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I am confused about the rationale behind the Fama Macbeth regression methodology. I understand how to practically perform the two steps but not why one should do so.

For instance, considering the Fama and French three-factor model:

$R_{it} - R_{ft} = \alpha_i + \beta_i(R_{mt}-R_{ft})+s_iSMB_t+h_iHML_t + \epsilon_{it}$

Why should the two step methodology be employed? Why is it not enough to just run a time series regression for each asset $i$ and estimate $\alpha_i$, $\beta_i$, $s_i$ and $h_i$? What is the economic meaning of the $\gamma_0$ and $\gamma_i$ coefficients that would be estimated from the second-step cross-sectional regressions at each point in time?

Edit: After further research, I understood that the FMB methodology is used to test the validity of CAPM. However, I still do not understand the meaning of the gamma coefficients found in the second-step regression.

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  • $\begingroup$ You may find this post interesting? quant.stackexchange.com/questions/37987/… $\endgroup$ – Kevin Jul 23 '19 at 10:25
  • $\begingroup$ I guess what is not clear to me is the difference between the factors used in a factor model such as FF and the risk premiums. Practically speaking, for instance, if $R_m - R_f$ is not a risk premium, then what is it? $\endgroup$ – Gianluca Jul 23 '19 at 10:53
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Clarification on the regression coefficients

Cochrane (Asset Pricing, rev. edition, 2005) states (p. 247):

It it easier to do this in a more standard setup, with left-hand variable $y$ and right-hand variable $x$. Consider a regression $$y_{it} = \beta´x_{it} + \epsilon_{it}$$ $$i = 1,2,..,N$$ $$t = 1,2,...,T$$ [...] In an expected return-beta asset pricing model, the $x_{it}$ stands for the $\beta_i$ and $\beta$ stands for $\lambda$.

Background

The Fama/MacBeth procedure is used to estimate consistent standard errors in the presence of cross-sectional correlation.

Fama-MacBeth (1973) - First step

The first step is a time series regression to get your right-hand variable $x_{it}$, i.e. the beta coefficients. As you are already aware of the technical details, let me just refer you to these answers [1], [2], [3] with further details on this step.

Fama-MacBeth (1973) - Second step

The gamma coefficients (here: $\lambda´_t$) are estimates for the risk-premium of your risk-factors $\beta´_t$. What does this mean? We apply a cross-sectional regression at each point of time $t$. If there is a (linear) relationship between your risk factors $\beta´_t$ and stock returns in period $t$, we would obtain a well-measured (i.e. statistical significant) positive factor risk-premium at $t$. The economic interpretation of $\lambda´_t$ is how much the expected stock return would rise, if this stocks risk-factor increases one unit.

We get estimates for the risk-premia $\lambda´_t$ at each point of time $t$. Due to limited computational power (and statistical methodologies) in 1973, we simply use the variation in $\lambda´_t$ over time to deduce its variation across samples.

You may look at this excellent answer on the technical details of this second step.

Fama-French three factor model

Your stated regression gives you the factor-loadings of a certain stock or portfolio. You may use these coefficients e.g. to calculate the expected return of this stock. However, the factor-returns are based on certain investment strategies (SMB/HML). As stated here,

you cannot interpret the average return for the factor as the risk premium.

but this needs further clarification, which follows now.

Conclusion

You may be confused by the term risk premium. The Fama/French factor time-series SMB or HML are indeed risk premiums (like the market-risk premium), but not in terms of the Fama/MacBeth procedure.

What Fama/French within their Three-factor model do, is to construct portfolios which follows certain investment strategies. These return series are risk-premia, because it measures how much a stock`s return should increase, if its beta for this factor increases one unit. We have strong empirical evidence, that these risk-factors drive stock returns.

Fama/MacBeth however start with risk-factors (like market-beta) and test, if there is any observable market-premium for this risk-factor in the cross-section of stock returns. If we would not see any significant and positive risk-premium, our risk-factor is not able to explain differences in the cross-section of stock returns.

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