A note on the tails of the GO‐GARCH process

Abstract

AbstractThe possibility of modeling heavy tails using generalized autoregressive conditional heteroskedasticity (GARCH) models has been rigorously established in the univariate case, and the consequences that this heavy tailedness has on the distributional limits of the sample autocovariance function are well known. In the multivariate case, however, the results have not as yet been provided, and the asymptotic properties of the autocovariance function are not fully understood. In this note, we focus on the generalized orthogonal GARCH (GO‐GARCH) model, a multivariate specification that has recently received some attention in the literature. We first show that marginal heavy tailedness is a simple consequence of the definition of the process and argue that all tail indexes should be equal. Next, we show that the finite‐dimensional distributions of the GO‐GARCH model possess some of the properties known to hold for univariate GARCH and comment on some implications of this fact, which are of practical import. Specifically, we show that the sample autocovariance function may either have a random limit or converge rather slowly to its population counterpart depending on how heavy the tails of the process are. Copyright © 2014 John Wiley & Sons Ltd.