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For the classical linear model with an endogenous variable estimated by themethod of instrumental variables (IVs) with multiple instruments, Masten andPoirier (2021) introduced the falsification adaptive set (FAS). When a model isfalsified, the FAS reflects the model uncertainty that arises fromfalsification of the baseline model. It is the set of just-identified IVestimands, where each relevant instrument is considered as the just-identifyinginstrument in turn, whilst all other instruments are included as controls. Ittherefore applies to the case where the exogeneity assumption holds and invalidinstruments violate the exclusion assumption only. We propose a generalized FASthat reflects the model uncertainty when some instruments violate theexogeneity assumption and/or some instruments violate the exclusion assumption.This FAS is the set of all possible just-identified IV estimands where thejust-identifying instrument is relevant. There are a maximum of$k_{z}2^{k_{z}-1}$ 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 itsatisfies the exogeneity and exclusion assumptions, then this generalized FASis guaranteed to contain $\beta$ and therefore to be the identified set for$\beta$.

The Falsification Adaptive Set in Linear Models with Instrumental Variables that Violate the Exogeneity or Exclusion Restriction