Testing the Joint Omission of Factors with the GRS Test
Summary
The note describes how to test whether multiple factors can be omitted from a multifactor asset-pricing model. Its example asks whether the size and value factors in the Fama-French three-factor model can be excluded while retaining the market factor. For one omitted factor, the described approach regresses that factor on the retained factors and examines whether the intercept is statistically significant.
For multiple omitted factors, the answer recommends testing their intercepts jointly with the GRS test. In the example, the candidate omitted factors serve as test assets, while the retained market factor is the explanatory factor. The note generalizes this setup to other groups of factors and points to its use in published factor-selection work. It supplies a testing procedure but no data, numerical result, or discussion of assumptions and implementation details, so it does not establish that any particular factor set is adequate.
Key ideas
- For a single candidate omitted factor, regress it on the retained factors and assess its intercept.
- For multiple candidate omitted factors, test their intercepts jointly with the GRS test.
- In the example, SMB and HML are the test assets and RMRF is the retained factor.
- The procedure tests factor omission; it does not establish a preferred model without empirical results.
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Full text
# Omitting more than one factor from a factor model # Omitting more than one factor from a factor model 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 (RMRF), 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? ## Answer by Richard Hardy (score 1, accepted) https://quant.stackexchange.com/a/82082 Following the posts linked in the OP, when considering an omission of one particular factor, we look at whether omitting it changes the "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. If we consider omitting two factors at once, as in the OP's example, we should 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. That is, we should run the GRS test with the test assets being SMB and HML and the factor being the RMRF. More generally, if we consider omitting multiple factors at once, we would run the GRS test with the test assets being the factors we consider omitting and the factors (the right hand side variables in the GRS test) being the factors we are going to keep in the model. This has been applied e.g. in section 3 of Fama & French Choosing factors (2018).
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