Sparse Multivariate ARCH Models: Finite Sample Properties

TL;DR

This paper analyzes the finite sample properties of sparse multivariate ARCH models, providing theoretical error bounds and variable selection consistency, supported by empirical evidence.

Contribution

It introduces non-asymptotic error bounds and variable selection consistency results for sparse multivariate ARCH models under certain regularity conditions.

Findings

Non-asymptotic error bounds established

Variable selection consistency proven

Empirical studies support theoretical results

Abstract

We provide finite sample properties of sparse multivariate ARCH processes, where the linear representation of ARCH models allows for an ordinary least squares estimation. Under the restricted strong convexity of the unpenalized loss function, regularity conditions on the penalty function, strict stationary and beta-mixing process, we prove non-asymptotic error bounds on the regularized ARCH estimator. Based on the primal-dual witness method of Loh and Wainwright (2017), we establish variable selection consistency, including the case when the penalty function is non-convex. These theoretical results are supported by empirical studies.