User:QRho/Fama-MacBeth regression

The Fama–MacBeth regression is a method used to estimate parameters for asset pricing models such as the capital asset pricing model (CAPM). The method estimates the betas and risk premia for any risk factors that are expected to determine asset prices.

Model

The method works with multiple assets across time (panel data). The parameters are estimated in two steps:

1. First regress each of n asset's returns against m proposed risk factors to determine each asset's beta exposures.
   ${\begin{array}{lcr}R_{1,t}=\alpha _{1}+\beta _{1,F_{1}}F_{1,t}+\beta _{1,F_{2}}F_{2,t}+\cdots +\beta _{1,F_{m}}F_{m,t}+\epsilon _{1,t}\\R_{2,t}=\alpha _{2}+\beta _{2,F_{1}}F_{1,t}+\beta _{2,F_{2}}F_{2,t}+\cdots +\beta _{2,F_{m}}F_{m,t}+\epsilon _{2,t}\\\vdots \\R_{n,t}=\alpha _{n}+\beta _{n,F_{1}}F_{1,t}+\beta _{n,F_{2}}F_{2,t}+\cdots +\beta _{n,F_{m}}F_{m,t}+\epsilon _{n,t}\end{array}}$
2. Then regress all asset returns for each of T time periods against the previously estimated betas to determine the risk premium for each factor.
   ${\begin{array}{lcr}R_{i,1}=\gamma _{1,0}+\gamma _{1,1}{\hat {\beta }}_{i,F_{1}}+\gamma _{1,2}{\hat {\beta }}_{i,F_{2}}+\cdots +\gamma _{1,m}{\hat {\beta }}_{i,F_{m}}+\epsilon _{i,1}\\R_{i,2}=\gamma _{2,0}+\gamma _{2,1}{\hat {\beta }}_{i,F_{1}}+\gamma _{2,2}{\hat {\beta }}_{i,F_{2}}+\cdots +\gamma _{2,m}{\hat {\beta }}_{i,F_{m}}+\epsilon _{i,2}\\\vdots \\R_{i,T}=\gamma _{T,0}+\gamma _{T,1}{\hat {\beta }}_{i,F_{1}}+\gamma _{T,2}{\hat {\beta }}_{i,F_{2}}+\cdots +\gamma _{T,m}{\hat {\beta }}_{i,F_{m}}+\epsilon _{i,T}\end{array}}$

History

Eugene F. Fama and James D. MacBeth (1973) first introduced the Fama–MacBeth regression procedure in order to test the relationship between average return and risk in the New York Stock Exchange. The method has been widely used in both academia and industry and in the fields of economics, corporate finance, and asset pricing.

Model

The first step of the procedure is to regress each of n asset's returns against the proposed risk factors to estimate beta exposures:

$\mathbf {R_{i}} =\mathbf {F} {\boldsymbol {\beta _{i}}}+{\boldsymbol {\varepsilon _{i}}}$ for each i of n assets,

where

$\mathbf {R_{i}} ={\begin{pmatrix}R_{i,1}\\R_{i,2}\\\vdots \\R_{i,T}\end{pmatrix}},$

$\mathbf {F} ={\begin{pmatrix}F_{1}^{\intercal }\\F_{2}^{\intercal }\\\vdots \\F_{T}^{\intercal }\end{pmatrix}}={\begin{pmatrix}1&F_{11}&\cdots &F_{1M}\\1&F_{21}&\cdots &F_{2M}\\\vdots &\vdots &\ddots &\vdots \\1&F_{T1}&\cdots &F_{TM}\end{pmatrix}},$

${\boldsymbol {\beta }}_{i}={\begin{pmatrix}\beta _{i,0}\\\beta _{i,1}\\\beta _{i,2}\\\vdots \\\beta _{i,M}\end{pmatrix}},\quad {\boldsymbol {\varepsilon }}_{i}={\begin{pmatrix}\varepsilon _{i,1}\\\varepsilon _{i,2}\\\vdots \\\varepsilon _{i,T}\end{pmatrix}}.$

The second step is to regress asset returns for each of T periods against the previously estimated betas to calculate risk premia:

$\mathbf {R_{t}} ={\boldsymbol {\hat {\beta _{t}}}}{\boldsymbol {\lambda _{t}}}+{\boldsymbol {\varepsilon _{t}}}$ for each t of T time periods,

where

$\mathbf {R_{t}} ={\begin{pmatrix}R_{1,t}\\R_{2,t}\\\vdots \\R_{n,t}\end{pmatrix}},$

${\boldsymbol {\hat {\beta _{t}}}}={\begin{pmatrix}{\hat {\beta _{t,1}}}^{\intercal }\\{\hat {\beta _{t,2}}}^{\intercal }\\\vdots \\{\hat {\beta _{t,n}}}^{\intercal }\end{pmatrix}}={\begin{pmatrix}1&{\hat {\beta _{t,11}}}&\cdots &{\hat {\beta _{t,1M}}}\\1&{\hat {\beta _{t,21}}}&\cdots &{\hat {\beta _{t,2M}}}\\\vdots &\vdots &\ddots &\vdots \\1&{\hat {\beta _{t,n1}}}&\cdots &{\hat {\beta _{t,nM}}}\end{pmatrix}},$

${\boldsymbol {\lambda }}_{t}={\begin{pmatrix}\lambda _{t,0}\\\lambda _{t,1}\\\lambda _{t,2}\\\vdots \\\lambda _{t,M}\end{pmatrix}},\quad {\boldsymbol {\varepsilon }}_{t}={\begin{pmatrix}\varepsilon _{t,1}\\\varepsilon _{t,2}\\\vdots \\\varepsilon _{t,n}\end{pmatrix}}.$

[Add estimates of lambdas by averaging over time periods]

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- Add history section

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- Explain intuition of model and comparison to previously used techniques

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Eugene F. Fama and James D. MacBeth (1973) demonstrated that the residuals of risk-return regressions and the observed "fair game" properties of the coefficients are consistent with an "efficient capital market" (quotes in the original).

Note that Fama MacBeth regressions provide standard errors corrected only for cross-sectional correlation. The standard errors from this method do not correct for time-series autocorrelation. This is usually not a problem for stock trading since stocks have weak time-series autocorrelation in daily and weekly holding periods, but autocorrelation is stronger over long horizons. This means Fama MacBeth regressions may be inappropriate to use in many corporate finance settings where project holding periods tend to be long. For alternative methods of correcting standard errors for time series and cross-sectional correlation in the error term look into double clustering by firm and year.