Equivariant online predictions of non-stationary time series

TL;DR

This paper establishes theoretical properties of online prediction methods for non-stationary time series, demonstrating that certain dynamic models achieve minimax predictive densities under model misspecification, with applications across various fields.

Contribution

It introduces a decision theoretic framework for analyzing online predictions in non-stationary time series and proves that random walk dynamic linear models are minimax optimal under this setting.

Findings

Random walk dynamic linear models produce exact minimax predictive densities.

Theoretical results extend from Gaussian to semi-martingale processes.

Applications confirm the theoretical predictions in epidemiology, climatology, and economics.

Abstract

We discuss the finite sample theoretical properties of online predictions in non-stationary time series under model misspecification. To analyze the theoretical predictive properties of statistical methods under this setting, we first define the Kullback-Leibler risk, in order to place the problem within a decision theoretic framework. Under this framework, we show that a specific class of dynamic models -- random walk dynamic linear models -- produce exact minimax predictive densities. We first show this result under Gaussian assumptions, then relax this assumption using semi-martingale processes. This result provides a theoretical baseline, under both non-stationary and stationary time series data, for which other models can be compared against. We extend the result to the synthesis of multiple predictive densities. Three topical applications in epidemiology, climatology, and economics, confirm and highlight our theoretical results.