Table Of ContentADVANCED TEXTBOOKS
IN ECONOMICS
VOLUME 7
Editors:
C. J. BLISS
M. D. INTRILIGATOR
Advisory Editors:
S. M. GOLDFELD
L. JOHANSEN
D. W. JORGENSON
M. C. KEMP
J.-C. MILLERON
1976
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · OXFORD
AMERICAN ELSEVIER PUBLISHING CO., INC.
NEW YORK
FOUNDATIONS
OF ECONOMETRICS
ALBERT MADANSKY
Graduate School of Business
University of Chicago
1976
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · OXFORD
AMERICAN ELSEVIER PUBLISHING CO., INC.
NEW YORK
© North-Holland Publishing Company, 1976
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior permission of the copyright owner.
North-Holland ISBN for this series: 0 7204 3600 1
North-Holland ISBN for this volume: 0 7204 3607 9 (cloth-bound)
North-Holland ISBN for this volume: 0 7204 3806 3 (paperback)
American Elsevier ISBN: 0 444 10906 4 (cloth-bound)
American Elsevier ISBN: 0 444 10943 9 (paperback)
Library of Congress Cataloging in Publication Data
Madansky, Albert, 1934-
Foundations of econometrics.
(Advanced textbooks in economics; v. 7)
An outgrowth of a one-year graduate course in
econometrics that the author taught at UCLA in
1962-65 and at Yale in 1972-73.
Includes bibliographies.
1. Econometrics. I. Title.
HB139.M33 330'.01'82 75-25610
ISBN 0-444-10906-4 (Elsevier)
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
NORTH-HOLLAND PUBLISHING COMPANY, LTD.-OXFORD
SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
PRINTED IN THE NETHERLANDS
Preface
This book is an outgrowth of a one-year graduate course in econometrics
that I taught at UCLA in 1962-65 and again at Yale in 1972-73. Though
in principle it is designed for students with no substantial background in
linear algebra, statistics, or econometrics, in practice the book should be
used as a second course in econometrics, for those students who are not
content with learning econometric practice and wish to understand the
underlying basis for the techniques taught in a modern first course in
econometrics.
Much of the material in this book was at the frontier of research in
econometrics in 1962. Today this material is more commonplace among
econometricians, though scattered through the book are some insights
and unifying themes which I have not yet seen in the econometric
literature. There are many more topics in econometrics which could have
been covered, e.g., spectral analysis, exact sampling distributions of
simultaneous equations estimates, and Bayesian econometrics, but the
topics and level of treatment in this book will amply fill a one-year
graduate course.
The reader should be forewarned that I am a bit idiosyncratic in my
notation. In chapter IV, I invent a new notation for simultaneous
equation models which, though imperfect, is of greater mnemonic value
than that of the two current sets of notation for such models. I also use
lower case letters to denote scalar variables, upper case letters to denote
matrix variables, regular type to denote non-random variables, and
bold-face type to denote random variables. Thus we can look at a symbol
and know whether or not it is a matrix and whether or not it is random.
(No longer need we write X for a random variable and x for its value,
equations like X = x, and no longer will we be confused about what
symbols to use for a random matrix and its value.) At the blackboard one
vi Preface
can easily denote randomness by using the printer's symbol ~ under the
letter to indicate randomness, e.g., what appears in this book as u would
be written at the blackboard as u.
Finally, I would like to acknowledge my debts to my teachers and my
students. The influence especially of courses taught by Ingram Olkin and
William H. Kruskal on the material in this book will be apparent to them.
My students, especially those of my 1972-73 Yale class, helped greatly
by finding errors, rooting out the unteachable, and asking stimulating
questions. I also gratefully acknowledge the encouragement of my
UCLA colleagues Karl Brunner, Jack Hirshleifer, and Michael In-
triligator to write up my lectures as a text and the stimulating influence of
the Cowles Foundation which gave me the impetus to complete this
book.
December, 1974 ALBERT MADANSKY
I
Matrix theory
The presentation of the small portion of matrix theory to which you will
be exposed in this chapter begins (section 1) with an admittedly
unmotivated presentation of the elementary formal operations with
matrices. Our aim in the beginning is merely to acquaint you with
matrices, and get you as used to manipulating them as you are to
manipulating real numbers arithmetically. We follow this introduction
with the theory of linear transformations in n-dimensional Euclidean
vector spaces (section 2) and then (section 3) motivate the elementary
operations with matrices by interpreting a matrix as a representation of a
linear transformation on such a space.
Next (section 4), we examine a special linear transformation, the
projection transformation, which will be used many times over in our
description of econometric methods. In section 5, we introduce in a
motivated fashion that object familiar to you from college algebra days
(though never motivated for you then), the determinant. We get back to
matrices in section 6, in which we study properties of a special type of
matrix, the symmetric matrix. In section 7, we introduce the generalized
inverse, a concept which will be found quite useful in expediting our
presentation of some aspects of both parameter estimation and
hypothesis testing in econometric models. Finally, in section 8, we
present the Dwyer-MacPhail notation so helpful in computing deriva-
tives of functions of matrices.
If you have already studied matrix theory, you would still be well
advised not to ignore this chapter. You may still find novel things in this
chapter, derivations if not facts. If you insist on ignoring this chapter, let
me at least warn you now of an idiosyncracy of this book. All vectors are
row vectors. I do this for two reasons, ease of typography and consis-
tency with the standard practice of describing points of the plane (i.e.,
vectors in Euclidean 2-space) by their Cartesian coordinates as (JC, y).
2 Foundations of econometrics
The price I will pay for this in later chapters is that standard econometric
models will at a superficial glance look a bit different from the way they
appear elsewhere. Be assured that this is only superficial. (And besides,
the hope is that, having read this book, you will have minimal need to
look elsewhere.)
1. Matrix operations
The rectangular m by n array of real numbers,
au al2- ' a,n
a \ a i - ' a n
2 2 2
A =
a \ a
m n
will be called an m x n (read "m by n") matrix. The order of the matrix
is the pair of numbers (m, n), where m, the first member of the pair, is the
number of rows and n, the second member of the pair, is the number of
columns of the matrix. The number a«, the element in the ith row and the
jth column of the matrix A, will be called the (i, j)th coordinate of A. A
description of all the a for i = 1,..., m, j = 1,..., n, is thus a descrip-
ih
tion of the matrix A, and vice versa. We will sometimes, for ease of
expression, say "A is the m x n matrix of ao-'s"; what we mean by this is
that A is the ra xn matrix whose (i, j)th coordinate is a for i =
ih
1,..., m, j = 1,..., n. All matrices will be denoted by upper case Latin
letters, and their coordinates by the corresponding lower case Latin
letters.
Two matrices A and B of the same order are said to be equal if a,·,· = bu
for all i and j. Equality of matrices is defined only for matrices of like
order.
Given an m x n matrix A, we denote by A ' the n x m matrix obtained
from A by defining its (i, j)th coordinate a'a as a . The matrix A ' is called
yi
A-transpose. Note that the rows of A are now the columns of A'.
Exercise: Show that (A')' = A.
If A is an m x n matrix with (i, j)th coordinate α and B is an m x n
0
Matrix theory 3
matrix with (/, j)th coordinate b then we define the matrix sum of A and
ih
J3, Λ + B, as the m x n matrix with (i, j)th coordinate a + ba. Notice that
(J
the matrix sum is defined only for matrices of the same order.
Exercise: Check that the following hold if A, B, and C are m x n
matrices:
A + B = B + A,
(A +B)+C= A +(B+C),
(A+B)' = A' + B'.
Suppose A is an m x n matrix and B is a p x q matrix. We define the
Kronecker product of A and B, A ® B, as the mp x rçq matrix,
abu aiibu'- · ai\bi a b a b ··· a b ··· a\ b
u q x2 n x2 X2 X2 Xq n iq
aubn a\ib2 - * · anb ai b i ^12^22 · · · ai^2q * · * a\b
2 2q 2 2 2 n 2q
aubpi aub · · · aub a\b \ a\b 2 · · · a\b · · · a\b
p2 pq 2 P 2 P 2 pq n pq
a\bu a\b\ · · · a b\ a b a b\ · · · a 2^ic, · * · a b
2 2 2 2X q 22 u 22 2 2 2n ïq
a\b\ a\bp · · · a\b a b a b · · · a b · · · a b
2 p 2 2 2 pq 22 pX 22 p2 22 pq 2n p
am\ bw am\b\2 - - - am\b\q am2bu am2b\2 · · · am2b\q · · · amnb\q
amibpX am\ bp amnbpq am2bp\ am2bp2 ami2hu pq amntyp
Notice that even when mp = nq, A ® B need not equal B ® A.
When A is a 1 x 1 matrix (i.e., A = a, a being a real number) it is usual
to suppress the symbol "®" in A ® B and write the Kronecker product
of A and £ as aB. We call this the scalar multiple of a and B.
Exercise : Check that the following hold if c and d are real numbers
4 Foundations of econometrics
and A and B are m x n matrices:
(c + d)A = cA + dA,
c(dA) = (cd)A,
(cA)' = cA\
c(A + B) = cA + c£.
With the operation of scalar multiple defined, the Kronecker product
of A and B described above can be expressed more succinctly as
a B a B • ülnB
u l2
a \B a B • a B
2 22 2n
A ®B =
[_a \B Q B ' ' * QmnB
m m2
Aided by the scalar multiple operation, the definition of matrix
difference of A and B, A - B is quite easy. Like the matrix sum, the
matrix difference is only defined for matrices of the same order, and is
given by A - B = A + (- \)B.
If A is an m x n matrix and B is an n x p matrix, then we define the
matrix product of A and B, AB, as the mxp matrix whose (i,j)th
coordinate is given by
n
Notice that the matrix product is defined only for matrices such that the
first member of the product has as many columns as the second member
has rows. Thus, though AB is defined, BA is not defined except when
m = p.
Exercise: (1) Check that, provided that the matrix products given
below are defined, the following hold:
(AB)C=A(BC),
A(cB) = c(AB), where c is a real number,
A(B+C) = AB+AC,
(A +B)C=AC+BQ
(AB)' = B'A'.
Matrix theory 5
(2) Let
1 6 -2 2 - 1 3]
4 0 4 0 2 1
A = B =
7 2 0 4 - 2 3'
6 3 3 2 -3 4j
Compute AB' and A'B.
The motivation behind this at-first-glance strange definition of the
product of two matrices will be given in the next section. To assure you
now that this strange definition of matrix multiplication is helpful,
consider the m simultaneous linear equations
011*1 + ai2*2+ * * ' + αΐπΧη = fei,
α ιΧι + a x + · · · + a Xn = fe ,
2 22 2 2n 2
û,mlX\ + Qm2X2 "I" * ' * + (ImnXn ~ fern,
in the n unknowns JCI, x ,..., x . The equations can be rewritten quite
2 n
succinctly in matricial form as the matrix equation
where A is the m xn matrix of ay's, X is the n-vector of JC.-'S, and B is
the m-vector of fe.'s.
Exercise: Show that, in general, there is no unique solution to m
simultaneous linear equations in n unknowns if n > m.
An n xm matrix A can be partitioned into blocks of submatrices
A A , A , A of order p x q, p x m - q, n - p x q, and n - p xm — q,
u 2 3 4
respectively, as follows:
Let
