A New Robust Inference for Predictive Quantile Regression

Abstract

For predictive quantile regressions with highly persistent regressors, a conventional test statistic suffers from a serious size distortion and its limiting distribution relies on the unknown persistence degree of predictors. This paper proposes a double-weighted approach to offer a robust inferential theory across all types of persistent regressors. We first estimate a quantile regression with an auxiliary regressor, which is generated as a weighted combination of an exogenous random walk process and a bounded transformation of the original regressor. With a similar spirit of rotation in factor analysis, one can then construct a weighted estimator using the estimated coefficients of the original predictor and the auxiliary regressor. Under some mild conditions, it shows that the self-normalized test statistic based on the weighted estimator converges to a standard normal distribution. Our new approach enjoys a nice property that it can reach the local power under the optimal rate T with nonstationary predictor and squared root of T for stationary predictor, respectively. More importantly, our approach can be easily used to characterize mixed persistence degrees in multiple regressions. Simulations and empirical studies are provided to demonstrate the effectiveness of the newly proposed approach. The heterogenous predictability of US stock returns at different quantile levels is reexamined.