Pivotal inference for linear predictions in stationary processes

TL;DR

This paper introduces pivotal inference methods for linear prediction errors in stationary processes, providing confidence intervals, hypothesis tests, and uncertainty quantification without estimating asymptotic variances.

Contribution

It develops self-normalizing, pivotal inference techniques for prediction errors, partial autocorrelation, and R-squared in stationary processes, avoiding variance estimation.

Findings

Pivotal confidence intervals for (R)FPE are constructed.

Estimates for minimal model order for desired accuracy are provided.

New inference tools for partial autocorrelation without AR assumptions are developed.

Abstract

In this paper we develop pivotal inference for the final (FPE) and relative final prediction error (RFPE) of linear forecasts in stationary processes. Our approach is based on a self-normalizing technique and avoids the estimation of the asymptotic variances of the empirical autocovariances. We provide pivotal confidence intervals for the (R)FPE, develop estimates for the minimal order of a linear prediction that is required to obtain a prespecified forecasting accuracy and also propose (pivotal) statistical tests for the hypotheses that the (R)FPE exceeds a given threshold. Additionally, we provide pivotal uncertainty quantification for the commonly used coefficient of determination $R^2$ obtained from a linear prediction based on the past $p \geq 1$ observations and develop new (pivotal) inference tools for the partial autocorrelation, which do not require the assumption of an autoregressive process.