Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

Abstract

This paper is about vector autoregressive-moving average (VARMA) models with time-dependent coefficients to represent non-stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length 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 marginally heteroscedastic with a correlation ranging from −0.8 to 0.8. In the 1 Preprint version of a paper that will appear in Scandinavian Journal of Statistics (2017) two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite-sample behavior is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

“…For estimation and testing of structural change in time series regression models, see Kristensen . Finally, the early work of Nicholls and Quinn and the recent work of Alj et al are of interest.…”

“…There is also a large literature on the existence of stationary random parameter, threshold, and Markov switching AR models, see Bougerol and Picard (1992), Cline (2007), and Francq and Zakoïan (2001). For estimation and testing of structural change in time series regression models, see Kristensen (2012).Finally, the early work of Nicholls and Quinn (1982) and the recent work of Alj et al (2016) are of interest.Our kernel estimation approach applied in this setting has a number of attractions as an alternative tool. First, comparable theoretical results on consistency and rates of convergence are not available for existing estimators, under the assumption that parameter processes follow a bounded random walk.…”

Measures of Cross‐Dependence for Bidimensional Periodic AR(1) Model with α‐Stable Distribution

“…They can be also treated as an example, similar to a one‐dimensional case, of the multivariate ARMA models with time‐varying coefficients (Peiris, ; Bertha and Golinval, ; Alj et al . ).…”

“…On the one hand, similar to the one-dimensional case, the multidimensional PARMA models are the extension of the multidimensional ARMA models (called also vector ARMA, VARMA) (Kang, 1981;Swift, 1990;Hallin and Saidi, 2005;Brockwell et al, 2012). They can be also treated as an example, similar to a one-dimensional case, of the multivariate ARMA models with time-varying coefficients (Peiris, 1988;Bertha and Golinval, 2017;Alj et al, 2017).…”

An indirect proof for the asymptotic properties of VARMA model estimators

“…In parametric approaches, it is assumed that there is a relatively small number of parameters. In Alj et al (2017), the authors have considered time-dependent VARMA (tdVARMA) models with possibly marginal heteroscedasticity (as opposed to conditional heteroscedasticity). Subject to a set of assumptions, they establish strong consistency and asymptotic normality of a Gaussian quasimaximum likelihood estimator (QMLE) for the parameters of a tdVARMA model.…”