Whittle estimation in a heavy-tailed GARCH(1,1) model.
type: Item
instance of: scholarly article
title: Whittle estimation in a heavy-tailed GARCH(1,1) model.
zbMATH Open document ID: 1057.62072
DOI: 10.1016/S0304-4149(02)00097-2
author: T. Mikosch
author: Daniel Straumann
published in: Stochastic Processes and their Applications
publication date: 2005-02-25T00:00:00Z
review text: 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.
reviewed by: Jiří Anděl
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 2138924
zbMATH Keywords: Whittle estimation
zbMATH Keywords: GARCH processes
zbMATH Keywords: heavy tails
zbMATH Keywords: stable limit distribution
describes a project that uses: itsmr
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2138924