On strong consistency and asymptotic normality of one-step Gauss-Newton estimators in ARMA time series models

Abstract

ABSTRACT One-step Gauss-Newton estimators for causal and invertible autoregressive moving-average (ARMA) time series models with an unknown mean are considered. A proof of the strong consistency and asymptotic normality of these estimators is provided. In a simulation study, their empirical properties are illustrated in ARMA models and compared to a other estimators: the estimators based on the innovations algorithm procedure, the estimators based on the spectral methods of [Krampe J, Kreiss J-P, Paparoditis E. Estimated Wold representation and spectral-density-driven bootstrap for time series. J R Stat Soc: Ser B (Statist Method). 2018;80:703–726.], and the one-step Gauss-Newton estimators using these methods as preliminary estimators. In the order-one moving-average model, we included the estimator based on the method of moments. The sample mean is used as a preliminary estimator to estimate the mean. These estimators are compared empirically to the maximum Gaussian likelihood estimator with respect to exact biases and mean squared errors.