Negative Moment Bounds for Sample Autocovariance Matrices of Stationary Processes Driven by Conditional Heteroscedastic Errors and Their Applications

TL;DR

This paper derives a negative moment bound for sample autocovariance matrices of stationary processes with heteroscedastic errors, enabling improved model selection and prediction error analysis in such contexts.

Contribution

It introduces a novel negative moment bound for autocovariance matrices of heteroscedastic processes, facilitating asymptotic MSPE expression and model selection.

Findings

The moment bound holds under heteroscedasticity.

The MSPE can be decomposed into model complexity, misspecification, and heteroscedasticity terms.

Numerical simulations confirm the theoretical results.

Abstract

We establish a negative moment bound for the sample autocovariance matrix of a stationary process driven by conditional heteroscedastic errors. This moment bound enables us to asymptotically express the mean squared prediction error (MSPE) of the least squares predictor as the sum of three terms related to model complexity, model misspecification, and conditional heteroscedasticity. A direct application of this expression is the development of a model selection criterion that can asymptotically identify the best (in the sense of MSPE) subset AR model in the presence of misspecification and conditional heteroscedasticity. Finally, numerical simulations are conducted to confirm our theoretical results.