The Falsification Adaptive Set in Linear Models with Instrumental Variables that Violate the Exogeneity or Exclusion Restriction
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For the classical linear model with an endogenous variable estimated by the method of instrumental variables (IVs) with multiple instruments, Masten and Poirier (2021) introduced the falsification adaptive set (FAS). When a model is falsified, the FAS reflects the model uncertainty that arises from falsification of the baseline model. It is the set of just-identified IV estimands, where each relevant instrument is considered as the just-identifying instrument in turn, whilst all other instruments are included as controls. It therefore applies to the case where the exogeneity assumption holds and invalid instruments violate the exclusion assumption only. We propose a generalized FAS that reflects the model uncertainty when some instruments violate the exogeneity assumption and/or some instruments violate the exclusion assumption. This FAS is the set of all possible just-identified IV estimands where the just-identifying instrument is relevant. There are a maximum of k_z2^k_z-1 such estimands, where k_z is the number of instruments. If there is at least one relevant instrument that is valid in the sense that it satisfies the exogeneity and exclusion assumptions, then this generalized FAS is guaranteed to contain β and therefore to be the identified set for β.
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