Restricted Likelihood Ratio Tests in Predictive Regression

Chen and Deo (2009a) proposed procedures based on restricted maximum likelihood (REML) for estimation and inference in the context of predictive regression. Their method achieves bias reduction in both estimation and inference which assists in overcoming size distortion in predictive hypothesis testing. This paper provides extensions of the REML approach to more general cases which allow for drift in the predictive regressor and multiple regressors. It is shown that without modification the REML approach is seriously oversized and can have unit rejection probability in the limit under the null when the drift in the regressor is dominant. A limit theory for the modified REML test is given under a localized drift specification that accommodates predictors with varying degrees of persistence. The extension is useful in empirical work where predictors typically involve stochastic trends with drift and where there are multiple regressors. Simulations show that with these modifications, the good performance of the restricted likelihood ratio test (RLRT) is preserved and that RLRT outperforms other predictive tests in terms of size and power even when there is no drift in the regressor.