General Estimation Results for tdVARMA Array Models

The paper is concerned with vector autoregressive-moving average (VARMA) models with time-dependent coe_cients (td) to represent some non-stationary time series. The coe_cients depend on time but can also depend on the length of the series n, hence the name tdVARMA(n) for the models. As a consequence of dependency of the model on n, we need to consider array processes instead of stochastic processes. Generalizing results for univariate time series combined with new results for array models, under appropriate assumptions, it is shown that a Gaussian quasi-maximum likelihood estimator is consistent in probability and asymptotically normal. The theoretical results are illustrated using two examples of bivariate processes, both with marginal heteroscedasticity. The first example is a tdVAR(n)(1) process while the second example is a tdVMA(n)(1) process. It is shown that the assumptions underlying the theoretical results apply. In the two examples, the asymptotic information matrix is obtained, not only in the Gaussian case. Finally, the finite-sample behaviour is checked via a Monte Carlo simulationstudy. The results con_rm the validity of the asymptotic properties even for small n and reveal that the asymptotic information matrix deduced from thetheory is correct.