Fama–MacBeth Standard Errors and First-Stage Estimation Uncertainty

Summary

The document examines the standard error calculation for second-stage coefficients in a Fama–MacBeth regression. It gives the estimator based on the dispersion of period-by-period coefficient estimates around their average, scaled by the square of the number of periods. The author questions whether this calculation assumes no time-series dependence and whether it accounts for estimation error in the first-stage estimates used to obtain each period’s coefficients.

The central issue is how cross-sectional dependence and time-series dependence affect consistency of the reported standard errors, and why the treatment may differ across the two. The text frames these as questions rather than resolving them: it provides no derivation, simulation, or proposed correction. Readers should therefore treat it as an econometric problem statement, not evidence that the stated standard errors are valid under particular dependence structures.

Key ideas

• The displayed Fama–MacBeth standard error uses variation across period-specific second-stage estimates.
• The author questions the role of time-series correlation in the estimator’s assumptions.
• The document asks whether first-stage estimation error must be propagated into second-stage uncertainty.
• It raises the distinct effects of cross-sectional and time-series dependence without answering them.

Tags

# Fama MacBeth regression standard errors: sampling variations in the first stage


# Fama MacBeth regression standard errors: sampling variations in the first stage












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?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.