Maximum Likelihood Unit Root Testing in the Presence of GARCH: A New Test with Increased Power

Abstract

The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. A new test based upon the combination of local-to-unity detrending and joint maximum likelihood estimation of the autoregressive parameter and GARCH process is presented. The finite sample distribution of the test is derived under alternative decisions regarding the deterministic terms employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit increased power of relative to rival tests. Finally, the empirical relevance of the simulation results is illustrated via an application to real GDP for the UK.

Keywords:

• GARCH
• GDP
• Monte Carlo simulation
• Maximum likelihood estimation
• Unit roots

Mathematics Subject Classification:

• 62F03
• 65C05

Notes

¹Note that while Cook (2008) explicitly includes deterministic terms in the testing equation of (2), Seo (1999) instead considers initial demeaning or detrending of the series of interest via preliminary regression.

²The stated values of the quasi-differencing parameters are employed also by Elliott et al. (1996) when extending the Dickey–Fuller test.

Notes: The figures in the above tables represent critical values for the test at the 10, 5, and 1% levels of significance using (8)–(11) for data generated using the DGP of (12)–(15). Estimation was performed using the BHHH algorithm and the Bollerslev–Wooldridge covariance matrix estimator under alternative decisions concerning inclusion of deterministic terms.

Notes: The figures in the above table represent empirical rejection frequencies for the τ_μ, t _β, and tests at the 5% level of significance using the DGP of (16)–(19) over 25,000 replications.

Notes: The figures in the above table represent empirical rejection frequencies for the τ_μ, t _β, and tests at the 5% level of significance using the DGP of (16)–(19) over 25,000 replications.

³The data were downloaded from the National Statistics Web site (http://www. statistics.gov.uk/). The series code is ABMI.

⁴The Monte Carlo experimentation employed to derive this critical value follows the approach adopted above, with the error process generated in the same manner and an identical number of replications used.