What No First Stage Can Detect: Functional-Form Contamination in Linear IV

Applied instrumental variables (IV) practice reports a first-stage F, now often the conditional F of Sanderson and Windmeijer (2016), and reads a large value as license to interpret the second stage. We show that no first-stage diagnostic can provide it. With a scalar instrument, a scalar treatment, and covariates entered linearly, the 2SLS estimand splits into a signal that a saturated specification would target and a contamination, the covariance between curvature in the instrument propensity and a covariate level function. The same nuisance sits in both terms, so it biases the estimand and inflates the reported strength at once. When the instrument is nearly collinear with the covariates the signal vanishes and the strength is manufactured. When the strength is honest the curvature still biases the estimand through the outcome, where no first-stage number reaches it. We prove that no functional of the joint distribution of instrument, treatment, and covariates can detect this second bias, and we give a directed test built from reduced-form regressions, a corrected estimator, and a reportable contamination share. Two applications show the modes. A husband's insurance instrument (Olson, 1998) has a conditional F above 36,000, yet correcting a linear income control more than doubles the estimate. The instrument of Nunn and Wantchekon (2011) loses most of its first stage once geography enters flexibly.