Whittle estimation in a heavy-tailed GARCH(1,1) model. (Q1766031)

Let \((Z_t)\) be a sequence of iid random variables with \(\text{var}\,Z_1=1\). Define \[ \sigma _t^2=\alpha _0 + \sum _{i=1}^p \alpha _i X_{t-i}^2 + \sum _{j=1}^q \beta _j\sigma _{t-j}^2. \] Then \(X_t=\sigma _t Z_t\) is a GARCH\((p,q)\) process. We have \[ X_t^2 - \sum _{i=1}^k \phi _i X_{t-i}^2 = \alpha _0+\nu _t-\sum _{j=1}^q \beta _j \nu _{t-j}, \] where \(k=\max (p,q)\), \(\phi _i=\alpha _i + \beta _i\), and \(\nu _t=\sigma _t^2(Z_t^2-1)\). If \((\sigma _t^2)\) is strictly stationary and \(\text{var}\,X_t^2<\infty \), then \((\nu _t)\) is a white noise, \((X_t^2)\) is an ARMA\((k,q)\) process and classical estimation theory for ARMA processes can be applied. The authors investigate the GARCH(1,1) model when the 8\,th moment of the marginal distributions is infinite. They prove that the Whittle estimator is consistent if the 4\,th moment is finite, and inconsistent in the opposite case. If the 4\,th moment is finite then the rate of convergence of the Whittle estimator is the slower, the fatter the tail of the distribution. The small, moderate, and large sample properties of the Whittle estimator and the conditional Gaussian quasi-maximum likelihood estimator are investigated in a simulation study. It is shown that the speed of convergence to the limiting distribution is relatively slow and that the quasi-maximum likelihood estimator is better for large sample sizes.