Asymptotic properties of QML estimators for VARMA models with time-dependent coefficients: Part I

Abstract

This paper is about vector autoregressive-moving average (VARMA) models with time-dependent coefficients to represent non-stationary time series. Contrarily to other papers in the univariate case, the coefficients depend on time but not on the length of the series $n$. Under appropriate assumptions, it is shown that a Gaussian quasi-maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example the innovations are also marginally heteroscedastic with a correlation ranging from -0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite-sample behaviour is checked via a Monte Carlo simulation study for $n$ going from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and reveal that the asymptotic information matrix deduced from the theory is correct.