Extremal behavior of the autoregressive process with ARCH(1) errors

We investigate the extremal behavior of a special class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation Xn = ffXn\Gamma1 + q fi + X 2 n\Gamma1 " n ; n 2 N ; where (" n ) n2N are i.i.d. random variables. The extremes of such processes occur typically in clusters. We give an explicit formula for the extremal index and the probabilities for the length of a cluster

Extremal behavior of the autoregressive process with ARCH(1) errors

We investigate the extremal behavior of a special class of autoregressive processes with ARCH(1) errors given by the stochastic difference equationwhere are i.i.d. random variables. The extremes of such processes occur typically in clusters. We give an explicit formula for the extremal index and the probabilities for the length of a cluster.ARCH model Autoregressive process Compound Poisson process Coupling Extremal behavior Extremal index Frechet distribution Heavy tail Heteroscedastic homogeneous Markov process Recurrent Harris chain Separating sequence Strong mixing

Asymptotic behavior of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1) errors

We study the sample autocovariance and autocorrelation function of the stationary AR(1) process with ARCH(1) errors. In contrast to ARCH and GARCH processes, AR(1) processes with ARCH(1) errors can not be transformed into solutions of linear stochastic recurrence equations. However, we show that they still belong to the class of stationary sequences with regular varying finite-dimensional distributions and therefore the theory of Davis and Mikosch (1998) can be applied