Testing for ARCH in the presence of a possibly misspecified conditional mean

Ever since the development of the Autoregressive Conditional Heteroskedasticity (ARCH) model (Engle, 1982), testing for the presence of ARCH has become a routine diagnostic. One popular method of testing for ARCH is T times the R2 from a regression of the squared residuals on p of its lags. This test has been shown to have a lagrange multiplier interpretation and is asymptotically distributed as a χ2(p) random variable. Underlying this test is the assumption of a correctly specified conditional mean. In this paper, we consider the properties of the ARCH test when there is a possibly misspecified conditional mean. Examples of misspecification include omitted variables, structural change, and parameter instability. We show that, in general, misspecification will lead to overrejection of the null hypothesis of conditional homoskedasticity. We demonstrate the use of recursive residuals to improve the fit of a first-stage conditional mean regression. We illustrate these results via Monte Carlo simulations and consider two empirical examples.