Embracing the Blessing of Dimensionality in Factor Models

TL;DR

This paper advocates for leveraging all available high-dimensional data in factor models to improve covariance matrix estimation, providing theoretical conditions, algorithms, and empirical evidence for the benefits of using more data.

Contribution

It introduces conditions under which additional data improves covariance estimation, proposes a divide-and-conquer algorithm, and demonstrates empirical benefits in high-dimensional settings.

Findings

Using more data enhances covariance estimation accuracy.

The divide-and-conquer algorithm maintains statistical accuracy.

Empirical results confirm the advantages of full data utilization.

Abstract

Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data is often ignored in constructing covariance matrix estimates. If our goal is to accurately estimate a covariance matrix of a set of targeted variables, shall we employ additional data, which are beyond the variables of interest, in the estimation? In this paper, we provide sufficient conditions for an affirmative answer, and further quantify its gain in terms of Fisher information and convergence rate. In fact, even an oracle-like result (as if all the factors were known) can be achieved when a sufficiently large number of variables is used. The idea of utilizing data as much as possible brings computational challenges. A divide-and-conquer algorithm is thus proposed to alleviate the computational burden, and also shown not to sacrifice any statistical accuracy in comparison with a pooled analysis. Simulation studies further confirm our advocacy for the use of full data, and demonstrate the effectiveness of the above algorithm. Our proposal is applied to a microarray data example that shows empirical benefits of using more data.