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Testing Fama–French Models with Joint Alpha Tests

Article Quant Q&A · Author: NewbieFinance
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Summary

The note clarifies how to interpret intercepts in separate Fama–French regressions for individual stocks. An alpha different from zero for any test asset is inconsistent with the model, but individual p-values address each asset separately. When the individual estimates are not statistically significant, that is not strong evidence against the model; it also does not establish that the model is valid.

The suggested next step is a joint test of whether all test-asset alphas equal zero. The response identifies the GRS test for this purpose. It notes that implementing this test in Excel may be tedious, but gives no formula, calculation steps, or regression results beyond the question's reported example. The lesson is about the distinction between individual significance tests and joint model evaluation, with conclusions limited by the small set of assets and the specific test setup.

Key ideas

  • A nonzero alpha for at least one test asset conflicts with the model's joint alpha restriction.
  • Individual alpha p-values do not test whether all asset alphas are jointly zero.
  • Insignificant individual estimates mean there is not solid evidence against the model, not proof that it holds.
  • The GRS test is suggested for testing the joint alpha restriction.

Tags

EquitiesFactor investingStatisticsBacktesting
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# Fama-French Regression Output Interpretation (Intercept/Alpha)


# Fama-French Regression Output Interpretation (Intercept/Alpha)












I am currently doing a report regarding Fama and French 3 and 5 factors model. I was provided 3 companies with each of its daily stock return from 2015-2020, and the values of all 5 factors during that period (Mkt-RF, SMB, HML, RMW, CMA, RF). I will be using Excel to compute a regression analysis with no sorting portfolios. I got the multiple regression output (5 independent var) already for each company individually. However, for the intercept/alpha values, I got coefficient 0.226 and the p-value 0.210. The p-value indicates here that it is not statistically significant (since its >0.05 sig level). So does it mean that the model is unreliable? How is the correct way to interpret the coefficient and p-value of intercept/alpha according to Fama French?

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/77485

If the model holds, $\alpha_1=\dots=\alpha_N=0$ for all the test assets $i=1,\dots,N$. (In your case $N=3$.) Conversely, if $\alpha_i\neq 0$ for at least one asset $i$, the model does not hold. Now, you report individual assessments of $\alpha_1$, $\alpha_2$ and $\alpha_3$. Individually, each $p$-value is greater than any conventional significance level (1%, 5%, 10%), so it would seem that you do not have solid evidence against the model. However, you want to test the joint hypothesis, not the individual ones. Thus, you want to run the GRS test. Unfortunately, it may be a little tedious to implement in Excel.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.