Oracally Efficient Estimation and Consistent Model Selection for Auto-Regressive Moving Average Time Series with Trend

Abstract

Summary Most time series that are encountered in practice contain non-zero trend, yet textbook approaches to time series analysis are typically focused on zero-mean stationary auto-regressive moving average (ARMA) processes. Trend is often estimated by ad hoc methods and subtracted from time series, and the residuals are used as the true ARMA noise for data analysis and inference, including parameter estimation, lag selection and prediction. We propose a theoretically justified two-step method to analyse time series consisting of a smooth trend function and ARMA error term, which is computationally efficient and easy for practitioners to implement. The trend is estimated by B-spline regression, and the maximum likelihood estimator based on residuals is shown to be oracally efficient in the sense that it is asymptotically as efficient as if the true trend function were known and then removed to obtain the ARMA errors. In addition, consistency of the Bayesian information criterion for model selection is established for the detrended residual sequence. Finite sample performance of the procedure is illustrated by simulation studies and real data analysis.