Asymptotic Theory for Econometricians_ Revised Edition

Cover photo credit: Copyright© 1999 This book is printed on acid-free paper Dynamic Graphics, Inc Copyright 19 2001, 1984 by ACADEMIC PRESS All Rights Reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777 Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NWI 7BY, UK http://www.academicpress.com Library of Congress Catalog Card Number: 00-107735 International Standard Book Number: 0-12-746652-5 PRINTED IN THE UNITED STATES OF AMERICA 00 01 02 03 04 05 QW I Contents IX Preface to the First Edition Preface to the Revised Edition References Xl Xlll The Linear Model and Instrumental Variables Estimators References 12 12 For Further Reading Consistency 2.1 2.2 2.3 2.4 Limits Almost Sure Convergence Convergence in Probability Convergence in rth Mean References Laws of Large Numbers 3.1 3.2 3.3 3.5 Independent Identically Distributed Observations Independent Heterogeneously Distributed Observations Dependent Identically Distributed Observations Dependent Heterogeneously Distributed Observations Martingale Difference Sequences References Vll 15 15 18 24 28 30 31 32 35 39 46 53 62 VIII Contents Asymptotic Normality 65 4.1 Convergence in Distribution 4.2 Hypothesis Testing 4.3 Asymptotic Efficiency References 65 74 83 111 Central Limit Theory 113 5.1 5.2 5.3 5.4 5.5 Independent Identically Distributed Observations Independent Heterogeneously Distributed Observations Dependent Identically Distributed Observations Dependent Heterogeneously Distributed Observations Martingale Difference Sequences References Estimating Asymptotic Covariance Matrices 6.1 General Structure of V n 6.2 Case 1: {Ztct} Uncorrelated 6.3 Case 2: {Ztct } Finitely Correlated 6.4 Case 3: { Ztct} Asymptotically Uncorrelated References Functional Central Limit Theory and Applications 7.1 7.2 7.3 7.4 7.5 7.6 Random Walks and Wiener Processes Weak Convergence Functional Central Limit Theorems Regression with a Unit Root Spurious Regression and Multivariate FCLTs Cointegration and Stochastic Integrals References Directions for FUrther Study 8.1 Extending the Data Generating Process 8.2 Nonlinear Models 8.3 Other Estimation Techniques 8.4 Model Misspecification References 114 117 122 130 133 136 137 137 139 147 154 164 167 167 171 175 178 184 190 204 207 207 209 209 211 211 Solution Set References 213 259 Index 261 Preface to the First Edition Within the framework of the classical linear model it is a fairly straight­ forward matter to establish the properties of the ordinary least squares (OLS) and generalized least squares (GLS) estimators for samples of any size Although the classical linear model is an excellent framework for de­ veloping a feel for the statistical techniques of estimation and inference that are central to econometrics, it is not particularly well adapted to the study of economic phenomena, because economists usually cannot conduct controlled experiments Instead, the data usually exist as the outcome of a stochastic process outside the control of the investigator For this rea­ son, both the dependent and the explanatory variables may be stochastic, and equation disturbances may exhibit nonnormality or heteroskedasticity and serial correlation of unknown form, so that the classical assumptions are violated Over the years a variety of useful techniques has evolved to deal with these difficulties Many of these amount to straightforward mod­ ifications or extensions of the OLS techniques (e.g., the Cochrane-Orcutt technique, two-stage least squares, and three-stage least squares) However, the finite sample properties of these statistics are rarely easy to establish outside of somewhat limited special cases Instead, their usefulness is jus­ tified primarily on the basis of their properties in large samples, because these properties can be fairly easily established using the powerful tools provided by laws of large numbers and central limit theory Despite the importance of large sample theory, it has usually received fairly cursory treatment in even the best econometrics textbooks This is IX X Preface to the First Edition really no fault of the textbooks, however, because the field of asymptotic theory has been developing rapidly It is only recently that econometricians have discovered or established methods for treating adequately and com­ prehensively the many different techniques available for dealing with the difficulties posed by economic data This book is intended to provide a somewhat more comprehensive and unified treatment of large sample theory than has been available previ­ ously and to relate the fundamental tools of asymptotic theory directly to many of the estimators of interest to econometricians In addition, because economic data are generated in a variety of different contexts (time series, cross sections, time series-cross sections ) , we pay particular attention to the similarities and differences in the techniques appropriate to each of these contexts That it is possible to present our results in a fairly unified manner high­ lights the similarities among a variety of different techniques It also allows us in specific instances to establish results that are somewhat more gen­ eral than those previously available We thus include some new results in addition to those that are better known This book is intended for use both as a reference and as a text book for graduate students taking courses in econometrics beyond the introductory level It is therefore assumed that the reader is familiar with the basic concepts of probability and statistics as well as with calculus and linear algebra and that the reader also has a good understanding of the classical linear model Because our goal here is to deal primarily with asymptotic theory, we not consider in detail the meaning and scope of econometric models per se Therefore, the material in this book can be usefully supplemented by standard econometrics texts, particularly any of those listed at the end of Chapter I would like to express my appreciation to all those who have helped in the evolution of this work In particular, I would like to thank Charles Bates, Ian Domowitz, Rob Engle, Clive Granger, Lars Hansen, David Hendry, and Murray Rosenblatt Particular thanks are due Jeff Wooldridge for his work in producing the solution set for the exercises I also thank the stu­ dents in various graduate classes at UCSD, who have served unwitting and indispensable guinea pigs in the development of this material I am deeply grateful to Annetta Whiteman, who typed this difficult manuscript with incredible swiftness and accuracy Finally, I would like to thank the National Science Foundation for providing financial support for this work under grant SESSl-07552 as Preface to the Revised Edition It is a gratifying experience to be asked to revise and update a book written over fifteen years previously Certainly, this request would be unnecessary had the book not exhibited an unusual tenacity in serving its purpose Such tenacity had been my fond hope for this book, and it is always gratifying to see fond hopes realized It is also humbling and occasionally embarrassing to perform such a revision Certain errors and omissions become painfully obvious Thoughts of "How could I have thought that?" or "How could I have done that?" arise with regularity Nevertheless, the opportunity is at hand to put things right, and it is satisfying to believe that one has succeeded in this (I know, of course, that errors still lurk, but I hope that this time they are more benign or buried more deeply, or preferably both ) Thus, the reader of this edition will find numerous instances where defini­ tions have been corrected or clarified and where statements of results have been corrected or made more precise or complete The exposition, too, has been polished in the hope of aiding clarity Not only is a revision of this sort an opportunity to fix prior shortcom­ ings, but it is also an opportunity to bring the material covered up-to-date In retrospect, the first edition of this book was more ambitious than origi­ nally intended The fundamental research necessary to achieve the intended scope and cohesiveness of the overall vision for the work was by no means complete at the time the first edition was written For example, the central limit theory for heterogeneous mixing processes had still not developed to XI XII Preface to the Revised Edition the desired point at that time, nor had the theories of optimal instrumental variables estimation or asymptotic covariance estimation Indeed, the attempt made in writing the first edition to achieve its in­ tended scope and coherence revealed a host of areas where work was needed, thus providing fuel for a great deal of my own research and (I like to think) that of others In the years intervening, the efforts of the econometrics re­ search community have succeeded wonderfully in delivering results in the areas needed and much more Thus, the ambitions not realized in the first edition can now be achieved If the theoretical vision presented here has not achieved a much better degree of unity, it can no longer be attributed to a lack of development of the field, but is now clearly identifiable as the author's own responsibility As a result of these developments, the reader of this second edition will now find much updated material, particularly with regard to central limit theory, asymptotically efficient instrumental variables estimation, and esti­ mation of asymptotic covariance matrices In particular, the original Chap­ ter (concerning efficient estimation with estimated error covariance ma­ trices) and an entire section of Chapter concerning efficient IV estimation have been removed and replaced with much more accessible and coherent results on efficient IV estimation, now appearing in Chapter There is also the progress of the field to contend with When the first edition was written, cointegration was a subject in its infancy, and the tools needed to study the asymptotic behavior of estimators for models of cointegrated processes were years away from fruition Indeed, results of De­ Jong and Davidson (2000) essential to placing estimation for cointegrated processes cohesively in place with the theory contained in the first six chap­ ters of this book became available only months before work on this edition began Consequently, this second edition contains a completely new Chapter devoted to functional central limit theory and its applications, specifically unit root regression, spurious regression, and regression with cointegrated processes Given the explosive growth in this area, we cannot here achieve a broad treatment of cointegration Nevertheless, in the new Chapter the reader should find all the basic tools necessary for entree into this fascinating area The comments, suggestions, and influence of numerous colleagues over the years have had effects both subtle and patent on the material pre­ sented here With sincere apologies to anyone inadvertently omitted, I ac­ knowledge with keen appreciation the direct and indirect contributions to the present state of this book by Takeshi Amemiya, Donald W K An­ drews, Charles Bates, Herman Bierens, James Davidson, Robert DeJong, References X ill Ian Domowitz, Graham Elliott, Robert Engle, A Ronald Gallant, Arthur Goldberger, Clive W J Granger, James Hamilton, Bruce Hansen, Lars Hansen, Jerry Hausman, David Hendry, S0ren Johansen, Edward Leamer, James Mackinnon, Whitney Newey, Peter C B Phillips, Eugene Savin, Chris Sims, Maxwell Stinchcombe, James Stock, Mark Watson, Kenneth West, and Jeffrey Wooldridge Special thanks are due Mark Salmon, who originally suggested writing this book UCSD graduate students who helped with the revision include Jin Seo Cho, Raffaella Giacomini, Andrew Pat­ ton, Sivan Ritz, Kevin Sheppard, Liangjun Su, and Nada Wasi I also thank sincerely Peter Reinhard Hansen, who has assisted invaluably with the cre­ ation of this revised edition, acting as electronic amanuensis and editor, and who is responsible for preparation of the revised set of solutions to the exercises Finally, I thank Michael J Bacci for his invaluable logistical sup­ port and the National Science Foundation for providing financial support under grant SBR-9811562 Del Mar, CA July, 2000 References DeJong R M and J Davidson (2000) "The functional central limit theorem and weak convegence to stochastic integrals I : Weakly dependent processes," forthcoming in Econometric Theory, 16 CHAPTER The Linear Model and Instrumental Variables Estimators The purpose of this book is to provide the reader with the tools and con­ cepts needed to study the behavior of econometric estimators and test statistics in large samples Throughout, attention will be directed to esti­ mation and inference in the framework of a linear stochastic relationship such as t 1, ,n , lt = X�/30 + t t: , = where we have n observations on the scalar dependent variable yt and the vector of explanatory variables Xt = ( Xt , Xt , , Xtk ) The scalar stochastic disturbance Et is unobserved, and {30 is an unknown k x vector of coefficients that we are interested in learning about, either through esti­ mation or through hypothesis testing In matrix notation this relationship is written as ' Y = X{30 + E, where Y is ann x vector, X is ann x k matrix with rows X�, and E is an n x vector with elements Et· ( Our notation embodies a convention we follow throughout: scalars will be represented in standard type, while vectors and matrices will be repre­ sented in boldface Throughout, all vectors are column vectors ) Most econometric estimators can be viewed as solutions to an optimiza­ tion problem For example, the ordinary least squares estimator is the value 250 Solution Set Now, given (ii'), (iv.a), and (iv.b) , Z'X/n - Qn E , by Corollary 3.48 and Theorem 2.24 The remaining results follow as before • Exercise 6.2 Proof The following conditions are sufficient: Yt= X� f3o + et , t = 1, 2, , f3o E �k ; {(Z�, X�, et) } is a mixing sequence with either ¢ of size -r/2(r - ) , r > 2, or o: of size -r/(r - 2), r > 2; (iii) (a) E(Ztgi€thi Ft - d = for all t, where {Ft } is adapted to { Ztgi cth }, h , ,p, i = , ' l; (b) EI Ztgi cthi r+ 0, h 1, i = , , l, and all t ; (c) E(et e�IZt) = a� Ip , t = 1, , n; (iv) (a) EI Zthi i r+ 0, h = , , p, i = , , l, j = , k, and all t; (b) Qn E(Z'X/n ) has full column rank uniformly in n for all n sufficiently large; (c) Ln E(Z'Z/n) has det (Ln) > > for all n sufficiently large (i) (ii) g, = g, = < oo < < < oo , p, < < oo < oo = = Given conditions (i)-(iv), the asymptotically efficient estimator is by Exercise 4.47 First, consider Z'Zj n Now {ZtZD is a mixing sequence with the same size as {(Z�,X�,et)} by Proposition 3.50 Hence, by Corol­ lary 8, Z'Z/n - Ln n- :Z::::�= ZtZ� - n- :Z::::�= E(ZtZD � given (iv.a) and Z'Z/n - Ln E , by Theorem 24 Next consider = (np) - (Y - Xj3n ) ' (Y - Xj3n ) (e - X(j3n - /30 )) ' (e - X(j3n - /30 ))/(np) e1 e/(np) - 2(j3n - f3o )'X'e/(np) + (i3n - f3o) ' (X'Xjn)(j3n - f3o ) fp As the conditions of Exercise 79 are satisfied, it follows that i3n -/30 Also, X'e/n = Oa s (1) by Corollary given (ii), (iii.b) , and (iv.a) Hence (j3n - f30 )X'e/n by Exercise 2.22 and Theorem 2.24 Similarly, {Xt XD a mixing sequence with size given in (ii) with elements satisfying a;, � E , Solution Set 251 = the moment condition of Corollary 3.48 given (iv.a), so that X'X/n Finally, consider Oa s (1) and therefore (f3n -{30 )' (X'X/n )(f3n -{30 ) - - p + n e 'e/(np) = p-1 L n -1 l:e�h· h= l t=l Now for any h = 1, { e�h } is a mixing sequence with ¢ of size /2 ( ) , > 2, or of size - / ( - 2) , > As {e�d satisfies the moment condition of Corollary 3.48 given (iii.b) and E(e�h ) = a� given (iii.c) , it follows that n n n n -1 l:e�h - n -1 LE(e�h ) n -1 l:e�h - � 0, t=l t=l t=l P a r -r ,p, r r r = Hence, e'e/(np) E , a �, r a E , and it follows that a-; E , a� by Exercise 2.35 • Exercise 6.6 Proof The proof is analogous to that of Theorem 6.3 Consider for sim­ plicity the case p = We decompose V V as follows: n- n n n n- l: e�ZtZ� - n-1 LE(e�ZtZ�) t=l n t= l -2n- 2:C e {3 Z Z )1X� t t � i3n t=l +n- l 2:Ci3n - f3o ) ' XtX�(,Bn - {30 )ZtZ� Now {e;Z t Z� } is a mixing sequence with either ¢ of size - / (2 - ) , or of size -r / (r - ) , r > , given (ii) with elements satisfying the moment condition of Corollary 3.48 given (iii.b) Hence r > 1, r a n n n-1 l:e�ZtZ� - n - LE(e�ZtZ�) t= l t=l r E , as The remaining terms converge to zero in probability in Theorem 6.3, where we now use results on mixing sequences in place of results on sta­ tionary, ergodic sequences For example, by the Cauchy-Schwarz inequality, 252 Solution Set given (iii.b) and (iv.a) As { Xt� 2, or of size -rj(r - 2), r > 2, with elements satisfying the moment condition of Corollary 3.48 given (iii.b) Hence a n n The remaining terms can be shown to converge to zero in probability in a manner similar to Theorem 6.3 • Exercise 6.13 Proof Immediate from Theorem 5.22 and Exercise 6.12 • Exercise 6.14 ensure that Exercise 6.12 holds for j3n and Vn Proof Conditions (i)-(iv) A A p Vn > Next set P n = V;;- in Exercise 6.13 Then Pn = V;;- and the result follows • Exercise 6.15 Proof Under 6.12 A A p it holds that p the conditions of Theorem 6.9 or Exercise Vn Vn > 0, so the result will follow if also Vn - Vn > Now - n A - - r e-'t z't · V - n - = �( L z etet, r z't r + z t - r et L Wnr 1) n - � t =r +1 r= Since for each T = , , m we have n � n - L z tetet, r z't r + z t r et t t = op (1) - - - r e-'z' t=r + rn _ - _ it follows that rn L Op(1)0p(1) = Op(1), rn r= p where we used that Wnr > • 254 Solution Set Exercise 7.2 Proof Since Xt = Xo + 2:�=1 Z8 , we have s=1 t + L E(Z8) = 0, s=1 and t var(Xt ) var(Xo t + s=L Zs ) + I": var(Zs ) = ta , s=1 using the independence between Zr and Zs for r =1- s Exercise 7.3 Proof Note that Zt4 Zt2 + Zt - + + Zt2 - + · · · · · · • + Zt3 +1 , + Zt1 +I · Since (Zt1 + l , Zt2 ) is independent of (Zt3 +1 , , Zt ) it follows from Proposition 3.2 (ii) that Xt4 - Xt3 and Xt2 - Xt1 are independent • Exercise 7.4 Proof By definition [nb] n - 1/2 L Zt t=[na]+1 n - 12 ([nb] - [na]) 12 [nb] x ( [nb] - [na]) - 1 L Zt t=[na]+l The last term ([nb] - [na]) - 12 I:��lna]+1 Zt � N(O, 1) by the central limit theorem, and n - 12 ([nb] - [naj) 12 = (([nb] - [na])jn) 12 > (b - a) 1 n > Hence Wn (b) - Wn (a) N(O, b - a) as oo d > • 255 Solution Set Exercise 7.22 Proof ( i) Let U E C, and let Un be a sequence of mappings in D such that Un U, i.e., bn = du (Un , U) = SUPaE[O, l ] IUn (a) - U(a) l as n -> 00 -> -> We have du(U� , U2 ) sup 1Un (a) - U(a)21 aE[O,l] sup IUn (a) - U(a) I IUn (a) + U(a) l aE[O, ] < bn sup IUn (a) + U(a) l , aE[O, l j where bn = du (Un , U) The boundedness of U and the fact that bn imply that for all n sufficiently large, Un is bounded on [0, 1] Hence, supaE[O, l ] IUn (a) + U(a) l = 0(1) and it follows that du (U� , U2) -> 0, which proves that M1 is continuous at U Now consider the function on [0, 1] given by -> V(a) = { int(0,l�a ) , for < a < for a = For < a < /2, this function is 1, then jumps to for a = /2 , to for a 2/3, and so forth Clearly V(a) is continuous from the right and has left limits everywhere, so V E D = D [O, 1] Next, define the sequence of functions Vn(a) = V(a) + c:fn, for some E > Then Vn E D and Vn -> V Since = sup IVn (a) - V(a) aE[O,l] sup I (Vn (a) - V(a)) (Vn (a) + V(a) ) l aE[O,l] (c:jn) sup I Vn (a) + V(a) l aE[O,l] is infinite for all n, we conclude that V� V2 Hence, M1 is not continuous everywhere in D (ii) Let U E C; then U is bounded and f01 IU(a) l da < Let {Un} be a sequence of functions in D such that Un -> U Define bn = du(Un , U) ; then bn -> and f01 IUn (a) lda < f01 IU(a) lda + bn -+> oo 256 Solution Set Since 1 Un(a)2 da - 1 U(a)2da l < 1 1Un(a)2 - U(a)2 ida 1 IUn(a) - U(a) I IUn (a) + U(a) ida < < 1 IUn (a) + U(a) ida bn(bn + 21 IU(a) ida) bn > M2 is continuous at U E C (iii) Let V(a) = a for all a E [0, 1] and let Vn(a) = for a 1/n and Vn = V(a) for a > /n Thus du(V, Vn) = 1/n and Vn V; however, J01 log(IVn (a) l )da f01 log(IV(a) l)da, since J01 log(IV(a) l )da = - whereas J; log(I Vn(a) l)da = - oo for all n Hence, it follows that M3 is not continuous < < > 17 • Exercise 7.23 Proof If { ct } satisfies the conditions of the heterogeneous mixing CLT and is globally covariance stationary, then (ii) of Theorem 7.21 follows directly from Theorem 7.18 Since {ct } is assumed to be mixing, then {c-Z } is also mixing of the same size (see Proposition 3.50) By Corollary 3.48 it also then follows that n- 2:.:� ct - n -1 L.:�= l E(c-t) = op(1) So if n -1 L.:�=l E(t:t) converges as oo , then (iii) of Theorem 7.21 holds For a2 = T2 it is required that lim n- 2:.:�=2 2:.:�=.; E(ct - s ct ) = It suffices that { c-t } is independent or that { ct, Ft } is a martingale difference sequence n > • Exercise 7.28 Proof From Donsker's theorem and the continuous mapping theorem => we have that n - L �= l x; J wl (a) da The multivariate version of Donsker's theorem states that Applying the continuous mapping theorem to the mapping (x, y) � > 1 x(a)y(a)da, Solution Set 257 we have that n � Hence n-1 I:: a W1 n(at _ l )a2 W2n(at - ) t= 1 1TI1T2 W1 n(a)W2n (a)da a a2 W1 (a)W2 (a)da x; ) n - t XtYt t t= t= -1 1 (a� W1 (a) da) a a2 W1 (a)W2 (a)da -1 1 ( a2 /a ) (1 W1 (a) da) W1 (a)W2 (a)da -1 (n - l3n � • 1 Exercise 7.31 Proof Since { 7Jt , ct } satisfies the conditions of Theorem 7.30, it holds that ( n -1 /2 x[na] ' n -1 /2 y;[na] ) ( 1T W1n( at -1 ), 1T2 W2n ( at -1 )) ( a W1 (a), a2 W2 (a)), where and a� are the diagonal elements of :E (the off diagonal elements are zero, due to the indepe?dence of 7Jt and ct) · Applying the continuous mapping theorem, we find f3n to have the same limiting distribution as we found in Exercise 7.28 • Exercise 7.43 Proof (i ) If {17t} and {et} are independent, then E(1J8 eD and t So n t -1 :E� /2n -1 2:::: 2:::: E(1J s e� ) :E� /2' t =2 s= An 0, and clearly A n A = = for all s 258 Solution Set (ii) If Tit = et -1 , then n t -1 :E � /2n -1 L L E(rys e:�) :E2 /21 An t=2 s= n t- -1 � /2 � -1 /21 '""' '""' u n-1 L L E(e s -1 e t ) u t=2 s= If the sequence { e:t} is covariance stationary such that Ph = E(e:te:� _ h ) is well defined, then n t -1 n t n - L L E(e: s -1 e:�) n -1 L L E(e:t - s e�) t= s= t=2 s=2 I t=2 s=2 � ( n + - s) L Ps , n s=2 so that ( ) n (n + - s ) -1 /2 L An - :E Ps :E 1-1 /21 A n s=2 where in this situation we have n n n -1 (n - s ) ( ) " n-1 L L Pt s = Po + lI" m L � + u = nl -+1m u = � Ps Ps n -+ CXJ s= n CXJ t = s= 1 = > ' I · Notice that when { e:t } is a sequence of independent variables such that Ps = for s > , then var (e: t ) = Po = :E and A = This remains true for the case where Tit = et, which is assumed in the likelihood analysis of cointegrated processes by Johansen (1988, 1991) • Exercise 7.44 Proof (a) This follows from Theorem 7.21 (a ) , or we have directly References (b) (c ) 259 Similarly, by Theorem 7.42 and using that A = since {et} and {17t } are inde­ pendent (see Exercise 7.43) n(�n - (i0) =? (d) • (n-2 �?tf (n- ' t, x,o,) (a2/a1 ) [1 W1 (a)da] -1 1 W1 (a) dW2 (a) From ( ) we have that n(/3n - {30) = Op(l), and by Exercise 2.35 we have that (fin - (30) = n - n(/3n - (30) = op (l)Op(l) = op(l) So ' p f3n f3o · c + References Dhrymes, P ( 980) Econometrics Springer-Verlag, New York Granger, C W J and P Newbold ( 977) Forecasting Economic Time Series Academic Press, New York Johansen, S ( 988) "Statistical analysis of cointegration vectors " Journal of Economic Dynamics and Control, 12, 23 1-254 ( 1991 ) "Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models." Econometrica, 59, 155 1-80 Laha, R G and V K Rohatgi ( 979) Probability Theory Wiley, New York Rao, C R ( 973) Linear Statistical Inference and Its Applications Wiley, New York Index Adapted mixingale, 124 Adapted stochastic sequence, 58 Approximatable stochastic function, 194 AR ( ) , 48, 49 ARCH ( q ) , 102 ARMA , 49 Asymptotic covariance matrix, 70, 73 estimator, 37 heteroskedasticity consistent, 1391 46 heteroskedasticity / autocorrelation consistent, 54-164 heteroskedasticity/ moving aver­ age consistent , 147-154 Newey-West estimator, 163 Asymptotic distribution, 66 Asymptotic efficiency, 83 Bates and White framework, 93 IV estimator, 84 two-stage least squares, 86 Asymptotic equivalence, 67 Asymptotic normality, of !Jn, 71 of !Jn, 73 Asymptotically uncorrelated process, 52, 139, 54 avar, 70, 83 261 Backshift operator, 42 Best linear unbiased estimator, Borel sets, a-field, 40 Boundary of a set, 173 Bounded in probability, 28 Brownian motion, 169 BM(�), 203 definition, 70 C[O, ] , 75 C[O, oo), 72 ck [o, ] , 187 ck [o, oo ) , 186 Cadlag function, 72 Cauchy-Schwarz inequality, 33 Central limit theorem, 1 functional, 75 Liapounov, 18 Lindeberg-Feller, 1 Lindeberg-Levy, 14 martingale difference, 133 mixing, 130 stationary ergodic adapted mixin­ gale, 25 Wooldridge-White, 30 262 Characteristic function, 67-71 continuity theorem, 69 uniqueness theorem, 68 Chebyshev inequality, 30 generalized, 29 Classical assumptions of least squares, generalized, Classical linear model, 1-8 Coint.egrat.ed processes, 90 likelihood analysis, 258 Cointegrated regression, 199, 201 Conditional expectation, 54-59 linearity of, 55 Consistency strong, super, 180 weak, 24 Continuity uniform , Continuity set, 73 Continuous function, Continuous mapping theorem, 78 Convergence almost sure, i n distribution, 65 in probability, 24 in rth mean, 28 of stochastic integral, 99, 201 weak, 71 Covariance stationary process, 52 globally, 77 Cramer-Wold device, 1 functional, 188 D[O, ] , 75 D[O, oo ) , 72 D k [O, ] , 187 D k [O, oo ) , 186 Data generating process, 94, 207 Dickey-Fuller distribution, 180 Dispersion matrix, 73 Donsker's theorem, 76 multivariate, 188 Ergodic process, 44 theorem, 44 Index Estimator asymptotic covariance matrix, 137 heteroskedasticity consistent , 139-146 heteroskedasticity / autocorrela­ tion consistent, 154-164 heteroskedasticity / moving aver­ age consistent, 147-154 generalized least squares, 5, 100 efficiency, 1 feasible, 102 feasible and efficient, 105 generalized method of moments, 85 instrumental variables, constrained, 86 efficient, 109 M-, 210 maximum likelihood, , 210 method of moments, 85 ordinary least squares, three-stage least squares, 1 two-stage instrumental variables, 144, 163 two-stage least squares, 10, 144 asymptotically efficient, 86 efficient, 1 Events, 39 Filtration, 58, 193 Finitely correlated process, 138, 147 Fractionally integrated process, 208 Functional central limit theorem, 75 Donsker, 76 heterogeneous mixing, 177 multivariate, 189 Liapounov, 77 Lindeberg-Feller, 77 martingale difference, 78 multivariate, 189 stationary ergodic, 177 GARCH ( , ) , 102 Gaussian AR ( ) , 48 263 Index Generalized least squares, estimator, efficiency, 101 feasible estimator, 102 efficient, 105 Generalized method of moments, 85 Heteroskedast.icit.y, 3, 38, 143 ARCH, 02 conditional, 130 GARCH, 102 unconditional, 30 HOlder inequality, 33 Hypothesis testing, 74-·83 Implication rule, 25 Inequalities Cauchy-Schwarz, 33 Chebyshev, 30 Cr, 35 Holder, 33 Jensen, 29 conditional, 56 Markov, 30 M inkowski, 36 Instrumental variables, estimator, asymptotic normality, 74, 143, 145, 150, 152 consistency, , 23, 26, 27, 34, 38, 46, , 61 constrained, 86 definition, efficient, 109 Integrated process, 79 fractionally, 208 Invariance principle, 76 Ito stochastic integral, 94 for random step functions, 93 Jensen inequality, 29 conditional, 56 Lagged dependent variable, Lagrange multiplier test., 77 Law of iterated expectations, 57 Law of large numbers martingale difference, 60 Laws of large numbers, ergodic theorem, 44 i.i.d ' 32 i.n.i.d , 35 mixing, 49 Levy device, 58 Likelihood analysis, cointegrated processes, 258 Likelihood ratio test, 80 Limit, almost sure, i n mean square, 29 in probability, 24 Limited dependent variable, 209 Limiting distribution, 66 Lindeberg condition, 1 M-est.imator, Markov condition, Markov inequality, 30 Martingale difference sequence, 53, 58 Maximum likelihood estimator, Mean value theorem, 80 Measurable function, 41 one-to-one transformation, 42 space, 39 Measure preserving t ransformation, 42 Measurement errors, Method of moments estimator, Metric, 73 space, 73 M etrized measurable space, 174 Metrized probability space, 74 M inkowski inequality, 36 M isspecification, 207 Mixing, 47 coefficients, conditions, 46 size, 49 strong, (a-mixing) , 47 uniform, (¢-mixing) , Mixingale, 125 Nonlinear model, 209 O(n>- ) , o(n>- ) , 264 Oa.s (n"' ) , 23 Oa.s (n"' ) , 23 Op(n"' ) , 28 op(n"' ) , 28 Ordinary least squares, estimator, definition, 72 asymptotic normality, 72 consistency, 20, 22, 26, 27, 33, 38, 45, 50, definition, Panel data, 1 Partial sum , 169 Probability measure, 39 space, 39 Product rule, 28 Quasi-maximum likelihood estimator, 79, 210 Random function, 169 step function, 93 Random walk, 167 multivariate, 187 Rcll function, 72 Serially correlated errors, Shift operator, 42 a-algebra, 39 a-field, 39 Borel, 40 a-fields increasing sequence, 58 Simultaneous systems, 1 for panel data, Spurious regression, 185, 188 Stationary process, 43 covariance stationary, 52 Stochastic function approximatable, 94 square integrable, 194 Stochastic integral, 192 Strong mixing, 47 Sum of squared residuals, Superconsistency, 180 Index Test statistics Lagrange multiplier, 77 Likelihood ratio, 80 Wald , 76, 81 Three-stage least squares estimator, 1 Two-stage instrumental variables estimator, 44, 163 Two-stage instrumental variables esti­ mator asymptotic normality, 46, , 53, 164 Two-stage least squares, 10 estimator, 144 efficient, 1 Uncorrelated process, 138, 139 asymptotically, 39, 54 Uniform asymptotic negligibility, 18 Uniform continuity, theorem, Uniform mixing, 47 Uniform nonsingularity, 22 Uniform positive definiteness, 22 Uniformly full column rank, 22 Uniqueness theorem, 68 Unit root, 79 regression, 78 Vee operator, 142 Wald test, 76, 81 Weak convergence, 71 definition, 74 Wiener measure, 76 Wiener process, 70 definition, 70 multivariate, 186 sample path, 171 ... at the time the first edition was written For example, the central limit theory for heterogeneous mixing processes had still not developed to XI XII Preface to the Revised Edition the desired point... this revised edition, acting as electronic amanuensis and editor, and who is responsible for preparation of the revised set of solutions to the exercises Finally, I thank Michael J Bacci for his... IX Preface to the First Edition Preface to the Revised Edition References Xl Xlll The Linear Model and Instrumental Variables Estimators References 12 12 For Further Reading Consistency