Computing and estimating information matrices of weak ARMA models

Abstract

Numerous time series admit weak autoregressive-moving average (ARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. The statistical inference of this general class of models requires the estimation of generalized Fisher information matrices. We give analytic expressions and propose consistent estimators of these matrices, at any point of the parameter space. Our results are illustrated by means of Monte Carlo experiments and by analyzing the dynamics of daily returns and squared daily returns of financial series.

Estimating the Marginal Law of a Time Series With Applications to Heavy-Tailed Distributions

“…The proof is based on a series of lemmas. Similar proofs can be found in the supplementary files of Francq, Roy and Zakoïan (2005) and Boubacar, Carbon and Francq (2011). We begin by proving thatĴ is a consistent estimator of J.…”

Estimating FARIMA models with uncorrelated but non-independent error terms

“…There are differences with [Ber74]: (H t (θ 0 )) t∈Z is multivariate, is not directly observed and is replaced by (Ĥ t ) t∈Z . It is shown that this result remains valid for the multivariate linear process (H t (θ 0 )) t∈Z with non-independent innovations (see [BMCF12,BMF11], for references in weak (multivariate) ARMA models). We will extend the results of [BMCF12] to weak FARIMA models.…”

“…Following the arguments developed in [BMCF12], the matrix I (θ 0 ) can be estimated using Berk's approach (see [Ber74]). More precisely, by interpreting I (θ 0 )/2π as the spectral density of the stationary process (H t (θ 0 )) t∈Z evaluated at frequency 0, we can use a parametric autoregressive estimate of the spectral density of (H t (θ 0 )) t∈Z in order to estimate the matrix I (θ 0 ).…”

Estimation of weak ARMA models with regime changes

“…In the literature, two types of estimators are generally employed: the nonparametric kernel estimator, also called Heteroskedasticity and Autocorrelation Consistent (HAC) estimators (see Andrews (1991) and Newey and West (1987) for general references, and Francq and Zakoïan (2007) for an application to testing strong linearity in weak ARMA models) and spectral density estimators (see e.g. Berk (1974) and den Haan and Levin (1997) for a general references and Boubacar Mainassara et al (2012) for estimating I when θ is not necessarily equal to θ 0 ).…”