Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information Nonparametric Econometrics This book systematically and thoroughly covers a vast literature on the non- parametric and semiparametric statistics and econometrics that has evolved over the past five decades. Within this framework this is the first book to dis- cuss the principles of the nonparametric approach to the topics covered in a first-year graduate course in econometrics, for example, regression function, heteroskedasticity, simultaneous equations models, logit-probit, and censored models. Nonparametric and semiparametric methods potentially offer consid- erable reward to applied researchers, owing to the methods’ ability to adapt to many unknown features of the data. Professors Pagan and Ullah provide intu- itive 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. Adrian Pagan is a Professor of Economics at the Institute of Advanced Stud- ies, Australian National University. A Fellow of the Econometric Society, Australian Academy of Social Sciences, and Journal of Econometrics, he is the coauthor or author of several books and numerous articles in economics, econometrics, and public policy. Professor Pagan has been coeditor of the Jour- nal of Applied Econometrics and Econometric Theory and associate editor of Econometrica and Journal of Econometrics. He is currently a member of the ed- itorial boards of Economic Record, Advances in Computational Economics, and Econometric Reviews and is coeditor of the Themes in Modern Econometrics se- ries for Cambridge University Press. He has also served as a visiting professor or scholar at UCLA, Johns Hopkins University, University of Rochester, Princeton and Yale Universities, and Institute of Advanced Studies, Vienna. Professor Pagan is also a member of the Board of Governors of the Reserve Bank of Australia. 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 Sci- ences (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, among others. He has taught at the University of Western Ontario, Canada, for several years and has served as a visiting professor or scholar at Southern Methodist University, UCLA, Stanford University, University of Illinois, Bilkent University, Turkey, Australian National University, Monash University, Tinbergen Institute, Holland, and CORE, Belgium, among others. © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information Themes in Modern Econometrics Managing editor PETER C.B. PHILLIPS, Yale University Series editors ADRIAN PAGAN, Australian National University CHRISTIAN GOURIEROUX, CREST and CEPREMAP, Paris MICHAEL WICKENS, University of York Themes in Modern Econometrics is designed to service the large and growing need for explicit teaching tools in econometrics. It will provide an organised sequence of textbooks in econometrics aimed squarely at the student popula- tion, and will be the first series in the discipline to have this as its express aim. Written at a level accessible to students with an introductory course in econo- metrics behind them, each book will address topics or themes that students and researchers encounter daily. While each book will be designed to stand alone as an authoritative survey in its own right, the distinct emphasis throughout will be on pedagogic excellence. Titles in the series Statistics and Econometric Models: Volumes 1 and 2 CHRISTIAN GOURIEROUX and ALAIN MONFORT Translated by QUANG VUONG Time Series and Dynamic Models CHRISTIAN GOURIEROUX and ALAIN MONFORT Translated and edited by GIAMPIERO GALLO Unit Roots, Cointegration, and Structural Change G.S. MADDALA and IN-MOO KIM Generalized Method of Moments Estimation Edited by LASZLO MATYAS © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information NONPARAMETRIC ECONOMETRICS ADRIAN PAGAN AMAN ULLAH Australian National University of California, University Riverside © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia C Adrian Pagan and Aman Ullah 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Typeset in Times Roman 10/12 pt. in LATEX2ε[TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Pagan, Adrian. Nonparametric econometrics / Adrian Pagan, Aman Ullah. p. cm. – (Themes in Modern Econometrics) Includes bibliographical references (p. ). ISBN 0-521-35564-8 (hardbound) 1. Econometrics. 2. Mathematical statistics. 3. Economics – Statistical methods. I. Ullah, Aman. II. Title. HB139.P34 1999 330.015195 – dc21 98-37218 CIP ISBN 0 521 35564 8 hardback ISBN 0 521 58611 9 paperback Transferred to digital printing 2004 © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information To my parents, Razia Begum and Ataullah Khan © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information Contents Preface page xvii 1 Introduction 1 2 Methods of Density Estimation 5 2.1 Introduction 5 2.2 Nonparametric Density Estimation 7 2.2.1 A “Local” Histogram Approach 7 ˆ ( ) 2.2.2 A Formal Derivation of f 1 x 9 2.2.3 Rosenblatt–Parzen Kernel Estimator 9 2.2.4 The Nearest Neighborhood Estimator 11 2.2.5 Variable Window-Width Estimators 12 2.2.6 Series Estimators 13 2.2.7 Penalized Likelihood Estimators 15 2.2.8 The Local Log-Likelihood Estimators 17 2.2.9 Summary 19 2.3 Estimation of Derivatives of a Density 19 2.4 Finite-Sample Properties of the Kernel Estimator 20 2.4.1 The Exact Bias and Variance of the Estimator fˆ 21 2.4.2 Approximations to the Bias and Variance and Choices of h and K 23 2.4.3 Reduction of Bias 29 2.5 Asymptotic Properties of the Kernel Density Estimator fˆ with Independent Observations 32 2.5.1 Asymptotic Unbiasedness 33 2.5.2 Consistency 34 2.5.3 Asymptotic Normality 39 2.5.4 Small-Sample Confidence Intervals 42 2.6 Sampling Properties of the Kernel Density Estimator with Dependent Observations 43 ix © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information x Contents 2.6.1 Unbiasedness 43 2.6.2 Consistency 43 2.6.3 Asymptotic Normality 48 2.6.4 Bibliographical Summary (Approximate and Asymptotic Results) 48 2.7 Choices of Window Width and Kernel: Further Discussion 49 2.7.1 Choice of h 49 2.7.2 Choice of Higher Order Kernels 54 2.7.3 Choice of h for Density Derivatives 56 2.8 Multivariate Density Estimation 57 2.9 Testing Hypotheses about Densities 60 2.9.1 Comparison with a Known Density Function 61 2.9.2 Testing for Symmetry 67 2.9.3 Comparison of Unknown Densities 68 2.9.4 Testing for Independence 69 2.10 Examples 71 2.10.1 Density of Stock Market Returns 71 2.10.2 Estimating the Dickey–Fuller Density 74 3 Conditional Moment Estimation 78 3.1 Introduction 78 3.2 Estimating Conditional Moments by Kernel Methods 79 3.2.1 Parametric Estimation 80 3.2.2 Nonparametric Estimation: A “Local” Regression Approach 81 3.2.3 Kernel-Based Estimation: A Formal Derivation 83 3.2.4 A General Nonparametric Estimator of m(x) 84 3.2.5 Unifying Nonparametric Estimators 86 3.2.6 Estimation of Higher Order Conditional Moments 95 3.3 Finite-Sample Properties 95 3.3.1 Approximate Results: Stochastic x 96 3.3.2 The Local Linear Regression Estimator 104 3.3.3 Combining Parametric and Nonparametric Estimators 106 3.4 Asymptotic Properties 108 3.4.1 Asymptotic Properties of the Kernel Estimator with Independent Observations 108 3.4.2 Asymptotic Properties of the Kernel Estimator with Dependent Observations 115 3.5 Bibliographical Summary (Asymptotic Results) 116 © Cambridge University Press www.cambridge.org Cambridge University Press 0521586119 - Nonparametric Econometrics Adrian Pagan and Aman Ullah Frontmatter More information Contents xi 3.6 Implementing the Kernel Estimator 118 3.6.1 Choice of Window Width 118 3.7 Robust Nonparametric Estimation of Moments 122 3.8 Estimating Conditional