A diagnostic criterion for approximate factor structure

TL;DR

This paper introduces a simple eigenvalue-based diagnostic criterion to assess approximate factor structures in large asset return datasets, capable of identifying the number of omitted factors and applicable to various models.

Contribution

It develops a practical, eigenvalue-based test for approximate factor structures that works with large, unbalanced panels and can determine the number of unobserved factors.

Findings

The criterion effectively distinguishes between models with different numbers of factors.

Empirical analysis on US stock returns demonstrates the criterion's ability to select appropriate factor models.

Monthly and quarterly return data show the criterion's flexibility in different settings.

Abstract

We build a simple diagnostic criterion for approximate factor structure in large cross-sectional equity datasets. Given a model for asset returns with observable factors, the criterion checks whether the error terms are weakly cross-sectionally correlated or share at least one unobservable common factor. It only requires computing the largest eigenvalue of the empirical cross-sectional covariance matrix of the residuals of a large unbalanced panel. A general version of this criterion allows us to determine the number of omitted common factors. The panel data model accommodates both time-invariant and time-varying factor structures. The theory applies to random coefficient panel models with interactive fixed effects under large cross-section and time-series dimensions. The empirical analysis runs on monthly and quarterly returns for about ten thousand US stocks from January 1968 to December 2011 for several time-invariant and time-varying specifications. For monthly returns, we can choose either among time-invariant specifications with at least four financial factors, or a scaled three-factor specification. For quarterly returns, we cannot select macroeconomic models without the market factor.