An indirect proof for the asymptotic properties of VARMA model estimators

Abstract

Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identifiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coefficients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.

Introduction

Vector autoregressive-moving average (VARMA) models are the multivariate generalization of univariate autoregressive-moving average (ARMA) models. ARMA models are used to represent stationary time series in almost all domains where a variable is measured at equidistant times. Its generalizations, ARIMA and SARIMA models, respectively for integrated and seasonal time series, can cope with a stochastic trend and stochastic seasonality. Vector autoregressive (VAR) models are widely used in econometrics. VARMA models are much less used, except in certain specific applications. For example, they are able to represent competitive behaviour between products in economics, see Jenkins (1979). VARMA models have been widely studied since the 1970’s, starting with Hannan (1969) and Tiao and Box (1981). See the books by Hannan and Deistler (2012), first published in 1988, Reinsel (1997), Lütkepohl (2005), Wei (2006), and Tsay (2014). New theoretical results are still being developed for model specification (Poskitt, 2005, Athanasopoulos, Vahid, 2008a, Boubacar Maïnassara, 2012, Brockwell, Lindner, Vollenbröker, 2012, Wegge, 2012), in particular in structural models (Gouriéroux, Monfort, Renne, 2017, Gouriéroux, Monfort, Renne, 2020), for estimation methods (Kascha, 2012, Boubacar Maïnassara, Francq, 2011, Dufour, Jouini, 2014, Hallin, La Vecchia, Liu, 2020, Roy, McElroy, Linton, 2019), also in the case of non-independent but uncorrelated errors (Boubacar Maïnassara, Saussereau, 2018, Dufour, Pelletier, 2019), for diagnostic checking (Boubacar Maïnassara, 2011), and for forecasting (Lütkepohl, 2006, Athanasopoulos, Vahid, 2008b, Kascha, Mertens, 2009). Work is also being carried out on sparse models (Wilms et al., 2019).

More recently, ARMA and VARMA models with time-dependent coefficients have been proposed and asymptotic theories have been developed in order to justify their use. In most cases, a parametric approach has been taken, but there are also semi-parametric approaches. 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 quasi-maximum likelihood estimator (QMLE) for the parameters of a tdVARMA model.

We present a new proof of the asymptotic properties of standard, i.e. homoscedastic, causal, invertible, and identifiable VARMA models. That proof is indirect and is based on Alj et al. (2017) and a new representation of tdVARMA models under a higher-dimensional tdVAR(1) model. This is done, on the basis of Lütkepohl (1991), by combining features from the two papers of Francq and Gautier (2004) and Francq and Zakoïan (2001); see also Boubacar Maïnassara and Rabehasaina (2020). This makes it possible to check the assumptions made by Alj et al. (2017). It is, therefore, an indirect proof. The paper is of course also applicable to ARMA models (see the Supplementary Material).

Since the proof of the asymptotic properties for tdVARMA models does not exploit ergodicity, a property that is certainly invalid for tdVARMA models, which are not usually stationary, our indirect proof for VARMA model does not use ergodicity. Although there exist elements of spectral theory for non-stationary time series, the Alj et al. (2017) approach does not make use of spectral theory. Consequently, the indirect new proof of the asymptotic properties for VARMA models is provided entirely in the time domain and does not refer to a frequency domain approach. Existing proofs of asymptotic properties in time series models, e.g. Hannan (1973), reported and expanded in Brockwell and Davis (1991) and Yao and Brockwell (2006), for scalar processes, or Dunsmuir and Hannan (1976), Deistler et al. (1978), Rissanen and Caines (1979), Kohn (1979), Hannan et al. (1980), and Ling and McAleer (2003) (the latter with GARCH errors), for vector processes, all make use of ergodicity. Several of these proofs also make use of spectral arguments, starting with Hannan (1973), but not those of Yao and Brockwell (2006) or Rissanen and Caines (1979). Several other comparative aspects will be discussed later, including the ability to handle structural models, the assumptions made about errors and their moments, and the compactness of the parameter space. It should be clear that our results do not require weaker assumptions than previous results. For example, we need the existence of fourth-order moments for the errors, while only the existence of the variance is needed in Yao and Brockwell (2006) for scalar models. The requirement is fourth-order moments for Ling and McAleer (2003) in the wider context of VARMA-ARCH models.

The proof for VARMA models is relatively short; it should be noted, however, that the Alj et al. (2017) paper relies on a law of large numbers and a central limit theorem for martingales and a law of large number for mixingales, and requires a long technical appendix to establish the required lemmas. The latter paper’s technical appendix is, however, not needed for scalar models, as the simpler theory for tdARMA models in Azrak and Mélard (2006) is then applicable. Moreover, in order to reduce the moment assumption on the model errors, the two papers need to be complemented by a result of Azrak and Mélard (2020).

In Section 2, we introduce VARMA and tdVARMA models, and quasi-maximum likelihood estimators (QMLE), and we state the assumptions for homoscedastic tdVARMA models. Section 3 is devoted to preliminaries, where tdVARMA models are put into a tdVAR(1) form that allows for easier treatment of the derivatives of the pure tdVMA representation. Section 4 contains the assumptions we need for standard VARMA models, a few remarks on those assumptions, a theorem of consistency and asymptotic normality of the QMLE estimators of a VARMA model, and its proof, which consists in the verification of the assumptions of Alj et al. (2017). As promised, the proof does not exploit ergodicity (even if the process is ergodic, given the assumptions being made) and does not require spectral arguments.