Cross sectional regression for monthly retuns wih time lag

Question

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?

Answer 1

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..."?