Testing for unit roots in the context of misspecified logarithmic random walks

Testing for unit roots has been among the most heavily researched topics in Econometrics for the last quarter of a century. Much less researched is the equally important issue of the appropriate transformation if any of the variable of interest which should preceed any such testing. In macroeconometrics and empirical finance (stock prices, exchange rates) there are often compelling arguments in favor of a logarithmic transformation. Elsewhere, for instance in the modelling of interest rates, a levels specification automatically suggests itself. In many applications, however, it is not a priori clear, given that one suspects a unit root, whether this unit root is present in the levels or the logs, so there is certainly some interest in the testing for unit roots in the context of an incompletely specified nonlinear transformation of the data . This issue can be approached from various angles: One is to check which transformations leave the I (1)-property of a time series intact, the presumption being that any such transformation could then do little damage to the null distribution of a test for unit roots (Granger and Hallmann 1991; Ermini and Granger 1993 Corradi 1995). A related one is to use tests whose null distribution is robust to monotonic transformations, whether the transformed data are I (1) or not (Granger and Hallmann 1991 , Burridge and Guerre 1996 , Gourieroux and Breitung 1999) or to embed the levels and log specifications respectively in a general Box-Cox-framework and to estimate the transformation parameter before testing (Franses and McAleer 1998, Franses and Koop 1998 , Kobayashi and McAleer 1999). The present paper continues along the lines of Granger and Hallmann (1991) by focussing on a conventional test procedure, the standard Dickey-Fuller-test, and by investigating its properties under a misspecified nonlinear transformation in particular, investigating whether an existing unit root is still detected, i.e. the null hypothesis of an existing unit root is not rejected when an inappropriate transformation is applied. Given that this test is widely employed and given that the choice between a linear and a log linear specification is often rather haphazardly done in applications, it is important to know the degree to which the acceptance of the null hypothesis depends on the correctness of the data transformation. Granger and Hallmann (1991) find through Monte Carlo that the standard Dickey-Fuller-test overrejects a correct null hypotheses of a random walk in the logs, when the test is instead applied to the levels. Below we prove analytically that the rejection probability can take arbitrary values between zero and one for any sample size. An analogous result obtains when the levels follow a random walk, but the Dickey-Fuller-test is applied to the logs. Again, the rejection probability is shown to be depend on both the sample size and the innovation variance, so the null distribution of the DF-test is extremely non-robust to improper data transformations.