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

This article is concerned by testing the nullity of coefficients in GARCH models. The problem is non standard because the quasi-maximum likelihood estimator is subject to positivity constraints. The paper establishes the asymptotic null and local alternative distributions of Wald, score, and quasi-likelihood ratio tests. Efficiency comparisons under fixed alternatives are also considered. Two cases of special interest are: (i) tests of the null hypothesis of one coefficient equal to zero and (ii) tests of the null hypothesis of no conditional heteroscedasticity. Finally, the proposed approach is used in the analysis of a set of financial data and leads to reconsider the preeminence of GARCH(1,1) among GARCH models.