Recursive identification for multidimensional ARMA processes with increasing variances

Abstract

In time series analysis, almost all existing results are derived for the case where the driven noise {w n } in the MA part is with bounded variance (or conditional variance). In contrast to this, the paper discusses how to identify coefficients in a multidimensional ARMA process with fixed orders, but in its MA part the conditional moment $$E(\parallel w_n \parallel ^\beta |\mathcal{F}_n - 1),\beta > 2$$ is possible to grow up at a rate of a power of log n. The well-known stochastic gradient (SG) algorithm is applied to estimating the matrix coefficients of the ARMA process, and the reasonable conditions are given to guarantee the estimate to be strongly consistent.