nonparametric instrumental variables estimation for efficiency frontier the paper investigates endogeneity issues in nonparametric frontier models it considers a nonseparable model for a cost function c y u where c and y are the cost and the output u is uniform in 0 1 and 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 and obtain an estimate of the quantile cost frontier y this involves the solution of a non linear integral equation if the true frontier y 0 is wanted it is then estimated by estimating the bias correction y 0 y under additional regularity conditions the procedure is illustrated through a simulated sample and with an empirical application to the efficiency of post offices

estimation of nonparametric models with simultaneity we introduce methods for estimating nonparametric nonadditive models with simultaneity the methods are developed by directly connecting the elements of the structural system to be estimated with features of the density of the observable variables such as ratios of derivatives or averages of products of derivatives of this density the estimators are therefore easily computed functionals of a nonparametric estimator of the density of the observable variables we consider in detail a model where to each structural equation there corresponds an exclusive regressor and a model with one equation of interest and one instrument that is included in a second equation for both models we provide new characterizations of observational equivalence on a set in terms of the density of the observable variables and derivatives of the structural functions based on those characterizations we develop two estimation methods in the first method the estimators of the structural derivatives are calculated by a simple matrix inversion and matrix multiplication analogous to a standard least squares estimator but with the elements of the matrices being averages of products of derivatives of nonparametric density estimators in the second method the estimators of the structural derivatives are calculated in two steps in a first step values of the instrument are found at which the density of the observable variables satisfies some properties in the second step the estimators are calculated directly from the values of derivatives of the density of the observable variables evaluated at the found values of the instrument we show that both pointwise estimators are consistent and asymptotically normal

identification and estimation of triangular simultaneous equations models without additivity this paper uses control variables to identify and estimate models with nonseparable multidimensional disturbances triangular simultaneous equations models are considered with instruments and disturbances that are independent and a reduced form that is strictly monotonic in a scalar disturbance here it is shown that the conditional cumulative distribution function of the endogenous variable given the instruments is a control variable also for any control variable identification results are given for quantile average and policy effects bounds are given when a common support assumption is not satisfied estimators of identified objects and bounds are provided and a demand analysis empirical example is given

instrumental variable estimation of nonseparable models there are many environments where knowledge of a structural relationship is required to answer questions of interest also nonseparability of a structural disturbance is a key feature of many models here we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance which disturbance is independent of instruments this model leads to conditional quantile restrictions we give local identification conditions for the structural equations from those quantile restrictions we find that a modified completeness condition is sufficient for local identification we also consider estimation via a nonparametric minimum distance estimator the estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments we show consistency of this estimator

testing a parametric quantile regression model with an endogenous explanatory variable against a nonparametric alternative abstract this paper is concerned with inference about a function g that is identified by a conditional quantile restriction involving instrumental variables the paper presents a test of the hypothesis that g belongs to a finite dimensional parametric family against a nonparametric alternative the test is not subject to the ill posed inverse problem of nonparametric instrumental variable estimation under mild conditions the test is consistent against any alternative model in large samples its power is arbitrarily close to 1 uniformly over a class of alternatives whose distance from the null hypothesis is proportional to n 1 2 where n is the sample size monte carlo simulations illustrate the finite sample performance of the test

efficient estimation in models with independence restrictions abstract unconditional and conditional independence restrictions are used in many econometric models to identify their parameters however there are few results about efficient estimation procedures for finite dimensional parameters under these independence restrictions this paper computes the efficiency bound for finite dimensional parameters under independence restrictions and proposes an estimator that is consistent asymptotically normal and which achieves the efficiency bound the estimator is based on a growing number of zero covariance conditions that are asymptotically equivalent to the independence restriction the results are illustrated with examples including an instrumental variables regression model and partially linear regression models a small monte carlo study is performed to investigate the estimator s small sample properties and to quantify the efficiency gains that can be made by using the proposed efficient estimator

finite sample inference for quantile regression models abstract under minimal assumptions finite sample confidence bands for quantile regression models can be constructed these confidence bands are based on the conditional pivotal property of estimating equations that quantile regression methods solve and provide valid finite sample inference for linear and nonlinear quantile models with endogenous or exogenous covariates the confidence regions can be computed using markov chain monte carlo mcmc methods we illustrate the finite sample procedure through two empirical examples estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling we find pronounced differences between asymptotic and finite sample confidence regions in cases where the usual asymptotics are suspect

applied nonparametric instrumental variables estimation 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

estimation of nonparametric conditional moment models with possibly nonsmooth moments this paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables it is a problem of nonparametric nonlinear instrumental variables iv estimation and a difficult nonlinear ill posed inverse problem with an unknown operator we first propose a penalized sieve minimum distance smd estimator of the unknown functions that are identified via the conditional moment models we then establish its consistency and convergence rate in strong metric allowing for possibly non compact function parameter spaces possibly non compact finite or infinite dimensional sieves with flexible lower semicompact or convex penalty or finite dimensional linear sieves without penalty under relatively low level sufficient conditions and for both mildly and severely ill posed problems we show that the convergence rates for the nonlinear ill posed inverse problems coincide with the known minimax optimal rates for the nonparametric mean iv regression we illustrate the theory by two important applications root n asymptotic normality of the plug in penalized smd estimator of a weighted average derivative of a nonparametric nonlinear iv regression and the convergence rate of a nonparametric additive quantile iv regression we also present a simulation study and an empirical estimation of a system of nonparametric quantile iv engel curves

nonparametric instrumental variables estimation of a quantile regression model we consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression error conditional on an instrumental variable to be zero the resulting estimating equation is a nonlinear integral equation of the first kind which generates an ill posed inverse problem the integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically we show that the estimator is mean square consistent derive its rate of convergence in probability and give conditions under which this rate is optimal in a minimax sense the results of monte carlo experiments show that the estimator behaves well in finite samples

efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals this paper considers semiparametric efficient estimation ofconditional moment models with possibly nonsmooth residuals in unknown parametric components and unknown functions h of endogenous variables we show that 1 the penalized sieve minimum distance psmd estima tor can simultaneously achieve root n asymptotic normality ofand nonparametric optimal convergence rate of allowing for noncompact function parameter spaces 2 a simple weighted bootstrap procedure consistently estimates the limiting distribution of the psmd 3 the semi parametric efficiency bound formula of ai and chen 2003 remains valid for conditional models with nonsmooth residuals and the optimally weighted psmd estimator achieves the bound 4 the centered profiled optimally weighted psmd criterion is asymptotically chi square distributed we illustrate our theories using a partially linear quantile instrumental variables iv regression a monte carlo study and an empirical estimation of the shape invariant quantile iv engel curves

local identification of nonparametric and semiparametric models in parametric nonlinear structural models a classical sufficient condition for local identification like fisher 1966 and rothenberg 1971 is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix we derive an analogous result for the nonparametric nonlinear structural models establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification importantly we show that additional conditions are often needed in nonlinear nonparametric models to avoid nonlinearities overwhelming linear effects we give restrictions on a neighborhood of the true value that are sufficient for local identification we apply these results to obtain new primitive identification conditions in several important models including nonseparable quantile instrumental variable iv models and semiparametric consumption based asset pricing models

instrumental variable treatment of nonclassical measurement error models while the literature on nonclassical measurement error traditionally relies on the availability of an auxiliary data set containing correctly measured observations we establish that the availability of instruments enables the identification of a large class of nonclassical nonlinear errors in variables models with continuously distributed variables our main identifying assumption is that conditional on the value of the true regressors some measure of location of the distribution of the measurement error e g its mean mode or median is equal to zero the proposed approach relies on the eigenvalue eigenfunction decomposition of an integral operator associated with specific joint probability densities the main identifying assumption is used to index the eigenfunctions so that the decomposition is unique we propose a convenient sieve based estimator derive its asymptotic properties and investigate its finite sample behavior through monte carlo simulations

direct instrumental nonparametric estimation of inverse regression functions abstract this paper treats the estimation of the inverse g 1 of a monotonic function g satisfying e y g x w 0 where x w is continuously distributed using instrumental restrictions many parameters of interest in econometrics can be expressed as inverses of functions satisfying such a conditional moment as far as i know consistent estimators are available only if g x e y x which precludes endogenous models this and other technical concerns motivates a methodology for estimating g 1 in one step from the data when w contains components excluded from x the presented estimator achieves this objective by identifying g 1 as the solution of a nonlinear integral equation whose sample analog can be estimated nonparametrically the existence of a well behaved joint density function for x w produces an ill posed problem namely the solution is discontinuous in the data to solve this the estimator is regularized with the tikhonov technique if g 1 is smooth in a certain sense then the mse rate of convergence in a global metric is equal to n r 2 r 1 where is a continuous function with 0 0 and r 1 denotes the number of partial derivatives for the joint density of x w with respect to its second argument the analytical expression of depends on a link function characterizing the smoothness of g 1 this result holds for a broad class of link functions belonging to a suitable space

tikhonov regularization for nonparametric instrumental variable estimators abstract we study a tikhonov regularized tir estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors the nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives after showing its consistency in the sobolev norm and uniform consistency under an embedding condition we derive the expression of the asymptotic mean integrated square error and the rate of convergence the optimal value of the regularization parameter is characterized in two examples we illustrate our theoretical findings and the small sample properties with simulation results finally we provide an empirical application to estimation of an engel curve and discuss a data driven selection procedure for the regularization parameter

iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression this paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression we propose a regularized newton type iteration and establish convergence and convergence rate results a particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption we demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model this is illustrated in computed examples with simulated data

estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals this paper studies nonparametric estimation of conditional moment restrictions 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 others 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 noncompact 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

injectivity of a class of integral operators with compactly supported kernels abstract injectivity of integral operators is related to completeness conditions of their corresponding kernel functions completeness provides a useful way of obtaining nonparametric identification in various models including nonparametric regression models with instrumental variables nonclassical measurement error models and auction models etc however the condition is quite abstract for empirical work and lacks a proper economic interpretation we rely on known results regarding the volterra equation to provide sufficient conditions for completeness conditions for densities with compact support our conditions include various smoothness assumptions and monotonously moving support assumptions on the kernel function of the operator we apply our results to establish nonparametric identification in nonparametric iv regression models nonclassical measurement error models and auction models with an accessible interpretation and without specific functional form restrictions

examples of l2 complete and boundedly complete distributions abstract completeness and bounded completeness conditions are used increasingly in econometrics to obtain nonparametric identification in a variety of models from nonparametric instrumental variable regression to non classical measurement error models however distributions that are known to be complete or boundedly complete are somewhat scarce in this paper we consider an l 2 completeness condition that lies between completeness and bounded completeness we construct broad nonparametric classes of distributions that are l 2 complete and boundedly complete the distributions can have any marginal distributions and a wide range of strengths of dependence examples of l 2 incomplete distributions also are provided

fourth order accurate scheme for the space fractional diffusion equations because of the nonlocal properties of fractional operators higher order schemes play a more important role in discretizing fractional derivatives than classical ones the striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy nowadays there are already two types of second order discretization schemes for space fractional derivatives the first type is given and discussed in sousa and li arxiv 1109 2345v1 2011 chen and deng appl math model 38 2014 pp 3244 3259 chen deng and wu appl numer math 70 2013 pp 22 41 and the second type is a class of schemes presented in tian zhou and deng math comp to appear also available online from arxiv 1201 5949 2012 the core object of this paper is to derive a class of fourth order approximations called the weighted and shifted lubich difference operators for space fractional derivatives then we use the derived schemes t

uniform confidence bands for functions estimated nonparametrically with instrumental variables abstract this paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables we show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed the asymptotic normality result holds in both mildly and severely ill posed cases we present methods to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values monte carlo experiments illustrate the finite sample performance of the uniform confidence band

posterior consistency of nonparametric conditional moment restricted models this paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g0 we estimate it in a quasi bayesian way based on the limited information likelihood and investigate the impact of three types of priors on the posterior consistency i truncated prior priors supported on a bounded set ii thin tail prior a prior that has very thin tail outside a growing bounded set and iii normal prior with nonshrinking variance in addition g0 is allowed to be only partially identified in the frequentist sense and the parameter space does not need to be compact the posterior is regularized using a slowly growing sieve dimension and it is shown that the posterior converges to any small neighborhood of the identified region we then apply our results to the nonparametric instrumental regression model finally the posterior consistency using a random sieve dimension parameter is studied

sieve wald and qlr inferences on semi nonparametric conditional moment models this paper considers inference on functionals of semi nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals which include all of the nonlinear nonparametric instrumental variables iv as special cases these models are often ill posed and hence it is difficult to verify whether a possibly nonlinear functional is root n estimable or not we provide computationally simple unified inference procedures that are asymptotically valid regardless of whether a functional is root n estimable or not we establish the following new useful results 1 the asymptotic normality of a plug in penalized sieve minimum distance psmd estimator of a possibly nonlinear functional 2 the consistency of simple sieve variance estimators for the plug in psmd estimator and hence the asymptotic chi square distribution of the sieve wald statistic 3 the asymptotic chi square distribution of an optimally weighted sieve quasi likelihood ratio qlr test under the null hypothesis 4 the asymptotic tight distribution of a non optimally weighted sieve qlr statistic under the null 5 the consistency of generalized residual bootstrap sieve wald and qlr tests 6 local power properties of sieve wald and qlr tests and of their bootstrap versions 7 asymptotic properties of sieve wald and sqlr for functionals of increasing dimension simulation studies and an empirical illustration of a nonparametric quantile iv regression are presented

testing for separability in structural equations abstract separability is an important feature of structural equations as it implies the absence of unobservable heterogeneity of effects and has significant implications for identification and efficiency of estimation this paper provides a nonparametric test for separability in structural equations the test is based on a conditional independence test recently developed by huang et al 2013 building on consistent procedures of bierens 1982 1990 and stinchcombe and white 1998 the test is easy to implement and achieves n local power we apply our test to study interest rate elasticities of loan demand in microfinance and the impact of education on wages

identification in nonseparable models weak nonparametric restrictions are developed sufficient to identify the values of derivatives of structural functions in which latent random variables are nonseparable these derivatives can exhibit stochastic variation in a microeconometric context this allows the impact of a policy intervention as measured by the value of a structural derivative to vary across people who are identical as measured by covariates when the restrictions are satisfied quantiles of the distribution of a policy impact across people can be identified the identification restrictions are local in the sense that they are specific to the values of the covariates and the specific quantiles of latent variables at which identification is sought the conditions do not include the commonly required independence of latent variables and covariates they include local versions of the classical rank and order conditions and local quantile insensitivity conditions values of structural derivatives are identified by functionals of quantile regression functions and can be estimated using the same functionals applied to estimated quantile regression functions

nonparametric instrumental regression the focus of this paper is the nonparametric estimation of an instrumental regression function f defined by conditional moment restrictions that stem from a structural econometric model e y f z w 0 and involve endogenous variables y and z and instruments w the function f is the solution of an ill posed inverse problem and we propose an estimation procedure based on tikhonov regularization the paper analyzes identification and overidentification of this model and presents asymptotic properties of the estimated nonparametric instrumental regression function

nonparametric estimation of nonadditive random functions we present estimators for nonparametric functions that are nonadditive in unobservable random terms the distributions of the unobservable random terms are assumed to be unknown we show that when a nonadditive nonparametric function is strictly monotone in an unobservable random term and it satisfies some other properties that may be implied by economic theory such as homogeneity of degree one or separability the function and the distribution of the unobservable random term are identified we also present convenient normalizations to use when the properties of the function other than strict monotonicity in the unobservable random term are unknown the estimators for the nonparametric function and for the distribution of the unobservable random term are shown to be consistent and asymptotically normal we extend the results to functions that depend on a multivariate random term the results of a limited simulation study are presented

a discontinuity test for identification in triangular nonseparable models this paper presents a test for the validity of control variable approaches to identification in triangular nonseparable models assumptions commonly imposed to justify such methods include full independence of instruments and disturbances and existence of a reduced form that is strictly monotonic in a scalar disturbance we show that if the data has a particular structure namely that the distribution of the endogenous variable has a mass point at the lower or upper boundary of its support validity of the control variable approach implies a continuity condition on an identified function which can be tested empirically

penalized sieve gel for weighted average derivatives of nonparametric quantile iv regressions abstract this paper considers estimation and inference for a weighted average derivative wad of a nonparametric quantile instrumental variables regression npqiv npqiv is a non separable and nonlinear ill posed inverse problem which might be why there is no published work on the asymptotic properties of any estimator of its wad we first characterize the semiparametric efficiency bound for a wad of a npqiv which unfortunately depends on an unknown conditional derivative operator and hence an unknown degree of ill posedness making it difficult to know if the information bound is singular or not in either case we propose a penalized sieve generalized empirical likelihood gel estimation and inference procedure which is based on the unconditional wad moment restriction and an increasing number of unconditional moments that are implied by the conditional npqiv restriction where the unknown quantile function is approximated by a penalized sieve under some regularity conditions we show that the self normalized penalized sieve gel estimator of the wad of a npqiv is asymptotically standard normal we also show that the quasi likelihood ratio statistic based on the penalized sieve gel criterion is asymptotically chi square distributed regardless of whether or not the information bound is singular

an iv model of quantile treatment effects the ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications however in observational studies the variables of interest e g education prices are often endogenous making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes in order to address this problem we develop a model of quantile treatment effects qte in the presence of endogeneity and obtain conditions for identification of the qte without functional form assumptions the principal feature of the model is the imposition of conditions that restrict the evolution of ranks across treatment states this feature allows us to overcome the endogeneity problem and recover the true qte through the use of instrumental variables the proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors where qte can be interpreted as the structural quantile effects sqe

convolution backprojection formulas for 3 d vector tomography with application to mri vector tomography is the reconstruction of vector fields from measurements of their projections in previous work it has been shown that the reconstruction of a general three dimensional 3 d vector field is possible from the so called inner product measurements it has also been shown how the reconstruction of either the irrotational or solenoidal component of a vector field can be accomplished with fewer measurements than that required for the full field the present paper makes three contributions first in analogy to the two dimensional 2 d approach of norton 1988 several 3 d projection theorems are developed these lead directly to new vector field reconstruction formulas that are convolution backprojection formulas it is shown how the local reconstruction property of these 3 d reconstruction formulas permits reconstruction of point flow or of regional flow from a limited data set second simulations demonstrating 3 d reconstructions both local and nonlocal are presented using the formulas derived herein and those derived in previous work these results demonstrate the reconstruction of the irrotational and solenoidal components their potential functions and the field itself from simulated inner product measurement data finally it is shown how 3 d inner product measurements can be acquired using a magnetic resonance scanner

overidentification in regular models in the unconditional moment restriction model of hansen specification tests and more efficient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest we show that a similar relationship between potential refutability of a model and existence of more efficient estimators is present in much broader settings specifically a condition we name local overidentification is shown to be equivalent to both the existence of specification tests with nontrivial local power and the existence of more efficient estimators of some smooth parameters in general semi nonparametric models under our notion of local overidentification various locally nontrivial specification tests such as hausman tests incremental sargan tests or optimally weighted quasi likelihood ratio tests naturally extend to general semi nonparametric settings we further obtain simple characterizations of local overidentification for general models of nonparametric conditional moment restrictions with possibly different conditioning sets the results are applied to determining when semi nonparametric models with endogeneity are locally testable and when nonparametric plug in and semiparametric two step gmm estimators are semiparametrically efficient examples of empirically relevant semi nonparametric structural models are presented

hierarchical spline models for conditional quantiles and the demand for electricity abstract methods for estimating nonparametric models for conditional quantiles are suggested based on the regression quantile methods of koenker and bassett spline parameterizations of the conditional quantile functions are used the methods are illustrated by estimating hierarchical models for household electricity demand using data from the chicago metropolitan area the empirical results show that lower quantiles of demand base load vary only slightly across residential households this variability is difficult to explain using household characteristics however upper quantiles of the demand distribution vary considerably and are systematically related to household characteristics and appliance ownership the implications of analyzing mean demand behavior rather than various quantiles of the distribution of demand are also discussed