Topics in Advanced Econometrics Volume II r l",', I I I Phoebus J Dhrymes Topics in Advanced Econometrics Volume II Linear and Nonlinear Simultaneous Equations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Phoebus J Dhryrnes Department of Economics Columbia University New York, NY 10027 USA Library of Congress Cataloging-in-Publication Data Dhrymes, Phoebus J Topics in advanced econometrics (v, 2: Linear and nonlinear simultaneous equations) Includes bibliographical references and index Contents: [II Probability foundations-v Linear and nonlinear simultaneous equations I Econometrics Probabilities I Title HB139.D49 1989 330' 01'5195 89-27330 ISBN 0-387-94156-8 Printed on acid-free paper © 1994Springer-Verlag New York, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Natalie Johnson; manufacturing supervised by Genieve Shaw Photocomposed copy produced using the author's LaTeX files Printed and bound by Braun-Brumfield, Inc., Ann Arbor, MI Printed in the United States of America 987654321 ISBN 0-387-94156-8 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-94156-8 Springer-Verlag Berlin Heidelberg New York To Ingram Olkin and Henri Theil, who stimulated my early interest in econometrics Preface This book is intended for second year graduate students and professionals who have an interest in linear and nonlinear simultaneous equations models It basically traces the evolution of econometrics beyond the general linear model (GLM), beginning with the general linear structural econometric model (GLSEM) and ending with the generalized method of moments (GMM) Thus, it covers the identification problem (Chapter 3), maximum likelihood (ML) methods (Chapters and 4), two and three stage least squares (28LS, 38L8) (Chapters and 2), the general nonlinear model (GNLM) (Chapter 5), the general nonlinear simultaneous equations model (GNLSEM), the special case of GNLSEM with additive errors, nonlinear two and three stage least squares (NL2SLS, NL3SLS), the GMM for GNLSEM, and finally ends with a brief overview of causality and related issues, (Chapter 6) There is no discussion either of limited dependent variables, or of unit root related topics It also contains a number of significant innovations In a departure from the custom of the literature, identification and consistency for nonlinear models is handled through the Kullback information apparatus, as well as the theory of minimum contrast (MC) estimators In fact, nearly all estimation problems handled in this volume can be approached through the theory of MC estimators The power of this approach is demonstrated in Chapter 5, where the entire set of identification requirements for the GLSEM, in an ML context, is obtained almost effortlessly, through the apparatus of Kullback information The limiting distribution of dynamic GLSEM is handled through various convergence theorems for dependent sequences and a martingale difference viii Preface central limit theorem on a step by step basis, so that the reader may appreciate the complexity of the problems and the manner in which such problems are resolved A simplified (two step) FIML estimator is derived whose computational complexity is quite analogous to that of 3SLS; this enables the reader to see precisely why the two estimators need not be numerically identical even if 3SLS is iterated The method of generalized moments (GMM) estimator is presented as a variant of a 3SLS-like estimator in the context of the GLSEM with additive errors Because notation has been a problem in this subject, I have maintained a consistent notation throughout the volume, so that one can read about FIML, LIML, 2SLS, 3SLS, and GMM in the same notation and mutatis mutandis with the same conventions and formulations This facilitates the teaching of the subject, and reduces the unproductive time devoted to reconciliation of alternative notations and conventions The material in this volume can be used as the basis for a variety of one semester or quarter courses, depending on the level of preparation of the class If students are conversant with a modicum of modern probability theory, the material may be covered for the most part in a semester course If not, one has the option of concentrating on Chapters 1, 3, and and those parts of Chapter that not delve too deeply into asymptotic theory Alternatively, one might devote a number of lectures on the probability background and let Topics in Advanced Econometrics: Probability Foundations (Volume I) serve as a reference for various convergence and central limit theorems needed in the development of asymptotic theory Thus, a semester course may be based on Chapter 1, parts of Chapter 2, and parts of Chapters and This basically leaves out the classic identification discussion and ML estimation, but covers nonlinear methods in the context of the general linear model as well as the GNLSEM with additive errors In my own teaching, I devote approximately two weeks to various convergence results from Topics in Advanced Econometrics: Probability Foundations (Volume I) and, by and large, let this as well as my other book Mathematics for Econometrics serve as reference material Normally, Chapter is never reached, and is covered in the follow-up course on Time Series, the discussion of GMM serving as a natural interface between these two strands of the literature I have developed the contents of this volume over several years, and nearly every part has been utilized, at one time or another, as class notes at Columbia University I wish to record here my appreciation for the many suggestions I have received from successive generations of students and It would not be an exaggeration to say that in reading the literature on this subject, perhaps more than half the effort involved is devoted to deciphering the particular notation and conventions of the material being studied Preface ix hope that their advice has made the presentation smoother and more easily comprehensible Finally, the general tenor of the presentation, as well as the selection of topics, invariably reflects in part the author's conceptual framework and the role envisioned for the subject in scientific pursuits It has always been my view that good empirical econometrics has to be informed by economic theory and, equally so, by econometric theory This requires practitioners to have a thorough grounding in the techniques employed for the purpose of empirical inference I deplore the employment of complex or opaque procedures when this is clearly not required by the problem at hand Equally important, when writing on theoretical issues it is highly desirable to be sufficiently well aware of first principles This enables the investigator to bring to bear the appropriate tools in the analysis of the issues under discussion and reduces excessive reliance on broad and general theorems to solve relatively straightforward problems, a feature not uncommon in the literature of econometric theory These concerns have led me, on one hand, to give perhaps too extensive a discussion of the underlying conceptual framework, notational conventions, and the motivation and rationalization of the assumptions made, and on the other, they have led me to pursue most proofs as explicitly as I could manage I hope I have succeeded in setting forth the richness of the literature on the subject as it was developed in the past fifty years or so, and that this volume will be equally useful to the advanced student, as well as the interested professional both in economics and in other disciplines as well Phoebus J Dhrymes Bronxville, NY July 1993 Bibliography 391 Hausman, J A and W E Taylor (1981), "Panel Data and Unobservable Individual Effects", Econometrica, vol 49, pp 1377-1398 Henderson, H V and S R Searle (1981), "The Vee-Permutation Matrix, the Vec Operator and Kronecker Products: A Review", Linear and Multilinear Algebra, vol 9, pp 271-288 Holly, A (1982), "A Remark on Hausman's Specification Test," Econometrica, vol 50, pp 749-759 Hood, W C and T C Koopmans (eds.) (1953), Studies in Econometric Method, Cowles Foundation Monograph No 14, New York: Wiley Hsiao, C (1983), "Identification", Ch in Griliches, Z and M D Intrilligator (eds.), (1983) Hsiao, C (1985), The Analysis of Panel Data, Cambridge: Cambridge University Press Hwang, H (1980), "A Comparison of Tests of Overidentifying Restrictions", Econometrica, vol 48, pp 1821-1825 Johnson N L., and S Katz (1970), Continuous Univariate Distributions-2, Boston: Houghton Mifflin Jorgenson, D W., and J Laffont (1974), "Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances", Annals of Economic and Social Measurement, vol 3, pp 615-640 Kadane, J B (1974), "Testing a Subset of the Overidentifying Restrictions", Econometrica, 42, pp 853-868 Kadane, J B and T.W Anderson (1977), "A Comment on the Test of Overidentifying Restrictions", Econometrica, vol 45, pp 1027-1032 Kalejian, H H (1971), "Two-Stage Least Squares and Econometric Systems Linear in Parameters but Nonlinear in the Endogenous Variables", Journal of the American Statistical Association, vol 66, pp 373-374 Khazzoom, J D (1976), "An Indirect Least Squares Estimator for Overidentified Equations", Econometrica, vol 44, pp 741-750 Klein, R (1950), Economic Fluctuations in the United States, 1921-1941, New York: Wiley Klein, R (1955), "On the Interpretation of Theil's Method of Estimating Economic Relationships", Metroeconomica, vol 7, pp 147-153 Klein, L R and H Barger (1954), "A Quarterly Model of the U.S Economy", Journal of the American Statistical Association, vol 49, pp 413-437 Klein, L R and A S Goldberger (1955), An Econometric Model of the United States, 1929-1952, Amsterdam: North Holland Koopmans, T C (ed.) (1950), Statistical Inference in Dynamic Economic Models, Monograph 10, Cowles Commission for Research in Economics, New York: Wiley 392 Bibliography Koopmans, T C and W C Hood (1953), "The Estimation of Simultaneous Linear Economic Relationships", Ch in Hood W C and T C Koopmans (eds.), (1953) Kullback, S (1968), Information Theory and Statistics, New York: Dover Liu, T C (1960), "Underidentification, Structural Estimation and Forecasting", Econometrica, vol 28, pp 855-865 Liu, T C and W J Breen (1969), "The Covariance Matrix of the Limited Information Estimator and the Identification Test" , Econometrica, vol 37, pp 222-227 Liu, T C and W J Breen (1972), "The Covariance Matrix of the Limited Information Estimator and the Identification Test: A Reply", Econometrica, vol 40, pp 905-906 Mann, H B and A Wald (1943), "On the Statistical Treatment of Linear Stochastic Difference Equations" , Econometrica, vol 11, pp 173-220 Nagar, A L (1962), "Double k -class Estimators of Parameters in Simultaneous Equations and Their Small Sample Properties" , International Economic Review, vol 3, pp 168-188 Newey, W K (1985), "Maximum Likelihood Specification Testing and Conditional Moment Tests", Econometrica, vol 53, pp 1047-1070 Rao, C R (1950), "Methods of Scoring Linkage Data Giving Simultaneous Segregation of Three Factors", Heredity, vol 4, pp 37-59 Rao, C R (1972), second edition, Linear Statistical Inference and Its Applications, New York: Wiley Ravenkar, N and P Mallela (1972), "The Power of an F-test in the Context of a Structural Equation", Econometrica, vol 40, pp 913-916 Rothenberg, T J and C T Leenders (1964), "Efficient Estimation of Simultaneous Systems", Econometrica, vol 32, pp 57-76 Sargan, J D (1964), "Three-Stage Least-Squares and Full Maximum Likelihood Estimates", Econometrica, vol 32, pp 77-81 Scheffe, H (1959), The Analysis of Variance, New York: Wiley Scheffe, H (1977), "A Note on a Formulation of the S-method of Multiple Comparison", Journal of the American Statistical Association, vol 72, pp 143-146 Sims, C A (1972), "Money, Income and Causality", American Economic Review, vol 62, pp 540-552 Sims, C A (1980), "Macroeconomics and Reality", Econometrica, vol 48, pp.1-48 Szroeter, J (1983), "Generalized Wald Methods for Testing Nonlinear Implicit and Overidentifying Restrictions", Econometrica, vol 51, pp 335-353 Bibliography 393 Theil, H (1953), "Estimation and Simultaneous Correlation in Complete Equation Systems", mimeograph, The Hague: Central Plan Bureau Theil, H (1958), Economic Forecasts and Policy, Amsterdam: North Holland Theil, H and A Zellner (1962), "Three Stage Least Squares: Simultaneous Estimation of Simultaneous Equations", Econometrica, vol vol 30, pp 54-74 Tinbergen, J (1939), Statistical Testing of Business Cycle Theories, vol II: Business Cycles in the USA, 1919-1932, Geneva: League of Nations Wald, A (1950), "Note on the Identification of Economic Relations", Ch in Koopmans, T C (ed.) (1950), pp 238-244; also reprinted in Anderson, T W (ed.) (1955), Selected Papers in Statistics and Probability by Abraham Wald, New York: McGraw-Hill Wegge, L (1978), "Constrained Indirect Least Squares Estimators", Econometrica, vol 46, pp 435-450 White, H (1982), "Maximum Likelihood Estimation of Misspecified Models", Econometrica, vol 50, pp 1-26 White, H (1983), "Corrigendum", Econometrica, vol 51, p 513 Zellner, A (1978), " Estimation of Functions of Population Means and Regression Coefficients Including Structural Coefficients", Journal of Econometrics, vol 8, pp 127-158 Zellner A., L Bauwens and H K Van Dijk (1988), "Bayesian Specification Analysis and Estimation of Simultaneous Equation Models Using Monte Carlo Methods", Journal of Econometrics, vol 38, pp 39-72 Index Absolutely continuous, 70, 268 Aitken estimator feasible, 32 (J" -algebra, 65, 265 Bicausality, Borel function, 373n space, 69 Canonical structural form, 31 see also GLSEM CSF Causality, 378, 381 Classic identifiability tests, 249 Characteristic root, 145 multiple, 236 largest, 145 smallest, 233 vector, 233 Consistent strongly, 301, 303 system of linear equations, 40, 184-185 cdf, see Distribution function Conformity test in GLSEM, 123, 129-136 in GNLM, 315 Contrast, 281 Contrast function, 281, 327n Cramer-Rao bound, 274 inequality, 276 Distribution function cumulative (cdf), 69, 267n probability, 69, 267n Estimators Aitken, 32 see also GLS estimators generalized least squares, see GLS estimators generalized method of momements, see GMM inconsistent, 322 indirect least squares, see ILS instrumental variables, see IV k-class, see k-class estimators 396 Index full information maximum likelihood, see FIML estimator limited information maximum likelihood, see LIML estimator minimum contrast, see MC minimum distance, 31 minimum variance, 53, 58 ordinary least squares, see OLS simplified FIML, see FIML Exclusion matrix, 16 augmented, 20 partial exclusion, 16 partially augmented, 20 Exogeneity strict, 379 weak,380 Filtration, 266 FIML estimator asymptotic equivalence to simplified FIML, 219 to 3SLS, 210, 219 consistency, 203, 219-220 defined, 196 limiting distribution, 203, 219 simplified FIML consistency, 208 defined, 203 Forecasting from dynamic GLSEM, 87-89 confidence intervals, 92 distributional aspects, 8992 restricted reduced form, 9396 unrestricted reduced form, 83 static GLSEM, 84 confidence intervals, 87 distributional aspects, 8587 efficiency, 85 restricted reduced form, 9396 unrestricted reduced form, 83 General linear model (GLM), 1-5 defined, dependent variables, independent variables, regressands, regressors, General linear structural econometric model (GLSEM) assumptions, 12, 13, behavioral equations in, 10 conventions, 15 CSF,31 defined, 10-12 final form of, 74-77 identities in, 10 inconsistency of OLS in, 2127 limiting distributions, 63-68, 77-82, 120-121 nature of, 6-9 nonstochastic equations in, 10 normalization, 15 notation, 15-21 recursive, 179 reduced form, 13, 159 restrictions on, a priori, 11, 12, 15 exclusion, 11, 12, 15 minimum number of, 126, 171 prior, 11, 12, 15 zero, 11, 12, 15 stochastic equations in, 10 structural form of, 13 tests of a priori restrictions, 4243 conformity, see Conformity test Index of exclusion restrictions, see Specification tests in GLSEM Hausman, see Specification tests in GLSEM identification, 250-254 specification; see Specification tests in GLSEM variables in dependent (jointly), 14 endogeneous (current), 14 exogenous 12, 14 lagged dependent, 14 lagged endogenous, 14 predetermined, 14 GMM estimators compared to NL3SLS, 368, 370 compared to NL2SLS, 368, 370 consistency, 370 convergence a.c., 372 (model) defined, 368 identification, 370 limiting distribution, 375-376 optimal, 375-376 orthogonality conditions, 376 restrictions, 376 tests of, 377 General nonlinear model (GNLM) consistency, 299 convergence a.c., 301 defined, 299 identification, 303 limiting distribution, 304-309 restricted, 310 convergence a.c., 313 limiting distribution, 313 tests of (restrictions) conformity, 315 equivalence of, 317-318 LM,317 LR,316 General nonlinear structural econometric model (GNLSEM), with additive errors, 322 defined, 322 397 identification, 323-324 Generalized inverse, 43 least squares, see GLS, variance, 109 Generalized method of moments, see GMM estimators GLS Estimator, see Aitken estimator Hypothesis maintained, 43 testable, 43 Identification conditions, 162, 167, 172 equivalence of, 176 order, 164, 175 rank, 164, 172 by covariance restrictions, 177179 by cross equation restrictions, 177 by exclusion restrictions, 157 global, 184 in GLSEM by KI, 287-291 by linear homogeneous restrictions, 169 by linear inhomogeneous restrictions, 166 by linear restrictions, 166 in a more general framework, 183 in panel data models, 181182 of parametric functions, 193 and parametric nonlinearities, 194-195 and the reduced form, 171 relation of rank condition and exclusion restrictions, 249-250 test, 122, 248-252 local, 184 398 Index in GNL8EM, 323-324 Identification problem, 10, 154-157 definition, 155 just identification, 35-37 overidentification, 35, 37 underidentification, 35 IlV estimators, 116 full information (FIIV) 117118 and insufficient sample size, 117 limited information (LIlV), 117118 IL8 estimators consistency, 241 defined, 237, 241 derivation, 241 equivalence to 28L8, 240, 244 limiting distribution, 242 relationship to LIML, 244 Inequality Cauchy, 151 Cramer-Rao, 274 Jensen, 267, 326n Information content Fisher, 269 Kullback (KI), 279 matrix, 270 alternative, 273 Instrumental Variables Estimators, see IV, Integral Lebesgue, 68 Riemann, 68 Riemann-8tieltjes, 68 Inverse image, 68, 69 Isomorphic, 71 Iterated IV estimators, see IlV IV estimators, 105 consistency, 105 defined, 104 and insufficient sample size, 115 28L8 and 38L8 as IV, 105-107 as optimal IV, 109-115 use of generalized inverse, 115 use of principal components, 116 k-class estimator 119 double k-class , 119 LIML as k-class , 248 Kullback information (KI), see Information Lag operators, 71-74 Lagrange Multiplier test, see LM test Least variance ratio, see also LIML, LVR Lebesgue integral, see Integral Likelihood function concentrated, 223 martingale properties of, 277 Likelihood ratio test, see LR test LIML estimator asymptotic equivalence to 28L8, 248 consistency, 234-236 defined, 196 derivation as least variance ratio (LVR), 238 limiting distribution, 248 relation to IL8, 246 relation to k-class , 248 single equation, 230 subset of structural equations, 223 Lindeberg condition, 66, 326, 358, 375 LM test in GL8EM, 122, 125-129, 132136 in GNLM, 317 Local alternative, 125 LR test in GL8EM, 123 in GNLM, 316 Index Mahalanobis distance, 31 Martingale, 277, 292, 296 difference, 78 square integrable, 297 Matrix block diagonal, 309 diagonal, 162 information, 270 Jacobian, 189, 326 norm of, 145 permutation, 215 second moment convergence of, 141-144 upper echelon, 185 Maximum likelihood full information, see FIML estimator limited information, see LIML estimator Mean value theorem, 189, 213 Measure admissible, 266 dominant, 267 probability, 266 Method of scoring, 206-208 MC estimator consistency, 282 strong, 284 convergence a.c., 284 defined, 282 ML as MC estimator, 285286 Minimum contrast estimator, 282 distance estimator, 31 variance estimator, 58 Misspecification tests, see Specification tests in GLSEM Model (definition), 158, 267 Moment generating function (mgf) , 328 Noncentrality parameter, 42, 126 Nonlinear ML estimator in GNLM, 399 consistency, 299 convergence a.c., 301 limiting distribution, 307310 Nonlinear ML estimator in GNLSEM consistency, 325 convergence a.c., 330 limiting distribution, 336 known covariance matrix, 341-348 unknown covariance matrix, 339, 342-348 in misspecified models, inconsistency, 348 limiting distribution, 349351 "pseudo"-ML, 333, 349 "quasi" -ML, 333-349 Nonlinear 3SLS consistency, 363 convergence a.c., 364 identification, 364 limiting distribution, 365-366 optimal, 366 Nonlinear 2SLS, convergence a.c., 356 identification, 355 limiting distribution, 359 as MC estimator, 355 optimal, 360 Norm of a matrix, 145 Null space column, 98, 167, 290 OLS estimator, inconsistency of, in GLSEM, 21-27 of restricted form, 27 Ordinary least squares, see OLS Orthogonal basis, 186 complement, 186 subspace, 186 400 Index sample, 65, 265 Specification tests in GLSEM Radon-Nikodym theorem, 70 conformity, 122, 123 Random elements, 144 Hausman's, 42, 134 convergence a.c., 146 comparison to LM, 134-137 convergence in probability, 146 Lagrange multiplier (LM) L P convergence, 146 based on 3SLS, 131 Rao-Blackwell theorem, 55 based on SE equation 2SLS, Rational expectations, 368, 369n 123-126 Recursive relation to conformity, block, 179 127-129 simply, 179 based on systemwide 2SLS Reduced form and 3SLS, 130-132 estimator, 197-201 relation to conformity, Restrictions in GLSEM, 132-134 a priori, 12, 227, 229 relation to Hausmans's, covariance, 177-178 134-137 cross equation, 177 defined, 122, 123 exclusion, 12, 15 likelihood ratio (LR), 122-123 linear, 166 Square integrable, 299, 299n, 371, linear homogeneous, 167 see also Martingale linear inhomogenous, 169 Statistic, minimum number of, 126, 171 complete, 55 prior, 12, 227, 229 complete sufficient, 56 zero, 12, 227, 229 sufficient, 54 Restrictions in GNLSEM Stochastic a priori, 323-324 basis, 266 orthogonality, 376 process Restrictions in VAR, 102, 103 covariance stationary, 368n, Riemann-Stieltjes integral, see In369n tegral purely deterministic, 374n purely nondeterministic, Scoring, see Method of scoring 376n Seemingly unrelated regressions (SUR), strictly stationary, 368n, 371 344n ergodic, 369n Selection matrix, 16, 18 weakly stationary, 368n, 369n augmented, 18 Structure partially augmented, 17 admissible, 159 partial, 16 defined, 158 Space global identification, 184 Borel,69 local indentification, 184 Euclidean, 145, 194n observationally equivalent, 158 measurable, 267 orthogonal subspace, 186 Three stage least squares (3SLS) probability, 53, 69 compared to 2SLS, 34 Panel data, 181 Index derivation from CSF as GLS, 32-34 from CSF by LM as restricted GLS, 48, 49 identical to 2SLS, 34 as IV estimator, 105-107 limiting distribution in a dynamic GLSEM, 7780 with jointly normal errors, 64 with nonnormal errors, 6567 as optimal IV estimator, 109115 restricted, 41-42 compared to unrestricted, 41-42 restricted reduced form, 9396 Transformation admissible, 159 trivial, 160 Two stage least squares (2SLS) compared to 3SLS, 34 derivation from CSF by LM as restricted OLS, 43-48 from CSF as OLS, 29-31 original, 27-28 equivalence to ILS, 242, 246 identical to 3SLS, and insufficient sample size, 115 401 as IV estimator, 105 limiting distribution in a dynamic GLSEM, 7780 with jointly normal errors, 64 with nonnormal errors, 6567 as optimal IV estimator, 109115 and use of generalized inverse, 115 and use of principal components, 116 restricted, 40-41 compared to unrestricted, 39-40 restricted reduced form, 9396 systemwide estimator, 30 Upper echelon matrix, 185 Vector autoregression (VAR) , 102103 reduced form, 103 restrictions on, 102, 103 structural form, 102 Wald test, see Conformity test Wold decomposition, 374 .. .Topics in Advanced Econometrics Volume II r l",', I I I Phoebus J Dhrymes Topics in Advanced Econometrics Volume II Linear and Nonlinear Simultaneous Equations Springer-Verlag New York Berlin... Congress Cataloging -in- Publication Data Dhrymes, Phoebus J Topics in advanced econometrics (v, 2: Linear and nonlinear simultaneous equations) Includes bibliographical references and index Contents:... 0-387-94156-8 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-94156-8 Springer-Verlag Berlin Heidelberg New York To Ingram Olkin and Henri Theil, who stimulated my early interest in econometrics