Fama-French 1992 Redux with Robust Statistics

Abstract

Robust statistical methods provide estimates that are not much influenced by a small percentage of outliers but perform in a nearly optimal manner for normally distributed data. Unfortunately, such methods are rarely used in quantitative finance research despite their potential utility, particularly in empirical asset pricing studies. As a means of stimulating the use of robust statistical methods in such studies and in quantitative finance in general, we demonstrate the efficacy of using a theoretically well-justified robust regression method in the cross-sectional regressions often used in empirical asset pricing studies. We compare the results of using both least squares and robust regression methods for the models presented in Fama and French (1992) (FF92), as well as some extensions to these models, over the time period 1963–2015 and subsets thereof. Our analysis clearly demonstrates that a very small fraction of outliers, in the returns and/or the factors, often distorts least squares cross-sectional regression estimates sufficiently enough to result in misleading conclusions as to whether a risk factor is priced. We reconfirm previously reported robust regression results demonstrating a positive relationship between average equity returns and firm size during the period 1963–1990 of the FF92 study, and show that this relationship continues to hold through 2015. Furthermore, we show that, once the impact of extreme outliers is eliminated by use of robust regression, the size effect is significant in most months, not just in January as was previously shown by other researchers. We confirm and extend the FF92 results and other previous work demonstrating a positive relationship between average returns and the book-to-market ratio, and clarify that the relationship is driven largely by small stocks. Contrary to FF92 and other previous empirical studies, we find a significant and negative relationship between average returns and beta for most U.S. equities between 1963 and 2015 when the influence of a small fraction of outliers is eliminated by robust regression. Furthermore, we show that there exists a highly significant interaction between size and beta that is needed to fully explain the dependence of returns on size and beta. Unlike FF92, who focused on positive earnings-to-price, we find that total earnings-to-price factor is highly significant in explaining the cross-section of returns for small to moderately-sized stocks both alone and when included along with the size and book-to-market factors. Finally, we also show that the time series of cross-sectional regression slopes contain large outliers, and that a robust location estimate provides a better representation of the “typical” slope than the sample mean.