Bootstrapping unit root tests with covariates

We consider the bootstrap method for the covariates augmented Dickey–Fuller (CADF) unit root test suggested in Hansen (1995 Hansen, B. E. (1995). Rethinking the univariate approach to unit root testing: Using covariates to increase power. Econometric Theory 11:1148–1171.[Crossref], [Web of Science ®] , [Google Scholar]) which uses related variables to improve the power of univariate unit root tests. It is shown that there are substantial power gains from including correlated covariates. The limit distribution of the CADF test, however, depends on the nuisance parameter that represents the correlation between the equation error and the covariates. Hence, inference based directly on the CADF test is not possible. To provide a valid inferential basis for the CADF test, we propose to use the parametric bootstrap procedure to obtain critical values, and establish the asymptotic validity of the bootstrap CADF test. Simulations show that the bootstrap CADF test significantly improves the asymptotic and the finite sample size performances of the CADF test, especially when the covariates are highly correlated with the error. Indeed, the bootstrap CADF test offers drastic power gains over the conventional unit root tests. Our testing procedures are applied to the extended Nelson and Plosser data set.