ANALYZING THE RANDOM COEFFICIENT MODEL NONPARAMETRICALLY

Linearity in a causal relationship between a dependent variable and a set of regressors is a common assumption throughout economics. In this paper we consider the case when the coefficients in this relationship are random and distributed independently from the regressors. Our aim is to identify and estimate the distribution of the coefficients nonparametrically. We propose a kernel-based estimator for the joint probability density of the coefficients. Although this estimator shares certain features with standard nonparametric kernel density estimators, it also differs in some important characteristics that are due to the very different setup we are considering. Most importantly, the kernel is nonstandard and derives from the theory of Radon transforms. Consequently, we call our estimator the Radon transform estimator (RTE). We establish the large sample behavior of this estimator—in particular, rate optimality and asymptotic distribution. In addition, we extend the basic model to cover extensions, including endogenous regressors and additional controls. Finally, we analyze the properties of the estimator in finite samples by a simulation study, as well as an application to consumer demand using British household data.

[1]  E. Mammen,et al.  EMPIRICAL RISK MINIMIZATION IN INVERSE PROBLEMS , 2010, 1001.2089.

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

[3]  Pradeep K. Chintagunta,et al.  Nonparametric Discrete Choice Models With Unobserved Heterogeneity , 2010 .

[4]  Winfried Stute,et al.  The Oscillation Behavior of Empirical Processes: The Multivariate Case , 1984 .

[5]  R. Z. Khasʹminskiĭ,et al.  Statistical estimation : asymptotic theory , 1981 .

[6]  A consistent semiparametric estimation of the consumer surplus distribution , 2000 .

[7]  E. Giné,et al.  Rates of strong uniform consistency for multivariate kernel density estimators , 2002 .

[8]  P. Hall,et al.  Kernel density estimation with spherical data , 1987 .

[9]  H. Ichimura,et al.  Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution , 1998 .

[10]  Enno Mammen,et al.  Identification of marginal effects in nonseparable models without monotonicity , 2007 .

[11]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[12]  Stefan Hoderlein.  How many consumers are rational , 2009 .

[13]  Jeremy T. Fox,et al.  Identifying Heterogeneity in Economic Choice and Selection Models Using Mixtures , 2009 .

[14]  Gerda Claeskens,et al.  Nonparametric Estimation , 2011, International Encyclopedia of Statistical Science.

[15]  Bernard W. Silverman,et al.  Speed of Estimation in Positron Emission Tomography , 1987 .

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

[17]  Enno Mammen,et al.  The Existence and Asymptotic Properties of a Backfitting Projection Algorithm Under Weak Conditions , 1999 .

[18]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[19]  E. Gautier,et al.  Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding , 2011, 1106.3503.

[20]  Bernard W. Silverman,et al.  Speed of Estimation in Positron Emission Tomography and Related Inverse Problems , 1990 .

[21]  Stéphane Bonhomme,et al.  Identifying distributional characteristics in random coefficients panel data models , 2009 .

[22]  Cheng Hsiao,et al.  Random Coe¢ cient Panel Data Models , 2007 .

[23]  C. Hildreth,et al.  Some Estimators for a Linear Model With Random Coefficients , 1968 .

[24]  E. Mammen,et al.  Time Series Modelling With Semiparametric Factor Dynamics , 2007 .

[25]  M. Pesaran,et al.  Random Coefficient Panel Data Models , 2004, SSRN Electronic Journal.

[26]  E. Mammen,et al.  Identification and Estimation of Local Average Derivatives in Non-Separable Models Without Monotonicity , 2009 .

[27]  Eric Gautier,et al.  Nonparametric Estimation in Random Coefficients Binary Choice Models , 2009, 0907.2451.

[28]  Jeremy T. Fox,et al.  Identifying Heterogeneity in Economic Choice Models , 2009 .

[29]  A. Lewbel,et al.  Demand Systems with and without Errors , 2001 .

[30]  Yehuda Vardi,et al.  Positron Emission Tomography and Random Coefficients Regression , 2000 .

[31]  E. Gautier,et al.  ESTIMATING THE DISTRIBUTION OF TREATMENT EFFECTS , 2011 .

[32]  Jeremy T. Fox,et al.  Identifying Demand with Multidimensional Unobservables: A Random Functions Approach , 2011 .

[33]  Ndrey,et al.  ON NONPARAMETRIC ESTIMATION OF INTERCEPT AND SLOPE DISTRIBUTIONS IN RANDOM COEFFICIENT REGRESSION , 1997 .

[34]  Gregory Connor,et al.  Efficient Estimation of a Semiparametric Characteristic-Based Factor Model of Security Returns , 2007 .

[35]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[36]  B. Silverman,et al.  Wavelet decomposition approaches to statistical inverse problems , 1998 .

[37]  A. Lewbel,et al.  Price Dimension Reduction in Demand Systems With Many Goods , 2009 .

[38]  Stefan Hoderlein,et al.  Local Partitioned Regression , 2006 .

[39]  R. Prado.  Time series modelling, . . . , 2005 .

[40]  Jussi Klemelä,et al.  Estimation of Densities and Derivatives of Densities with Directional Data , 2000 .

[41]  Rudolf Beran,et al.  Estimating Coefficient Distributions in Random Coefficient Regressions , 1992 .

[42]  James J. Heckman,et al.  Instrumental Variables Methods for the Correlated Random Coefficient Model: Estimating the Average Rate of Return to Schooling When the Return is Correlated with Schooling , 1998 .

[43]  Jeffrey M. Woodbridge.  Econometric Analysis of Cross Section and Panel Data , 2002 .

[44]  Dag Tjøstheim,et al.  NONPARAMETRIC ADDITIVE MODELS FOR PANELS OF TIME SERIES , 2009, Econometric Theory.