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Following up on these questions of mine (1), (2), (3), one could ask whether we could leave more than one factor out of a multi-factor model. E.g., consider the Fama-French 3-factor (FF3f) model. Suppose we suspect that SMB and HML can be omitted from the model, and that the only factor that matters is the market factor, which leaves us with the CAPM. How could this be tested?

As the previous posts show, in the case of considering omission of one particular factor, we look at whether omitting it changes "alpha" in the cross-sectional regression. That can be done by regressing the factor we are going to omit on the other factors and examining the statistical significance of the intercept in that regression. This is a recipe for dealing with a single factor. But what if we consider omitting two factors at once, as in the example above? Should we run a system of two regressions, one of RMRF on SMB and the other on RMRF on HML, and test the joint significance of the two intercepts?

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  • $\begingroup$ Is that why one runs Fama MacBeth regressions? $\endgroup$
    – phdstudent
    Commented Dec 10 at 15:52
  • $\begingroup$ @phdstudent, I do not think this is their primary purpose, and it is not obvious to me they would answer the question, but maybe? What would be the logic? $\endgroup$
    – Richard Hardy
    Commented Dec 10 at 16:00
  • $\begingroup$ If SMB and HML are statistically significant, wouldn't it make sense to keep the factors in the model? Not quite understanding what is the purpose of omitting these 2 factors. Because if you omitted SMB and HML $despite$ their statistical significance, the model (CAPM in this case) and the coefficient of MKT would be misspecified. $\endgroup$
    – KaiSqDist
    Commented Dec 10 at 16:09
  • $\begingroup$ @KaiSqDist, I explain in the quoted posts why your suggestion does not make sense if we care about model validity (though not so much explanatory power). Or actually, I quote John Cochrane's explanation. He calls this a common misconception. This has to do with estimation of betas under different models. $\endgroup$
    – Richard Hardy
    Commented Dec 10 at 19:43

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