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This paper is about vector autoregressive-moving average (VARMA) models withtime-dependent coefficients to represent non-stationary time series. Contrarilyto other papers in the univariate case, the coefficients depend on time but noton the length of the series $n$. Under appropriate assumptions, it is shownthat a Gaussian quasi-maximum likelihood estimator is almost surely consistentand asymptotically normal. The theoretical results are illustrated by means oftwo examples of bivariate processes. It is shown that the assumptionsunderlying the theoretical results apply. In the second example the innovationsare also marginally heteroscedastic with a correlation ranging from -0.8 to0.8. In the two examples, the asymptotic information matrix is obtained in theGaussian case. Finally, the finite-sample behaviour is checked via a MonteCarlo simulation study for $n$ going from 25 to 400. The results confirm thevalidity of the asymptotic properties even for short series and reveal that theasymptotic information matrix deduced from the theory is correct.

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