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 clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases. KEY WORDS: Nonparametric instrumental regression; Nonparametric indirect regression; Statistical ill-posed inverse problems; Minimax risk lower bound; Optimal rate.

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.

We propose an estimator for discrete choice models, such as the logit, with a nonparametric distribution of random coefficients. The estimator is linear regression subject to linear inequality constraints and is robust, simple to program and quick to compute compared to alternative estimators for mixture models. We discuss three methods for proving identification of the distribution of heterogeneity for any given economic model. We prove the identification of the logit mixtures model, which, surprisingly given the wide use of this model over the last 30 years, is a new result. We also derive our estimator’s non-standard asymptotic distribution and demonstrate its excellent small sample properties in a Monte Carlo. The estimator we propose can be extended to allow for endogenous prices. The estimator can also be used to reduce the computational burden of nested fixed point methods for complex models like dynamic programming discrete choice.

Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used identify and estimate these parameters. However, linear and other finite-dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators.

Abstract. I analyze the optimal taxation problem in a closed economy under incomplete markets allowing for default on the debt. If the government defaults, it will go to temporary financial autarky and it can only exit by paying a given fraction of the defaulted debt. The possibility of paying may not arrive immediately; thus, in the meantime, households trade the defaulted debt in secondary markets. The equilibrium price in this market is used to price the debt during the default period. Households predict the possibility of default, and this generates endogenous debt limits, which hinder the government’s ability to smooth shocks using debt. I characterize the optimal default decision, optimal government policy, and the set of implementable allocations. Quantitative exercises match various qualitative features observed in the data for emerging economies.

This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.

In nonparametric instrumental variables estimation, the function being estimated is the solution to an integral equation. A solution may not exist if, for example, the instrument is not valid. This paper discusses the problem of testing the null hypothesis that a solution exists against the alternative that there is no solution. We give necessary and sufficient conditions for existence of a solution and show that uniformly consistent testing of an unrestricted null hypothesis is not possible. Uniformly consistent testing is possible, however, if the nullhypothesis is restricted by assuming that any solution to the integral equation is smooth. Many functions of interest in applied econometrics, including demand functions and Engel curves, are expected to be smooth. The paper presents a statistic for testing the null hypothesis that a smooth solution exists. The test is consistent uniformly over a large class of probability distributions of the observable random variables for which the integral equation has no smooth solution. The finite-sample performance of the test is illustrated through Monte Carlo experiments.

In parametric models a sufficient condition for local identification is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We show that there are corresponding sufficient conditions for nonparametric models. A nonparametric rank condition and differentiability of the moment conditions with respect to a certain norm imply local identification. It turns out these conditions are slightly stronger than needed and are hard to check, so we provide weaker and more primitive conditions. We extend the results to semiparametric models. We illustrate the sufficient conditions with endogenous quantile and single index examples. We also consider a semiparametric habit-based, consumption capital asset pricing model. There we find the rank condition is implied by an integral equation of the second kind having a one-dimensional null space.

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