Oracle-efficient estimation and trend inference in non-stationary time series with trend and heteroscedastic ARMA error

The non-stationary time series often contain an unknown trend and unobserved error terms. The error terms in the proposed model consist of a smooth variance function and the latent stationary ARMA series, which allows heteroscedasticity at different time points. The theoretically justified two-step B-spline estimation method is proposed for the trend and variance function in the model, and then residuals are obtained by removing the trend and variance function estimators from the data. The maximum likelihood estimator (MLE) for the latent ARMA error coefficients based on the residuals is shown to be oracally efficient in the sense that it has the same asymptotic distribution as the infeasible MLE if the trend and variance function were known. In addition to the oracle efficiency, a kernel estimator is obtained for the trend function and shown to converge to the Gumbel distribution. It yields an asymptotically correct simultaneous confidence band (SCB) for the trend function, which can be used to test the specific form of trend. A simulation-based procedure is proposed to implement the SCB, and simulation and real data analysis illustrate the finite sample performance.