Inference for Tail Index of GARCH(1,1) Model and AR(1) Model with ARCH(1) Errors

For a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors, it is known that the observations have a heavy tail and the tail index is determined by an estimating equation. Therefore, one can estimate the tail index by solving the estimating equation with unknown parameters replaced by quasi maximum likelihood estimation (QMLE), and profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index . However, this requires that the errors of such a model have at least finite fourth moment to ensure asymptotic normality with √ n rate of convergence and Wilks theorem. In this paper, we show that the finite fourth moment can be relaxed by employing a least absolute deviations estimate (LADE) instead of QMLE for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model. The proposed tail index estimators have a