Structural Inference for Dynamic Panel Quantile Regression: Asymptotic Theory of a Two-Stage Estimator Using a Single Lagged-Difference Instrument

Abstract

In dynamic panel quantile regression, the lagged dependent variable is the principal source of endogeneity. This paper develops the theory of a two-stage quantile estimator (TSQRDPD) for the first-order dynamic panel model, instrumenting the endogenous lag with its own lagged first difference. The instrument is generated internally by the dynamic process, so no external instrument is needed: first differencing annihilates the time-invariant individual effect, purging the instrument of the heterogeneity that drives endogeneity while preserving correlation with the lagged level under stationarity. The first stage projects the lag onto the instrument vector — the lagged difference and the strictly exogenous covariates — via a quantile specification; the second substitutes fitted values into the structural equation, both stages estimated at a common quantile. We establish consistency and asymptotic normality under fixed- , large-  asymptotics, derive a closed-form sandwich covariance matrix, and prove plug-in consistency. Because individual effects are neutralized rather than estimated, no incidental-parameters problem arises. We further provide per-quantile first-stage strength diagnostics with practical thresholds, discuss extensions to clustered dependence, common time effects, and unbalanced panels, and position the estimator’s assumptions, computational cost, and inference against instrumental-variable quantile regression (IVQR)-type, fixed-effects, and recent short-panel quantile alternatives. Identification weakens as the autoregressive coefficient approaches unity, and this weak-identification region is characterized explicitly and flagged as the method’s principal limitation.   Keywords: quantile regression; dynamic panel data; lagged-difference instrument; two-stage estimation; fixed effects; asymptotic normality; weak identification.