Oracle model selection for correlated data via residuals

This paper concerns model selection for autoregressive time series when the observations are contaminated with trend. We propose an adaptive least absolute shrinkage and selection operator (LASSO) type model selection method, in which the trend is estimated by B-splines, the detrended residuals are calculated, and then the residuals are used as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an adaptive LASSO model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The intensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements.