Mathematics for Econometrics

142 Pages·1978·4.569 MB·English

by  Phoebus J. Dhrymes (auth.)

Phoebus J. Dhrymes
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., 1932-
Mathematics for econometrics.
Bibliography: p.
Includes index.
I. Algebras, Linear. 2. Econometrics.
1. Title.
QA184.D53 512'.5 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
1.2 Vectors 4
1.3 Vector Spaces 6
Chapter 2
Matrix Algebra 8
2.1 Basic Definitions 8
2.2 Basic Operations 10
2.3 Rank and Inverse of a Matrix 12
2.4 Hermite Forms and Rank Factorization 18
2.5 Trace and Determinants 25
2.6 Computation of the Inverse 32
2.7 Partitioned Matrices 34
2.8 Kronecker Products of Matrices 40
2.9 Characteristic Roots and Vectors 42
2.10 Orthogonal Matrices 54
2.11 Symmetric Matrices 57
2.12 Indempotent Matrices 64
2.13 Semidefinite and Definite Matrices 65
vii
viii Contents
Chapter 3
Linear Systems of Equations and Generalized
Inverses of Matrices 77
3.1 Introduction 77
3.2 Conditional, Least Squares, and Generalized Inverses of Matrices 78
3.3 Properties of the Generalized Inverse 81
3.4 Solutions of Linear Systems of Equations and Pseudoinverses 88
3.5 Approximate Solutions of Systems of Linear Equations 92
Chapter 4
Vectorization of Matrices and Matrix Functions:
Matrix Differentiation 98
4.1 Introduction 98
4.2 Vectorization of Matrices 98
4.3 Vector and Matrix Differentiation 103
Chapter 5
Systems of Difference Equations with Constant
Coefficients 121
5.1 The Scalar Second-order Equation 121
5.2 Vector Difference Equations 128
References 132
Index 134
Vectors and Vector Spaces 1
In nearly all of the discussion to follow we shall deal with the set of real
numbers. Occasionally, however, we shall deal with complex numbers as
well. In order to avoid cumbersome repetition we shall denote the set we are
dealing with by F and let the context elucidate whether we are speaking of
real or complex numbers or both.
1.1 Complex Numbers
A complex number, say z, is denoted by
z = x + iy,
where, x and yare real numbers and the symbol i is defined by
i2 = -1. (1)
All other properties of the entity denoted by i are derivable from the basic
definition in (1). For example,
i4 = (i2)(i2) = ( -1)( -1) = 1.
Similarly,
and so on.
It is important for the reader to grasp, and bear in mind, that a complex
number is describable in terms of (an ordered) pair of real numbers.
2 I Vectors and Vector Spaces
Let
Zj = Xj + iYj' j = 1,2
be two complex numbers. We say
if and only if
Xl = X2 and Yt = Y2·
Operations with complex numbers are as follows.
Addition:
Zl + Z2 = (Xl + X2) + i(Yl + Y2)·
Multiplication by a real scalar:
= +
CZl (CX1) i(CY1)·
Multiplication of two complex numbers:
= + +
ZlZ2 (XIX2 - YIY2) i(XlY2 X2Yl)·
Addition and multiplication are, evidently, associative and commutative,
i.e., for complex Zj,j = 1,2,3,
Zl + Z2 + Z3 = (Zl + Z2) + Z3 and ZlZ2Z3 = (ZlZ2)Z3,
Zl + Z2 = Z2 + Zl and ZlZ2 = Z2Zl,
and so on.
z
The conjugate of a complex number Z is denoted by and is defined by
z =
X - iy.
Associated with each complex number is its modulus or length or absolute
value, which is a real number denoted by IZ I and defined by
Izi +
= (ZZ)1/2 = (X2 y2)1/2.
For the purpose of carrying out multiplication and division (an operation
which we have not, as yet, defined) of complex numbers it is convenient to
express them in polar form.
Polar Form Of Complex Numbers
Let z, a complex number, be represented in Figure 1 by the point (Xl' Yl)-its
coordinates.
It is easily verified that the length of the line from the origin to the point
(x 1, Y 1) represents the modulus of Z 1, which for convenience let us denote by
r Let the angle described by this line and the abscissa be denoted by
1• (Jl.
1.1 Complex Numbers 3
Yt •.... --.-.-.-----.--.--.. ---...•..
Figure I
As is well known from elementary trigonometry we have
. (J Yl
sm 1 =-. (2)
rl
Thus we may write the complex number as
Zl = Xl + iYl = r1 cos (Jl + irl sin (Jl = rl(cos (Jl + i sin (Jl).
Further, we may define the quantity
(3)
and thus write the complex number in the standard polar form
(4)
In the representation above rl is the modulus and (Jl the argument of the
complex number Zl. It may be shown that the quantity ei61 as defined in (3)
has all the properties of real exponentials in so far as the operations of
multiplication and division are concerned. If we confine the argument of a
complex number to the range [0, 2n) we have a unique correspondence
between the (x, y) coordinates of a complex number and the modulus and
argument needed to specify its polar form. Thus, for any complex number z
the representations
Z = x + iy,
where
x . (J Y
cos(J=-, SIn =-,
r r
are completely equivalent.
4 I Vectors and Vector Spaces
Addition and division of complex numbers, in polar form, are extremely
simple. Thus,
provided Z 2 i= O.
We may extend our discussion to complex vectors, i.e., ordered n-tuples
of complex numbers. Thus
Z = x + iy
where x and yare n-element (real) vectors (a concept to be defined
immediately below) is a complex vector. As in the scalar case, two complex
vectors Zl' Z2 are equal if and only if
Xl = X2, Yl = Y2'
where now Xi' Yi' i = 1,2, are n-element (column) vectors. The complex
conjugate of the vector Z is given by .
Z = x - iy
and the modulus of the complex vector is defined by
(Z'Z)l/2 = [(x + iy)'(x - iy)]l/2 = (x'x + y'y)l/2,
the quantities x' x, y' y being ordinary scalar products of two vectors. Addition
and multiplication of complex vectors are defined by
Zl + Z2 = (Xl + X2) + i(Yl + Y2),
Z~Z2 = (X~X2 - y~Y2) + i(y'lX2 + X~Y2)'
ZlZ; = (XlX; - YlY;) + i(Ylx; + xlY;),
where Xi' Yi' i = 1,2, are real n-element column vectors. The notation, for
example, x~ or Y; means that the vectors are written in row form, rather than
the customary column form. Thus XlX; is a matrix, while X~X2 is a scalar.
These concepts (vector, matrix) will be elucidated below. It is somewhat
awkward to introduce them now; still, it is best to set forth what we need
regarding complex numbers at the beginning.
1.2 Vectors
Let ai E F, i = 1,2, ... , n; then the ordered n-tuple
is said to be an n-dimensional vector.