I am trying to understand the cross-sectional regression methology of Fama MacBeth in the setting of monthly returns with time lags. I was reading "Seasonality in the cross-section, Steven L. Heston and Ronnie Sadka". On page 4 they write: We apply the cross-sectional regression methodology (Fama and MacBeth, 1973; Fama and French, 1992) to monthly returns $$r_{i,t}=\alpha_{k,t}+\gamma_{k,t}r_{i,t-k}+e_{i,t}.$$ If I understood correctly cross-sectional regression uses the same point in time. Indeed this is the case but according to https://en.wikipedia.org/wiki/Fama%E2%80%93MacBeth_regression the independent variable should be the same for each row which is not fulfilled. Consider a fixed time lag, let's say k=1, and we have $N$ assets and $T$ is the last time. Then we have $$ r_{1,t}=\alpha_{1,t}+\gamma_{1,t}r_{1,t-1}+e_{1,t}\\ r_{2,t}=\alpha_{1,t}+\gamma_{1,t}r_{2,t-1}+e_{2,t}\\ \vdots\\ r_{N,t}=\alpha_{1,t}+\gamma_{1,t}r_{N,t-1}+e_{N,t} $$ for each time $t=1,...,T$. As $r_{j,t-1}\neq r_{l,t-1}$ for $j\neq l$ (in general) we cannot apply cross-sectional regression or am I wrong? Does anybody have some helpful advice how to do it?
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$\begingroup$ The purpose of FM cross-sectional regression is to "explain" the returns of N stocks on day i using "explanatory variables" known (computable) on that day for each stock. In the original FM study the explanatory variable was the Beta of each stock, but it could be any stock specific known "fact" about each stock, for example the return on some other past date j for that stock (or to be absurd even the age of the CEO or the number of letters in the company's name). $\endgroup$– nbbo2Mar 4, 2022 at 13:07
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$\begingroup$ Thanks for your answer! Fixing some day $t$ I want to estimate $\gamma_{1,t}$. I tried to do this by using the least square method. Do I have to divide it by $N$ or is this already my estimator I want to have ? $\endgroup$– user61342Mar 4, 2022 at 13:20
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$\begingroup$ You estimate the $\gamma$s on a given day by OLS and that is it. (Later on you will average the gammas for different days (add them up and divide by T the number of periods) but that is a different step, after you have completed all the cross-sectional regressions). $\endgroup$– nbbo2Mar 4, 2022 at 13:25
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For any time $t$ I did an OLS so that for any $t$ I got $\gamma_{1,t}$. After doing this for all times I computed $$\hat{\gamma}_1 = \frac{1}{T}\sum_{t=1}^T \gamma_{1,t}.$$
I can do the same computation for several monthly time lags. Do they give me some kind of interpretation, so if I plot $\hat{\gamma}_1$,...,$\hat{\gamma}_k$ can I say something like "every six months it seems like monthly returns do have peeks due to..."?