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Statistica Sinica 18(2008), 647-665



Guodong Li and Wai Keung Li

Nanyang Technological University and University of Hong Kong

Abstract: The recent paper by Ling and Tong (2005) considered a quasi-likelihood ratio test for the threshold in moving average models with $i.i.d.$ errors. This article generalizes their results to the case with GARCH errors, and a new quasi-likelihood ratio test is derived. The generalization is not direct since the techniques developed for TMA models heavily depend on the property of $p$-dependence that is no longer satisfied by the time series models with conditional heteroscedasticity. The new test statistic is shown to converge weakly to a functional of a centered Gaussian process under the null hypothesis of no threshold, and it is also proved that the test has nontrivial asymptotic power under local alternatives. Monte Carlo experiments demonstrate the necessity of our test when a moving average time series has a time varying conditional variance. As further support, two data examples are reported.

Key words and phrases: Conditional heteroscedasticity, Gaussian process, likelihood ratio test, MA-GARCH model, threshold MA-GARCH model.

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