDP growth.
- Findings: The FESV method selected parsimonious models (3–7 variables) compared to the full 10-variable specification.
  - Developed: Selected Trade Openness, FDI, Population Growth, Inflation, Education, Unemployment, and Government Effectiveness.
  - Developing: Selected FDI, Exchange Rate, and Population Growth.
  - Pooled: Selected FDI, Population Growth, Trade Openness, and Debt.
- Performance: The selected models achieved lower BIC scores than the full models while retaining the primary determinants. The adjusted $R^2$ decreased only marginally.
- Comparison: The FESV rankings aligned closely with Bayesian Panel Variable Selection (BPVS) models regarding the top-ranked variables (e.g., FDI and Population Growth were consistently top predictors). However, FESV produced more parsimonious models, whereas BPVS retained all variables due to high posterior inclusion probabilities.

Significance and Claims
The paper claims that FESV offers a transparent, computationally efficient, and theoretically grounded tool for covariate selection in panel models. Its primary significance lies in:

Transparency: It converts the "black box" of variable selection into an auditable attribution of explanatory power, where the contribution of each variable is explicitly calculated and sums exactly to the model's total fit.
Parsimony without Shrinkage: It achieves model sparsity through a game-theoretic penalty on degrees of freedom rather than coefficient shrinkage, allowing for standard post-selection inference.
Robustness: The method is shown to be consistent with Bayesian approaches in identifying key drivers but superior in producing a single, defensible, and parsimonious specification.

Limitations Acknowledged
The authors note that the exact enumeration of $2^p$ coalitions limits the method to moderate numbers of covariates (though sampling approximations exist). The current framework is limited to linear models with continuous outcomes, and the paper does not provide a formal theory for post-selection inference (e.g., valid confidence intervals after selection), pointing to existing literature on penalized selection for future directions.

Technical Summary: Fixed-Effects Shapley-Value Covariate Selection in Panel Models

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