Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves

This paper studies a shape-invariant Engel curve system with endogenous total expenditure, in which the shape-invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identification and estimation of both the nonparametric shapes of the Engel curves and the parametric specification of the demographic scaling parameters. The identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions, allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support. Root-n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of "low-level" sufficient conditions. Our empirical application using the U.K. Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of Engel curves.

[1]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[2]  E. Lehmann,et al.  Testing Statistical Hypothesis. , 1960 .

[3]  C. Leser.  Forms of Engel functions , 1963 .

[4]  W. Hoeffding.  Some Incomplete and Boundedly Complete Families of Distributions. , 1977 .

[5]  M. Birman,et al.  ESTIMATES OF SINGULAR NUMBERS OF INTEGRAL OPERATORS , 1977 .

[6]  J. Muellbauer,et al.  An Almost Ideal Demand System , 1980 .

[7]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[8]  Dale W. Jorgenson,et al.  The Transcendental Logarithmic Model of Aggregate Consumer Behavior , 1982 .

[9]  H. Triebel.  Theory Of Function Spaces , 1983 .

[10]  Embedding maps between Hölder spaces over metric compacta and eigenvalues of integral operators , 1985 .

[11]  P. Robinson.  ROOT-N-CONSISTENT SEMIPARAMETRIC REGRESSION , 1988 .

[12]  Richard Blundell,et al.  Consumer Behaviour: Theory and Empirical Evidence--A Survey , 1988 .

[13]  James Stephen Marron,et al.  Semiparametric Comparison of Regression Curves , 1990 .

[14]  Arthur Lewbel,et al.  The Rank of Demand Systems: Theory and Nonparametric Estimation , 1991 .

[15]  W. Härdle,et al.  Cross Section Engel Curves over Time , 1991, Recherches économiques de Louvain.

[16]  Herman J. Bierens,et al.  Specification of household engel curves by nonparametric regression , 1991 .

[17]  Hidehiko Ichimura,et al.  Identification and estimation of polynomial errors-in-variables models , 1991 .

[18]  P. Hall.  The Bootstrap and Edgeworth Expansion , 1992 .

[19]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[20]  X. M. Yu,et al.  Monotone and probabilistic wavelet approximation , 1992 .

[21]  L. Mattner,et al.  Some incomplete but boundedly complete location families , 1993 .

[22]  Y. Meyer.  Wavelets and Operators , 1993 .

[23]  A. Tsybakov,et al.  Minimax theory of image reconstruction , 1993 .

[24]  C. Blackorby,et al.  The Measurement of Household Welfare: Measuring the cost of children: a theoretical framework , 1994 .

[25]  A. Timan.  Theory of Approximation of Functions of a Real Variable , 1994 .

[26]  Rosa L. Matzkin.  Restrictions of economic theory in nonparametric methods , 1994 .

[27]  P. Robinson,et al.  Pooling nonparametric estimates of regression functions with similar shape , 1995 .

[28]  W. Newey,et al.  Nonparametric Estimation of Exact Consumers Surplus and Deadweight Loss by , 2009 .

[29]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[30]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[31]  H. Triebel,et al.  Function Spaces, Entropy Numbers, Differential Operators: Function Spaces , 1996 .

[32]  A. Deaton.  The Analysis of Household Surveys : A Microeconometric Approach to Development Policy , 1997 .

[33]  W. Newey,et al.  Convergence rates and asymptotic normality for series estimators , 1997 .

[34]  J. Banks,et al.  Quadratic Engel Curves and Consumer Demand , 1997, Review of Economics and Statistics.

[35]  Xiaotong Shen,et al.  Sieve extremum estimates for weakly dependent data , 1998 .

[36]  Richard Blundell,et al.  Kernel Regression in Empirical Microeconomics , 1998 .

[37]  R. Blundell,et al.  Semiparametric estimation and consumer demand , 1998 .

[38]  Jianhua Z. Huang.  Projection estimation in multiple regression with application to functional ANOVA models , 1998 .

[39]  K. Pendakur,et al.  Semiparametric Estimates and Test of Base-Independent Equiv-alence Scales , 1999 .

[40]  R. Eubank.  Nonparametric Regression and Spline Smoothing , 1999 .

[41]  W. Newey,et al.  Nonparametric estimation of triangular simultaneous equations models , 1999 .

[42]  J. Robin,et al.  TESTS OF RANK , 2000, Econometric Theory.

[43]  Saad T. Bakir,et al.  Nonparametric Regression and Spline Smoothing , 2000, Technometrics.

[44]  Timothy G. Conley,et al.  A new semiparametric spatial model for panel time series , 2001 .

[45]  Vladimir Koltchinskii,et al.  On inverse problems with unknown operators , 2001, IEEE Trans. Inf. Theory.

[46]  Whitney K. Newey.  Flexible Simulated Moment Estimation of Nonlinear Errors-in-Variables Models , 2001, Review of Economics and Statistics.

[47]  Jonathan H. Wright,et al.  A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments , 2002 .

[48]  Jean-Pierre Florens,et al.  Instrumental variables, local instrumental variables and control functions , 2002 .

[49]  A. Tsybakov,et al.  Oracle inequalities for inverse problems , 2002 .

[50]  J. Florens,et al.  Nonparametric Instrumental Regression , 2010 .

[51]  Martin Browning,et al.  Nonparametric Engel curves and revealed preference , 2003 .

[52]  J. Florens.  Advances in Economics and Econometrics: Inverse Problems and Structural Econometrics: The Example of Instrumental Variables , 2003 .

[53]  J. Florens,et al.  Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization , 2003 .

[54]  W. Newey,et al.  Instrumental variable estimation of nonparametric models , 2003 .

[55]  Xiaohong Chen,et al.  Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , 2003 .

[56]  Advances in Economics and Econometrics: Endogeneity in Nonparametric and Semiparametric Regression Models , 2003 .

[57]  P. Hall,et al.  Nonparametric methods for inference in the presence of instrumental variables , 2003, math/0603130.

[58]  Albert Cohen,et al.  Adaptive Wavelet Galerkin Methods for Linear Inverse Problems , 2004, SIAM J. Numer. Anal..