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Maximum likelihood estimator (MLE) and Bayesian Information Criterion (BIC) order selection are examined for ARMA time series with slowly varying trend to validate the well-known detrending technique of moving average [Section 1.4, Brockwell, P.J., and Davis, R.A. (1991),  Time Series: Theory and Methods , New York: Springer-Verlag]. In step one, a moving average equivalent to local linear regression is fitted to the raw data with a data-driven lag number, and subtracted from raw data to produce a sequence of residuals. The residuals are used in step two as substitutes of the latent ARMA series for MLE and BIC procedures. It is shown that with second order smooth trend and correctly chosen lag number, the two-step MLE is oracally efficient, i.e. it is asymptotically as efficient as the would-be MLE based on the unobserved ARMA series. At the same time, the two-step BIC consistently selects the orders as well. Simulation experiments corroborate the theoretical findings.

Statistical inference for ARMA time series with moving average trend