The notion of completeness between two random elements has been considered recently to provide identification in nonparametric instrumental problems. This condition is quite abstract, however, and characterizations have been obtained only in special cases. This paper considers a nonparametric model between the two variables with an additive separability and a large support condition. In this framework, different versions of completeness are obtained, depending on which regularity conditions are imposed. This result allows one to establish identification in an instrumental nonparametric regression with limited endogenous regressor, a case where the control variate approach breaks down.

The nonparametric estimation of a regression function from conditional moment restrictions involving instrumental variables is considered. The rate of convergence of penalized estimators is studied in the case where the regression function is not identified from the conditional moment restriction. We also study the gain of modifying the penalty in the estimation, considering derivatives in the penalty. We analyze the effect of this modification on the identification of the regression function and the rate of convergence of its estimator.

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 NPIV models under two basic regularity conditions: the approximation number and the link condition. 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.

This paper studies the estimation of a nonparametric function ϕ from the inverse problem r = Tϕ given estimates of the function r and of the linear transform T. We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The two assumptions quantify the prior regularity of ϕ and the prior link existing between T and the Hilbert scale. The approach provides a unified framework that allows us to compare various sets of structural assumptions found in the econometric literature. Moreover, general upper bounds are also derived for the risk of the estimator of the structural function ϕ as well as that of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Two important applications are also studied. The first is the blind nonparametric deconvolution on the real line, and the second is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.

We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the density is identified even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.

This paper describes an estimator of the additive components of a nonparametric additive model with an unknown link function. When the additive components and link function are twice differentiable with sufficiently smooth second derivatives, the estimator is asymptotically normally distributed with a rate of convergence in probability of n−2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, the estimator has no curse of dimensionality. Moreover, the asymptotic distribution of the estimator of each additive component is the same as it would be if the link function and the other components were known with certainty. Thus, asymptotically there is no penalty for not knowing the link function or the other components.

In this paper we are concerned with analyzing the behavior of a semiparametric estimator that corrects for endogeneity in a nonparametric regression by assuming mean independence of residuals from instruments only. Because it is common in many applications, we focus on the case where endogenous regressors and additional instruments are jointly normal, conditional on exogenous regressors. This leads to a severely ill-posed inverse problem. In this setup, we show first how to test for conditional normality. More importantly, we then establish how to exploit this knowledge when constructing an estimator, and we derive the large sample behavior of such an estimator. In addition, in a Monte Carlo experiment we analyze its finite sample behavior. Our application comes from consumer demand. We obtain new and interesting findings that highlight both the advantages and the difficulties of an approach that leads to ill-posed inverse problems. Finally, we discuss the somewhat problematic relationship between endogenous nonparametric regression models and the recently emphasized issue of unobserved heterogeneity in structural models.

We propose a semiparametric IGARCH model that allows for persistence in variance but also allows for more flexible functional form. We assume that the difference of the squared process is weakly stationary. We propose an estimation strategy based on the nonparametric instrumental variable method. We establish the rate of convergence of our estimator.

This note discusses identification in nonparametric, continuous triangular systems. It provides conditions that are both necessary and sufficient for the existence of control functions satisfying conditional independence and support requirements. Confirming a commonly noticed pattern, these conditions restrict the admissible dimensionality of unobserved heterogeneity in the first-stage structural relation, or more generally the dimensionality of the family of conditional distributions of second-stage heterogeneity given explanatory variables and instruments. These conditions imply that no such control function exists without assumptions that seem hard to justify in most applications. In particular, none exists in the context of a generic random coefficient model.