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A two-step model selection procedure is proposed for autoregressive and moving-average (ARMA) model class. It is an adaptive least absolute shrinkage and selection operator (adLASSO) type model selection method that simultaneously chooses both the orders and significant lagged variables when the autoregressive and moving-average (ARMA) time series is contaminated with a nonlinear trend. The adLASSO is applied not to the observations, but to the detrended residuals. The proposed two-step adLASSO model selection procedure under some conditions 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 simulation studies show that the proposed method performs well with sample size as small as 50. The application of the method is illustrated by the annual varve thickness data collected from a location in Massachusetts.

Model selection for time series with nonlinear trend