Nonparametric Econometrics

Preface xvii PART I: Nonparametric Kernel Methods 1 Chapter 1: Density Estimation 3 1.1 Univariate Density Estimation 4 1.2 Univariate Bandwidth Selection: Rule-of-Thumb and Plug-In Methods 14 1.3 Univariate Bandwidth Selection: Cross-Validation ZMethods 15 1.3.1 Least Squares Cross-Validation 15 1.3.2 Likelihood Cross-Validation 18 1.3.3 An Illustration of Data-Driven Bandwidth Selection 19 1.4 Univariate CDF Estimation 19 1.5 Univariate CDF Bandwidth Selection: Cross- Validation Methods 23 1.6 Multivariate Density Estimation 24 1.7 Multivariate Bandwidth Selection: Rule-of-Thumb and Plug-In Methods 26 1.8 Multivariate Bandwidth Selection: Cross-Validation Methods 27 1.8.1 Least Squares Cross-Validation 27 1.8.2 Likelihood Cross-Validation 28 1.9 Asymptotic Normality of Density Estimators 28 1.10 Uniform Rates of Convergence 30 1.11 Higher Order Kernel Functions 33 1.12 Proof of Theorem 1.4 (Uniform Almost Sure Convergence) 35 1.13 Applications 40 1.13.1 Female Wage Inequality 41 1.13.2 Unemployment Rates and City Size 43 1.13.3 Adolescent Growth 44 1.13.4 Old Faithful Geyser Data 44 1.13.5 Evolution of Real Income Distribution in Italy, 1951-1998 45 1.14 Exercises 47 Chapter 2: Regression 57 2.1 Local Constant Kernel Estimation 60 2.1.1 Intuition Underlying the Local Constant Kernel Estimator 64 2.2 Local Constant Bandwidth Selection 66 2.2.1 Rule-of-Thumb and Plug-In Methods 66 2.2.2 Least Squares Cross-Validation 69 2.2.3 AICc 72 2.2.4 The Presence of Irrelevant Regressors 73 2.2.5 Some Further Results on Cross-Validation 78 2.3 Uniform Rates of Convergence 78 2.4 Local Linear Kernel Estimation 79 2.4.1 Local Linear Bandwidth Selection: Least Squares Cross-Validation 83 2.5 Local Polynomial Regression (General pth Order) 85 2.5.1 The Univariate Case 85 2.5.2 The Multivariate Case 88 2.5.3 Asymptotic Normality of Local Polynomial Estimators 89 2.6 Applications 92 2.6.1 Prestige Data 92 2.6.2 Adolescent Growth 92 2.6.3 Inflation Forecasting and Money Growth 93 2.7 Proofs 97 2.7.1 Derivation of (2.24) 98 2.7.2 Proof of Theorem 2.7 100 2.7.3 Definitions of Al,p+1 and Vl Used in Theorem 2.10 106 2.8 Exercises 108 Chapter 3: Frequency Estimation with Mixed Data 115 3.1 Probability Function Estimation with Discrete Data 116 3.2 Regression with Discrete Regressors 118 3.3 Estimation with Mixed Data: The Frequency Approach 118 3.3.1 Density Estimation with Mixed Data 118 3.3.2 Regression with Mixed Data 119 3.4 Some Cautionary Remarks on Frequency Methods 120 3.5 Proofs 122 3.5.1 Proof of Theorem 3.1 122 3.6 Exercises 123 Chapter 4: Kernel Estimation with Mixed Data 125 4.1 Smooth Estimation of Joint Distributions with Discrete Data 126 4.2 Smooth Regression with Discrete Data 131 4.3 Kernel Regression with Discrete Regressors: The Irrelevant Regressor Case 134 4.4 Regression with Mixed Data: Relevant Regressors 136 4.4.1 Smooth Estimation with Mixed Data 136 4.4.2 The Cross-Validation Method 138 4.5 Regression with Mixed Data: Irrelevant Regressors 140 4.5.1 Ordered Discrete Variables 144 4.6 Applications 145 4.6.1 Food-Away-from-Home Expenditure 145 4.6.2 Modeling Strike Volume 147 4.7 Exercises 150 Chapter 5: Conditional Density Estimation 155 5.1 Conditional Density Estimation: Relevant Variables 155 5.2 Conditional Density Bandwidth Selection 157 5.2.1 Least Squares Cross-Validation: Relevant Variables 157 5.2.2 Maximum Likelihood Cross-Validation: Relevant Variables 160 5.3 Conditional Density Estimation: Irrelevant Variables 162 5.4 The Multivariate Dependent Variables Case 164 5.4.1 The General Categorical Data Case 167 5.4.2 Proof of Theorem 5.5 168 5.5 Applications 171 5.5.1 A Nonparametric Analysis of Corruption 171 5.5.2 Extramarital Affairs Data 172 5.5.3 Married Female Labor Force Participation 175 5.5.4 Labor Productivity 177 5.5.5 Multivariate Y Conditional Density Example: GDP Growth and Population Growth Conditional on OECD Status 178 5.6 Exercises 180 Chapter 6: Conditional CDF and Quantile Estimation 181 6.1 Estimating a Conditional CDF with Continuous Covariates without Smoothing the Dependent Variable 182 6.2 Estimating a Conditional CDF with Continuous Covariates Smoothing the Dependent Variable 184 6.3 Nonparametric Estimation of Conditional Quantile Functions 189 6.4 The Check Function Approach 191 6.5 Conditional CDF and Quantile Estimation with Mixed Discrete and Continuous Covariates 193 6.6 A Small Monte Carlo Simulation Study 196 6.7 Nonparametric Estimation of Hazard Functions 198 6.8 Applications 200 6.8.1 Boston Housing Data 200 6.8.2 Adolescent Growth Charts 202 6.8.3 Conditional Value at Risk 202 6.8.4 Real Income in Italy, 1951-1998 206 6.8.5 Multivariate Y Conditional CDF Example: GDP Growth and Population Growth Conditional on OECD Status 206 6.9 Proofs 209 6.9.1 Proofs of Theorems 6.1, 6.2, and 6.4 209 6.9.2 Proofs of Theorems 6.5 and 6.6 (Mixed Covariates Case) 214 6.10 Exercises 215 PART II: Semiparametric Methods 219 Chapter 7: Semiparametric Partially Linear Models 221 7.1 Partially Linear Models 222 7.1.1 Identification of 222 7.2 Robinson's Estimator 222 7.2.1 Estimation of the Nonparametric Component 228 7.3 Andrews's MINPIN Method 230 7.4 Semiparametric Efficiency Bounds 233 7.4.1 The Conditionally Homoskedastic Error Case 233 7.4.2 The Conditionally Heteroskedastic Error Case 235 7.5 Proofs 238 7.5.1 Proof of Theorem 7.2 238 7.5.2 Verifying Theorem 7.3 for a Partially Linear Model 244 7.6 Exercises 246 Chapter 8: Semiparametric Single Index Models 249 8.1 Identification Conditions 251 8.2 Estimation 253 8.2.1 Ichimura's Method 253 8.3 Direct Semiparametric Estimators for 258 8.3.1 Average Derivative Estimators 258 8.3.2 Estimation of g() 262 8.4 Bandwidth Selection 263 8.4.1 Bandwidth Selection for Ichimura's Method 263 8.4.2 Bandwidth Selection with Direct Estimation Methods 265 8.5 Klein and Spady's Estimator 266 8.6 Lewbel's Estimator 267 8.7 Manski's Maximum Score Estimator 269 8.8 Horowitz's Smoothed Maximum Score Estimator 270 8.9 Han's Maximum Rank Estimator 270 8.10 Multinomial Discrete Choice Models 271 8.11 Ai's Semiparametric Maximum Likelihood Approach 272 8.12 A Sketch of the Proof of Theorem 8.1 275 8.13 Applications 277 8.13.1 Modeling Response to Direct Marketing Catalog Mailings 277 8.14 Exercises 281 Chapter 9: Additive and Smooth (Varying) Coefficient Semiparametric Models 283 9.1 An Additive Model 283 9.1.1 The Marginal Integration Method 284 9.1.2 A Computationally Efficient Oracle Estimator 286 9.1.3 The Ordinary Backfitting Method 289 9.1.4 The Smoothed Backfitting Method 290 9.1.5 Additive Models with Link Functions 295 9.2 An Additive Partially Linear Model 297 9.2.1 A Simple Two-Step Method 299 9.3 A Semiparametric Varying (Smooth) Coefficient Model 301 9.3.1 A Local Constant Estimator of the Smooth Coefficient Function 302 9.3.2 A Local Linear Estimator of the Smooth Coefficient Function 303 9.3.3 Testing for a Parametric Smooth Coefficient Model 306 9.3.4 Partially Linear Smooth Coefficient Models 308 9.3.5 Proof of Theorem 9.3 310 9.4 Exercises 312 Chapter 10: Selectivity Models 315 10.1 Semiparametric Type-2 Tobit Models 316 10.2 Estimation of a Semiparametric Type-2 Tobit Model 317 10.2.1 Gallant and Nychka's Estimator 318 10.2.2 Estimation of the Intercept in Selection Models 319 10.3 Semiparametric Type-3 Tobit Models 320 10.3.1 Econometric Preliminaries 320 10.3.2 Alternative Estimation Methods 323 10.4 Das, Newey and Vella's Nonparametric Selection Model 328 10.5 Exercises 330 Chapter 11: Censored Models 331 11.1 Parametric Censored Models 332 11.2 Semiparametric Censored Regression Models 334 11.3 Semiparametric Censored Regression Models with Nonparametric Heteroskedasticity 336 11.4 The Univariate Kaplan-Meier CDF Estimator 338 11.5 The Multivariate Kaplan-Meier CDF Estimator 341 11.5.1 Nonparametric Regression Models with Random Censoring 343 11.6 Nonparametric Censored Regression 345 11.6.1 Lewbel and Linton's Approach 345 11.6.2 Chen, Dahl and Khan's Approach 346 11.7 Exercises 348 III Consistent Model Specification Tests 349 Chapter 12: Model Specification Tests 351 12.1 A Simple Consistent Test for Parametric Regression Functional Form 354 12.1.1 A Consistent Test for Correct Parametric Functional Form 355 12.1.2 Mixed Data 360 12.2 Testing for Equality of PDFs 362 12.3 More Tests Related to Regression Functions 365 12.3.1 Hardle and Mammen's Test for a Parametric Regression Model 365 12.3.2 An Adaptive and Rate Optimal Test 367 12.3.3 A Test for a Parametric Single Index Model 369 12.3.4 A Nonparametric Omitted Variables Test 370 12.3.5 Testing the Significance of Categorical Variables 375 12.4 Tests Related to PDFs 378 12.4.1 Testing Independence between Two Random Variables 378 12.4.2 A Test for a Parametric PDF 380 12.4.3 A Kernel Test for Conditional Parametric Distributions 382 12.5 Applications 385 12.5.1 Growth Convergence Clubs 385 12.6 Proofs 388 12.6.1 Proof of Theorem 12.1 388 12.6.2 Proof of Theorem 12.2 389 12.6.3 Proof of Theorem 12.5 389 12.6.4 Proof of Theorem 12.9 391 12.7 Exercises 394 Chapter 13: Nonsmoothing Tests 397 13.1 Testing for Parametric Regression Functional Form 398 13.2 Testing for Equality of PDFs 401 13.3 A Nonparametric Significance Test 401 13.4 Andrews's Test for Conditional CDFs 402 13.5 Hong's Tests for Serial Dependence 404 13.6 More on Nonsmoothing Tests 408 13.7 Proofs 409 13.7.1 Proof of Theorem 13.1 409 13.8 Exercises 410 PART IV: Nonparametric Nearest Neighbor and Series Methods 413 Chapter 14: K-Nearest Neighbor Methods 415 14.1 Density Estimation: The Univariate Case 415 14.2 Regression Function Estimation 419 14.3 A Local Linear k-nn Estimator 421 14.4 Cross-Validation with Local Constant k-nn Estimation 422 14.5 Cross-Validation with Local Linear k-nn Estimation 425 14.6 Estimation of Semiparametric Models with k-nn Methods 427 14.7 Model Specification Tests with k-nn Methods 428 14.7.1 A Bootstrap Test 431 14.8 Using Different k for Different Components of x 432 14.9 Proofs 432 14.9.1 Proof of Theorem 14.1 435 14.9.2 Proof of Theorem 14.5 435 14.9.3 Proof of Theorem 14.10 440 14.10 Exercises 444 Chapter 15: Nonparametric Series Methods 445 15.1 Estimating Regression Functions 446 15.1.1 Convergence Rates 449 15.2 Selection of the Series Term K 451 15.2.1 Asymptotic Normality 453 15.3 A Partially Linear Model 454 15.3.1 An Additive Partially Linear Model 455 15.3.2 Selection of Nonlinear Additive Components 461 15.3.3 Estimating an Additive Model with a Known Link Function 463 15.4 Estimation of Partially Linear Varying Coefficient Models 466 15.4.1 Testing for Correct Parametric Regression Functional Form 471 15.4.2 A Consistent Test for an Additive Partially Linear Model 474 15.5 Other Series-Based Tests 479 15.6 Proofs 480 15.6.1 Proof of Theorem 15.1 480 15.6.2 Proof of Theorem 15.3 484 15.6.3 Proof of Theorem 15.6 488 15.6.4 Proof of Theorem 15.9 492 15.6.5 Proof of Theorem 15.10 497 15.7 Exercises 502 PART V: Time Series, Simultaneous Equation, and Panel Data Models 503 Chapter 16: Instrumental Variables and Efficient Estimation of Semiparametric Models 505 16.1 A Partially Linear Model with Endogenous Regressors in the Parametric Part 505 16.2 A Varying Coefficient Model with Endogenous Regressors in the Parametric Part 509 16.3 Ai and Chen's Efficient Estimator with Conditional Moment Restrictions 511 16.3.1 Estimation Procedures 511 16.3.2 Asymptotic Normality for 513 16.3.3 A Partially Linear Model with the Endogenous Regressors in the Nonparametric Part 515 16.4 Proof of Equation (16.16) 517 16.5 Exercises 520 Chapter 17: Endogeneity in Nonparametric Regression Models 521 17.1 A Nonparametric Model 521 17.2 A Triangular Simultaneous Equation Model 522 17.3 Newey-Powell Series-Based Estimator 527 17.4 Hall and Horowitz's Kernel-Based Estimator 529 17.5 Darolles, Florens and Renault's Estimator 532 17.6 Exercises 533 Chapter 18: Weakly Dependent Data 535 18.1 Density Estimation with Dependent Data 537 18.1.1 Uniform Almost Sure Rate of Convergence 541 18.2 Regression Models with Dependent Data 541 18.2.1 The Martingale Difference Error Case 541 18.2.2 The Autocorrelated Error Case 544 18.2.3 One-Step-Ahead Forecasting 546 18.2.4 d-Step-Ahead Forecasting 547 18.2.5 Estimation of Nonparametric Impulse Response Functions 548 18.3 Semiparametric Models with Dependent Data 551 18.3.1 A Partially Linear Model with Dependent Data 551 18.3.2 Additive Regression Models 552 18.3.3 Varying Coefficient Models with Dependent Data 553 18.4 Testing for Serial Correlation in Semiparametric Models 554 18.4.1 The Test Statistic and Its Asymptotic Distribution 554 18.4.2 Testing Zero First Order Serial Correlation 555 18.5 Model Specification Tests with Dependent Data 556 18.5.1 A Kernel Test for Correct Parametric Regression Functional Form 556 18.5.2 Nonparametric Significance Tests 557 18.6 Nonsmoothing Tests for Regression Functional Form 558 18.7 Testing Parametric Predictive Models 559 18.7.1 In-Sample Testing of Conditional CDFs 559 18.7.2 Out-of-Sample Testing of Conditional CDFs 562 18.8 Applications 564 18.8.1 Forecasting Short-Term Interest Rates 564 18.9 Nonparametric Estimation with Nonstationary Data 566 18.10 Proofs 567 18.10.1 Proof of Equation (18.9) 567 18.10.2 Proof of Theorem 18.2 569 18.11 Exercises 572 Chapter 19: Panel Data Models 575 19.1 Nonparametric Estimation of Panel Data Models: Ignoring the Variance Structure 576 19.2 Wang's Efficient Nonparametric Panel Data Estimator 578 19.3 A Partially Linear Model with Random Effects 584 19.4 Nonparametric Panel Data Models with Fixed Effects 586 19.4.1 Error Variance Structure Is Known 587 19.4.2 The Error Variance Structure Is Unknown 590 19.5 A Partially Linear Model with Fixed Effects 592 19.6 Semiparametric Instrumental Variable Estimators 594 19.6.1 An Infeasible Estimator 594 19.6.2 The Choice of Instruments 595 19.6.3 A Feasible Estimator 597 19.7 Testing for Serial Correlation and for Individual Effects in Semiparametric Models 599 19.8 Series Estimation of Panel Data Models 602 19.8.1 Additive Effects 602 19.8.2 Alternative Formulation of Fixed Effects 604 19.9 Nonlinear Panel Data Models 606 19.9.1 Censored Panel Data Models 607 19.9.2 Discrete Choice Panel Data Models 614 19.10 Proofs 618 19.10.1 Proof of Theorem 19.1 618 19.10.2 Leading MSE Calculation of Wang's Estimator 621 19.11 Exercises 624 Chapter 20: Topics in Applied Nonparametric Estimation 627 20.1 Nonparametric Methods in Continuous-Time Models 627 20.1.1 Nonparametric Estimation of Continuous-Time Models 627 20.1.2 Nonparametric Tests for Continuous-Time Models 632 20.1.3 Ait-Sahalia's Test 632 20.1.4 Hong and Li's Test 633 20.1.5 Proofs 636 20.2 Nonparametric Estimation of Average Treatment Effects 639 20.2.1 The Model 640 20.2.2 An Application: Assessing the Efficacy of Right Heart Catheterization 642 20.3 Nonparametric Estimation of Auction Models 645 20.3.1 Estimation of First Price Auction Models 645 20.3.2 Conditionally Independent Private Information Auctions 648 20.4 Copula-Based Semiparametric Estimation of Multivariate Distributions 651 20.4.1 Some Background on Copula Functions 651 20.4.2 Semiparametric Copula-Based Multivariate Distributions 652 20.4.3 A Two-Step Estimation Procedure 653 20.4.4 A One-Step Efficient Estimation Procedure 655 20.4.5 Testing Parametric Functional Forms of a Copula 657 20.5 A Semiparametric Transformation Model 659 20.6 Exercises 662 A Background Statistical Concepts 663 1.1 Probability, Measure, and Measurable Space 663 1.2 Metric, Norm, and Functional Spaces 672 1.3 Limits and Modes of Convergence 680 1.3.1 Limit Supremum and Limit Infimum 680 1.3.2 Modes of Convergence 681 1.4 Inequalities, Laws of Large Numbers, and Central Limit Theorems 688 1.5 Exercises 694 Bibliography 697 Author Index 737 Subject Index 744