Interpreting Risk Premia in Fama–MacBeth Regressions

# Rationale of Fama Macbeth procedure


# Rationale of Fama Macbeth procedure












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.

## Answer by skoestlmeier (score 15, accepted)

https://quant.stackexchange.com/a/46764

#### 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.