Empirical Risk Minimization for Time Series: Nonparametric Performance Bounds for Prediction

TL;DR

This paper investigates the use of empirical risk minimization for one-step-ahead prediction in univariate time series, establishing that it asymptotically achieves optimal predictive performance within a broad class of recursive algorithms.

Contribution

It provides a theoretical framework showing that empirical risk minimization leads to asymptotically optimal predictions for time series forecasting without assuming a specific data-generating process.

Findings

ERM achieves asymptotic optimality in predictive performance.

Framework covers various forecasting applications.

Results are applicable to recursive algorithms in time series prediction.

Abstract

Empirical risk minimization is a standard principle for choosing algorithms in learning theory. In this paper we study the properties of empirical risk minimization for time series. The analysis is carried out in a general framework that covers different types of forecasting applications encountered in the literature. We are concerned with 1-step-ahead prediction of a univariate time series generated by a parameter-driven process. A class of recursive algorithms is available to forecast the time series. The algorithms are recursive in the sense that the forecast produced in a given period is a function of the lagged values of the forecast and of the time series. The relationship between the generating mechanism of the time series and the class of algorithms is unspecified. Our main result establishes that the algorithm chosen by empirical risk minimization achieves asymptotically the optimal predictive performance that is attainable within the class of algorithms.