Nonparametric conditional predictive regions for time series

Several nonparametric predictors based on the Nadaraya-Watson kernel regression estimator have been proposed in the literature. They include the conditional mean, the conditional median, and the conditional mode. In this paper, we consider three types of predictive regions for these predictors — the conditional percentile interval (CPI), the shortest conditional modal interval (SCMI), and the maximum conditional density region (MCDR). Further, we introduce a data-driven method for the choice of the optimal bandwidth. This method is based on the minimization of a cross-validation criterion given three different types of predictors. When the underlying conditional distribution is multi-modal, we show that the MCDR is much shorter in length than the CPI or SCMI irrespective of the type of predictor used. This point is illustrated using both a simulated and a real data set.