This paper studies how standard measures of uncertainty behave when linear regression models include a very large number of fixed effects. Fixed effects are dummy variables that control for group-level factors (for example, a firm or year). When these fixed effects “saturate” the data, some popular variance estimators give misleading results. The authors show which estimators work, which fail, and under what conditions.

The authors examine an asymptotic setting where the fraction of parameters coming from fixed effects, rho_n = d_K/n, converges to a limit rho in [0,1), and a scaled residual treatment variance, tau_n^2 = n Q_K, converges to a finite positive value. Here Q_K is the population residual variance of the treatment after projecting out the fixed effects (formally, E[Var(X | G_K)]). Under standard regularity assumptions (independent observations, bounded leverage, and some moment conditions) they derive the joint limiting distribution of the regression score and residual sums of squares and use this to study several common variance estimators.

Their main technical findings are concrete. First, a degrees-of-freedom corrected homoskedastic estimator — the Cattaneo–Jansson–Newey (CJN) correction that divides the residual sum of squares by (n − d_K − 1) — gives asymptotically exact t-statistics under strict exogeneity and conditional homoskedasticity. In fact, with Gaussian errors the CJN t-statistic has the exact t_{n−d_K−1} distribution conditional on the fixed-effect design. Second, the naive homoskedastic estimator that divides by n (or the plain Eicker–White HC0 heteroskedasticity-robust estimator in saturated settings) is biased downward by a factor (1 − rho). That downward bias makes tests over-reject and the problem grows as the fixed-effect space saturates. Third, the commonly recommended HC3 estimator over-corrects in the other direction, inflating variance by a factor 1/(1 − rho) and producing overly conservative tests. The leave-one-out estimator (HC2), which removes one observation’s influence when estimating variance, removes the first-order leverage distortion and is asymptotically exact under homoskedasticity or under a balanced heteroskedastic design. Under general heteroskedasticity with non-uniform leverage, HC2 retains a smaller additional bias of order rho times a characterized quantity.