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Following the notation of this post, the standard errors of the second stage coefficients is computed as $$\sigma^{2}(\hat{\lambda})=\frac{1}{T^{2}} \sum_{t=1}^{T}\left(\hat{\lambda}_{t}-\hat{\lambda}\right)^{2}.$$

I think the assumption behind this is that the time-series correlation is supposed to be zero (no stock effect), so each $\hat{\lambda}_t$ represents a draw from population and $\hat{\lambda}$ is (approximately) the true mean. However, both $\hat{\lambda}_t$ and $\hat{\lambda}$ are estimated in the first stage as the hat denotes, so I thought that the sampling variation that arises in the first stage should be taken into account, as is the case in any two-step econometrics methods such as 2SLS.

What am I missing? Why is the Fama-MacBeth regression coefficients SE consistent for data with cross-sectional correlations but not time-series correlations?

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  • $\begingroup$ Isn't sampling variation implicitly accounted for in the formula as this affects variability of the estimates? I don't remember 2SLS too well though and this is not necessarily the most efficient estimation approach. Your formula assumes that lambda estimates are uncorrelated in time, otherwise these autocorrelations would appear in the standard error formula. $\endgroup$
    – fes
    Feb 20, 2022 at 20:11

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