Unified Inference for Predictive Mean and Quantile Regressions via Empirical Likelihood

We develop an empirical likelihood framework for testing return predictability in the conditional mean and conditional quantiles. A unified chi-square limit theory is established across a broad spectrum of predictor persistence, including stationary, mildly integrated, nearly integrated, unit-root, and mildly explosive cases. We provide two complementary approaches to handle the unknown intercept: (i) a sample-splitting approach under relaxed regularity conditions and (ii) a new two-stage method that improves efficiency and accommodates quantile inference, where sample-splitting is infeasible. We examine the finite-sample bias of the two-stage method, and propose a bias-correction scheme and gradually saturated weights that improve performance under high persistence. Simulation evidence demonstrates that our tests exhibit competitive size and power across persistence classes, with notable gains in quantile predictability. An empirical application to the U.S. stock market shows modest evidence of mean predictability, whereas quantile-based inference reveals stronger and economically relevant predictability in the tails of the return distribution.