Exact and Asymptotic Tests on a Factor Model in Low and Large Dimensions with Applications

TL;DR

This paper develops and evaluates three statistical tests for the validity of factor models applicable to both small and large datasets, with theoretical derivations, simulations, and real data applications.

Contribution

It introduces three new tests for factor model validation with exact and asymptotic distributions, applicable across different data dimensions, and provides practical calibration and application methods.

Findings

Tests perform well under classical and high-dimensional regimes.

Simulation results align with theoretical predictions.

Empirical analysis challenges the adequacy of the considered factors.

Abstract

In the paper, we suggest three tests on the validity of a factor model which can be applied for both small dimensional and large dimensional data. Both the exact and asymptotic distributions of the resulting test statistics are derived under classical and high-dimensional asymptotic regimes. It is shown that the critical values of the proposed tests can be calibrated empirically by generating a sample from the inverse Wishart distribution with identity parameter matrix. The powers of the suggested tests are investigated by means of simulations. The results of the simulation study are consistent with the theoretical findings and provide general recommendations about the application of each of the three tests. Finally, the theoretical results are applied to two real data sets, which consist of returns on stocks from the DAX index and on stocks from the S&P 500 index. Our empirical results do not support the hypothesis that all linear dependencies between the returns can be entirely captured by the factors considered in the paper.