Estimating Return Predictability with Fama–MacBeth Cross-Sectional Regressions
Summary
The document asks how to apply Fama–MacBeth regressions to monthly asset returns when each asset’s predictor is its own lagged return. For each month, the proposed procedure runs a cross-sectional regression of that month’s returns on the corresponding lagged returns across assets, producing a time series of slope estimates. It then averages those estimates across months and considers repeating the process for different lags.
The setup clarifies that a cross-sectional predictor need not be identical across assets: it varies by asset at a given date, while the regression is estimated across assets for that date. The discussion does not provide standard-error calculations, controls, or evidence that any lag has predictive power. Its proposed interpretation of lag-specific averages as seasonal return patterns would require further statistical analysis, including attention to dependence across dates, multiple lag selection, and the study’s sample and specification.
Key ideas
- A monthly cross-sectional regression can use each asset’s own lagged return as its predictor.
- Estimate a separate slope for each month, then average the slopes over time.
- Repeating the estimation for different lags gives lag-specific average coefficients.
- A plotted coefficient pattern alone does not establish a seasonal effect or its cause.
- Inference requires attention to time dependence, controls, and the choice of lags.
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# Cross sectional regression for monthly retuns wih time lag
# Cross sectional regression for monthly retuns wih time lag
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 by user61342 (score 1)
https://quant.stackexchange.com/a/70080
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..."?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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