AN ESTIMATION METHOD IN TIME SERIES ERRORS-IN-VARIABLES MODELS

Abstract

Suppose that we observe(x_t, y_t)from the errors-in-variables model : x_t=ξ_t+δ_t, y_t=βξ_t+ε_t, where{δ_t}and{ε_t}are i.i.d.measurement errors. Here we assume that{ξ_t}is a non-Gaussian stationary process with zero mean and spectral density f_ξ(λ). For this model, some estimators for β have been proposed in the literature. However, they are constructed under the assumption that the data are independent normal variates. Thus they do not contain the dependent structure of the data(e.g., time-lagged sample covariances, etc.). In this paper we propose a new class Λ of estimators of β, which is defined under consideration for dependent structures of (x_t, y_t, ξ_t). Then the asymptotic distribution of β^^^∈Λ is derived. We give an asymptotically optimal estimator in this class. Comparison with the existing estimators is also discussed. Since the asymptotic variance of β^^^ is complicated we have illuminated some aspect of the asymptotics numerically using Mathematica.