Nonparametric Econometrics: Theory and Practice

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 H¨ardle 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