On Tests For Double Differencing: Methods Of Demeaning And Detrending And The Role Of Initial Values

In this paper we investigate the impact that starting values have upon the double differencing tests of Hasza and Fuller (1979, Annals of Statistics 7, 1106–1120) and Sen and Dickey (1987, Journal of Business and Economic Statistics 5, 463–473), based on ordinary least squares (OLS) and simple symmetric least squares (SSLS) estimation, respectively. We demonstrate that, contrary to what is observed for conventional unit root tests, when based on data that have been demeaned, either directly or by the inclusion of a constant in the test regression, these tests are not exact similar to the starting values of the process, except where they are fixed and equal. We show that such test statistics can also fail to be asymptotically pivotal, contrary to claims made previously in the literature. We demonstrate that OLS tests based on data that have been detrended, either directly or by the inclusion of a constant and linear time trend in the test regression, do provide exact similar inference. In the context of the SSLS-based approach, we demonstrate that one obtains exact similar tests only where direct detrending is used. We highlight another error that appears in the literature, demonstrating that the two demeaned OLS-based test statistics do not coincide, even asymptotically, and that the same holds for the OLS- and SSLS-based test statistics for detrended data. We use Monte Carlo methods to quantify the finite-sample dependence of these tests on the starting values. These results suggest that the SSLS-based tests are considerably more sensitive to the nature of the starting values than are the OLS-based tests.We are grateful to Pentti Saikkonen and two anonymous referees for their helpful and constructive comments on an earlier version of this paper. We are especially grateful to one of the referees for the suggestion implemented in Remark 3.1 of the paper.