Chen Zhong

A certain dependence is imposed on the innovation of a heteroscedastic autoregressive moving average (ARMA) time series with a trend. When the trend and variance functions were known, the infeasible maximal likelihood estimator (MLE) of the ARMA coefficients is shown to be asymptotically normal with a covariance structure that is more intricate than that of the classical ARMA model with independent and identically distributed (i.i.d.) innovations. We further estimate the trend and variance functions through B‐spline and kernel smoothing, and obtain the ARMA residuals by removing these two estimators from the data. The MLE based on the residuals enjoys oracle efficiency in the sense that it is asymptotically equivalent to the infeasible estimator. Furthermore, we employ the cumulative sum (CUSUM) and moving‐sum test statistics to detect a single structural break and multiple structural breaks, respectively, in the trend function. The asymptotic theory for the break‐date estimator under different jump magnitudes is also investigated. Finite‐sample performance is demonstrated through the numerical studies including the simulation and the WTI crude oil price data.

The analysis of nonlinearity in spatial and spatial‐temporal data continues to be a challenging topic. This article introduces a two‐step estimation procedure for modeling nonstationary spatial processes that comprise a smooth bivariate trend function and a spatial autoregressive and moving average (SRAMA) error term. To remove the bivariate trend from the observed process, we apply the cutting‐edge bivariate penalized spline method. The modified maximum likelihood estimator based on the residuals, as suggested by Yao and Brockwell (2006), is shown to be consistent and oracle efficient, achieving the same efficiency asymptotically as if the true trend function were known and removed to obtain the SARMA errors. Furthermore, we establish the consistency of Bayesian information criteria for model selection concerning the residual sequence. The finite sample performance of the proposed approach is evaluated with simulations and real data.