Breakdown Reliability for Saturated Fixed-Effect Inference

Abstract

Fixed-effect saturation does not itself distort conventional inference, but classical measurement error does. Under a local noise drift ${\sigma }_{\nu }^2=c^2/n$, the FE-OLS $t$-statistic converges to a non-central normal; saturation contributes a common $\sqrt{1-\rho }$ scaling rather than preferentially destroying signal or noise. Inverting the size distortion gives a Stock--Yogo-style critical value. Self-consistency of the within-reliability-corrected pilot yields a breakdown reliability ${\lambda }^{†}=|t|/(|t|+{\eta }^{†})$ --- the minimum within reliability at which conventional inference retains nominal size within the chosen tolerance --- computable from the reported $t$-statistic alone and algebraically $\rho$-free conditional on it; ${\eta }^{†}\approx 0.65$ at $5\%$ size and a 5-point tolerance. Replacing $|t|$ by $|t|+z_{1-{\gamma }_{\beta }}$ gives a certified breakdown reliability; with a lower-reliability bound whose coverage error is ${\gamma }_{\lambda }$, false certification is at most ${\gamma }_{\beta }+{\gamma }_{\lambda }$. Under a checkable projection-compatibility condition, a cluster-level score CLT and consistency of the Arellano variance estimator in the many-fixed-effect regime justify applying the same map to the reported cluster-robust $t$-statistic; clustering can reverse a verdict. In a saturated democracy--growth panel, aggregate V-Dem polyarchy is certified at ${\gamma }_{\beta }=0.05$, conditional on the supplied measurement model, while its judicial-constraints sub-index is flagged under i.i.d.\ and clustered standard errors. In a twin-pair wage design, the specification is flagged under both independent and correlated reporting-error models, although implied coverage of the nominal-$95\%$ interval ranges from $8\%$ to $68\%$. The diagnostic covers classical error in a continuous regressor, not binary-treatment misclassification.