In this paper, we develop a projected instrumental variable framework that relaxes conventional assumptions for causal identification with instrumental variables. Building on the interpretation of conditional expectation as an orthogonal projection in a Hilbert space, the framework replaces the conditional expectation operator with a general projection operator. We provide an alternative inverse problem formulation based on the Riesz representation. Drawing on semiparametric efficiency theory, we further develop debiased and adaptive machine learning methods for estimation and inference. The proposed framework connects invalid instrumental variables to instrumented difference-in-differences. We also examine invalid projected instrumental variables and discuss their relationship to existing methods. Simulation studies and real data analysis demonstrate that the proposed methods perform favorably compared with existing approaches.

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