A Hausman Type Test for Differences between Least Squares and Robust Time Series Factor Model Betas

Abstract

Robust regression is playing an increasingly important role in fitting time series and cross-section factor models for stock returns. We introduce and study the properties of a Hausman type test for comparing factor model regression coefficients computed with LS, which is fully efficient under idealized normal data distributions, and a Robust MM-estimate, which is highly efficient for normally distributed data but also controls variance inflation and bias for outlier generating non-normal data distributions. The test is based on the asymptotic distribution of the difference between the two estimators, one of which is fully efficient. The test can detect a significant difference between the LS and Robust estimate due to the inefficiency of the LS estimator under outlier-generating non-normal error distributions, and due to bias of the LS estimator relative to the Robust estimator caused by bias inducing distributions. The applications efficacy of the new test is demonstrated for comparison of LS and Robust estimates of both CAPM betas and Fama-French three-factor model betas. Monte Carlo studies of the finite sample level and power of the test reveal good performance for sample sizes of at least 100 to 200, which are typical for weekly and daily returns for such models.