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Fama–MacBeth and GMM in Asset Pricing Tests

Article Quant Q&A · Author: Richard Hardy
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Summary

The document compares the Fama–MacBeth two-stage procedure with generalized method of moments for estimating factor risk premia and testing asset pricing models. It outlines a typical workflow: estimate asset betas, form portfolios, estimate test-portfolio betas, then run cross-sectional regressions. The author asks why Fama and French appeared to favor this approach despite GMM's ability to estimate betas, premia, and model tests using a shared sample.

The discussion proposes simplicity and beta dynamics as possible explanations. Fama–MacBeth can use recent rolling windows for betas, while a basic GMM specification often assumes constant betas; GMM can be extended to allow variation. The document also notes that the practical efficiency advantage may be modest when sample length is sufficient. It offers no cited explanation from Fama or French and presents the reasons as hypotheses, not established conclusions. It distinguishes years lost to initial portfolio construction from additional years potentially unused by the Fama–MacBeth procedure.

Key ideas

  • Fama–MacBeth estimates risk premia through repeated cross-sectional regressions using estimated portfolio betas.
  • GMM can jointly estimate betas, risk premia, and asset-pricing restrictions within a sample.
  • A basic GMM setup commonly treats betas as constant, while rolling estimation can represent changing exposures.
  • Simplicity and beta flexibility are proposed as possible reasons for choosing Fama–MacBeth, rather than documented explanations from Fama or French.
  • The sample-efficiency comparison depends on which stages are shared and how betas are estimated.

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Factor investingStatisticsBacktesting
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# Why did Fama & French prefer Fama-MacBeth to GMM?


# Why did Fama & French prefer Fama-MacBeth to GMM?












In their asset pricing papers that I have seen (including some late ones from 2010s), Fama & French used the two stage procedure of Fama-MacBeth (FM) from 1973. Since 1982, GMM was available as an alternative, but I have not seen it used by Fama & French. FM seems to be less efficient than GMM because the first few (around 5?) years of data are used for estimating the betas but not for estimating the factor risk premia or testing the asset pricing model. In contrast, GMM enables estimation of betas and factor risk premia as well as testing the model from the same sample, so no initial years are lost.* So why not use GMM?

I can see two reasons right off the bat. The first is simplicity. While FM is a bit cumbersome, it consists of simple steps (though once we add Shanken correction, it does not look as simple anymore). GMM is elegant but not quite as simple. Second, the simplest version of GMM keeps the betas constant, while FM has them time varying, as they are based on the most recent 5 years instead of all past years. GMM is of course very flexible, so time-varying betas can be implemented with it, too – though not the exact same kind as in FM lest we wish to effectively lose the first few years just as we do there. Also, if the length of the time series is not a problem, the efficiency gain from GMM need not be all that large.

But what are the actual reasons? Have Fama or French addressed this question anywhere?

*The FM approach often consists of three stages: (0) Estimate betas for individual assets using time-series regressions and group the stocks into portfolios (test portfolios as well as ones that are used to define factors such as SMB and HML). (1) Estimate test portfolio betas using time-series regressions. (2) Estimate cross-sectional regressions with test portfolio betas as regressors. I suppose stage (0) could be used as the initial step noth only for FM but also for the GMM approach. If not, I am not sure GMM could deal well with thousands of stocks and just hundreds of time periods (months). Due to stage (0), about 5 years of data would not be used for estimating the factor risk premia or testing the asset pricing model in either approach. However, the FM approach "throws away" another 5 years worth of data. In what follows, you may consider GMM to replace stages (1) and (2) but not (0).

(I have used past simple instead of present perfect tense referring to Fama's research, as I have noticed he has not published much lately. But perhaps I am too early with my choice of tense...)

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