Nonparametric Econometrics

by Adrian Pagan , Aman Ullah

• ISBN13: 9780521355643
• ISBN10: 0521355648
• Format: Hardcover
• Copyright: 1999-06-28
• Publisher: Cambridge University Press

Summary

This book systematically and thoroughly covers the vast literature on the nonparametric and semiparametric statistics and econometrics that has evolved over the last five decades. Within this framework this is the first book to discuss the principles of the nonparametric approach to the topics covered in a first year graduate course in econometrics, e.g. regression function, heteroskedasticity, simultaneous equations models, logit-probit and censored models. Nonparametric and semiparametric methods potentially offer considerable reward to applied researchers, owing to the methods' ability to adapt to many unknown features of the data. Professors Pagan and Ullah provide intuitive explanations of difficult concepts, heuristic developments of theory, and empirical examples emphasizing the usefulness of the modern nonparametric approach. The book should provide a new perspective on teaching and research in applied subjects in general and econometrics and statistics in particular.

Author Biography

Adrian Pagan is a Professor of Economics at the Institute of Advanced Studies, Australian National University. A Fellow of the Econometric Society and Australian Academy of Social Sciences, he is the coauthor or author of several books and numerous articles in economics, econometrics, and public policy. Professor Pagan has been coeditor of Journal of Applied Econometrics and Econometric Theory, and associate editor of Econometrica and Journal of Econometrics. He is currently a member of the editorial boards of Economic Record, Advances in Computational Economics, and Econometric Reviews and is coeditor of the Themes in Modern Econometrics series for Cambridge University Press. Aman Ullah is a Professor and Chair in the Department of Economics at the University of California, Riverside. A Fellow of the National Academy of Sciences (India), he is the coauthor, editor, or coeditor of six books and over ninety professional papers in economics, econometrics, and statistics. Professor Ullah is a coeditor of the journal Econometric Reviews and associate editor of Journal of Nonparametric Statistics, Journal of Quantitative Economics, and Empirical Economics.

Table of Contents

Preface	xvii
1 Introduction	1	(4)
2 Methods of Density Estimation	5	(73)
2.1 Introduction	5	(2)
2.2 Nonparametric Density Estimation	7	(12)
2.2.1 A "Local" Histogram Approach	7	(2)
2.2.2 A Formal Derivation of f(1)(x)	9	(1)
2.2.3 Rosenblatt -- Parzen Kernel Estimator	9	(2)
2.2.4 The Nearest Neighborhood Estimator	11	(1)
2.2.5 Variable Window-Width Estimators	12	(1)
2.2.6 Series Estimators	13	(2)
2.2.7 Penalized Likelihood Estimators	15	(2)
2.2.8 The Local Log-Likelihood Estimators	17	(2)
2.2.9 Summary	19	(1)
2.3 Estimation of Derivatives of a Density	19	(1)
2.4 Finite-Sample Properties of the Kernel Estimator	20	(12)
2.4.1 The Exact Bias and Variance of the Estimator f	21	(2)
2.4.2 Approximations to the Bias and Variance and Choices of h and K	23	(6)
2.4.3 Reduction of Bias	29	(3)
2.5 Asymptotic Properties of the Kernel Density Estimator f with Independent Observations	32	(11)
2.5.1 Asymptotic Unbiasedness	33	(1)
2.5.2 Consistency	34	(5)
2.5.3 Asymptotic Normality	39	(3)
2.5.4 Small-Sample Confidence Intervals	42	(1)
2.6 Sampling Properties of the Kernel Density Estimator with Dependent Observations	43	(6)
2.6.1 Unbiasedness	43	(1)
2.6.2 Consistency	43	(5)
2.6.3 Asymptotic Normality	48	(1)
2.6.4 Bibliographical Summary (Approximate and Asymptotic Results)	48	(1)
2.7 Choices of Window Width and Kernel: Further Discussion	49	(8)
2.7.1 Choice of h	49	(5)
2.7.2 Choice of Higher Order Kernels	54	(2)
2.7.3 Choice of h for Density Derivatives	56	(1)
2.8 Multivariate Density Estimation	57	(3)
2.9 Testing Hypotheses about Densities	60	(11)
2.9.1 Comparison with a Known Density Function	61	(6)
2.9.2 Testing for Symmetry	67	(1)
2.9.3 Comparison of Unknown Densities	68	(1)
2.9.4 Testing for Independence	69	(2)
2.10 Examples	71	(7)
2.10.1 Density of Stock Market Returns	71	(3)
2.10.2 Estimating the Dickey-Fuller Density	74	(4)
3 Conditional Moment Estimation	78	(82)
3.1 Introduction	78	(1)
3.2 Estimating Conditional Moments by Kernel Methods	79	(16)
3.2.1 Parametric Estimation	80	(1)
3.2.2 Nonparametric Estimation: A "Local" Regression Approach	81	(2)
3.2.3 Kernel-Based Estimation: A Formal Derivation	83	(1)
3.2.4 A General Nonparametric Estimator of m(x)	84	(2)
3.2.5 Unifying Nonparametric Estimators	86	(9)
3.2.6 Estimation of Higher Order Conditional Moments	95	(1)
3.3 Finite-Sample Properties	95	(13)
3.3.1 Approximate Results: Stochastic x	96	(8)
3.3.2 The Local Linear Regression Estimator	104	(2)
3.3.3 Combining Parametric and Nonparametric Estimators	106	(2)
3.4 Asymptotic Properties	108	(8)
3.4.1 Asymptotic Properties of the Kernel Estimator with Independent Observations	108	(7)
3.4.2 Asymptotic Properties of the Kernel Estimator with Dependent Observations	115	(1)
3.5 Bibliographical Summary (Asymptotic Results)	116	(2)
3.6 Implementing the Kernel Estimator	118	(4)
3.6.1 Choice of Window Width	118	(4)
3.7 Robust Nonparametric Estimation of Moments	122	(1)
3.8 Estimating Conditional Moments by Series Methods	123	(3)
3.9 Asymptotic Properties of Series Estimators with Independent Observations	126	(7)
3.10 Asymptotic Properties of Series Estimators with Dependent Observations	133	(1)
3.11 Implementing the Estimator	133	(4)
3.12 Imposing Structure on the Conditional Moments	137	(4)
3.12.1 Generalized Additive Models	137	(2)
3.12.2 Projection Pursuit Regression	139	(1)
3.12.3 Neural Networks	140	(1)
3.13 Measuring the Affinity of Parametric and Nonparametric Models	141	(9)
3.14 Examples	150	(10)
3.14.1 A Model of Strike Duration	150	(2)
3.14.2 Earnings-Age Profiles	152	(5)
3.14.3 Review of Applied Work on Nonparametric Regression	157	(3)
4 Nonparametric Estimation of Derivatives	160	(36)
4.1 Introduction	160	(1)
4.2 The Model and Partial Derivative Formulae	161	(3)
4.3 Estimation	164	(9)
4.3.1 Estimation of Partial Derivatives by Kernel Methods	164	(3)
4.3.2 Estimation of Partial Derivatives by Series Methods	167	(1)
4.3.3 Estimation of Average Derivatives	167	(3)
4.3.4 Local Linear Derivative Estimators	170	(2)
4.3.5 Pointwise Versus Average Derivatives	172	(1)
4.4 Restricted Estimation and Hypothesis Testing	173	(4)
4.4.1 Imposing Linear Equality Restriction on Partial Derivatives	174	(1)
4.4.2 Imposing Linear Inequality Restrictions	175	(1)
4.4.3 Hypothesis Testing	176	(1)
4.5 Asymptotic Properties of Partial Derivative Estimators	177	(7)
4.5.1 Asymptotic Properties of Kernel-Based Estimators	178	(4)
4.5.2 Series-Based Estimators	182	(1)
4.5.3 Higher Order Derivatives	182	(1)
4.5.4 Local Linear Estimators	183	(1)
4.6 Asymptotic Properties of Kernel-Based Average Derivative Estimators	184	(5)
4.7 Implementing the Derivative Estimators	189	(1)
4.8 Illustrative Examples	190	(6)
4.8.1 A Monte Carlo Experiment with a Production Function	190	(2)
4.8.2 Earnings-Age Relationship	192	(2)
4.8.3 Review of Applied Work	194	(2)
5 Semiparametric Estimation of Single-Equation Models	196	(58)
5.1 Introduction	196	(2)
5.2 Semiparametric Estimation of the Linear Part of a Regression Model	198	(16)
5.2.1 General Results	198	(10)
5.2.2 Diagnostic Tests after Nonparametric Regression	208	(2)
5.2.3 Semiparametric Estimation of Some Macro Models	210	(2)
5.2.4 The Asymptotic Covariance Matrix of SP Estimators without Asymptotic Independence	212	(2)
5.3 Efficient Estimation of Semiparametric Models in the Presence of Heteroskedasticity of Unknown Form	214	(3)
5.4 Conditions for Adaptive Estimation	217	(8)
5.5 Efficient Estimation of Regression Parameters with Unknown Error Density	225	(9)
5.5.1 Efficient Estimation by Likelihood Approximation	225	(2)
5.5.2 Efficient Estimation by Kernel-Based Score Approximation	227	(3)
5.5.3 Efficient Estimation by Moment-Based Score Approximation	230	(4)
5.6 Estimation of Scale Parameters	234	(1)
5.7 Optimal Diagnostic Tests in Linear Models	234	(1)
5.8 Adaptive Estimation with Dependent Observations	235	(2)
5.9 M-Estimators	237	(8)
5.9.1 Estimation	237	(5)
5.9.2 Diagnostic Tests with M-Estimators	242	(1)
5.9.3 Sequential M-Estimators	243	(2)
5.10 The Semiparametric Efficiency Bound for Moment-Based Estimators	245	(3)
5.10.1 Approximating the SP Efficiency Bound by a Conditional Moment Estimator	246	(2)
5.11 Applications	248	(6)
5.11.1 Semiparametric Estimation of a Heteroskedastic Model	248	(2)
5.11.2 Adaptive Estimation of a Model of House Prices	250	(1)
5.11.3 Review of Other Applications	251	(3)
6 Semiparametric and Nonparametric Estimation of Simultaneous Equation Models	254	(18)
6.1 Introduction	254	(1)
6.2 Single-Equation Estimators	255	(5)
6.2.1 Parametric Estimation	256	(2)
6.2.2 Rilstone's Semiparametric Two-Stage Least Squares Estimator	258	(2)
6.3 Systems Estimation	260	(7)
6.3.1 A Parametric Estimator	260	(1)
6.3.2 The SP3SLS Estimator	261	(1)
6.3.3 Newey's Estimator	262	(2)
6.3.4 Newey's Efficient Distribution-Free Estimators	264	(3)
6.4 Finite-Sample Properties	267	(2)
6.5 Nonparametric Estimation	269	(3)
6.5.1 Identification	269	(1)
6.5.2 Nonparametric Two-Stage Least Squares (2SLS) Estimation	270	(2)
7 Semiparametric Estimation of Discrete Choice Models	272	(28)
7.1 Introduction	272	(1)
7.2 Parametric Estimation of Binary Discrete Choice Models	273	(2)
7.3 Semiparametric Efficiency Bounds for Binary Discrete Choice Models	275	(4)
7.4 Semiparametric Estimation of Binary Discrete Choice Models	279	(7)
7.4.1 Ichimura's Estimator	280	(3)
7.4.2 Klein and Spady's Estimator	283	(2)
7.4.3 The SNP Maximum Likelihood Estimator	285	(1)
7.4.4 Local Maximum Likelihood Estimation	286	(1)
7.5 Alternative Consistent SP Estimators	286	(10)
7.5.1 Manski's Maximum Score Estimator	286	(1)
7.5.2 Horowitz's Smoothed Maximum Score Estimator	287	(4)
7.5.3 Han's Maximum Rank Correlation Estimator	291	(1)
7.5.4 Cosslett's Approximate MLE	292	(1)
7.5.5 An Iterative Least Squares Estimator	293	(1)
7.5.6 Derivative-Based Estimators	294	(1)
7.5.7 Models with Discrete Explanatory Variables	295	(1)
7.6 Multinomial Discrete Choice Models	296	(1)
7.7 Some Specification Tests for Discrete Choice Models	297	(2)
7.8 Applications	299	(1)
8 Semiparametric Estimation of Selectivity Models	300	(17)
8.1 Introduction	300	(1)
8.2 Some Parametric Estimators	301	(3)
8.3 Some Sequential Semiparametric Estimators	304	(6)
8.3.1 Cosslett's Dummy Variable Method	306	(1)
8.3.2 Powell's Kernel Estimator	306	(2)
8.3.3 Newey's Series Estimator	308	(2)
8.3.4 Newey's GMM Estimator	310	(1)
8.4 Maximum Likelihood-Type Estimators	310	(5)
8.4.1 Gallant and Nychka's Estimator	310	(1)
8.4.2 Newey's Estimator	311	(4)
8.5 Estimation of the Intercept in Selection Models	315	(1)
8.6 Applications of the Estimators	315	(1)
8.7 Conclusions	316	(1)
9 Semiparametric Estimation of Censored Regression Models	317	(22)
9.1 Introduction	317	(2)
9.2 Some Parametric Estimators	319	(3)
9.3 Semiparametric Efficiency Bounds for the Censored Regression Model	322	(2)
9.4 The Kaplan-Meier Estimator of the Distribution Function of a Censored Random Variable	324	(2)
9.5 Semiparametric Density-Based Estimators	326	(3)
9.5.1 The Semiparametric Generalized Least Squares Estimator (SGLS)	327	(1)
9.5.2 Estimators Replacing Part of the Sample	328	(1)
9.5.3 Maximum Likelihood Type Estimators	329	(1)
9.6 Semiparametric Nondensity-Based Estimators	329	(8)
9.6.1 Powell's Censored Least Absolute Deviation (CLAD) Estimator	330	(3)
9.6.2 Powell's (1986a) Censored Quantile Estimators	333	(1)
9.6.3 Powell's Symmetrically Censored Least Squares Estimators	333	(3)
9.6.4 Newey's Efficient Estimator under Conditional Symmetry	336	(1)
9.7 Comparative Studies of the Estimators	337	(2)
10 Retrospect and Prospect	339	(3)
A Statistical Methods	342	(41)
A.1 Probability Concepts	342	(10)
A.1.1 Random Variable and Distribution Function	345	(2)
A.1.2 Conditional Distribution and Independence	347	(1)
A.1.3 Borel Measurable Functions	348	(2)
A.1.4 Inequalities Involving Expectations	350	(1)
A.1.5 Characteristic Function (c.f.)	351	(1)
A.2 Results on Convergence	352	(13)
A.2.1 Weak and Strong Convergence of Random Variables	352	(2)
A.2.2 Laws of Large Numbers	354	(1)
A.2.3 Convergence of Distribution Functions	355	(2)
A.2.4 Central Limit Theorems	357	(3)
A.2.5 Further Results on the Law of Large Numbers and Convergence in Moments and Distributions	360	(1)
A.2.6 Convergence in Moments	361	(4)
A.3 Some Probability Inequalities	365	(3)
A.4 Order of Magnitudes (Small o and Large O)	368	(2)
A.5 Asymptotic Theory for Dependent Observations	370	(13)
A.5.1 Ergodicity	371	(1)
A.5.2 Mixing Sequences	372	(4)
A.5.3 Near-Epoch Dependent Sequences	376	(1)
A.5.4 Martingale Differences and Mixingales	377	(2)
A.5.5 Rosenblatt's (1970) Measure of Dependence Beta(n)	379	(1)
A.5.6 Stochastic Equicontinuity	379	(4)
References	383	(36)
Index	419