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 $ρ_n = d_{K_n}/n \to ρ\in [0,1)$ and the residual treatment variance $τ_n^2 = nQ_{K_n} \to τ^2 \in (0, \infty]$ 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 $τ^2 > 0$: there is no Stock--Yogo-style threshold in $τ^2$. Second, the Eicker--White HC0 estimator is biased downward by a fixed factor $(1-ρ)$, producing over-rejection that grows with saturation. Third, HC3 over-corrects in the opposite direction by a factor $1/(1-ρ)$. 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 $ρ|μ- ω^2|$ that we characterize. An empirical application to Piotroski F-Score returns in CEE markets illustrates the predicted variance hierarchy in real data.