Adaptive Inference with Weak Instruments

Inference with weak instruments presents a choice between the two-stage least squares (2SLS) yielding tight confidence intervals with low coverage, and the test-based methods, like Anderson–Rubin, producing vacuously wide but valid intervals. We show that this trade-off is unnecessary. First, we show that a convex combination of a bias-controlled estimator, like ordinary least squares, and a valid estimator, like 2SLS, preserves coverage. Second, to yield tight intervals, we propose StAR that uses e-process diagnostics to adaptively set these weights. StAR's weights involve a multivariate e-value aggregating relevance of all the instruments, and the confidence width of the e-process gating against premature reliance on any instrument. We prove three finite-sample results: (i) StAR's coverage, (ii) StAR's width that transitions from OLS-like at weak instruments to IV-efficient at strong instruments at an explicit rate, and (iii) an oracle inequality for StAR showing that it achieves an error rate within a constant factor of the oracle rate. Our experiments on synthetic data shows that StAR maintains $\ge 96\%$ coverage and produces 6× tighter intervals than Anderson–Rubin at weak instruments. Our experiments on clinical AI and econometric data shows that StAR yields valid and informative CIs, and accurately detects collective instrument strength.