Smoothed Quantile Regression for Panel Data

This paper studies fixed effects estimation of quantile regression (QR) models with panel data. Previous studies show that there are two important difficulties with the standard QR estimation. First, the estimator can be biased because of the well-known incidental parameters problem. Secondly, the non-smoothness of the objective function significantly complicates the asymptotic analysis of the estimator, especially in panel data models. We overcome the latter problem by smoothing the objective function. Under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate, we show that the fixed effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean, and provide the analytic form of the asymptotic bias. We propose a one-step bias correction to the fixed effects estimator based on the analytic bias formula obtained from our asymptotic analysis. Importantly, our results cover the case that the observations are dependent in the time dimension. We illustrate the effect of the bias correction to the estimator through simulations