This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features 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 x_(H-1)^2and x_H^2, where H is the number of moment conditions, as opposed to a x_(H-1)^2. With more than two assets, this distribution lies between the x_(H-p)^2 and x_H^2 (p, the number of parameters). 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.