Testing the nullity of GARCH coefficients: Correction of the standard tests and relative efficiency comparisons

This article is concerned with testing the nullity of coefficients in generalized autoregressive conditionally heteroscedastic (GARCH) models. The problem is nonstandard because the quasi-maximum likelihood estimator is subject to positivity constraints. This article establishes the asymptotic null and local alternative distributions of Wald, score, and quasi-likelihood ratio tests. Efficiency comparisons under fixed alternatives are considered. Two cases of special interest are tests of the null hypothesis of one coefficient equal to zero and tests of the null hypothesis of no conditional heteroscedasticity. Finally, the proposed approach is used in the analysis of financial data and suggests reconsidering the preeminence of GARCH(1,1) among GARCH models.