On Tests for Double Differencing: Some Extensions and the Role of Initial Values

In this paper we investigate the impact which starting values have upon the double differencing tests of Hasza and Fuller (1979, An. Stats.) and Sen and Dickey (1987, Jn.Bus.Ec.Stats.). We demonstrate that when bases on data which has been de-meaned, these tests are not exact similar to the starting values of the process, except when they are fixed and equal. We show that such tests can also fail to be asymptotically pivotal. We demonstrate that OLS test based on data which has been de-trended, yield exact similar inference correcting a further mistake which appears in the literature. We extend the SSLS-based tests in a similar direction and show that here one obtains exact similar tests only if direct de-trending is used. We highlight another error which appears in the literature, demonstrating that the two de-meaned OLS-based tests do not coincide, even asymptotically, and the same holds for the OLS- and SSLS-based tests for detrended data. We use Monte Carlo methods to quantify the finite sample dependence of these tests on the starting values.