Predictive Quantile Regressions with Persistent and Heteroskedastic Predictors: A Powerful 2SLS Testing Approach

We develop new tests for predictability at a given quantile, based on the Lagrange Multiplier [LM] principle, in the context of quantile regression [QR] models which allow for persistent and endogenous predictors driven by heteroskedastic errors. Of the extant predictive QR tests in the literature, only the moving blocks bootstrap implementation, due to Fan and Lee (2019), of the Wald-type test of Lee (2016) can allow for conditionally heteroskedastic errors in the context of a QR model with persistent predictors. In common with all other tests in the literature these tests cannot, however, allow for unconditionally heteroskedastic behaviour in the errors. The LM- based approach we adopt in this paper is obtained from a simple auxiliary linear test regression which facilitates inference based on established instrumental variable methods. We demonstrate that, as a result, the tests we develop, based on either conventional or heteroskedasticity- consistent standard errors in the auxiliary regression, are robust under the null hypothesis of no predictability to conditional heteroskedasticity and to unconditional heteroskedasticity in the errors driving the predictors, with no need for bootstrap implementation. We also propose tests for joint predictability across a set of multiple distinct quantiles. Simulation results for both conditionally and unconditionally heteroskedastic errors highlight the superior finite sample properties of our proposed LM tests over the tests of Lee (2016) and Fan and Lee (2019) and the recent variable addition tests of Cai et al. (2023). An empirical application to the equity premium for the S&P 500 highlights the practical usefulness of our proposed tests, uncovering significant evidence of predictability in the left and right tails of the returns distribution for a number of predictors containing information on market or firm risk.