Parameter Estimation of Gaussian Stationary Processes using the Generalized Method of Moments

TL;DR

This paper develops a GMM estimator for stationary Gaussian processes, proving its consistency and asymptotic normality, and applies it to estimate parameters of a fractional Ornstein-Uhlenbeck process, with numerical validation.

Contribution

It introduces a GMM estimation method for stationary Gaussian processes with proven theoretical properties and applies it to the fractional Ornstein-Uhlenbeck process.

Findings

The GMM estimator is consistent and asymptotically normal.

The estimator performs well in moderate data scenarios.

Applicability extends to processes driven by fractional Brownian motion.

Abstract

We consider the class of all stationary Gaussian process with explicit parametric spectral density. Under some conditions on the autocovariance function, we defined a GMM estimator that satisfies consistency and asymptotic normality, using the Breuer-Major theorem and previous results on ergodicity. This result is applied to the joint estimation of the three parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. The asymptotic normality of its GMM estimator applies for any H in (0,1) and under some restrictions on the remaining parameters. A numerical study is performed in the fOU case, to illustrate the estimator's practical performance when the number of datapoints is moderate.