Identification via Distributional Shifts without Exclusion Restrictions

This paper studies identification and inference in a triangular system with an endogenous regressor when exclusion restrictions are unavailable and the dependence between structural disturbances is modeled through an unrestricted control function. In this setting, standard orthogonality conditions do not deliver point identification, as the unknown control function can rationalize a wide range of structural coefficients. We show that identifying information can be extracted from distributional shifts in the first-stage disturbance induced by an auxiliary variable that may directly affect the outcome. Imposing a local restriction on the log density ratio, together with an explicit bound on the sieve approximation error of the control function, we derive moment inequalities that restrict the structural parameter. We develop a practical inference procedure based on test inversion and multiplier bootstrap that accommodates generated regressors, cross-fitted sieve estimation, and locally estimated density-ratio nuisances. The results clarify how identification can be recovered from weak local distributional structure in the absence of classical instruments.