On the Performance of Popular Unit-Root Tests Against Various Nonlinear Dynamic Models: A Simulation Study

ABSTRACT

There is a widespread perception that standard unit-root tests have poor discriminatory power when they are applied to time series with nonlinear dynamics. Via Monte Carlo simulations this study re-examines the finite sample properties of selected univariate tests for unit-root and stationarity under a broad class of nonlinear dynamic models. Our simulation experiments produce a couple of interesting findings. First, performance of tests is driven by the degree of underlying persistence rather than the nonlinear dynamics per se. Tests under study exhibit reasonable performance for nonlinear models with mild persistence, while the accuracy of inference deteriorates substantially when the models are highly persistent regardless of the linearity. Second, when it comes to deciding which one to identify first between linearity and stationarity, our results suggest to conduct linearity test first to enhance the reliability of test inference.

Keywords:

• Monte Carlo simulation
• Nonlinear dynamic models
• Unit-root tests

Mathematics Subject Classification:

• Primary 62F03
• Secondary 62M10

Acknowledgment

We thank an anonymous referee for useful comments that helped to improve the article.

Notes

e.g. Note: I(0) and I(1) respectively, denote stationary and unit-root processes.

¹Another class of model that has been popular in the literature on unit-root testing is models with structural breaks. In the present study, we consider the models only for I(1) processes partly because considerable work has been already devoted to them when series with structural breaks are stationary and more because the general conclusions on the poor performance of usual tests in the presence of structural breaks or regime shifts are largely repeated in our analysis that are not reported here. For further discussion on this issue, the reader is referred to recent studies by Sen (2003) for the DF-type test and Cook and Manning (2004) for the ADF-GLS test.

²In a recent analysis of the impact of the initial condition on the problem of testing for unit-root, Müller and Elliott (2003) report that the power of unit-root tests is sensitive to the deviation of the initial observations from its modeled deterministic part. This motivated us to conduct a separate simulation to explore the robustness of our findings to the initial condition issue. Our unreported simulation results show that the test performances are qualitatively similar except for the ADF-GLS test which turns out to be highly sensitive to the initial condition. To be concrete, the ADF-GLS test exhibits poorer performances when the initial observations are more deviant from the modeled deterministic part. We thank an anonymous referee for bringing this issue to our attention.

³We present only the results from i.i.d. error terms here because qualitatively similar results are obtained from non i.i.d. error terms only with a slight deterioration in power performance. They are available from the authors upon request.

Notes: Rejection rates are calculated by the number of times out of 5,000 simulations that corresponding statistics are greater than the critical values of 10% significance level. The optimal lag lengths for the ADF and ADF-GLS tests are selected based on the sequential t-test criterion due to Hall (1994) with the maximum lag length set as integer [8(T/100)^1/4].

⁴Given that the KSS test tends to reject the unit-root null quite substantively when it is applied to other nonlinear models than the STAR models under the alternative hypothesis, care should be exercised in interpreting rejection of the null as conclusive evidence for ESTAR process because the rejection can be driven by other nonlinear models or linear model.

⁵Cook and Manning (2004) report poor finite sample properties of the ADF-GLS test in the presence of structural breaks.

Note: The sum of AR coefficients (SARC) are estimated using the recursive demeaning Cauchy estimator due to So and Shin (1999) to address the downward bias embedded in the LS estimation.

⁶According to Dahl (2002) and Dahl and Gonzalez-Rivera (2003), these tests have good power performance in detecting many types of nonlinear model. Consequently we do not re-examine their performance under nonlinear stationary models here.

Note: Hamilton refers to the LM-type test developed by Hamilton (2001), λ ^A is a modified version of Hamilton test due to Dahl and Gonzalez-Rivera (2003), McLeod–Li refers to the tests developed by McLeod and Li (1983), and V23 is a LM-type version of the neural network test suggested by Teräsvirta et al. (1993). Rejection rates are computed by the number of times out of 5,000 simulations that corresponding p-values are less than 0.05 for 5% and 0.1 for 10%.