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.