A Companion toTheoretical

Econometrics

Edited by

BADI H. BALTAGITexas A & M University

A COMPANION TO THEORETICAL ECONOMETRICS

Blackwell Companions to Contemporary Economics

The Blackwell Companions to Contemporary Economics are reference volumesaccessible to serious students and yet also containing up-to-date material fromrecognized experts in their particular fields. They focus on basic, bread-and-butter issues in economics as well as popular contemporary topics often notcovered in textbooks. Coverage avoids the overly technical, is concise, clear,and comprehensive. Each Companion features an introductory essay by theeditor, bibliographical reference sections, and an index.

A Companion to Theoretical Econometrics edited by Badi H. BaltagiA Companion to Business Forecasting edited by Michael P. Clements and DavidF. Hendry

Forthcoming:

A Companion to the History of Economic Thought edited by Warren J. Samuels,Jeff E. Biddle, and John B. Davis

A Companion toTheoretical

Econometrics

Edited by

BADI H. BALTAGITexas A & M University

© 2001, 2003 by Blackwell Publishing Ltd

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A companion to theoretical econometrics / edited by Badi H. Baltagi.p. cm. — (Blackwell companions to contemporary economics)

A collection of articles by an international group of scholars.Includes bibliographical references and index.ISBN 0–631–21254–X (hb : alk. paper) — ISBN 1–4051–0676–X (pb. : alk.

paper)1. Econometrics. I. Title: Theoretical econometrics. II. Baltagi, Badi H.

(Badi Hani) III. Series.

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Contents

List of Figures viii

List of Tables ix

List of Contributors x

Preface xii

List of Abbreviations xiv

Introduction 1

1 Artificial Regressions 16Russell Davidson and James G. MacKinnon

2 General Hypothesis Testing 38Anil K. Bera and Gamini Premaratne

3 Serial Correlation 62Maxwell L. King

4 Heteroskedasticity 82William E. Griffiths

5 Seemingly Unrelated Regression 101Denzil G. Fiebig

6 Simultaneous Equation Model Estimators: Statistical Propertiesand Practical Implications 122Roberto S. Mariano

7 Identification in Parametric Models 144Paul Bekker and Tom Wansbeek

8 Measurement Error and Latent Variables 162Tom Wansbeek and Erik Meijer

9 Diagnostic Testing 180Jeffrey M. Wooldridge

10 Basic Elements of Asymptotic Theory 201Benedikt M. Pötscher and Ingmar R. Prucha

11 Generalized Method of Moments 230Alastair R. Hall

12 Collinearity 256R. Carter Hill and Lee C. Adkins

13 Nonnested Hypothesis Testing: An Overview 279M. Hashem Pesaran and Melvyn Weeks

14 Spatial Econometrics 310Luc Anselin

15 Essentials of Count Data Regression 331A. Colin Cameron and Pravin K. Trivedi

16 Panel Data Models 349Cheng Hsiao

17 Qualitative Response Models 366G.S. Maddala and A. Flores-Lagunes

18 Self-Selection 383Lung-fei Lee

19 Random Coefficient Models 410P.A.V.B. Swamy and George S. Tavlas

20 Nonparametric Kernel Methods of Estimation and Hypothesis Testing 429Aman Ullah

21 Durations 444Christian Gouriéroux and Joann Jasiak

22 Simulation Based Inference for Dynamic Multinomial Choice Models 466John Geweke, Daniel Houser, and Michael Keane

23 Monte Carlo Test Methods in Econometrics 494Jean-Marie Dufour and Lynda Khalaf

24 Bayesian Analysis of Stochastic Frontier Models 520Gary Koop and Mark F.J. Steel

25 Parametric and Nonparametric Tests of Limited Domain andOrdered Hypotheses in Economics 538Esfandiar Maasoumi

vi CONTENTS

26 Spurious Regressions in Econometrics 557Clive W.J. Granger

27 Forecasting Economic Time Series 562James H. Stock

28 Time Series and Dynamic Models 585Aris Spanos

29 Unit Roots 610Herman J. Bierens

30 Cointegration 634Juan J. Dolado, Jesús Gonzalo, and Francesc Marmol

31 Seasonal Nonstationarity and Near-Nonstationarity 655Eric Ghysels, Denise R. Osborn, and Paulo M.M. Rodrigues

32 Vector Autoregressions 678Helmut Lütkepohl

Index 700

CONTENTS vii

Figures

19.1 Short-term interest rate elasticity for RCM1 (without concomitants) 42519.2 Short-term interest rate elasticity for RCM2 (with concomitants) 42521.1 Censoring scheme: unemployment spells 45521.2 Truncation scheme 45621.3 Hazard functions for accelerated hazard models 45721.4 Hazard functions for proportional hazard models 45821.5 (Under) Overdispersion of intertrade durations 46322.1 Marginal posterior densities of first log-wage equation’s parameters

from data set 3-EMAX 48522.2 EMAX and polynomial future components evaluated at mean values of

state variables at each period 48727.1 US unemployment rate, recursive AR(BIC)/unit root pretest forecast, and

neural network forecast 56827.2 Six-month US CPI inflation at an annual rate, recursive AR(BIC)/unit root

pretest forecast, and neural network forecast 56827.3 90-day Treasury bill at an annual rate, recursive AR(BIC)/unit root pretest

forecast, and neural network forecast 56927.4 Six-month growth of US industrial production at an annual rate, recursive

AR(BIC)/unit root pretest forecast, and neural network forecast 56927.5 Six-month growth of total real US manufacturing and trade inventories at

an annual rate, recursive AR(BIC)/unit root pretest forecast, and neuralnetwork forecast 570

28.1 US industrial production index 58628.2 De-trended industrial production index 58729.1 Density of r0 61729.2 Density of τ0 compared with the standard normal density 61829.3 Density of r1 61929.4 Density of τ1 compared with the standard normal density 62029.5 Density of r2 63029.6 Density of τ2 compared with the standard normal density 631

Tables

12.1 Matrix of variance proportions 26112.2 Harmful collinearity decision matrix 26719.1 Long-run elasticities 42419.2 Long-run elasticities and direct effects from RCM2 42422.1 Quality of polynomial approximation to the true future component 47722.2 Choice distributions and mean accepted wages in the data generated

with true and OLS polynomial future components 480–122.3 Descriptive statistics for posterior distributions of the model’s

structural parameters for several different data sets generatedusing polynomial future component 482

22.4 Log-wage equation estimates from OLS on observed wages generatedunder the polynomial future component 483

22.5 Descriptive statistics for posterior distributions of the model’s structuralparameters for several different data sets generated using true futurecomponent 484

22.6 Wealth loss when posterior polynomial approximation is used in placeof true future component 486

23.1 IV-based Wald/Anderson–Rubin tests: empirical type I errors 49923.2 Kolmogorov–Smirnov/Jarque–Bera residuals based tests:

empirical type I errors 50123.3 Empirical type I errors of multivariate tests: uniform linear hypotheses 50326.1 Regression between independent AR(1) series 56027.1 Comparison of simulated out-of-sample linear and nonlinear forecasts

for five US macroeconomic time series 57527.2 Root mean squared forecast forecast errors of VARs, relative to AR(4) 57532.1 Models and LR type tests 687

Contributors

Lee C. Adkins, Oklahoma State UniversityLuc Anselin, University of IllinoisPaul Bekker, University of GroningenAnil K. Bera, University of IllinoisHerman J. Bierens, Pennsylvania State UniversityA. Colin Cameron, University of California – DavisRussell Davidson, Queen’s University, Ontario, and GREQAM, MarseillesJuan Dolado, Universidad Carlos III de MadridJean-Marie Dufour, University of MontrealDenzil G. Fiebig, University of SydneyA. Flores-Lagunes, University of ArizonaJohn Geweke, University of Minnesota and University of lowaEric Ghysels, Pennsylvania State UniversityJesús Gonzalo, Universidad Carlos III de MadridChristian Gouriéroux, CREST and CEPREMAP, ParisClive W.J. Granger, University of California – San DiegoWilliam E. Griffiths, University of MelbourneAlastair R. Hall, North Carolina State UniversityR. Carter Hill, Louisiana State UniversityDaniel Houser, University of ArizonaCheng Hsiao, University of Southern CaliforniaJoann Jasiak, York UniversityMichael Keane, University of Minnesota and New York UniversityLynda Khalaf, University of LavalMaxwell L. King, Monash UniversityGary Koop, University of GlasgowLung-fei Lee, Hong Kong University of Science & TechnologyHelmut Lütkepohl, Humboldt UniversityEsfandiar Maasoumi, Southern Methodist University, DallasJames MacKinnon, Queen’s University, Ontario

G.S. Maddala, Ohio State UniversityRoberto S. Mariano, University of PennsylvaniaFrancesc Marmol, Universidad Carlos III de MadridErik Meijer, University of GroningenDenise R. Osborn, University of ManchesterM. Hashem Pesaran, Cambridge UniversityBenedikt M. Pötscher, University of ViennaGamini Premaratne, University of IllinoisIngmar R. Prucha, University of MarylandPaulo M.M. Rodrigues, University of AlgarveAris Spanos, Virginia Polytechnic Institute and State UniversityMark F.J. Steel, University of KentJames H. Stock, Harvard UniversityP.A.V.B. Swamy, Department of the Treasury, WashingtonGeorge S. Tavlas, Bank of GreecePravin K. Trivedi, Indiana UniversityAman Ullah, University of California – RiversideTom Wansbeek, University of GroningenMelvyn Weeks, Cambridge UniversityJeffrey M. Wooldridge, Michigan State University

LIST OF CONTRIBUTORS xi

Preface

This companion in theoretical econometrics is the first in a series of companionsin economics published by Blackwell. The emphasis is on graduate students ofeconometrics and professional researchers who need a guide, a friend, a compan-ion to lead them through this exciting yet ever growing and expanding field. Thisis not a handbook of long chapters or exhaustive surveys on the subject. Theseare simple chapters, written by international experts who were asked to give abasic introduction to their subject. These chapters summarize some of the wellknown results as well as new developments in the field and direct the readerto supplementary reading. Clearly, one single volume cannot do justice to thewide variety of topics in theoretical econometrics. There are five handbooksof econometrics published by North-Holland and two handbooks of appliedeconometrics published by Blackwell, to mention a few. The 32 chapters in thiscompanion give only a sample of the important topics in theoretical econometrics.We hope that students, teachers, and professionals find this companion useful.I would like to thank Al Bruckner who approached me with this idea and whoentrusted me with the editorial job, the 50 authors who met deadlines and pagelimitations.

I would also like to thank the numerous reviewers who read these chapters andcommented on them. These include Seung Ahn, Paul Bekker, Anil Bera, HermanBierens, Erik Biorn, Siddahrtha Chib, James Davidson, Francis Diebold, JuanDolado, Jean-Marie Dufour, Neil Ericsson, Denzil Fiebig, Philip Hans Franses,John Geweke, Eric Ghysels, David Giles, Jesús Gonzalo, Clive Granger, WilliamGreene, William Griffith, Alastair Hall, Bruce Hansen, R. Carter Hill, Cheng Hsiao,Hae-Shin Hwang, Svend Hylleberg, Michael Keane, Lynda Khalaf, Gary Koop,Lung-fei Lee, Qi Li, Oliver Linton, Helmut Lütkepohl, Essie Maasoumi, JamesMacKinnon, G.S. Maddala, Masao Ogaki, Denise Osborn, Pierre Perron, PeterPhillips, Ingmar Prucha, Peter Schmidt, Mark Steel, James Stock, Pravin Trivedi,Aman Ullah, Marno Verbeek, Tom Wansbeek, Rainer Winkelmann, and JeffreyWooldridge.

On a sad note, G.S. Maddala, a contributing author to this volume, died beforethis book was published. He was a leading figure in econometrics and a prolificresearcher whose writings touched students all over the world. He will be sorelymissed.

Finally, I would like to acknowledge the support and help of Blackwell Pub-lishers and the secretarial assistance of Teri Bush at various stages of the prepa-ration of this companion.

BADI H. BALTAGI

Texas A&M UniversityCollege Station, Texas

PREFACE xiii

Abbreviations

2SLS two-stage least squares3SLS three-stage least squaresa.s. almost sureACD Autoregressive Conditional DurationADF Augmented Dickey–FullerAE asymptotically equivalentAIC Aikake’s information criteriaAIMA asymptotically ideal modelAIMSE average integrated mean square errorAR autoregressiveAR(1) first-order autoregressiveARCH autoregressive conditional heteroskedasticityARFIMA autoregressive fractionally integrated moving averageARIMA autoregressive integrated moving averageARMA autoregressive moving averageBDS Brock, Dechert, and ScheinkmanBIC Bayesian information criteriaBKW Belsley, Kuh, and WelschBLUE best, linear unbiased estimatorBMC bound Monte CarloCAPM capital asset pricing modelCAPS consistent adjusted least squaresCDF (or cdf ) cumulative distribution functionCES constant elasticity of substitutionCFI comparative fit indexCG matrix matrix of contributions to the gradientCI confidence intervalCLT central limit theoremCM conditional momentCME conditional mean encompassing

CMT conditional moment testCPI consumer price indexCPS current population surveyCUAN consistent and uniformly asymptotic normalDEA data envelopment analysisDF Dickey–FullerDGLS dynamic generalized least squaresDGM data generating mechanismDGP data generating processDHF Dickey, Hasza, and FullerDLR double-length artificial regressionDOLS dynamic ordinary least squaresDW Durbin–WatsonDWH Durbin–Wu–HausmanEBA elimination-by-aspectsECM expectation conditional maximizationEM expectation maximizationEPE estimated prediction errorESS explained sums of squaresESSR restricted sum of squaresESSU unrestricted error sum of squaresEWMA exponentially weighted moving averageFCLT functional central limit theoremFGLS feasible generalized least squaresFIML full information maximum likelihoodFIVE full information instrumental variables efficientFM-OLS fully modified ordinary least squares estimatorFSD first-order stochastic dominateFWL Frisch–Waugh–LovellGARCH generalized autoregressive conditional heteroskedasticGEV generalized extreme valueGHK Geweke, Hajivassiliou, and KeaneGHM Gouriéoux, Holly, and MontfortGIS geographic information systemsGL generalized LorenzGLM generalized linear modelGLN Ghysels, Lee, and NohGLS generalized least squaresGML generalized maximum likelihoodGMM generalized method of momentsGNR Gauss–Newton regressionGSUR generalized seemingly unrelated regressionHAC heteroskedasticity and autocorrelation consistentHEBA hierarchical elimination-by-aspectsHEGY Hylleberg, Engle, Granger, and YooH–K Honoré and Kyriazidou

ABBREVIATIONS xv

HRGNR heteroskedasticity-robust Gauss–Newton regressioni.p. in probabilityIC information criteriaID independently distributedIIA independence of irrelevant alternativesIID independently identically distributedIIV iterated instrumental variableILS indirect least squaresIM information matrixIMSE integrated mean square errorINAR integer autoregressiveIP industrial productionIV instrumental variableJB Jarque–BeraKLIC Kullback–Leibler information criterionKPSS Kwiatkowski, Phillips, Schmidt, and ShinKS Kolmogorov–SmirnovKT Kuhn–TuckerLBI locally best invariantLCLS local constant least squaresLEF linear exponential familyLI limited informationLIML limited information maximum likelihoodLIVE limited information instrumental variables efficientLL local linearLLLS local linear least squaresLLN law of large numbersLLS local least squaresLM Lagrange multiplierLMC local Monte CarloLMP locally most powerfulLMPU locally most powerful unbiasedLPLS local polynomial least squaresLR likelihood ratioLS least squaresLSE least squares estimationLSTAR logistic smooth transition autoregressionM2SLS modified two-stage least squaresMA moving averageMA(1) first-order moving averageMC Monte CarloMCMC Markov Chain Monte CarloMD martingale differenceMDML multivariate dynamic linear regressionMIMIC multiple indicators-multiple causesML maximum likelihood

xvi ABBREVIATIONS

MLE maximum likelihood estimationMLR multivariate linear regressionMM method of momentsMMC maximized Monte CarloMML maximum marginal likelihoodMNL multinomial logitMNP multinomial probitMP most powerfulMS maximum scoreMSE mean square errorMSFE mean squared forecast errorMSL method of simulated likelihoodMSM method of simulated momentsMSS method of simulated scoresNB negative binomialNFI normed fit indexNLS nonlinear least squaresNMNL nested multinomial logitNN neural networkN–P Neyman–PearsonNPRSS nonparametric residual sum of squaresN–W Nadaraya–WatsonNYSE New York Stock ExchangeOLS ordinary least squaresOPG outer-product-of-the-gradientPDF probability distribution functionPLS predictive least squaresPML pseudo-MLPP Phillips–PerronPR probabilistic reductionPRSS parametric residual sum of squarespsd positive semi-definitePSP partial sum processQML quasi-MLQP quadratic programmingQRM qualitative response modelRCM random coefficient modelsRESET regression error specification testRIS recursive importance samplingRLS restricted least squaresRMSE root mean squared errorRMSFE root mean squared forecast errorRNI relative noncentrality indexRRR reduced rank regressionRS Rao’s scoreRSS residual sum of squares

ABBREVIATIONS xvii

s/n signal-to-noiseSA simulated annealingSAR spatial autoregressiveSD stochastic dominanceSEM simultaneous equations modelSET score encompassing testSMA spatial moving averageSML simulated maximum likelihoodSNP semi-nonparametricSP semiparametricSSD second-order stochastic dominateSSE sum of square errorSSR sum of squared residualsSTAR smooth transition autoregressionSUR(E) seemingly unrelated regressionSVD Stochastic Volatility DurationTAR transition autoregressionTSD third-order stochasticUI union intersectionUL uniform linearULLN uniform law of large numbersUMP uniformly most powerfulUMPI uniformly most powerful invariantUMPU uniformly most powerful unbiasedVAR vector autoregressionVECM vector error correction modelVIF variance-inflation factorVNM von Neumann–MorgensternW WaldWET Wald encompassing testwrt with respect to

xviii ABBREVIATIONS

INTRODUCTION 1

IntroductionBadi H. Baltagi

This is the first companion in econometrics. It covers 32 chapters written byinternational experts in the field. The emphasis of this companion is on “keepingthings simple” so as to give students of econometrics a guide through the mazeof important topics in econometrics. These chapters are helpful for readers andusers of econometrics who are not looking for exhaustive surveys on the subject.Instead, these chapters give the reader some of the basics and point to furtherreadings on the subject. The topics covered vary from basic chapters on serialcorrelation and heteroskedasticity, which are found in standard econometricstexts, to specialized topics that are covered by econometric society monographsand advanced books on the subject like count data, panel data, and spatial corre-lation. The authors have done their best to keep things simple. Space and timelimitations prevented the inclusion of other important topics, problems and exer-cises, empirical applications, and exhaustive references. However, we believethat this is a good start and that the 32 chapters contain an important selection oftopics in this young but fast growing field.

Chapter 1 by Davidson and MacKinnon introduces the concept of an artificialregression and gives three conditions that an artificial regression must satisfy.The widely used Gauss–Newton regression (GNR) is used to show how artificialregressions can be used for minimizing criterion functions, computing one-stepestimators, calculating covariance matrix estimates, and more importantly com-puting test statistics. This is illustrated for testing the null hypothesis that asubset of the parameters of a nonlinear regression model are zero. It is shownthat the test statistic can be computed as an explained sum of squares of the GNRdivided by a consistent estimate of the residual variance. Two ways of comput-ing this statistic are: (i) the sample size times the uncentered R2 of the GNR, or (ii)an ordinary F-statistic testing the subset of the parameters are zero from theGNR. The two statistics are asymptotically equivalent under the null hypothesis.The GNR can be used for other types of specification tests, including serial corre-lation, nonnested hypothesis and obtaining Durbin–Wu–Hausman type tests.This chapter also shows how to make the GNR robust to heteroskedasticity of

2 B.H. BALTAGI

unknown form. It also develops an artificial regression for the generalized methodof moments (GMM) estimation. The outer-product-of-the-gradient (OPG) regres-sion is also discussed. This is a simple artificial regression which can be usedwith most models estimated by maximum likelihood. It is shown that the OPGsatisfies the three conditions of an artificial regression and has the usual uses ofan artificial regression. It is appealing because it requires only first derivatives.However, it is demonstrated that the OPG regression yields relatively poor esti-mates of the covariance matrices and unreliable test statistics in small samples.In fact, test statistics based on the OPG regression tend to overreject, often veryseverely. Davidson and MacKinnon also discuss double-length or triple-lengthartificial regressions where each observation makes two or three contributionsto the criterion function. In this case, the artificial regression has twice or threetimes the sample size. This artificial regression can be used for many purposes,including tests of models with different functional forms. Finally, this chapterextends the GNR to binary response models such as the logit and probit models.For further readings on this subject, see Davidson and MacKinnon (1993).

Chapter 2 by Bera and Premaratne gives a brief history of hypothesis testing instatistics. This journey takes the reader through the basic testing principles lead-ing naturally to several tests used by econometricians. These tests are then linkedback to the basic principles using several examples. This chapter goes throughthe Neyman–Pearson lemma and the likelihood ratio test. It explains what ismeant by locally most powerful tests and it gives the origins of the Rao-scoretest. Next, locally most powerful unbiased tests and Neyman’s smooth test arereviewed. The interrelationship among the holy trinity of test statistics, i.e., theWald, likelihood ratio, and Rao-score tests is brought home to the reader by anamusing story. Neyman’s C(α) test is derived and motivated. This approachprovides an attractive way of dealing with nuisance parameters. Next, this chap-ter goes through some of the application of testing principles in econometrics.Again a brief history of hypothesis testing in econometrics is given beginningwith the work of Ragnar Frisch and Jan Tinbergen, going into details throughthe Durbin–Watson statistic linking its origins to the Neyman–Pearson lemma.The use of the popular Rao-score test in econometrics is reviewed next, empha-sizing that several tests in econometrics old and new have been given a score testinterpretation. In fact, Bera and Premaratne consider the conditional moment testdeveloped by Newey (1985) and Tauchen (1985) and derive its score test inter-pretation. Applications of Neyman’s C(α) test in econometrics are cited and someof the tests in econometrics are given a smooth test interpretation. The chapterfinishes with a double warning about testing: be careful how you interpret thetest results. Be careful what action you take when the null is rejected.

Chapter 3 by King surveys the problem of serial correlation in econometrics.Ignoring serial correlation in the disturbances can lead to inefficient parameterestimates and a misleading inference. This chapter surveys the various waysof modeling serial correlation including Box–Jenkins time series models. Auto-regressive and moving average (ARMA) models are discussed, highlighting thecontributions of Cochrane and Orcutt (1949) for the AR(1) model; Thomas andWallis (1971) for the restricted AR(4) process and Nichols, Pagan, and Terrell

INTRODUCTION 3

(1975) for the MA(1) process. King argues that modeling serial correlation in-volves taking care of the dynamic part of model specification. The simple versionof a dynamic model includes the lagged value of the dependent variable amongthe regressors. The relationship between this simple dynamic model and theAR(1) model is explored. Estimation of the linear regression model with ARMAdisturbances is considered next. Maximum likelihood estimation (MLE) undernormality is derived and a number of practical computation issues are discussed.Marginal likelihood estimation methods are also discussed which work well forestimating the ARMA process parameters in the presence of nuisance parameters.In this case, the nuisance parameters include the regression parameters and theresidual variance. Maximizing marginal likelihoods were shown to reduce theestimation bias of maximum likelihood methods. Given the nonexperimentalnature of economic data and the high potential for serial correlation, King arguesthat it is important to test for serial correlation. The von Neuman as well asDurbin–Watson (DW) tests are reviewed and their statistical properties are dis-cussed. For example, the power of the DW test tends to decline to zero whenthe AR(1) parameter ρ tends to one. In this case, King suggests a class of pointoptimal tests that provide a solution to this problem. LM, Wald and LR tests forserial correlation are mentioned, but King suggests constructing these tests usingthe marginal likelihood rather than the full likelihood function. Testing for AR(1)disturbances in the dynamic linear regression model is also studied, and thedifficulty of finding a satisfactory test for this model is explained by showing thatthe model may suffer from a local identification problem. The last section of thischapter takes up the problem of deciding what lags should be used in the ARIMAmodel. Model selection criteria are recommended rather than a test of hypothesesand the Bayesian information criteria is favored because it is consistent. Thismeans that as the sample size goes to infinity, this criteria selects the correctmodel from a finite number of models with probability one.

Chapter 4 by Griffiths gives a lucid treatment of the heteroskedasticity prob-lem. The case of a known variance covariance term is treated first and general-ized least squares is derived. Finite sample inference under normality as well aslarge sample inference without the normality assumption are summarized in thecontext of testing linear restrictions on the regression coefficients. In addition,inference for nonlinear restrictions on the regression coefficients is given and theconsequences of heteroskedasticity on the least squares estimator are explained.Next, the case of the unknown variance covariance matrix is treated. In thiscase, several specifications of the form of heteroskedasticity are entertained andmaximum likelihood estimation under normality is derived. Tests of linearrestrictions on the regression coefficients are then formulated in terms of theML estimates. Likelihood ratio, Wald and LM type tests of heteroskedasticity aregiven under normality of the disturbances. Other tests of heteroskedasticity aswell as Monte Carlo experiments comparing these tests are cited. Adaptiveestimators that assume no form of heteroskedasticity are briefly surveyed as wellas several other miscellaneous extensions. Next, this chapter discusses Bayesianinference under heteroskedasticity. The joint posterior probability density func-tion is specified assuming normality for the regression and uninformative priors

4 B.H. BALTAGI

on heteroskedasticity. An algorithm for obtaining the marginal posterior prob-ability density function is given.

Chapter 5 by Fiebig, surveys the most recent developments on seemingly unre-lated regressions (SUR) including both applied and theoretical work on the speci-fication, estimation and testing of SUR models. This chapter updates the surveyby Srivastava and Dwivedi (1979) and the book by Srivastava and Giles (1987). Abasic introduction of the SUR model introduced by Zellner (1962), is given alongwith extensions of the model to allow for more general stochastic specifications.These extensions are driven in part by diagnostic procedures as well as theoret-ical economic arguments presented for behavioral models of consumers and pro-ducers. Here the problem of testing linear restrictions which is important fortesting demand systems or estimating say a cost function with share equations isstudied. In addition, tests for the presence of contemporaneous correlation inSUR models as well as spatial autocorrelation are discussed. Next, SUR with miss-ing observations and computational matters are reviewed. This leads naturallyto a discussion of Bayesian methods for the SUR model and improved estima-tion methods for SUR which include several variants of the Stein-rule family andthe hierarchical Bayes estimator. Finally, a brief discussion of misspecification,robust estimation issues, as well as extensions of the SUR model to time seriesmodeling and count data Poisson regressions are given.

Chapter 6 by Mariano, considers the problem of estimation in the simultane-ous equation model. Both limited as well as full information estimators are dis-cussed. The inconsistency of ordinary least squares (OLS) is demonstrated. Limitedinformation instrumental variable estimators are reviewed including two-stageleast squares (2SLS), limited information instrumental variable efficient (LIVE),Theil’s k-class, and limited information maximum likelihood (LIML). Full in-formation methods including three-stage least squares (3SLS), full informationinstrumental variables efficient (FIVE) and full information maximum likeli-hood (FIML) are studied next. Large sample properties of these limited and fullinformation estimators are summarized and conditions for their consistency andasymptotic efficiency are stated without proof. In addition, the finite sampleproperties of these estimators are reviewed and illustrated using the case of twoincluded endogenous variables. Last, but not least, practical implications of thesefinite sample results are given. These are tied up to the recent literature on weakinstruments.

Chapter 7 by Bekker and Wansbeek discusses the problem of identification inparametric models. Roughly speaking, a model is identified when meaningfulestimates of its parameters can be obtained. Otherwise, the model is under-identified. In the latter case, different sets of parameter values agree well withthe statistical evidence rendering scientific conclusions based on any estimates ofthis model void and dangerous. Bekker and Wansbeek define the basic conceptsof observational equivalence of two parameter points and what is meant bylocal and global identification. They tie up the notion of identification to that ofthe existence of a consistent estimator, and provide a link between identifica-tion and the rank of the information matrix. The latter is made practically usefulby presenting it in terms of the rank of a Jacobian matrix. Although the chapter

INTRODUCTION 5

is limited to the problem of parametric identification based on sample informa-tion and exact restrictions on the parameters, extensions are discussed and thereader is referred to the book by Bekker, Merckens, and Wansbeek (1994) forfurther analysis.

Chapter 8 by Wansbeek and Meijer discusses the measurement error problemin econometrics. Many economic variables like permanent income, productivityof a worker, consumer satisfaction, financial health of a firm, etc. are latent vari-ables that are only observed with error. This chapter studies the consequences ofmeasurement error and latent variables in econometric models and possible solu-tions to these problems. First, the linear regression model with errors in variablesis considered, the bias and inconsistency of OLS is demonstrated and the attenu-ation phenomenon is explained. Next, bounds on the parameters of the modelare obtained by considering the reverse regression. Solutions to the errors invariables include restrictions on the parameters to identify the model and henceyield consistent estimators of the parameters. Alternatively, instrumental vari-ables estimation procedures can be employed, or nonnormality of the errors maybe exploited to obtain consistent estimates of these parameters. Repeated meas-urements like panel data on households, firms, regions, etc. can also allow theconsistent estimation of the parameters of the model. The second part of thischapter gives an extensive discussion of latent variable models including factoranalysis, the multiple indicators-multiple causes (MIMIC) model and a frequentlyused generalization of the MIMIC model known as the reduced rank regressionmodel. In addition, general linear structural equation models estimated by LISRELare considered and maximum likelihood, generalized least squares, test statistics,and model fit are studied.

Chapter 9 by Wooldridge provides a comprehensive account of diagnostictesting in econometrics. First, Wooldridge explains how diagnostic testing differsfrom classical testing. The latter assumes a correctly specified parametric modeland uses standard statistics to test restrictions on the parameters of this model,while the former tests the model for various misspecifications. This chapter con-siders diagnostic testing in cross section applications. It starts with diagnostictests for the conditional mean in the linear regression model. Conditional meandiagnostics are computed using variable addition statistics or artificial regres-sions (see Chapter 1 by Davidson and MacKinnon). Tests for functional form aregiven as an example and it is shown that a key auxiliary assumption neededto obtain a usable limiting distribution for the usual nR2 (LM) test statistic ishomoskedasticity. Without this assumption, the limiting distribution of the LMstatistic is not χ2 and the resulting test based on chi-squared critical values maybe asymptotically undersized or oversized. This LM statistic is adjusted to allowfor heteroskedasticity of unknown form under the null hypothesis. Next, testingfor heteroskedasticity is considered. A joint test of the conditional mean and con-ditional variance is an example of an omnibus test. However, if this test rejectsit is difficult to know where to look. A popular omnibus test is White’s (1982)information matrix test. This test is explicitly a test for homoskedasticity, condi-tional symmetry and homokurtosis. If we reject, it may be for any of these reasonsand it is not clear why one wants to toss out a model because of asymmetry, or

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because its fourth and second moments do not satisfy the same relationship asthat for a normal distribution. Extensions to nonlinear models are discussed nextas well as diagnostic tests for completely specified parametric models like limiteddependent variable models, probit, logit, tobit, and count data type models.The last section deals with diagnostic testing in time series models. In this case,one can no longer assume that the observations are independent of one anotherand the discussion of auxiliary assumptions under the null is more complicated.Wooldridge discusses different ways to make conditional mean diagnostics robustto serial correlation as well as heteroskedasticity. Testing for heteroskedasticityin time series contexts and omnibus tests on the errors in time series regressions,round up the chapter.

Chapter 10 by Pötscher and Prucha gives the basic elements of asymptotictheory. This chapter discusses the crucial concepts of convergence in probabilityand distribution, the convergence properties of transformed random variables,orders of magnitude of the limiting behavior of sequences of random variables,laws of large numbers both for independent and dependent processes, and cen-tral limit theorems. This is illustrated for regression analysis. Further readingsare suggested including the recent book by Pötscher and Prucha (1997).

Chapter 11 by Hall provides a thorough treatment of the generalized methodof moments (GMM) and its applications in econometrics. Hall explains that themain advantage of GMM for econometric theory is that it provides a generalframework which encompasses many estimators of interest. For econometric ap-plications, it provides a convenient method of estimating nonlinear dynamicmodels without complete knowledge of the distribution of the data. This chaptergives a basic definition of the GMM estimation principle and shows how it ispredicated on the assumption that the population moment condition providessufficient information to uniquely determine the unknown parameters of thecorrectly specified model. This need not be the case, and leads naturally to adiscussion of the concepts of local and global identification. In case the modelis overidentified, Hall shows how the estimation effects a decomposition on thepopulation moment condition into identifying restrictions upon which the estima-tion is based, and overidentifying restrictions which are ignored in the estima-tion. This chapter describes how the estimated sample moment can be used toconstruct the overidentification restrictions test for the adequacy of the modelspecification, and derives the consistency and asymptotic distribution of the esti-mator. This chapter also characterizes the optimal choice of the weighting matrixand shows how the choice of the weight matrix impacts the GMM estimator viaits asymptotic variance. MLE is shown to be a special case of GMM. However,MLE requires that we know the distribution of the data. GMM allows one tofocus on the information used in the estimation and thereby determine the conse-quences of choosing the wrong distribution. Since the population moment condi-tion is not known in practice, the researcher is faced with a large set of alternativesto choose from. Hall focuses on two extreme scenarios where the best and worstchoices are made. The best choice is the population moment condition whichleads to the estimator with the smallest asymptotic variance. The worst choice iswhen the population moment condition does not provide enough information to

INTRODUCTION 7

identify the unknown parameters of our model. This leads to a discussion ofnearly uninformative population moment conditions, their consequence and howthey might be circumvented.

Chapter 12 by Hill and Adkins gives a lucid discussion of the collinearityproblem in econometrics. They argue that collinearity takes three distinct forms.The first is where an explanatory variable exhibits little variability and thereforemakes it difficult to estimate its effect in a linear regression model. The second iswhere two explanatory variables exhibit a large correlation leaving little inde-pendent variation to estimate their separate effects. The third is where there maybe one or more nearly exact linear relationships among the explanatory variables,hence obscuring the effects of each independent variable on the dependent vari-able. Hill and Adkins examine the damage that multicollinearity does to estima-tion and review collinearity diagnostics such as the variance decomposition ofBelsley, Kuh, and Welsch (1980), the variance-inflation factor and the determi-nant of X′X, the sum of squares and cross-product of the regressor matrix X. Oncecollinearity is detected, this chapter discusses whether collinearity is harmful byusing Belsley’s (1982) test for adequate signal to noise ratio in the regressionmodel and data. Next, remedies to harmful collinearity are reviewed. Here thereader is warned that there are only two safe paths. The first is obtaining moreand better data which is usually not an option for practitioners. The secondis imposing additional restrictions from economic theory or previous empiricalresearch. Hill and Adkins emphasize that although this is a feasible option, onlygood nonsample information should be used and it is never truly known whetherthe information introduced is good enough. Methods of introducing exact andinexact nonsample information including restricted least squares, Stein-ruleestimators, inequality restricted least squares, Bayesian methods, the mixedestimation procedure of Theil and Goldberger (1961) and the maximum entropyprocedure of Golan, Judge, and Miller (1996) are reviewed. In addition, twoestimation methods designed specifically for collinear data are discussed if onlyto warn the readers about their use. These are ridge regression and principalcomponents regression. Finally, this chapter extends the collinearity analysis tononlinear models.

Chapter 13 by Pesaran and Weeks gives an overview of the problem of non-nested hypothesis testing in econometrics. This problem arises naturally whenrival economic theories are used to explain the same phenomenon. For example,competing theories of inflation may suggest two different sets of regressors nei-ther of which is a special case of the other. Pesaran and Weeks define non-nestedmodels as belonging to separate families of distributions in the sense that none ofthe individual models may be obtained from the remaining either by impositionof parameter restrictions or through a limiting process. This chapter discusses theproblem of model selection and how it relates to non-nested hypothesis testing.By utilizing the linear regression model as a convenient framework, Pesaranand Weeks examine three broad approaches to non-nested hypotheses testing:(i) the modified (centered) log-likelihood ratio procedure also known as the Coxtest; (ii) the comprehensive models approach, whereby the non-nested modelsare tested against an artificially constructed general model that includes the

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non-nested models as special cases; and (iii) the encompassing approach, wherethe ability of one model to explain particular features of an alternative model istested directly. This chapter also focuses on the Kullback-Leibler divergence meas-ure which has played a pivotal role in the development of a number of non-nested test statistics. In addition, the Vuong (1989) approach to model selection,viewed as a hypothesis testing problem is also discussed. Finally, practical prob-lems involved in the implementation of the Cox procedure are considered. Thisinvolves finding an estimate of the Kullback-Leibler measure of closeness of thealternative to the null hypothesis which is not easy to compute. Two methods arediscussed to circumvent this problem. The first examines the simulation approachand the second examines the parametric bootstrap approach.

Chapter 14 by Anselin provides an excellent review of spatial econometrics.These methods deal with the incorporation of spatial interaction and spatial struc-ture into regression analysis. The field has seen a recent and rapid growth spurredboth by theoretical concerns as well as the need to apply econometric models toemerging large geocoded databases. This chapter outlines the basic terminologyand discusses in some detail the specification of spatial effects including theincorporation of spatial dependence in panel data models and models with quali-tative variables. The estimation of spatial regression models including maximumlikelihood estimation, spatial 2SLS, method of moments estimators and a numberof other approaches are considered. In addition, specification tests for spatialeffects as well as implementation issues are discussed.

Chapter 15 by Cameron and Trivedi gives a brief review of count data regres-sions. These are regressions that involve a dependent variable that is a count,such as the number of births in models of fertility, number of accidents in studiesof airline safety, hospital or doctor visits in health demand studies, number oftrips in models of recreational demand, number of patents in research and devel-opment studies or number of bids in auctions. In these examples, the sample isconcentrated on a few discrete values like 0, 1 and 2. The data is skewed to theleft and the data is intrinsically heteroskedastic with its variance increasing withthe mean. Two methods of dealing with these models are considered. The first isa fully parametric approach which completely specifies the distribution of thedata and restricts the dependent variable to nonnegative integer values. Thisincludes the Poisson regression model which is studied in detail in this chapterincluding its extensions to truncated and censored data. Limitations of the Poissonmodel, notably the excess zeros problem and the overdispersion problem areexplained and other parametric models, superior to the Poisson are presented.These include continuous mixture models, finite mixture models, modified countmodels and discrete choice models. The second method of dealing with countdata is a partial parametric method which focuses on modeling the data via theconditional mean and variance. This includes quasi-maximum likelihood estima-tion, least squares estimation and semiparametric models. Extensions to othertypes of data notably time series, multivariate, and panel data are discussed andthe chapter concludes with some practical recommendations. For further read-ings and diagnostic procedures, the reader is referred to the recent econometricsociety monograph on count data models by Cameron and Trivedi (1998).

INTRODUCTION 9

Chapter 16 by Hsiao gives a selected survey of panel data models. First, thebenefits from using panels are discussed. This includes more degrees of freedom,controlling for omitted variable bias, reducing the problem of multicollinearityand improving the accuracy of parameter estimates and predictions. A generalencompassing linear panel data model is provided which includes as specialcases the error components model, the random coefficients model and the mixedfixed and random coefficients model. These models assume that some variablesare subject to stochastic constraints while others are subject to deterministic con-straints. In practice, there is little knowledge about which variables are subject tostochastic constraints and which variables are subject to deterministic constraints.Hsiao recommends the Bayesian predictive density ratio method for selectingbetween two alternative formulations of the model. Dynamic panel data modelsare studied next and the importance of the initial observation with regards to theconsistency and efficiency of the estimators is emphasized. Generalized methodof moments estimators are proposed and the problem of too many orthogonalityconditions is discussed. Hsiao suggests a transformed maximum likelihood estim-ator that is asymptotically more efficient than GMM. Hsiao also reports thatthe mean group estimator suggested by Pesaran and Smith (1995) does not per-form well in finite samples. Alternatively, a hierarchical Bayesian approach per-forms well when T is small and the initial value is assumed to be a fixed constant.Next, the existence of individual specific effects in nonlinear models is discussedand the conditional MLE approach of Chamberlain (1980) is given. The problembecomes more complicated if lagged dependent variables are present. T ≥ 4 isneeded for the identification of a logit model and this conditional method willnot work with the presence of exogenous variables. In this case, a consistent andasymptotically normal estimator proposed by Honoré and Kyriazidou (1997) issuggested. An alternative semiparametric approach to estimating nonlinear panelmodels is the maximum score estimator proposed by Manski (1975). This appliessome data transformation to eliminate the individual effects if the nonlinear modelis of the form of a single index model with the index possessing a linear struc-ture. This estimator is consistent but not root n consistent. A third approachproposed by Lancaster (1998) finds an orthogonal reparametrization of the fixedeffects such that the new fixed effects are independent of the structural para-meters in the information matrix sense. Hsiao discusses the limitations of allthree methods, emphasizing that none of these approaches can claim generalapplicability and that the consistency of nonlinear panel data estimators must beestablished on a case by case basis. Finally, Hsiao treats missing observations inpanels. If individuals are missing randomly, most estimators in the balancedpanel case can be easily generalized to the unbalanced case. With sample selec-tion, Hsiao emphasizes the dependence of the MLE and Heckman’s (1979) two-step estimators on the exact specification of the joint error distribution. If thisdistribution is misspecified, then these estimators are inconsistent. Alternativesemiparametric methods are discussed based on the work of Ahn and Powell(1993), Kyriazidou (1997), and Honoré and Kyriazidou (1998).

Chapter 17 by Maddala and Flores-Lagunes gives an update of the econometricsof qualitative response models. First, a brief introduction to the basic material on

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the estimation of binary and multinomial logit and probit models is given andthe reader is referred to Maddala (1983) for details. Next, this chapter reviewsspecification tests in qualitative response models and the reader is referred tothe recent review by Maddala (1995). Panel data with qualitative variables andsemiparametric estimation methods for qualitative response models are reviewedincluding Manski’s maximum score, quasi-maximum likelihood, generalizedmaximum likelihood and the semi-nonparametric estimator. Maddala and Flores-Lagunes comment on the empirical usefulness and drawbacks of the differentmethods. Finally, simulation methods in qualitative response models are reviewed.The estimation methods discussed are the method of simulated moments, themethod of simulated likelihood and the method of simulated scores. Someexamples are given comparing these simulation methods.

Chapter 18 by Lee gives an extensive discussion of the problem of self-selectionin econometrics. When the sample observed is distorted and is not representativeof the population under study, sample selection bias occurs. This may be due tothe way the sample was collected or it may be due to the self-selection decisionsby the agents being studied. This sample may not represent the true populationno matter how large. This chapter discusses some of the conventional sampleselection models and counterfactual outcomes. A major part of this chapterconcentrates on the specification, estimation, and testing of parametric modelsof sample selection. This includes Heckman’s two-stage estimation procedure aswell as maximum likelihood methods, polychotomous choice sample selectionmodels, simulation estimation methods, and the estimation of simultaneousequation sample selection models. Another major part of this chapter focuseson semiparametric and nonparametric approaches. This includes semiparametrictwo-stage estimation, semiparametric efficiency bound and semiparametricmaximum likelihood estimation. In addition, semiparametric instrumental vari-able estimation and conditional moments restrictions are reviewed, as well assample selection models with a tobit selection rule. The chapter concludeswith the identification and estimation of counterfactual outcomes.

Chapter 19 by Swamy and Tavlas describes the purpose, estimation, and use ofrandom coefficient models. Swamy and Tavlas distinguish between first gen-eration random coefficient models that sought to relax the constant coefficientassumption typically made by researchers in the classical tradition and secondgeneration random coefficient models that relax the assumptions made regard-ing functional forms, excluded variables, and absence of measurement error. Theauthors argue that the latter are useful approximations to reality because theyprovide a reasonable approximation to the underlying “true” economic relation-ship. Several model validation criteria are provided. Throughout, a demand formoney model is used as a backdrop to explain random coefficient models and anempirical application to United Kingdom data is given.

Chapter 20 by Ullah provides a systematic and unified treatment of estima-tion and test of hypotheses for nonparametric and semiparametric regressionmodels. Parametric approaches to specifying functional form in econometricsmay lead to misspecification. Nonparametric and semiparametric approachesprovide alternative estimation procedures that are more robust to functional form