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) 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 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 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 ADF test. We apply our testing procedures to the extended Nelson-Plosser data set for the post-1929 samples as well as postwar annual CPI-based real exchange rates for 14 OECD countries.
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