Stationary Errors and Quantile Regression in Short Panels

Abstract

This paper studies a linear panel model with an unrestricted individual effect and a time- stationary idiosyncratic disturbance. We first show that stationarity is a strong restriction in a quantile model. In a linear conditional quantile specification with quantile-dependent slopes, equality of the conditional residual distributions across periods generically forces the slope coefficient to be constant over the quantile index. Thus, a stationary-error model identifies a common location coefficient rather than a collection of quantile-specific slope effects. We then develop a fixed-T estimator of this common coefficient. For each period, we run a cross- sectional quantile regression of the outcome on the full history of regressors. Stationarity makes the quantile projection of the composite individual effect and disturbance common across the period-specific regressions. Differences between diagonal and off-diagonal blocks of the resulting projection coefficients therefore identify the common slope whenever T>=2. We combine all such restrictions by a two-step minimum-distance estimator. The estimator is root-n-consistent and asymptotically normal with fixed T, permits unrestricted dependence across periods within an individual, and does not estimate the individual effects. We provide a consistent analytic covariance estimator, a cluster bootstrap, and an overidentification test of the projection restrictions implied by stationarity. Extensive Monte Carlo experiments show adequate performance under various designs.