How Do Nonlinear Unit Root Tests Perform with Non Normal Errors?

Abstract

This article examines how popular nonlinear unit root tests perform in the presence of non normal errors. Non normal errors normally do not pose a problem in the usual linear unit root tests since the least squares estimator will still be the most efficient under certain ideal conditions regardless of normal or non normal errors. Whether similar results will carry over to nonlinear unit root tests with non normal errors is a question that merits examination. We find that in contrast to the linear tests, the presence of non normal errors in nonlinear unit root tests will lead to a significant loss of power.

Keywords:

• Nonlinear processes
• Non normal errors
• Unit root

Mathematics Subject Classification:

• 62F03
• 62F05

Notes

In particular, the popular literature documents how to control for serially correlated errors in unit root tests. The augmented Dickey-Fuller tests (1979) and the nonparametric tests of Phillips and Perron (1988) have been very popular in this regard. Subsequent studies examined the cases where certain heterogeneity exists in the error term. The usual unit root tests, if properly modified, can be valid asymptotically if certain regularity conditions are satisfied. Hall and Heyde (1980) noted the required condition that the errors need to be martingale differences, under which the martingale central limit theorem will apply. In this regard, Kim and Schmidt (1993) examined the performance of the DF tests with IGARCH errors.

The simulation results with T = 1, 000 are similar to those with T = 500 and are omitted to conserve space. These results are available from the authors upon request.

Indeed, nonlinear tests can induce more a problem than a solution when a proper nonlinear form is not adopted. For example, the exponential smooth transition autoregressive (ESTAR) unit root tests can be less powerful than the linear DF tests or other nonlinear tests if the true functional form does not necessarily follow the ESTAR models; for example, see Choi and Moh (2007).

The usual maximum likelihood estimators in cross-sectional nonlinear models, such as in Tobit or Probit models, are inconsistent when the error term is non normal. Perhaps, the present article might be the first to demonstrate that this same analogy holds for nonlinear unit root tests.