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 (This abstract was borrowed from another version of this item.)