Mean-Squared Error Analysis of Kernel Regression Estimator for Time Series

Because of a lack of a priori information, the minimum mean-squared error predictor, the conditional expectation, is often not known for a non-Gaussian time series. We show that the nonparametric kernel regression estimator of the conditional expectation is mean-squared consistent for a time series: When used as a predictor, the estimator asymptotically matches the mean-squared error produced by the true conditional expectation. We also describe a more computationally eecient predictor based on the recursive kernel regression estimator, and show it can asymptotically achieve mean-squared errors arbitrarily close to the conditional expectation. Numerical examples are provided to demonstrate the eeectiveness of nonparametric prediction.