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The Cross-Sectional Step in Fama–MacBeth Regression

Article Quant Q&A · Author: user3672855
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Summary

The document explains the second stage of the Fama–MacBeth procedure after estimating each stock’s factor betas from a time-series regression. In each period, the cross-sectional regression uses that period’s returns across the stocks as dependent observations and their previously estimated betas as explanatory variables. The betas can remain fixed over the estimation window even as the monthly returns change.

Each period’s regression produces a factor-return estimate. Averaging those period-specific factor returns gives the estimated risk premium for each factor. The exchange clarifies the role of monthly cross sections, but it does not discuss implementation details such as beta estimation choices, standard errors, missing data, or how changing beta windows may affect inference. The explanation is consequently a concise account of the two-stage logic rather than a full econometric treatment.

Key ideas

  • The first Fama–MacBeth stage estimates asset betas using time-series returns and factors.
  • The second stage runs a separate cross-sectional regression for each period.
  • In each cross section, asset returns are dependent observations and estimated betas are explanatory variables.
  • The second-stage factor coefficients vary by period, and their average estimates factor risk premia.

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StatisticsFactor investingEquities
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# Fama MacBeth 2 step regression (2nd part)


# Fama MacBeth 2 step regression (2nd part)












I get the first part of the regression, basically it is a time series regression of returns on the proposed factors.

So, I need to run the regression of monthly return on the monthly time series of the factors from 2000 to 2005. So from this, we get the alpha and beta for this stock from the period 2000 to 2005. Now, let’s say we have 10 stocks, so we have 10 alphas and 10 betas.

Now, the part I don’t get is the second part, the cross section regression.

I don’t understand the dependent variables for this regression.

So, do I regress the monthly return of each stock on its beta over and over again? Because for each stock we have one beta and one alpha, so each month from 2000 to 2005, do I run a regression of each stock on its beta? This beta from that time period won’t be changing though?

## Answer by Tim Wilding (score 3, accepted)

https://quant.stackexchange.com/a/55146

Yes, the second step of the Fama MacBeth procedure requires you to run a cross-sectional regression of the monthly returns of each stock against their betas for each month. This regression gives you a return for each factor for each period. The average factor return is the risk premium for the factor - see Rationale of Fama Macbeth procedure for a good description of the overall procedure, and Interpreting the coefficients of Fama-MacBeth regression for a discussion about what these second-stage coefficients mean.

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.