Finite Sample Theory of QMLEs in ARCH Models with an Exogenous Variable in the Conditional Variance Equation

In this paper we provide simulation and theoretical results concerning the finite sample theory of QML estimators in ARCH models when we include an exogenous variable in the conditional variance equation. In this setting, we find theoretical and simulation support to suggest that if we consider two exogenous variables with the same variance, the one that has the larger sample mean is more likely to produce a larger bias in the QML estimators, in such a way that can be quite misleading in practical applications. 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. Finally, we generalize the Lumsdaine (1995) invariance properties for the biases in these situations. An empirical application shows the usefulness of our theoretical results.