We provide simulation and theoretical results concerning the finite-sample theory of quasi-maximum-likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)-ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t-tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean-squared error of the maximum-likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)-ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1-10] invariance properties for the biases in these situations. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
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