An Exponential Chi-Squared QMLE for Log-GARCH Models Via the ARMA Representation*

Abstract

We propose an ARMA-based quasi-maximum likelihood estimator for log-generalized autoregressive conditional heteroscedasticity (GARCH) models that is efficient when the conditional error is normal, and prove its consistency and asymptotic normality under mild assumptions. A study of efficiency shows the estimator can provide major improvements, both asymptotically and in finite samples. Next, two empirical applications illustrate the usefulness of our estimator. The first shows how it can be used to obtain volatility estimates in the presence of zeros, that is, inliers, since ARMA-based log-GARCH estimators enable a practical and straightforward solution to the inlier problem—even when the zero-generating process is non-stationary. Our study shows volatility estimates can be substantially underestimated if zeros are not handled appropriately. In our second empirical application, we show how our estimator can readily be used to model high-order volatility dynamics where one or more squared error autocorrelations are negative, a characteristic that is not compatible with ordinary (i.e., non-exponential) GARCH models.

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