Multi‐step‐ahead prediction interval for locally stationary time series with application to air pollutant concentration data

Abstract

Locally stationary time series frequently appears in both finance and environmental sciences (e.g., daily air pollutant concentration or financial returns). However, constructing the multi‐step‐ahead prediction interval for such time series remains an open question. Hence, we extend the nonparametric regression model with autoregressive errors for equally spaced designs to the time series setup. We propose a B‐spline estimator for the trend function and a kernel estimator for the variance function to implement the model. The prediction interval of multi‐step‐ahead future observations is also constructed after fitting the autoregressive model of errors and obtaining the quantile of prediction residuals. The proposed method is illustrated by various simulation studies and an example of air pollutant data, containing 8 years of daily air pollutant concentrations in Xi'an. Our results demonstrate that our method outperforms others owing to its higher prediction accuracy and versatility.