Estimation of a Multiplicative Correlation Structure in the Large Dimensional Case

TL;DR

This paper introduces a Kronecker product model for high-dimensional correlation matrices, providing estimators with proven convergence properties and demonstrating their effectiveness in portfolio optimization.

Contribution

It develops a novel Kronecker product model for large-dimensional correlation matrices and proposes efficient estimators with theoretical guarantees and practical tools for model selection.

Findings

Estimators show superior performance in Monte Carlo simulations.

Kronecker models effectively approximate covariance matrices in empirical data.

The approach enables scalable inference in high-dimensional settings.

Abstract

We propose a Kronecker product model for correlation or covariance matrices in the large dimensional case. The number of parameters of the model increases logarithmically with the dimension of the matrix. We propose a minimum distance (MD) estimator based on a log-linear property of the model, as well as a one-step estimator, which is a one-step approximation to the quasi-maximum likelihood estimator (QMLE). We establish rates of convergence and central limit theorems (CLT) for our estimators in the large dimensional case. A specification test and tools for Kronecker product model selection and inference are provided. In a Monte Carlo study where a Kronecker product model is correctly specified, our estimators exhibit superior performance. In an empirical application to portfolio choice for SP500 daily returns, we demonstrate that certain Kronecker product models are good approximations to the general covariance matrix.