Variance Estimation for Saturated Fixed-Effect Specifications

Abstract

We characterize the asymptotic behavior of conventional variance estimators in linear regression with high-dimensional fixed effects under a drift in which both the proportional fixed-effect dimension $\ddot{I}{}_n=d_{K_n}/n\to \ddot{I}\in [0,1)$ and the residual treatment variance $\ddot{I}{„}_n^2=n{Q}_{K_n}\to \ddot{I}{„}^2\in (0,\infty ]$ are non-degenerate. Three findings emerge. First, under strict exogeneity and conditional homoskedasticity, the Cattaneo--Jansson--Newey-corrected $t$-statistic is asymptotically exact for any $\ddot{I}{„}^2>0$: there is no Stock--Yogo-style threshold in $\ddot{I}{„}^2$. Second, the Eicker--White HC0 estimator is biased downward by a fixed factor $(1-\ddot{I})$, producing over-rejection that grows with saturation. Third, HC3 over-corrects in the opposite direction by a factor $1/(1-\ddot{I})$. The leave-one-out estimator (HC2) removes the first-order leverage distortion and is asymptotically exact under homoskedasticity or design-balanced heteroskedasticity; under general heteroskedasticity with non-uniform leverage, HC2 retains an additional bias of order $\ddot{I}|\hat{I}¼-\ddot{I}{‰}^2|$ that we characterize. An empirical application to Piotroski F-Score returns in CEE markets illustrates the predicted variance hierarchy in real data.

Submitted to The Econometrics Journal for consideration