Efficient inference for autoregressive coefficients in the presence of trends

AMS 2000 subject classifications: primary 62F12 62M10 secondary 62G08 62G20 Keywords: Autoregressive time series Local polynomial Oracle efficiency Yule–Walker estimator Moving average a b s t r a c t Time series often contain unknown trend functions and unobservable error terms. As is known, Yule–Walker estimators are asymptotically efficient for autoregressive time series. The focus of this article is the Yule–Walker estimators for time series with trends. A nonparametric detrending procedure is proposed. It is concluded that the asymptotic properties of the Yule–Walker estimators of autoregressive coefficients are not altered by the detrending procedure. The results of the simulation studies and real data application corroborate the asymptotic theory. Typically, the first step in time series analysis is to ''separate'' the deterministic trend and seasonality components from the stochastic noise component. One of the common approaches taught by textbooks is to apply a moving average filter to ''remove'' the slowly varying trend and the seasonality from the time series data, and then proceed to make inference based on the residual series which is used as a substitute of the unobserved time series without trends. Although much has been done for such residual based inference, little attention has been paid to the appropriateness of substituting the residual sequence for the unobservable time series except for trends with known parametric forms; see for example, [9] and Chapter 9 of [4]. There are two most relevant works pertaining to the asymptotic property of the Yule–Walker estimators for autoregressive coefficients of time series with nonparametric trends. In particular, Truong [8] tackled the issue and established the asymptotics of Yule–Walker estimators when the trend was estimated by moving average or kernel regression techniques under the restrictive assumption of Gaussian noise; Shao and Yang [6] showed the oracle efficiency of Yule–Walker estimators when the trend was estimated by B-splines under mild moment assumptions on the noise. By ''oracle efficiency'' we refer to the asymptotic equivalence of the autoregressive coefficient estimators based on the unobserved stationary noise sequence and the computed residual sequence. In other words, Yule–Walker estimators from 41 the residual sequence with the trend estimated and removed are as efficient as those from the time series with the trend known by ''oracle'' and deleted. Both of these two articles, however, did not provide data-driven smoothing parameter selection. In this paper we will propose a nonparametric detrending procedure. This procedure constructed from local linear regression is a …