Mathematics for Econometrics
This booklet was begun as an appendix to Introductory Econometrics. As it progressed, requirements of consistency and completeness of coverage seemed to make it inordinately long to serve merely as an appendix, and thus it appears as a work in its own rig
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Mathematics for Econometrics
[I Springer Science+Business Media, LLC
Phoebus J. Dhrymes Department of Economics Columbia University New York, New York 10027 USA
Library of Congress Cataloging in Publication Data Dhrymes, Phoebus J., 1932Mathematics for econometrics. Bibliography: p. Includes index. I. Algebras, Linear. 1. Title. QA184.D53 512'.5
2. Econometrics. 78-17362
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media,LLC.
© 1978 by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1978
9 8 7 6 543 2 I ISBN 978-0-387-90316-3 ISBN 978-1-4757-1691-7 (eBook) DOI 10.1007/978-1-4757-1691-7
To my Mother and the memory of my Father
Preface
This booklet was begun as an appendix to Introductory Econometrics. As it progressed, requirements of consistency and completeness of coverage seemed to make it inordinately long to serve merely as an appendix, and thus it appears as a work in its own right. Its purpose is not to give rigorous instruction in mathematics. Rather it aims at filling the gaps in the typical student's mathematical training, to the extent relevant for the study of econometrics. Thus, it contains a collection of mathematical results employed at various stages of Introductory Econometrics. More generally, however, it would be a useful adjunct and reference to students of econometrics, no matter what text is being employed. In the vast majority of cases, proofs are provided and there is a modicum of verbal discussion of certain mathematical results, the objective being to reinforce the reader's understanding of the formalities. In certain instances, however, when proofs are too cumbersome, or complex, or when they are too obvious, they are omitted. Phoebus J. Dhrymes
New York, New York May 1978 v
Contents
Chapter 1
Vectors and Vector Spaces 1.1 Complex Numbers 1.2 Vectors 1.3 Vector Spaces
1 4 6
Chapter 2
Matrix Algebra 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13
Basic Definitions Basic Operations Rank and Inverse of a Matrix Hermite Forms and Rank Factorization Trace and Determinants Computation of the Inverse Partitioned Matrices Kronecker Products of Matrices Characteristic Roots and Vectors Orthogonal Matrices Symmetric Matrices Indempotent Matrices Semidefinite and Definite Matrices
8 8 10 12 18 25 32 34
40 42
54 57
64 65
vii
viii
Contents
Chapter 3
Linear Systems of Equations and Generalized Inverses of Matrices
77
3.1 3.2 3.3 3.4 3.5
77 78 81 88 92
Introduction Conditional, Least Squares, and Generalized Inverses of Matrices Properties of the Generalized Inverse Solutions of Linear Systems of Equations and Pseudoinverses Approximate Solutions of Systems of Linear Equations
Chapter 4
Vectorization of Matrices and Matrix Functions: Matrix Differentiation 4.1 Introduction 4.2 Vectorization of Matrices 4.3 Vector and Matrix Differentiation
98 98 98
103
Chapter 5
Systems of Difference Equations with Constant Coeffi
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