Background

Fama Macbeth procedure is proposed by Fama and Macbeth(1973). It is an alternative way to ran cross-section regression.

Procedure

There are three steps to obtain the Fama Macbeth regression estimates and test statistics: obtain betas with time-series regression for each firm, do cross-sectional regression for each date, and conduct newey-west adjustments for standard error. The following three subsection describe them, respetively.

Rolling Betas

Time-series regression to get betas[1]

$R_{it} = \hat a_i+\hat\beta_{it}f_{it}+\epsilon_{it}$

Where $R_{it}$ refers to the stock return of firm $i$ at time $t$. $f_{it}$ represents a factor that affect $R_{it}$, and $\hat\beta_{it}$ is the estimated coefficient on factor $f_{ijt}$.

Genernally, estimating the model involves using rolling regressions with historic data. For example,a dataset includes 50 stocks with each stock has 36-month observations with variables: stock return, factor 1, 2, 3. If we require the number of observations for estimating beta1,2,3 is 24. Then for each firm $i$ at month $t$, the monthly observation used to estimate the beta is from $t-24$ to $t-1$.

Now, How do we achieve calculating So the input dataset looks like

Calculate Rolling Betas with SAS

a. Rolling beta estimation with Macro. In chapter 4 of Boehmer,Broussard, and Kallunki(2002)[2], they employ macro to achieve rolling regression and get betas. The first step is to construct a dataset called good data with additional variable n

data gooddata;
set msf8;
by gvkey;
n+1;
if first.gvkey then n =1;
run;

%MACRO estim; /* begin macro definition */
%do X = 50 %to 66;
DATA temp;
SET gooddata;
if &X - 49 <= N <= &X;
per = &x;
PROC REG DATA = temp NOPRINT OUTEST = results;
MODEL r = rm;
BY firm per;
quit;
PROC APPEND BASE = betas1 DATA = results;
%end;
%mend estim; /* end macro definition */
%estim; /* execute macro */

b. Rolling regressions without macros. André de Souza provides a approach to do rolling regression without invoke macros. The basic idea is to construct a dataset that can be used with proc reg directly. You click the link for more detail.

Calculate Rolling Betas with Python

Step2: Cross-sectional regressions

Cross-ection regression at each time period

$R_{it} = \hat\alpha_i+\hat\lambda_{it}\beta_{it}+\delta_{it}$

How many firms are needed for cross sectional regression.

Averaging across time to get the final estimates $\hat\lambda = 1/T\sum^T_{t=1}\hat\lambda_{t}$ $\hat\alpha = 1/T\sum^T_{t=1}\hat\lambda_{t}$

Step3: Newey West Adjusted Standard Error

Reference

[1] This procedure part is constructed based on Johan Cochrane Teaching Notes.

[2]Boehmer, E., Broussard, J. P., & Kallunki, J. P. (2002). Using SAS in financial research. SAS Institute.