Quadratic Prediction Methodology and Calibration of Prediction Intervals Based on Subsampling

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We consider nonlinear prediction of a stationary time series using quadratic functions of the past data. We derive explicit formulae for the best quadratic predictor and its MSPE. We also give conditions under which the quadratic approach improves over the standard linear case and provide a complete characterization for such processes. We next consider the problem of constructing asymptotically valid prediction intervals based on a general point predictor. While much of the existing literature either assumes a parametric time series model or makes specific distributional assumptions (e.g., Gaussian), this work relaxes both and develops a nonparametric method that is applicable to a general stationary sequence. Specifically, we propose a Subsampling method for constructing distribution free prediction intervals for linear and nonlinear prediction methods and establish its validity. For the case of best linear predictor, we also derive the optimal rate of the subsample block size. The results in the prediction context are very nonstandard when compared with the known results on optimal block sizes for the Block Bootstrap/Subsampling in standard variance estimation problems. Finite sample properties of the proposed method are illustrated with simulation. This is joint work with Dhrubajyoti Ghosh, Tucker McElroy and Daniel Nordman. You can attend the seminar on Zoom: https://lse.zoom.us/j/81827052506?pwd=-ZKSJTlQIyNmiHstDDmSeaUWxq8mhQ.1