A residual-prediction test with user-chosen ML and sample splitting delivers asymptotic type I error control for linear IV model specification under mean independence of structural errors and instruments, with extensions to weak instruments.
Hence, (27) follows using Lemma 8 and assertion 3 of Assumption 2
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Machine-Learning-Powered Specification Testing in Linear Instrumental Variable Models
A residual-prediction test with user-chosen ML and sample splitting delivers asymptotic type I error control for linear IV model specification under mean independence of structural errors and instruments, with extensions to weak instruments.