Adaptive, rate-optimal testing in instrumental variables models

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This paper proposes simple, data-driven, optimal rate-adaptive inferences on a structural function in semi-nonparametric conditional moment restrictions. We consider two types of hypothesis tests based on leave-one-out sieve estimators. A structure-space test (ST) uses a quadratic distance between the structural functions of endogenous variables; while an image-space test (IT) uses a quadratic distance of the conditional moment from zero. For both tests, we analyze their respective classes of nonparametric alternative models that are separated from the null hypothesis by the \textit{minimax rate of testing}. That is, the sum of the type I and the type II errors of the test, uniformly over the class of nonparametric alternative models, cannot be improved by any other test. Our new minimax rate of ST differs from the known minimax rate of estimation in nonparametric instrumental variables (NPIV) models. We propose computationally simple and novel exponential scan data-driven choices of sieve regularization parameters and adjusted chi-squared critical values. The resulting tests attain the minimax rate of testing, and hence optimally adapt to the unknown smoothness of functions and are robust to the unknown degree of ill-posedness (endogeneity). Data-driven confidence sets are easily obtained by inverting the adaptive ST. Monte Carlo studies demonstrate that our adaptive ST has good size and power properties in finite samples for testing monotonicity or equality restrictions in NPIV models. Empirical applications to nonparametric multi-product demands with endogenous prices are presented.