Gaussian pseudo-maximum likelihood estimation of fractional time series models

TL;DR

This paper develops a Gaussian pseudo-maximum likelihood estimation method for fractional time series models, addressing unknown memory parameters and stationarity regions, and proves its consistency and asymptotic normality.

Contribution

It introduces a consistent and asymptotically normal estimation approach for both univariate and multivariate fractional time series models without assuming Gaussianity.

Findings

Proves consistency of the estimator in general models.

Establishes asymptotic normality of the estimator.

Extends results to multivariate models with a one-step estimate.

Abstract

We consider the estimation of parametric fractional time series models in which not only is the memory parameter unknown, but one may not know whether it lies in the stationary/invertible region or the nonstationary or noninvertible regions. In these circumstances, a proof of consistency (which is a prerequisite for proving asymptotic normality) can be difficult owing to nonuniform convergence of the objective function over a large admissible parameter space. In particular, this is the case for the conditional sum of squares estimate, which can be expected to be asymptotically efficient under Gaussianity. Without the latter assumption, we establish consistency and asymptotic normality for this estimate in case of a quite general univariate model. For a multivariate model, we establish asymptotic normality of a one-step estimate based on an initial $\sqrt{n}$-consistent estimate.