Regular variation of GARCH processes.
Let a \(d\)-dimensional time series \(({\mathbf X}_t)\) be defined by a stochastic recurrence equation (SRE) \({\mathbf X}_t={\mathbf A}_t{\mathbf X}_{t-1}+{\mathbf B}_t\) where \({\mathbf A}_t\) are \(d\times d\) random matrices and \({\mathbf B}_t\) are \(d\)-dimensional vectors such that \((({\mathbf A}_t,{\mathbf B}_t))\) is an iid sequence. A generalized autoregressive conditionally heteroscedastic process \((X_t)\) of order \((p,q)\) (GARCH\((p,q)\)) is given by \(X_t=\sigma _t Z_t\), \(\sigma _t^2=\alpha _0+\sum _{i=1}^p \alpha _i X_{t-i}^2 + \sum _{j=1}^q \beta _j\sigma _{t-j}^2\) where \((Z_t)\) is an iid sequence and \(\alpha _i\)'s, \(\beta _j\)'s are nonnegative constants. The squared processes \((X_t^2)\) and \((\sigma _t^2)\) satisfy a SRE \({\mathbf X}_t={\mathbf A}_t{\mathbf X}_{t-1}+{\mathbf B}_t\) with \({\mathbf X}_t=(\sigma _{t+1}^2,\dots ,\sigma _{t-q+2}^2, X_t^2, \dots , X_{t-p+2}^2)'\). The authors start with some introduction to basic theory for SRE. This part is a valuable review of results concerning SRE. Using the connection between GARCH and SRE the authors show that the finite-dimensional distributions of a GARCH process are regularly varying. This implies the moment properties of the process and the dependence structure between neighboring observations. Regular variation enables to establish the large sample behavior of statistics from GARCH process such as the sample mean, sample autocovariance and sample autocorrelation function. For example, if the fourth moment of the process does not exist, then the rate of convergence of the sample autocorrelations is extremely slow.
scientific article; zbMATH DE number 2138964
scientific article