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This paper studies nonparametric estimation of conditional moment models in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables. This is a nonparametric nonlinear instrumental variables (IV) problem. We propose a class of penalized sieve minimum distance (PSMD) estimators which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite dimensional function parameter space. Some of the PSMD procedures use slowly growing finite dimensional sieves with flexible penalties or without any penalty; some use large dimensional sieves with lower semicompact and/or convex penalties. We establish their consistency and the convergence rates in Banach space norms (such as a sup-norm or a root mean squared norm), allowing for possibly non-compact infinite dimensional parameter spaces. For both mildly and severely ill-posed nonlinear inverse problems, our convergence rates in Hilbert space norms (such as a root mean squared norm) achieve the known minimax optimal rate for the nonparametric mean IV regression. We illustrate the theory with a nonparametric additive quantile IV regression. We present a simulation study and an empirical application of estimating nonparametric quantile IV Engel curves.

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For semi/nonparametric conditional moment models containing unknown parametric components (θ) and unknown functions of endogenous variables (h), Newey and Powell (2003) and Ai and Chen (2003) propose sieve minimum distance (SMD) estimation of (θ, h) and derive the large sample properties. This paper greatly extends their results by establishing the followings: (1) The penalized SMD (PSMD) estimator ( ˆ θ, ˆ h) can simultaneously achieve root-n asymptotic normality of ˆ θ and nonparametric optimal convergence rate of ˆ h, allowing for models with possibly nonsmooth residuals and/or noncompact infinite dimensional parameter spaces. (2) A simple weighted bootstrap procedure can consistently estimate the limiting distribution of the PSMD ˆ θ. (3) The semiparametric efficiency bound results of Ai and Chen (2003) remain valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bounds. (4) The profiled optimally weighted PSMD criterion is asymptotically Chi-square distributed, which implies an alternative consistent estimation of confidence region of the efficient PSMD estimator of θ. All the theoretical results are stated in terms of any consistent nonparametric estimator of conditional mean functions. We illustrate our general theories using a partially linear quantile instrumental variables regression, a Monte Carlo study, and an

In this paper we develop two nonparametric tests of treatment effect heterogeneity. The first test is for the null hypothesis that the treatment has a zero average effect for all subpopulations defined by covariates. The second test is for the null hypothesis that the average effect conditional on the covariates is identical for all subpopulations, i.e., that there is no heterogeneity in average treatment effects by covariates. We derive tests that are straightforward to implement and illustrate the use of these tests on data from two sets of experimental evaluations of the effects of welfare-to-work programs.

Game-theoretic models are frequently employed to study strategic interaction between agents. Empirical research has focused on estimating payoff functions while maintaining strong assumptions regarding the information structure of the game. I show how to relax informational assumptions to enhance the credibility of empirical analysis in discrete games. I propose the use of a flexible information structure that nests the most common assumptions used in the discrete games literature. Despite the added flexibility, the parameters of the players ’ payoff functions remain point identified under standard rich support assumptions. As is common in discrete-choice games, multiple equilibria complicate identification and inference in empirical applications. I introduce an approach to equilibrium selection that avoids parametric assumptions and lets selection depend on public information that is not observed by the econometrician. I apply the method to data on the entry and exit patterns of grocery stores. The model provides useful bounds to equilibrium outcomes. In addition, the empirical analysis indicates that more restrictive informational assumptions commonly used in empirical applications models can generate

This paper studies the nonparametric identification of the first-price auction model with risk averse bidders within the private value paradigm. First, we show that the benchmark model is nonindentified from observed bids. We also derive the restrictions imposed by the model on observables and show that these restrictions are weak. Second, we establish the nonparametric identification of the bidders ’ utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders ’ participation, leading to a latent distribution of private values that is independent of the number of bidders. The key idea is to exploit the property that the bid distribution varies with the number of bidders while the private value distribution does not. We then extend these results to endogenous bidders ’ participation when the exclusion restriction takes the form of instruments that do not affect the bidders ’ private value distribution. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values, and asymmetric bidders. Last, possible estimation methods are proposed.

The goal of this paper is to develop techniques to simplify semiparametric inference. We do this by deriving a number of numerical equivalence results. These illustrate that in many cases, one can obtain estimates of semiparametric variances using standard formulas derived in the already-well-known parametric literature. This means that for computational purposes, an empirical researcher can ignore the semiparametric nature of the problem and do all calculations “as if” it were a parametric situation. We hope that this simplicity will promote the use of semiparametric procedures.

This paper develops methods for hypothesis testing in a nonparametric instrumental vari-ables (IV) setting within a partial identification framework. We construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction. The same test statistic can be employed under identification, in which case the hypothesis is of whether the true model satisfies the posited property. An almost sure consistent bootstrap procedure is provided for obtaining critical val-ues. Possible applications include testing for semiparametric specifications as well as building confidence regions for certain functionals on the identified set. As an illustration we obtain confidence intervals for the level and slope of fuel Engel curves in Brazil. A Monte Carlo study examines finite sample performance.