A COMPANION TO Theoretical Econometrics

Multiple tests and levels of significance

It is notable that many tests of the seasonal unit root null hypothesis involve tests on multiple coefficients. In particular, for the application of the HEGY test (31.44),

Hylleberg et al. (1990) recommend that one-sided tests of п1 and п 2 should be applied, with (п 3, п4) either tested sequentially or jointly. The rationale for apply­ing one-sided tests for п 1, п 2, and п 3 is that it permits a test against stationarity, which is not the case when a joint F-type test is applied. Thus, the null hypo­thesis is rejected against stationarity only if the null hypothesis is rejected for each of these three tests. Many applied researchers have followed HEGY's advice, apparently failing to recognize the implications of this strategy for the overall level of significance for the implied joint test of п 1 = п 2 = п 3 = п4 = 0.

Let us assume that separate tests are applied to п 1 and п 2, with a joint test applied to (п 3, п4), with each of these three tests applied at the same level of significance, a. Conveniently, these tests are mutually independent, due to the asymptotic orthogonality of the regressors, as discussed in Section 3.3. Therefore, the overall probability of not rejecting the SI(1) null hypothesis when it is true is (1 - a)3 ~ 1 - 3a for a small. Thus, with a = .05, the implied level of significance for the overall test is 1 - .953 = .14, or approximately three times that of each individual test. With monthly data the issue is even more important of course.

In conclusion, the impact of multiple tests must be borne in mind when applying seasonal unit root tests. To date, however, these issues have received relatively little attention in this literature.