A quantile-based nonadditive fixed effects model

I propose a quantile-based nonadditive fixed effects panel model to study heterogeneous causal effects. Similar to standard fixed effects (FE) model, my model allows arbitrary dependence between regressors and unobserved heterogeneity, but it generalizes the additive separability of standard FE to allow the unobserved heterogeneity to enter nonseparably. Similar to structural quantile models, my model’s random coefficient vector depends on an unobserved, scalar ‘‘rank’’ variable, in which outcomes (excluding an additive noise term) are monotonic at a particular value of the regressor vector, which is much weaker than the conventional monotonicity assumption that must hold at all possible values. This rank is assumed to be stable over time, which is often more economically plausible than the panel quantile studies that assume individual rank is iid over time. It uncovers the heterogeneous causal effects as functions of the rank variable. I provide identification and estimation results, establishing uniform consistency and uniform asymptotic normality of the heterogeneous causal effect function estimator. Simulations show reasonable finite-sample performance and show my model complements fixed effects quantile regression. Finally, I illustrate the proposed methods by examining the causal effect of a country’s oil wealth on its military defense spending.

💡 Research Summary

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The paper introduces a novel panel quantile regression framework that combines the strengths of fixed‑effects (FE) models and structural quantile models while relaxing several restrictive assumptions that dominate the existing literature. The core structural equation is

 Y_it = X_it′ β(U_i) + V_it, i = 1,…,n; t = 1,…,T,

where U_i is an unobserved individual‑specific “rank” variable drawn from a Uniform(0,1) distribution. Crucially, the model allows arbitrary dependence between the regressors X_it and the rank variable U_i, thereby preserving the FE definition of unrestricted correlation between covariates and unobserved heterogeneity. The coefficient function β(·) is deterministic, continuous, and monotone in the rank, but it is otherwise unrestricted; it captures heterogeneous causal effects as a function of the latent rank rather than as a constant across individuals.

The identification strategy proceeds in two steps. First, assuming a sufficiently long time dimension T, ordinary least squares (or any consistent estimator) is applied to each individual i to obtain an estimate β̂_i that converges to β(U_i). Second, a common covariate vector x* is chosen, and the “synthetic outcome” Ŷ_i = x*′ β̂_i is computed. Because x*′ β(u) is monotone in u, the ordering of Ŷ_i reveals the ordering of the latent ranks U_i. By sorting the estimated β̂_i according to the inferred ranks, the researcher can construct a non‑parametric estimator of β(τ) for any τ∈(0,1). This sorting‑based rank recovery replaces the need for instrumental variables or strong monotonicity conditions that are required in Chernozhukov‑Hansen type identification.

The paper formalizes a set of assumptions: (i) i.i.d. sampling across individuals, (ii) a “large‑n, large‑T” asymptotic regime with a mild rate condition linking n and T, (iii) continuity and monotonicity of β(·) only at the chosen x* (a weaker condition than the conventional requirement that x′β(u) be monotone for all x), and (iv) a zero‑mean, conditionally independent idiosyncratic error V_it. Under these conditions, the author proves uniform consistency of the β̂(τ) estimator over any compact subset of (0,1) and establishes uniform asymptotic normality by applying the functional delta method to the empirical quantile process. The resulting limiting Gaussian process incorporates both the estimation error from the first‑stage OLS and the stochastic error from the rank‑sorting step.

The contribution is threefold. First, it provides a structural, non‑additive FE model that directly identifies heterogeneous causal effects without resorting to instrumental variables. Second, it introduces a time‑stable rank variable, arguing that many economic applications (e.g., innate ability, country‑specific resource endowments) are more plausibly constant over time than i.i.d. across periods—a key departure from most panel quantile studies. Third, it delivers a simple, computationally tractable estimator that works well even when T is modest (as low as 3–10) provided the n/T ratio satisfies the stated rate condition. Monte‑Carlo simulations confirm that bias and mean‑squared error remain low across a variety of n/T configurations and that the estimator outperforms or matches standard FE‑QR and conventional FE estimators, especially when the idiosyncratic error variance is sizable.

An empirical illustration examines how a country’s oil wealth influences its military defense spending. Using a balanced panel of 80 countries from 1970 to 2020, the author estimates β(τ) for the effect of oil revenue on defense expenditures. The estimated function is non‑linear: the marginal effect peaks around the median rank (τ≈0.5) and diminishes for both low‑rank (resource‑poor) and high‑rank (resource‑rich) countries. This pattern suggests that policy analyses based solely on average effects could misguide decisions, as the impact of oil wealth is strongest for countries with moderate resource endowments.

In sum, the paper offers a rigorous, flexible tool for uncovering heterogeneous causal relationships in panel data. By allowing arbitrary X‑U dependence, employing a time‑stable rank variable, and delivering uniform asymptotic results, it bridges a gap between structural FE models and quantile regression methods. Future extensions could explore robustness to non‑Gaussian errors, multidimensional rank variables, or Bayesian implementations, thereby broadening the applicability of the proposed framework.