Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power

Abstract

In the context of testing for a unit root in a univariate time series, the convention is to ignore information in related time series. This paper shows that this convention is quite costly, as large power gains can be achieved by including correlated stationary covariates in the regression equation. The paper derives the asymptotic distribution of ordinary least-squares estimates of the largest autoregressive root and its t-statistic. The asymptotic distribution is not the conventional Dickey-Fuller distribution, but a convex combination of the Dickey-Fuller distribution and the standard normal, the mixture depending on the correlation between the equation error and the regression covariates. The local asymptotic power functions associated with these test statistics suggest enormous gains over the conventional unit root tests. A simulation study and empirical application illustrate the potential of the new approach.