Long-run predictability tests are even worse than you thought

Abstract

We derive asymptotic results for the long-horizon ordinary least squares (OLS) estimator and corresponding $t$-statistic for stationary autoregressive predictors. The $t$-statistic—formed using the correct asymptotic variance—together with standard-normal critical values result in a correctly-sized test for exogenous predictors. For endogenous predictors, the test is size distorted regardless of the persistence in the predictor and adjusted critical values are necessary. The endogeneity problem stems from the long-run estimation and is distinct from the ordinary persistence-dependent “Stambaugh” bias. The bias for fully stationary predictors appears not to have been previously noted and adds further difficulty to inference in long-run predictive regressions.