Identification and Information after Nuisance Projection

Empirical work often removes fixed effects, latent factors, or high-dimensional controls before estimating structural relationships. These transformations reduce confounding but may also remove identifying variation. We study linear panel IV after one equation-compatible nuisance projection under two-way dependence. The projected Jacobian determines which structural directions remain visible; the projected-score law determines their precision; and, on Gaussian fixed-rank strata, they combine in a Projected Information Matrix. We derive weak-identification limits with dimension-specific information accumulation, feasible factor-transfer conditions, identification-robust tests, bootstrap procedures for non-Gaussian interaction limits, and inference for the projected spectrum, rank, subspaces, and information matrix. Simulations show that a raw first-stage statistic above 500 can support the wrong sign while projected diagnostics reveal weak valid information. In an international monetary application, common projection substantially attenuates apparent foreign-output persistence, while Gaussian-reference Anderson--Rubin sets remain unbounded. Identification should therefore be assessed after nuisance removal.