Whittle estimation in a heavy-tailed GARCH(1,1) model

The squares of a GARCH(p,q) process satisfy an ARMA equation with white noise innovations and parameters which are derived from the GARCH model. Moreover, the noise sequence of this ARMA process constitutes a strongly mixing stationary process with geometric rate. These properties suggest to apply classical estimation theory for stationary ARMA processes. We focus on the Whittle estimator for the parameters of the resulting ARMA model. Giraitis and Robinson (2000) show in this context that the Whittle estimator is strongly consistent and asymptotically normal provided the process has finite 8th moment marginal distribution.

We focus on the GARCH(1,1) case when the 8th moment is infinite. This case corresponds to various real-life log-return series of financial data. We show that the Whittle estimator is consistent as long as the 4th moment is finite and inconsistent when the 4th moment is infinite. Moreover, in the finite 4th moment case rates of convergence of the Whittle estimator to the true parameter are the slower, the fatter the tail of the distribution.

These findings are in contrast to ARMA processes with iid innovations. Indeed, in the latter case it was shown by Mikosch et al. (1995) that the rate of convergence of the Whittle estimator to the true parameter is the faster, the fatter the tails of the innovations distribution. Thus the analogy between a squared GARCH process and an ARMA process is misleading insofar that one of the classical estimation techniques, Whittle estimation, does not yield the expected analogy of the asymptotic behavior of the estimators.