Multivariate Markov-switching ARMA processes with regularly varying noise
The author considers a stationary multivariate process \(\{X_t,t\in \mathbb{Z}\}\) which satisfies the MS-ARMA\((p,q)\) equation \[ X_t-\Phi_{1t}X_{t-1}-\cdots- \Phi_{pt}X_{t-p}=Z_t+\Theta_{1t}Z_{t-1}+\cdots+\Theta_{qt}Z_{t-q},\;t\in \mathbb{Z}, \] where \(p,q\in \mathbb{N}_{0}\), \(p+q\geq 1\), \(Z_t=\Sigma_t\epsilon_t\), \(\{\epsilon_t,t\in\mathbb{Z}\}\) is a sequence of i.i.d. random vectors independent of \(\Delta=\{\Sigma_t,\Phi_{1t},\dots,\Phi_{pt},\Theta_{1t},\dots,\Theta_{qt},t\in\mathbb{Z}\}\), and \(\Delta\) is a stationary and ergodic Markov chain. This process is called a multivariate Markov-switching ARMA (MS-ARMA) process. The author first provides some results on multivariate regular variations and uses them to analyse the tail behaviour of the MS-ARMA process. The author shows, that under some conditions on the coefficients, there is a unique stationary ergodic solution of the MS-ARMA\((p,q)\) equation and that the MS-ARMA process is regularly varying and has the same index of regular variation as the sequence \(\{\epsilon_t\}\). The author also gives some examples which describe the behaviour of the Markov-switching ARMA processes. The MS-AR(1) process is considered as a special case of the MS-ARMA process.
scientific article; zbMATH DE number 5287851
scientific article