Nonparametric Econometrics : Theory and Practice

A comprehensive, up-to-date textbook on nonparametric methods for students and researchersUntil now, students and researchers in nonparametric and semiparametric statistics and econometrics have had to turn to the latest journal articles to keep pace with these emerging methods of economic analysis. Nonparametric Econometrics fills a major gap by gathering together the most up-to-date theory and techniques and presenting them in a remarkably straightforward and accessible format. The empirical tests, data, and exercises included in this textbook help make it the ideal introduction for graduate students and an indispensable resource for researchers.Nonparametric and semiparametric methods have attracted a great deal of attention from statisticians in recent decades. While the majority of existing books on the subject operate from the presumption that the underlying data is strictly continuous in nature, more often than not social scientists deal with categorical data—nominal and ordinal—in applied settings. The conventional nonparametric approach to dealing with the presence of discrete variables is acknowledged to be unsatisfactory.This book is tailored to the needs of applied econometricians and social scientists. Qi Li and Jeffrey Racine emphasize nonparametric techniques suited to the rich array of data types—continuous, nominal, and ordinal—within one coherent framework. They also emphasize the properties of nonparametric estimators in the presence of potentially irrelevant variables.Nonparametric Econometrics covers all the material necessary to understand and apply nonparametric methods for real-world problems. Contents Preface I Nonparametric Kernel Methods 1 Density Estimation 1.1 Univariate Density Estimation 1.2 Univariate Bandwidth Selection: Rule-of-Thumb and Plug-In Methods 1.3 Univariate Bandwidth Selection: Cross-Validation Methods 1.3.1 Least Squares Cross-Validation 1.3.2 Likelihood Cross-Validation 1.3.3 An Illustration of Data-Driven Bandwidth Selection 1.4 Univariate CDF Estimation 1.5 Univariate CDF Bandwidth Selection: Cross-Validation Methods 1.6 Multivariate Density Estimation 1.7 Multivariate Bandwidth Selection: Rule-of-Thumb and Plug-In Methods 1.8 Multivariate Bandwidth Selection: Cross-Validation Methods 1.8.1 Least Squares Cross-Validation 1.8.2 Likelihood Cross-Validation 1.9 Asymptotic Normality of Density Estimators 1.10 Uniform Rates of Convergence 1.11 Higher Order Kernel Functions 1.12 Proof of Theorem 1.4 (Uniform Almost Sure Convergence) 1.13 Applications 1.13.1 Female Wage Inequality 1.13.2 Unemployment Rates and City Size 1.13.3 Adolescent Growth 1.13.4 Old Faithful Geyser Data 1.13.5 Evolution of Real Income Distribution in Italy, 1951-1998 1.14 Exercises 2 Regression 2.1 Local Constant Kernel Estimation 2.1.1 Intuition Underlying the Local Constant Kernel Estimator 2.2 Local Constant Bandwidth Selection 2.2.1 Rule-of-Thumb and Plug-In Methods 2.2.2 Least Squares Cross-Validation 2.2.3 AICc 2.2.4 The Presence of Irrelevant Regressors 2.2.5 Some Further Results on Cross-Validation 2.3 Uniform Rates of Convergence 2.4 Local Linear Kernel Estimation 2.4.1 Local Linear Bandwidth Selection: Least Squares Cross-Validation 2.5 Local Polynomial Regression (General pth Order) 2.5.1 The Univariate Case 2.5.2 The Multivariate Case 2.5.3 Asymptotic Normality of Local Polynomial Estimators 2.6 Applications 2.6.1 Prestige Data 2.6.2 Adolescent Growth 2.6.3 Inflation Forecasting and Money Growth 2.7 Proofs 2.7.1 Derivation of (2.24) 2.7.2 Proof of Theorem 2.7 2.7.3 Definitions of Al,p+1 and Vl Used in Theorem 2.10 2.8 Exercises 3 Frequency Estimation with Mixed Data 3.1 Probability Function Estimation with Discrete Data 3.2 Regression with Discrete Regressors 3.3 Estimation with Mixed Data: The Frequency Approach 3.3.1 Density Estimation with Mixed Data 3.3.2 Regression with Mixed Data 3.4 Some Cautionary Remarks on Frequency Methods 3.5 Proofs 3.5.1 Proof of Theorem 3.1 3.6 Exercises 4 Kernel Estimation with Mixed Data 4.1 Smooth Estimation of Joint Distributions with Discrete Data 4.2 Smooth Regression with Discrete Data 4.3 Kernel Regression with Discrete Regressors: The Irrelevant Regressor Case 4.4 Regression with Mixed Data: Relevant Regressors 4.4.1 Smooth Estimation with Mixed Data 4.4.2 The Cross-Validation Method 4.5 Regression with Mixed Data: Irrelevant Regressors 4.5.1 Ordered Discrete Variables 4.6 Applications 4.6.1 Food-Away-from-Home Expenditure 4.6.2 Modeling Strike Volume 4.7 Exercises 5 Conditional Density Estimation 5.1 Conditional Density Estimation: Relevant Variables 5.2 Conditional Density Bandwidth Selection 5.2.1 Least Squares Cross-Validation: Relevant Variables 5.2.2 Maximum Likelihood Cross-Validation: Relevant Variables 5.3 Conditional Density Estimation: Irrelevant Variables 5.4 The Multivariate Dependent Variables Case 5.4.1 The General Categorical Data Case 5.4.2 Proof of Theorem 5.5 5.5 Applications 5.5.1 A Nonparametric Analysis of Corruption 5.5.2 Extramarital Affairs Data 5.5.3 Married Female Labor Force Participation 5.5.4 Labor Productivity 5.5.5 Multivariate Y Conditional Density Example: GDP Growth and Population Growth Conditional on OECD Status 5.6 Exercises 6 Conditional CDF and Quantile Estimation 6.1 Estimating a Conditional CDF with Continuous Covariates without Smoothing the Dependent Variable 6.2 Estimating a Conditional CDF with Continuous Covariates Smoothing the Dependent Variable 6.3 Nonparametric Estimation of Conditional Quantile Functions 6.4 The Check Function Approach 6.5 Conditional CDF and Quantile Estimation with Mixed Discrete and Continuous Covariates 6.6 A Small Monte Carlo Simulation Study 6.7 Nonparametric Estimation of Hazard Functions 6.8 Applications 6.8.1 Boston Housing Data 6.8.2 Adolescent Growth Charts 6.8.3 Conditional Value at Risk 6.8.4 Real Income in Italy, 1951-1998 6.8.5 Multivariate Y Conditional CDF Example: GDP Growth and Population Growth Conditional on OECD Status 6.9 Proofs 6.9.1 Proofs of Theorems 6.1, 6.2, and 6.4 6.9.2 Proofs of Theorems 6.5 and 6.6 (Mixed Covariates Case) 6.10 Exercises II Semiparametric Methods 7 Semiparametric Partially Linear Models 7.1 Partially Linear Models 7.1.1 Identification of ?? 7.2 Robinson???s Estimator 7.2.1 Estimation of the Nonparametric Component 7.3 Andrews???s MINPIN Method 7.4 Semiparametric Efficiency Bounds 7.4.1 The Conditionally Homoskedastic Error Case 7.4.2 The Conditionally Heteroskedastic Error Case 7.5 Proofs 7.5.1 Proof of Theorem 7.2 7.5.2 Verifying Theorem 7.3 for a Partially Linear Model 7.6 Exercises 8 Semiparametric Single Index Models 8.1 Identification Conditions 8.2 Estimation 8.2.1 Ichimura???s Method 8.3 Direct Semiparametric Estimators for (3 8.3.1 Average Derivative Estimators 8.3.2 Estimation of g(???) 8.4 Bandwidth Selection 8.4.1 Bandwidth Selection for Ichimura???s Method 8.4.2 Bandwidth Selection with Direct Estimation Methods 8.5 Klein and Spady???s Estimator 8.6 Lewbel???s Estimator 8.7 Manski???s Maximum Score Estimator 8.8 Horowitz???s Smoothed Maximum Score Estimator 8.9 Han???s Maximum Rank Estimator 8.10 Multinomial Discrete Choice Models 8.11 Ai???s Semiparametric Maximum Likelihood Approach 8.12 A Sketch of the Proof of Theorem 8.1 8.13 Applications 8.13.1 Modeling Response to Direct Marketing Catalog Mailings 8.14 Exercises 9 Additive and Smooth (Varying) Coefficient Semiparametric Models 9.1 An Additive Model 9.1.1 The Marginal Integration Method 9.1.2 A Computationally Efficient Oracle Estimator 9.1.3 The Ordinary Backfitting Method 9.1.4 The Smoothed Backfitting Method 9.1.5 Additive Models with Link Functions 9.2 An Additive Partially Linear Model 9.2.1 A Simple Two-Step Method 9.3 A Semiparametric Varying (Smooth) Coefficient Model 9.3.1 A Local Constant Estimator of the Smooth Coefficient Function 9.3.2 A Local Linear Estimator of the Smooth Coefficient Function 9.3.3 Testing for a Parametric Smooth Coefficient Model 9.3.4 Partially Linear Smooth Coefficient Models 9.3.5 Proof of Theorem 9.3 9.4 Exercises 10 Selectivity Models 10.1 Semiparametric Type-2 Tobit Models 10.2 Estimation of a Semiparametric Type-2 Tobit Model 10.2.1 Gallant and Nychka???s Estimator 10.2.2 Estimation of the Intercept in Selection Models 10.3 Semiparametric Type-3 Tobit Models 10.3.1 Econometric Preliminaries 10.3.2 Alternative Estimation Methods 10.4 Das, Newey and Vella???s Nonparametric Selection Model 10.5 Exercises 11 Censored Models 11.1 Parametric Censored Models 11.2 Semiparametric Censored Regression Models 11.3 Semiparametric Censored Regression Models with Nonparametric Heteroskedasticity 11.4 The Univariate Kaplan-Meier CDF Estimator 11.5 The Multivariate Kaplan-Meier CDF Estimator 11.5.1 Nonparametric Regression Models with Random Censoring 11.6 Nonparametric Censored Regression 11.6.1 Lewbel and Linton???s Approach 11.6.2 Chen, Dahl and Khan???s Approach 11.7 Exercises III Consistent Model Specification Tests 12 Model Specification Tests 12.1 A Simple Consistent Test for Parametric Regression Functional Form 12.1.1 A Consistent Test for Correct Parametric Functional Form 12.1.2 Mixed Data 12.2 Testing for Equality of PDFs 12.3 More Tests Related to Regression Functions 12.3.1 Hardie and Mammen???s Test for a Parametric Regression Model 12.3.2 An Adaptive and Rate Optimal Test 12.3.3 A Test for a Parametric Single Index Model 12.3.4 A Nonparametric Omitted Variables Test 12.3.5 Testing the Significance of Categorical Variables 12.4 Tests Related to PDFs 12.4.1 Testing Independence between Two Random Variables 12.4.2 A Test for a Parametric PDF 12.4.3 A Kernel Test for Conditional Parametric Distributions 12.5 Applications 12.5.1 Growth Convergence Clubs 12.6 Proofs 12.6.1 Proof of Theorem 12.1 12.6.2 Proof of Theorem 12.2 12.6.3 Proof of Theorem 12.5 12.6.4 Proof of Theorem 12.9 12.7 Exercises 13 Nonsmoothing Tests 13.1 Testing for Parametric Regression Functional Form 13.2 Testing for Equality of PDFs 13.3 A Nonparametric Significance Test 13.4 Andrews???s Test for Conditional CDFs 13.5 Hong???s Tests for Serial Dependence 13.6 More on Nonsmoothing Tests 13.7 Proofs 13.7.1 Proof of Theorem 13.1 13.8 Exercises IV Nonparametric Nearest Neighbor and Series Methods 14 K-Nearest Neighbor Methods 14.1 Density Estimation: The Univariate Case 14.2 Regression Function Estimation 14.3 A Local Linear k-nn Estimator 14.4 Cross-Validation with Local Constant k-nn Estimation 14.5 Cross-Validation with Local Linear k-nn Estimation 14.6 Estimation of Semiparametric Models with k-nn Methods 14.7 Model Specification Tests with k-nn Methods 14.7.1 A Bootstrap Test 14.8 Using Different k for Different Components of x 14.9 Proofs 14.9.1 Proof of Theorem 14.1 14.9.2 Proof of Theorem 14.5 14.9.3 Proof of Theorem 14.10 14.10 Exercises 15 Nonparametric Series Methods 15.1 Estimating Regression Functions 15.1.1 Convergence Rates 15.2 Selection of the Series Term K 15.2.1 Asymptotic Normality 15.3 A Partially Linear Model 15.3.1 An Additive Partially Linear Model 15.3.2 Selection of Nonlinear Additive Components 15.3.3 Estimating an Additive Model with a Known Link Function 15.4 Estimation of Partially Linear Varying Coefficient Models 15.4.1 Testing for Correct Parametric Regression Functional Form 15.4.2 A Consistent Test for an Additive Partially Linear Model 15.5 Other Series-Based Tests 15.6 Proofs 15.6.1 Proof of Theorem 15.1 15.6.2 Proof of Theorem 15.3 15.6.3 Proof of Theorem 15.6 15.6.4 Proof of Theorem 15.9 15.6.5 Proof of Theorem 15.10 15.7 Exercises V Time Series, Simultaneous Equation, and Panel Data Models 16 Instrumental Variables and Efficient Estimation of Semiparametric Models 16.1 A Partially Linear Model with Endogenous Regressors in the Parametric Part 16.2 A Varying Coefficient Model with Endogenous Regressors in the Parametric Part 16.3 Ai and Chen???s Efficient Estimator with Conditional Moment Restrictions 16.3.1 Estimation Procedures 16.3.2 Asymptotic Normality for 0 16.3.3 A Partially Linear Model with the Endogenous Regressors in the Nonparametric Part 16.4 Proof of Equation (16.16) 16.5 Exercises 17 Endogeneity in Nonparametric Regression Models 17.1 A Nonparametric Model 17.2 A Triangular Simultaneous Equation Model 17.3 Newey-Powell Series-Based Estimator 17.4 Hall and Horowitz???s Kernel-Based Estimator 17.5 Darolles, Florens and Renault???s Estimator 17.6 Exercises 18 Weakly Dependent Data 18.1 Density Estimation with Dependent Data 18.1.1 Uniform Almost Sure Rate of Convergence 18.2 Regression Models with Dependent Data 18.2.1 The Martingale Difference Error Case 18.2.2 The Autocorrelated Error Case 18.2.3 One-Step-Ahead Forecasting 18.2.4 d-Step-Ahead Forecasting 18.2.5 Estimation of Nonparametric Impulse Response Functions 18.3 Semiparametric Models with Dependent Data 18.3.1 A Partially Linear Model with Dependent Data 18.3.2 Additive Regression Models 18.3.3 Varying Coefficient Models with Dependent Data 18.4 Testing for Serial Correlation in Semiparametric Models 18.4.1 The Test Statistic and Its Asymptotic Distribution 18.4.2 Testing Zero First Order Serial Correlation 18.5 Model Specification Tests with Dependent Data 18.5.1 A Kernel Test for Correct Parametric Regression Functional Form 18.5.2 Nonparametric Significance Tests 18.6 Nonsmoothing Tests for Regression Functional Form 18.7 Testing Parametric Predictive Models 18.7.1 In-Sample Testing of Conditional CDFs 18.7.2 Out-of-Sample Testing of Conditional CDFs 18.8 Applications 18.8.1 Forecasting Short-Term Interest Rates 18.9 Nonparametric Estimation with Nonstationary Data 18.10 Proofs 18.10.1 Proof of Equation (18.9) 18.10.2 Proof of Theorem 18.2 18.11 Exercises 19 Panel Data Models 19.1 Nonparametric Estimation of Panel Data Models: Ignoring the Variance Structure 19.2 Wang???s Efficient Nonparametric Panel Data Estimator 19.3 A Partially Linear Model with Random Effects 19.4 Nonparametric Panel Data Models with Fixed Effects 19.4.1 Error Variance Structure Is Known 19.4.2 The Error Variance Structure Is Unknown 19.5 A Partially Linear Model with Fixed Effects 19.6 Semiparametric Instrumental Variable Estimators 19.6.1 An Infeasible Estimator 19.6.2 The Choice of Instruments 19.6.3 A Feasible Estimator 19.7 Testing for Serial Correlation and for Individual Effects in Semiparametric Models 19.8 Series Estimation of Panel Data Models 19.8.1 Additive Effects 19.8.2 Alternative Formulation of Fixed Effects 19.9 Nonlinear Panel Data Models 19.9.1 Censored Panel Data Models 19.9.2 Discrete Choice Panel Data Models 19.10 Proofs 19.10.1 Proof of Theorem 19.1 19.10.2 Leading MSE Calculation of Wang???s Estimator 19.11 Exercises 20 Topics in Applied Nonparametric Estimation 20.1 Nonparametric Methods in Continuous-Time Models 20.1.1 Nonparametric Estimation of Continuous-Time Models 20.1.2 Nonparametric Tests for Continuous-Time Models 20.1.3 Ait-Sahalia???s Test 20.1.4 Hong and Li???s Test 20.1.5 Proofs 20.2 Nonparametric Estimation of Average Treatment Effects 20.2.1 The Model 20.2.2 An Application: Assessing the Efficacy of Right Heart Catheterization 20.3 Nonparametric Estimation of Auction Models 20.3.1 Estimation of First Price Auction Models 20.3.2 Conditionally Independent Private Information Auctions 20.4 Copula-Based Semiparametric Estimation of Multivariate Distributions 20.4.1 Some Background on Copula Functions 20.4.2 Semiparametric Copula-Based Multivariate Distributions 20.4.3 A Two-Step Estimation Procedure 20.4.4 A One-Step Efficient Estimation Procedure 20.4.5 Testing Parametric Functional Forms of a Copula 20.5 A Semiparametric Transformation Model 20.6 Exercises A Background Statistical Concepts 1.1 Probability, Measure, and Measurable Space 1.2 Metric, Norm, and Functional Spaces 1.3 Limits and Modes of Convergence 1.3.1 Limit Supremum and Limit Infimum 1.3.2 Modes of Convergence 1.4 Inequalities, Laws of Large Numbers, and Central Limit Theorems 1.5 Exercises Bibliography Author Index Subject Index A comprehensive, up-to-date textbook on nonparametric methods for students and researchers Until now, students and researchers in nonparametric and semiparametric statistics and econometrics have had to turn to the latest journal articles to keep pace with these emerging methods of economic analysis. Nonparametric Econometrics fills a major gap by gathering together the most up-to-date theory and techniques and presenting them in a remarkably straightforward and accessible format. The empirical tests, data, and exercises included in this textbook help make it the ideal introduction for graduate students and an indispensable resource for researchers. Nonparametric and semiparametric methods have attracted a great deal of attention from statisticians in recent decades. While the majority of existing books on the subject operate from the presumption that the underlying data is strictly continuous in nature, more often than not social scientists deal with categorical datanominal and ordinalin applied settings. The conventional nonparametric approach to dealing with the presence of discrete variables is acknowledged to be unsatisfactory. This book is tailored to the needs of applied econometricians and social scientists. Qi Li and Jeffrey Racine emphasize nonparametric techniques suited to the rich array of data typescontinuous, nominal, and ordinalwithin one coherent framework. They also emphasize the properties of nonparametric estimators in the presence of potentially irrelevant variables. Nonparametric Econometrics covers all the material necessary to understand and apply nonparametric methods for real-world problems. Until now, students and researchers in nonparametric and semiparametric statistics and econometrics have had to turn to the latest journal articles to keep pace with these emerging methods of economic analysis. "Nonparametric Econometrics" fills a major gap by gathering together the most up-to-date theory and techniques and presenting them in a remarkably straightforward and accessible format. The empirical tests, data, and exercises included in this textbook help make it the ideal introduction for graduate students and an indispensable resource for researchers. Nonparametric and semiparametric methods have attracted a great deal of attention from statisticians in recent decades. While the majority of existing books on the subject operate from the presumption that the underlying data is strictly continuous in nature, more often than not social scientists deal with categorical data-nominal and ordinal - in applied settings. The conventional nonparametric approach to dealing with the presence of discrete variables is acknowledged to be unsatisfactory. This book is tailored to the needs of applied econometricians and social scientists. Qi Li and Jeffrey Racine emphasize nonparametric techniques suited to the rich array of data types -continuous, nominal, and ordinal - within one coherent framework. They also emphasize the properties of nonparametric estimators in the presence of potentially irrelevant variables. "Nonparametric Econometrics" covers all the material necessary to understand and apply nonparametric methods for real-world problems