. This model is used as an example to illustrate the practice of the estimation by solving linear functional equations. This paper is specially focused on the data-driven selection of the regularization parameter and of the bandwidths. Simulations experiments illustrate the relevance of this approach. Copyright (C) 2010 The Author(s). The Econometrics Journal (C) 2010 Royal Economic Society

We present a review on the implementation of regularization methods for the estimation of additive nonparametric regression models with instrumental variables. We consider various versions of Tikhonov, Landweber-Fridman and Sieve (Petrov-Galerkin) regularization. We review data-driven techniques for the sequential choice of the smoothing and the regularization parameters. Through Monte Carlo simulations, we discuss the finite sample properties of each regularization method for different smoothness properties of the regression function. Finally, we present an application to the estimation of the Engel curve for food in a sample of rural households in Pakistan, where a partially linear specification is described that allows one to embed other exogenous covariates.

leopold.simar@uclouvain.be. October 17, 2014Abstract The paper investigates endogeneity issues in nonparametric frontier models. It considers a non separable model for a cost function C = ϕ(Y, U ) where C and Y are the cost and the output, U is uniform in [0, 1] and ϕ is increasing with respect to U . The cost frontier corresponds to U = 0 and U can be interpreted as a normalized level of inefficiency. The endogeneity issue arises when Y is dependent of U . For identification and estimation, we use a nonparametric instrumental variables estimator of the model for fixed value U = α, and obtain an estimate of the α-quantile cost frontier ϕ(Y, α). This involves the solution of a non linear integral equation. If the true frontier ϕ(Y, 0) is wanted, it is then estimated by estimating the bias correction ϕ(Y, 0) − ϕ(Y, α) under additional regularity conditions. The procedure is illustrated through a simulated sample and with an empirical application to the efficiency of post offices.

Consider a nonparametric nonseparable regression model Y = ϕ(Z, U), where ϕ(Z, U) is strictly increasing in U and U ∼ U[0, 1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z and W, all one-dimensional. We construct test statistics for the hypothesis that Z is exogenous, i.e. that U is independent of Z. The test statistics are based on the observation that Z is exogenous if and only if V = F Y|Z (Y|Z) is independent of W, and hence they do not require the estimation of the function ϕ. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to be consistent and to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. As a by-product of our results we obtain the asymptotic properties of a kernel estimator of the distribution of V, which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n −1/2 -rate.