On independence conditions in nonseparable models: Observable and unobservable instruments

This paper develops identification results employing independence conditions among unobservable variables. The independence conditions are used to derive first-stage nonseparable reduced form functions. Once constructed, these reduced form functions are employed to express the derivatives of nonseparable structural functions in terms of the derivatives of the reduced form functions. For models with simultaneity, we obtain the new results by combining the independence assumptions together with parametric specifications and exclusion restrictions. For models with triangularity, we allow all functions to be nonparametric and nonseparable in unobservable random terms. For the latter, we provide several equivalence results and discuss some of the trade-offs between observable and unobservable instruments.