Testing for Common Conditionally Heteroskedastic Factors

Econometrica · 2013 · Volume 81 · Issue 6

Abstract

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 Hansen's (1982)J-test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty–fifty mixture ofχ2H−1andχ2H, whereHis the number of moment conditions, as opposed to aχ2H−1. With more than two assets, this distribution lies between theχ2H−pandχ2H(pdenotes 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 a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings.

Keywords

Common features, GARCH factors, nonstandard asymptotics, GMM, GMM overidentification test, identification, first-order identification

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