Testing for Common GARCH Factors

This paper proposes a test for common GARCH factors in asset returns. Following Engle and Kozicki (1993), the common GARCH factors property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that the Hansen's (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain empirical process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a half-half mixture of chi-squares with H-1 and H degrees of freedom, where H is the number of moment conditions, as opposed to a chi-square with H-1 degree of freedom. With more than two assets, this distribution lies between the chi-square with H-p and the chi-square with H degrees of freedom (p, the number of parameters) and both bounds are conditionally sharp. These results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with possibly increasing over-rejection rate with the number of assets. A Monte Carlo study illustrates these findings.