We propose a new adaptive hypothesis test for polyhedral cone (e.g., monotonicity, convexity) and equality (e.g., parametric, semiparametric) restrictions on a structural function in a nonparametric instrumental variables (NPIV) model. Our test statistic is based on a modified leave-one-out sample analog of a quadratic distance between the restricted and unrestricted sieve NPIV estimators. We provide computationally simple, data-driven choices of sieve tuning parameters and adjusted chi-squared critical values. Our test adapts to the unknown smoothness of alternative functions in the presence of unknown degree of endogeneity and unknown strength of the instruments. It attains the adaptive minimax rate of testing in L2. That is, the sum of its type I error uniformly over the composite null and its type II error uniformly over nonparametric alternative models cannot be improved by any other hypothesis test for NPIV models of unknown regularities. Data-driven confidence sets in L2 are obtained by inverting the adaptive test. Simulations con rm that our adaptive test controls size and its nite-sample power greatly exceeds existing non-adaptive tests for monotonicity and parametric restrictions in NPIV models. Empirical applications to test for shape restrictions of differentiated products demand and of Engel curves are presented.
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 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.
This paper develops a new method informed by data and models to recover information about investor beliefs. Our approach uses information embedded in forward-looking asset prices in conjunction with asset pricing models. We step back from presuming rational expectations and entertain potential belief distortions bounded by a statistical measure of discrepancy. Additionally, our method allows for the direct use of sparse survey evidence to make these bounds more informative. Within our framework, market-implied beliefs may differ from those implied by rational expectations due to behavioral/psychological biases of investors, ambiguity aversion, or omitted permanent components to valuation. Formally, we represent evidence about investor beliefs using a novel nonlinear expectation function deduced using model-implied moment conditions and bounds on statistical divergence. We illustrate our method with a prototypical example from macro-finance using asset market data to infer belief restrictions for macroeconomic growth rates.