General estimation results for tdVARMA array models

Abstract

The article will focus on vector autoregressive-moving average (VARMA) models with time-dependent coefficients (td) to represent general nonstationary time series, not necessarily Gaussian. The coefficients depend on time, possibly on the length of the series $n$, hence the name tdVARMA${\hspace{0.17em}}^{\left(n\right)}$ for the models, but not necessarily on the rescaled time $t/n$. As a consequence of the dependency on $n$ of the model, we need to consider array processes instead of stochastic processes. 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 three examples of bivariate processes, the first two with marginal heteroscedasticity. The first example is a tdVAR${\hspace{0.17em}}^{\left(n\right)}$(1) process while the second example is a tdVMA${\hspace{0.17em}}^{\left(n\right)}$(1) process. In these two cases, the finite-sample behavior is checked via a Monte Carlo simulation study. The results are compatible with the asymptotic properties even for small $n$. A third example shows the application of the tdVARMA${\hspace{0.17em}}^{\left(n\right)}$ models for a real time series.