STRUCTURAL QUANTILE INFERENCE IN DYNAMIC PANEL DATA: ASYMPTOTIC AND DISTRIBUTIONAL PROPERTIES OF AN EXTENDED TWO-STAGE ESTIMATOR UNDER FIXED \(T\) AND LARGE \(N\)

Abstract

In quantile regression for dynamic panel data (QRDPD), lagged dependent variables are the principal source of endogeneity and estimation bias. This paper develops the theoretical foundations of an extended two-stage quantile regression for dynamic panel data (ETSQRDPD) estimator that attenuates endogeneity-induced bias while preserving the structural interpretability of the conditional quantile function. Unlike instrumental-variable remedies, which require strong identification conditions and are fragile to weak instruments, the proposed framework accommodates two lagged dependent variables together with strictly exogenous covariates. In the first stage, each endogenous lag is projected onto the full instrument set through a quantile-based autoregressive specification; in the second, the fitted values are substituted into the structural quantile regression, with both stages estimated at a common quantile level. We establish consistency and asymptotic normality under fixed- \(T\), large- \(N\) asymptotics, derive a closed-form sandwich variance-covariance matrix, and prove consistency of the plug-in estimator. Because individual effects are not estimated directly, the procedure is free of the incidental-parameters problem and imposes no restriction on the relative growth of \(N\) and \(T\). The estimator is equivariant, monotoneinvariant, and robust to outliers. Finite-sample behaviour is left to a companion study.