Inference for ARMA time series with mildly varying trend

Abstract

Statistical inference is studied for ARMA time series with a mildly varying smooth trend. Following properly calibrated B-spline estimation of the trend, approximates of the unobserved ARMA series are used instead to estimate ARMA parameters by maximum likelihood. The parameter estimates are shown to be oracally efficient in the sense that they are asymptotically as efficient as if the true trend function was known and removed to obtain the true ARMA series. The mildly varying trend function is estimated by Nadaraya–Watson method with asymptotically correct simultaneous confidence band (SCB). The SCB sprawls over an interval that eventually covers the half real line, instead of a bounded interval in the existing literature, with asymptotic width of the SCB dependent on ARMA coefficients. Simulation experiments corroborate the theoretical findings.

Keywords:

• Autoregressive moving average
• B-spline estimator
• mildly varying trend
• oracle efficiency
• simultaneous confidence band

AMS Subject Classifications:

• 62G05
• 62G20
• 62M10

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. The helpful comments by an associate editor and two reviewers are gratefully acknowledged.