Inference in Predictive Quantile Regressions

This paper studies inference in predictive quantile regressions when the predictive regressor has a near-unit root. We derive asymptotic distributions for the quantile regression estimator and its heteroskedasticity and autocorrelation consistent (HAC) t-statistic in terms of functionals of Ornstein-Uhlenbeck processes. We then propose a switching-fully modified (FM) predictive test for quantile predictability with persistent regressors. The proposed test employs an FM style correction with a Bonferroni bound for the local-to-unity parameter when the predictor has a near unit root. It switches to a standard predictive quantile regression test with a slightly conservative critical value when the largest root of the predictor lies in the stationary range. Simulations indicate that the test has reliable size in small samples and particularly good power when the predictor is persistent and endogenous, i.e., when the predictive regression problem is most acute. We employ this new methodology to test the ability of three commonly employed, highly persistent and endogenous lagged valuation regressors - the dividend price ratio, earnings price ratio, and book to market ratio - to predict the median, shoulders, and tails of the stock return distribution.