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Why Fama–French Uses a Second-Stage Regression

Article Quant Q&A · Author: JOHN
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Summary

The note explains why Fama–French portfolio returns are regressed on estimated factor loadings in a second stage. Sorting portfolios by size or value can produce return differences, but those differences alone do not show that the corresponding factor earns a distinct risk premium. A portfolio built on value, for example, may also have substantial market exposure, so its higher return could reflect market risk rather than value risk.

The second-stage regression relates portfolio returns to their estimated exposures to multiple factors, helping distinguish the premium associated with each factor from returns linked to other exposures. The explanation is conceptual and gives no empirical results or details about estimation choices, statistical uncertainty, or model limitations. Its main lesson is that portfolio sorting can reveal patterns, while a multifactor regression helps assess whether those patterns are independently associated with a particular risk factor.

Key ideas

  • Portfolio returns sorted by value do not by themselves identify a value risk premium.
  • A sorted portfolio can have exposure to several factors at once.
  • The second-stage regression relates returns to estimated factor loadings.
  • Multifactor analysis helps separate a factor's premium from confounding exposures.

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Factor investingStatistics
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# Fama French paper regression questions


# Fama French paper regression questions












I am reading the paper and get the following question. I think here is how the regression is constructed:



The first step, for each portfolio, regress the portfolio return $R_t$ on three factors, and get $\beta, \gamma, \nu$.

However, this portfolio is formed based on Size, Value. So, you would expect the loading to factors are pretty high.



with the second regression, you get the risk premium for each factor loading.

Here is what confuse me: Using Value as an example, you form the portfolios based on Value, and find higher value has higher expected return. Then you regress portfolio returns on Value, of course the higher return has higher factor beta, and the higher factor beta has higher expected return. Isn't the regression redundant? Isn't comparing the portfolio returns made the point?

## Answer by Tim Wilding (score 1)

https://quant.stackexchange.com/a/41469

No, looking at the returns of the value factor does not make the point. The second stage regression of returns on $\beta^i$, $\gamma^i$ and $\nu^i$ allows you to separate out whether the returns of the Value portfolio are due to a Value risk premium, or due to another confounding factor.

For example, it is quite possible that building the portfolios based on Value could be giving you a market exposure. Given that, we cannot be sure whether the higher return of the Value portfolio is due to the market or the Value factor. The second stage regression separates those effects.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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