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?
