The Asymptotic Distribution of Unit Root Tests of Unstable Autoregressive Processes

UNIT ROOT TESTING has been developed through numerous papers since the work of Ž . Dickey and Fuller 1979 . The idea is to test the hypothesis that the differences of an observed time series do not depend on its levels, or in other words, the levels of the time series have a unit root that can be removed by differencing. While it is in general possible to have multiple unit roots, only the hypothesis of exactly one unit root is considered Ž . here. The available tests therefore hinge on two assumptions: i the levels of the time Ž . series have exactly one unit root which can be removed by differencing, and ii the remaining characteristic roots of the time series are stationary roots. In this paper it is proved that for the likelihood ratio test and a number of other likelihood based statistics Ž . Ž . the assumption ii is redundant whereas i is necessary. It is also shown that for some tests that are not likelihood based it is indeed necessary to assume that the differences have stationary roots. The consequences of the result are perhaps best understood from the implications of Ž . condition i . For autoregressive models of order two or higher, that condition is not satisfied in the entire parameter space and the asymptotic distribution of the likelihood ratio test for a unit root depends on unknown nuisance parameters. In this situation the test statistic is not pivotal; hence the test is not similar, and this complicates the testing. Ž . For non-likelihood based tests the necessity of condition ii implies an additional similarity problem. The practitioner is therefore faced with a trade off between likelihood based tests with fewer similarity problems and other tests that may have other advantageous properties. There are thus two empirical implications of the result. First, when analyzing time series with stationary roots that have modulus close to one so that Ž . condition ii is nearly violated, then the likelihood based tests are preferable and other tests should be used cautiously. Secondly, if explosive roots are found in an application, most of the statistical analysis is actually valid and should not necessarily be disregarded because of the presence of explosive roots. Section 2 presents a Gaussian autoregressive model along with its statistical analysis Ž . and the result showing that condition ii is redundant for likelihood based tests. Robustness with respect to innovations that are martingale difference is also discussed. The results of Section 2 are given for a model without deterministic trends. In Section 3 these are generalized to models with deterministic terms. The mathematical proofs Ž . following in two Appendices are based on the work of Lai and Wei 1983 and Chan and Ž . Wei 1988 .