Predictive Inference for Locally Stationary Time Series With an Application to Climate Data

The model-free prediction principle of Politis has been successfully applied to general regression problems, as well as problems involving stationary time series. However, with long time series, for example, annual temperature measurements spanning over 100 years or daily financial returns spanning several years, it may be unrealistic to assume stationarity throughout the span of the dataset. In this article, we show how model-free prediction can be applied to handle time series that are only locally stationary, that is, they can be assumed to be stationary only over short time-windows. Surprisingly, there is little literature on point prediction for general locally stationary time series even in model-based setups, and there is no literature whatsoever on the construction of prediction intervals of locally stationary time series. We attempt to fill this gap here as well. Both one-step-ahead point predictors and prediction intervals are constructed, and the performance of model-free is compared to model-based prediction using models that incorporate a trend and/or heteroscedasticity. Both aspects of the article, model-free and model-based, are novel in the context of time-series that are locally (but not globally) stationary. We also demonstrate the application of our model-based and model-free prediction methods to speleothem climate data which exhibits local stationarity and show that our best model-free point prediction results outperform that obtained with the RAMPFIT algorithm previously used for analysis of this type of data. Supplementary materials for this article are available online.