# Interpreting the coefficients of Fama-MacBeth regression

According to Fama & MacBeth (1973) two-step regression, you start with estimating the beta factors. When applying the Fama-French 3-Factor model, you first run the linear regression

$$r_{i,t}=α_i+β_{i,MktRf}MktRf_t+β_{i,SMB}SMB_t+β_{i,HML}HML_t+ϵ_{i,t}$$

The second step is a cross-section regression for each t : $$r_{i,t}=λ_0+\hat{β}_iλ_t+α_{i,t}$$ with $\hat{β}_i≡[β_{i,MktRf},β_{i,SMB},β_{i,HML}]′$ as the estimated factor loadings from the first step.

The Wikipedia article describes the second step as follows:

Then regress all asset returns for a fixed time period against the estimated betas to determine the risk premium for each factor.

So in fact, the average value of the estimated $λ_t$ can be interpreted as the corresponding risk premium for each $β_{i,MktRf}$, $β_{i,SMB}$ and $β_{i,HML}$.

Question

I use data from Kenneth Frenchs website on the Fama-French portfolios for estimating the factor loadings in the first step of the regression. As far as i know, the data from Kenneth French are already the risk premium of the factors $MktRf$, $SMB$ and $HML$.

Can i just use the time-series data from Kenneth French, as they already are risk premiums on the corresponding portfolios, and interpret their average value as the estimated values of $λ_t$ following Fama & MacBeth regression?

Why should the results be different, if using Kenneth French data as input in the first step of Fama & MacBeth regression (when estimating the factor loadings following Fama & French 3 factor model) and then estimating the risk premiums or directly using Kenneth French data and calculate the average value of risk premiums?

• What are you trying to do? – Matthew Gunn Jan 31 '18 at 2:46
• Just a typical Fama/MacBeth regression on a test of the Fama-French-3-factor model. As common, i test the null hypothesis, if the average $λ_t$ is statistically different from zero. As i am using the Fama-French model for estimating betas on the first step, i assume that the final values of $λ_t$ should be the same as the published risk premia on Kenneth Frenchs website. – skoestlmeier Jan 31 '18 at 8:55

No, you cannot interpret the average return for the factor as the risk premium. The second stage regression is equivalent to building a set of portfolios that have no net investment, a unit exposure to one factor and 0 exposure to all others. These unit exposure portfolios are then used to estimate the risk premia for those factors ($\lambda_t$). In that sense, $\lambda_t$ is how much someone can earn for exposure to that risk factor alone and $\lambda_t$ will not necessarily match the average returns of the factor.
In examples I have run with Kenneth French’s data, the average of a specific factor can be very different from the risk premium. French’s returns factors have not been adjusted for the returns of the zero beta portfolio ($\lambda_0$) and I suspect this will cause the most significant differences.
I do think Fama MacBeth regression is a little confusing because the Kenneth French portfolios and the risk premia are both estimated portfolio returns and so the intuition is that they should have similar values. However, the process makes a little more sense when you remember that Fama MacBeth regression may also be used for factors that are not directly investable portfolios. For example, we could specify that the factor is any time series such as the number of tins of beans sold in your local supermarket. In that case, the second regression more clearly converts any factor exposures ($\beta_{i,SMB}$, etc.) into an investable strategy that would earn the risk premium for that factor in the marketplace.