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

TL;DR

This paper establishes the consistency and asymptotic normality of QML estimators for non-stationary VARMA models with time-dependent coefficients, supported by theoretical proofs, examples, and simulations.

Contribution

It extends asymptotic theory for QML estimators to VARMA models with time-dependent coefficients, not relying on series length.

Findings

QML estimators are consistent and asymptotically normal under certain conditions.

The asymptotic information matrix is accurately derived in Gaussian cases.

Finite-sample simulations confirm theoretical asymptotic properties.

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.