Fixed-Effect Saturation Is Not Weak Identification: Certifying Inference under Measurement Error

Abstract

Fixed-effect saturation alone is not weak identification. In the baseline model, fixed-effect--residualized OLS is unbiased and conventional inference is asymptotically exact at every level of residual treatment variation $\tau^2=nQ_K>0$: unlike a weak first stage in IV, a small $\tau^2$ produces no size distortion by itself. Classical measurement error in the treatment changes this. Under a local noise drift $\sigma_\nu^2=c^2/n$, the FE-OLS $t$-statistic converges to a non-central normal whose non-centrality $\eta$ falls with $\tau^2$ and, once the within reliability $\lambda$ is held apart from the fixed-effect dimension $\rho$, is $\rho$-free: saturation rescales the whole problem by $\sqrt{1-\rho}$ rather than preferentially destroying signal or noise. Inverting the resulting size distortion gives a closed-form Stock--Yogo-style critical value for $\tau^2$, and the reliability below which conventional inference breaks down has a fixed-point form computable from the reported $t$-statistic alone, with no auxiliary regression needed. Because the diagnostic only needs a lower bound on reliability, where correcting the point estimate needs its exact value, we separate a descriptive \emph{point pass} from a conservative \emph{certificate} evaluated at an upper confidence bound, with false-certification probability at most $\gamma$; a parallel cluster-robust theory extends both to standard clustered inference. In a saturated democracy--growth panel, the diagnostic tells apart two measures of the same underlying construct: aggregate V-Dem polyarchy is certified at $\gamma=0.05$, while its judicial-constraints sub-index, coded with far less inter-rater agreement, is flagged under both i.i.d.\ and clustered standard errors. The diagnostic covers classical error in a continuous regressor; it does not extend to binary-treatment misclassification, where the error is nonclassical by construction.