Statistical inference for ARMA time series with moving average trend

Abstract

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.

Keywords:

• Asymptotic normality
• local linear estimator
• maximum likelihood
• moving average
• oracle efficiency

AMS Subject Classifications:

• 62G05
• 62G20
• 62M10

Acknowledgements

Constructive and insightful comments from two Reviewers and the Associate Editor have led to substantial improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is part of the first author's doctoral dissertation under the supervision of the last author and has been supported exclusively by National Natural Science Foundation of China awards [NSFC12171269 and NSFC11771240].