THEORIES AND TECHNIQUES
IN HOUSING MARKET ANALYSIS
by
Joseph A Langsam
B.Sc. Massachusetts Institue of Technology
(1968)
M.Sc. University of Michigan, Ann Arbor
(1981)
Ph.D. (Mathematics) University of Michigan, Ann Arbor
(1982)
Submitted to the Department of
Urban Studies and Planning
and to the
Department of Economics
in Partial Fulfillment of the
Requirements of the
Degree of
DOCTORATE IN PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MAY 1983
Joseph A Langsam 1983
The author hereby grants to M.I.T. ermission to reproduce and
distribute copies of this thesis
cument in whole or in part.
Signature of Author:
ep
tment of Vrban Studies and Planning
Certified by:
Thesis Supervisor
Certified by
Second Reader
Certified by
Thie) Reader
Accepted by:
Head Ph.D. Com1 htee, Depattment of Urban
Studies and Planning
Accepted by:
Head Ph.D. Cbmmittee, Department of Economics
SS. INST. TEC,
RA R
Rotch
THEORIES AND TECHNIQUES
IN HOUSING MARKET ANALYSIS
by
JOSEPH A. LANGSAM
Submitted to the Department of Urban Studies and
Planning and Economics on May 1, 1982 in partial
fulfillment of the requirements for the
Degree of Doctor of Philosophy
ABSTRACT
Housing market analysis whether from the.vantage of
urban planning or economics presents both methodological and theoretical problems. The housing market
is characterized by search, while market data is frequently only available in cross sectional and aggregated formats. This dissertation contains three
principle results which should be for use in housing
market analysis. In the area of search theory, it
is shown that a search model can have an equilibrium
price vector where a commodity can have a nondegenerate equilibrium price distribution. A simple one
period urn type search model is analysized and the
conditions under which buyers or sellers are made
better off by market replication are determined. The
buyers bid problem is analysized and it is shown that
the bid structure need not be monotonic with respect
to time.
In the area of estimation and hypothesis testing two
results are developed. It is shown that an iterative
weighted least squares estimator converges in the
sense that for a fixed sample the iterates converge
almost surely and also in the sense that the estimator
constructed by taking these limit points converges to
the true.value of the parameters being estimated and
possesses other optimal properties. This analysis
corrects an error that appears in the article by
Oberhoffer and Kmenta., The final result is the analysis of a multistage heteroskedastic estimator which
enables the consistent estimation and hypothesis testing on the structure of the heteroskedasticity. This
2
procedure is a computationall1y simple procedure for
performing estimation and hypothesis testing on both
the underlying model and on the parameters generating
the heteroskedastic structure. The procedure presented in this essay, unlike that which appears in
Glejser and Parks papers leads to consistent estimation and consistent hypothesis tests.
The dissertation begins with a short introduction to
the problems in housing to which the theories and
methodologics developed in the thesis can be directed,
The principle results are presented in the second and
third chapters without their proofs. The mathematical
proofs are separated out and presented in the fourth
chapter so that the thesis can be used by those
researchers whose mathematical interests are minimal.
The fourth chapter should be of interest to those who
are interested in the application of functional analytic tools to regression theory.
Thesis Supervisor:
Professor William Wheaton
3
TO MY PARENTS
4
TABLE OF CONTENTS
Page
ABSTRACT...................
DEDICATION.................
..4
ACKNOWLEDGEMENTS...........
.. 7
CHAPTER I - INTRODUCTION...
.. 9
1.
Introduction......
.. 9
2.
Discrimination in hous ing
.14
3.
Estimation of hous ing
l e ti;-~ itias ±es..............
rel
ed
......o.. ... 25
MARKET SEARCH.............
...
37
1.
Introduction..................
...
37
2.
Review of search theory litera ture.... ... 38
3.
Equilibrium in search models..
...
4.
A housing search model........
... 50
5.
An optimal bidding problem in a search
... 64
model.........................
CHAPTER I I
CHAPTER I II
-
-
44
ESTIMATION AND HYPOTHESIS TESTING
IN THE PRESENCE OF HETERO SKEDAS-
TICITY...................
...76
1.
Introduction..................
...
2.
A theorem of Halbert White....
...80
3.
Block scalar covariance matrix
...85
4.
Linear variance model.........
.. 102
5
76
TABLE OF CONTENTS
Page
CHAPTER IV - MATHEMATICAL PROOFS ...................116
1.
Introduction.............................116
2.
Market search............................117
3.
The housing search model..................124
4.
Block scalar variance covariance matrix..133
5.
Linear variance model....................162
CHAPTER V - SUMMARY AND CONCLUSION................186
BIBLIOGRAPHY......................................189
ACKNOWLEDGEMENTS
I would like to thank a number of people who over
a ten year span have assisted and supported my efforts
to complete this dissertation.
My parents, Sid and
Helen Langsam gave both moral and financial support to
a degree beyond that even expected by a hopeful son.
They remained supportive even, when after five years
of study in economics and urban studies, I switched
fields to earn a Ph.D. in Mathematics.
To them, I owe
a great debt that cannot be repaid.
I am indebted to my advisor Professors Wheaton and
Hausmann and to Professor Diamond.
They have been
most helpful in assisting me to improve the thesis and
to navigate the various rules and regulations associated with earning a joint degree.
I would also like
to acknowledge kindness and the assistance of Professor
Eckaus.
That this dissertation is too wordy is my
fault; I am indebted to Professor Fisher that it is
not even more so.
I would like to gratefully acknowledge the many
mathematicians at M.I.T. and the University of Michigan
who spent the time to teach me the value of mathematical analysis and rigorous thinking.
7 *
I would publically
like to thank my colleagues in the Department of Mathematics at Case Western Reserve University who have
listened with understanding to my stories about writing
a second dissertation.
I want to thank and to acknowledge the contributions of my wife, Betty and my children, Daniel and
Jessica.
Without them, this thesis would never have
been written and my life could never have been as
happy.
CHAPTER I
INTRODUCTION
1.
The core of the dissertation is
Introduction.
comprised of -three distinct investigations.
The first
of these is directed towards problems arising in search
theoretic economic models.
A reasonable definition of
equilibrium is presented and an existence theorem is
proven.
A simple one period search model is described
and the welfare implications associated with increasing
the market size through replication are analysized.
The question of optimal bidding in a sequential search
model is discussed and an existence theorem for optimal
bids is proven.
An example is then given which shows
that with costly sampling the optimal bid profile need
not be monotonic.
The second and third investigations are concerned
with estimation and hypothesis testing in a linear
model in the presence of heteroskedasticity.
In the
second investigation an iterative weighted least square
estimation for a linear model with block scalar variance convariance matrix is described and analysized.
A theorem giving sufficient conditions for the
successive iterates associated with a fixed sample to
9 ,
10
form a Cauchy sequence almost surely as the sample
size increases is proven.
This theorem corrects an
error in the paper by Oberhofer and Kmenta [33] where
the argument showing convergence of the successive
iterate is faulty.
The estimator constructed by
taking the limits of the successive iterates is shown
to have desirable asymptotic properties.
The final investigation is directed towards estimation and hypothesis testing when the variances
follow a linear model.
A simple procedure is described
and analysized for estimating and performing hypothesis
tests on the variance model and then for estimating
and performing hypothesis tests on the original model.
Unlike the procedure given in Glejser [//],
the
asymptotic distribution of the estimator for the variance model can be computed.
The proofs of the theorems
showing consistency of this estimator and hypothesis
tests is an extension of the work given in White [51]
It is not an immediately consequence of White's
procedure in that his procedure would require inputs
that are not observable.
This proposed multiple step
procedure is then shown to generate an estimator for
the underlying linear model with optimal asymptotic
properties and with easily computed asymptotic
11
distributions.
It is a simple consequence of the
theorems proven, that this procedure generates a
simple test for heteroskedasticity.
The motivation for these essays comes from
problems encountered in urban planning, most particularly in housing and manpower planning.
The labor
and housing markets are ones in which search plays an
important role.
They are also markets in which metro-
politan and regional data. is collected in a cross
sectional rather than a time series formating.
Thus
heteroskedasticity is likely to be a major problem in
the analysis of the regional and metropolitan data.
This dissertation is intended to be a contribution in
both urban studies and economics.
For th.is reason,
the dissertation contains a brief discussion of urban
problems towards which the theoretical ideas developed
in the main body can be applied.
A brief discussion
of housing discrimination is given since programs
directed at housing desegregation tend to aggregate
the market and market aggregation or replication is
a subject of the first investigation.
The attempts
to estimate housing demand elasticities is briefly
reviewed since this estimation will often be done in
models with heteroskedasticity present.
Because of
12
the varied purposes towards which the dissertation is
directed, the thesis is organized to present the
major results first qualitatively,then quanitatively
with proofs only sketched, and finally with full
mathematical rigour with complete proofs.
While
this leads to unfortunate redundancy, it does provide the policy maker, technician, and theoretician
with the level of generality appropriate for their
needs.
The remainder of this chapter contains the above
mentioned discourse on housing discrimination and on
estimation of housing demand elasticities.
The
second chapter contains the search related results.
It begins with a brief review of known results
followed by a description of equilibrium in search
models and a theorem giving conditions for the equilibrium to existence.
A simple consequence of this
theorem is that in a search model identical commodities need not have the same equilibrium price.
A
simple housing search model is presented and the
welfare implications of market replication are
analysized.
The chapter ends with a discussion of
optimal bidding in a search model 'containing an
existence proof and an example showing that the
13
optimal bid profile need not be monotone with time.
The third chapter begins with a general discussion of
the nature of the heteroskedasticity problem.
The
iterative least squares estimator for the block
scalar variance covariance matrix is then analysized.
Theorems giving condtion for the convergence of
successive iterates and for the optimal asymptotic
properties of the estimator are stated.
White t s
procedure is briefly described followed by the description of a new procedures for estimation and
hypothesis tests for models where the variance
structure follows a linear model.
From this pro-
cedure an estimator and hypothesis tests for the
variance modelare developed.
properties of
A theorem giving the
these tests and estimator is stated.
As a corollary a simple new test for heteroskedasticity is presented.
Also from this procedure a
multiple stage estimator and hypothesis tests for the
basic model are developed.
A theorem giving condi-
tions for these tests and estimator to have desirable
optimal asymptotic properties is then stated.
The
fourth chapter contains a restatement of the theorem
of the first two chapters together with complete
proofs.
It should be of interest to those interested
14
in the application of functional analytic technique
to statistical problems.
The last chapter contains
concluding remarks and suggestions for future applied
and theoretical research.
2.
Discrimination in Housing.
The nation has adopted
as policy goals the desegregation of residential
neighborhoods and, the increase of housing consumption
by low income families.
Federal, state, and local
governments have instituted a variety of programs
seeking these goals including programs which provide:
rent subsidies, subsidized new construction, legislative relief through zoning, and market information
services.
These programs are designed to relieve
some barriers that the planner perceives to generate
segregated housing patterns.
Underlying the choice
of programmatic relief must be a theory of market
behavior.
Since resources are limited, one naturally
tries to select those programs which are most cost
effective.
To do this requires a sufficient knowledge
of economic theory and of empirical techniques.
[/o3,
In
p.49D], Stevens outlines the problem when she
states:
15
Studies of housing demand in the United
States have found significant differences between the behavior of white majority and black
In particular, blacks
minority households.
choose housing of a different tenure class mix,
quality, and location from white households.
These differences in demands have been said to
depend on any or all of the following:
(1) blacks' preferences differ from those of
(2) blacks consume different quantities
whites;
and qualities of housing than do whites because
blacks face price and entry discrimination in
the housing market; and (3) blacks' income, both
current and permanent, is lower than whites',
causing blacks to consume less housing services
than do whites.
Identification of the most important cause
of the difference between black and white housing
consumption patterns is important in devising
a housing policy to meet national housing goals.
If discrimination is the most important cause of
low housing consumption by blacks, then an open
housing policy is indicated. If differences in
consumption patterns are mainly attributable to
differences in current or permanent income, then
transfers or policies to increase job skills,
labor mobility and employment quality will
achieve housing goals.
Finally, if tastes
differ, housing vouchers or other housing subsidies may be the only feasible way to induce
some households to consume an "adequate" level
of housing services, however such level of
services is defined.
Most previous work has found that price and
entry discrimination exists in the housing market.
Blacks on the average earn smaller incomes than
do whites, whether income is measured on a
current or a permanent basis. There is, however,
disagreement on the portion of demand differences
which can be attributed to each of these factors.
The possible barriers which preclude integrated
neighborhoods as a result of market forces include:
16
differences in tastes between blacks and whites
resulting in differing preferences for housing consumption, differing tastes resulting in whites strongly
preferring self association, disciminatory practices
on the part of sellers and brokers, historically
generated endowment differences between blacks and
whites, transportation network limitations that result
in costly commuting between certain residences and
certain job sites, and historically generated housing
and work place locational differences between blacks
and whites that preclude the free flow of market information.
Clearly any subset of these may cause
segregated housing patterns.
The planner is
confronted with the problem of selecting those which
make the greatest contribution to market segregation.
To choose among these factors those that are dominant,
requires both a market structure theory and a means
of empirically estimating market parameters.
In the
area of residential segregation, much effort has been
made in determining the importance of these barriers
and one can find some of the results in
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17
A cental issue is planning to desegregate residential neighborhoods is wherer segregation results
from pure preference considerations or from discriminatory practices in the market place.
Pure preference
considerations can generate segregated housing if
either there are housing consumption preference and
endowment differences between blacks and whites or if
racial considerations enter directly into preference
structures.
Kern in [?/] using an equilibrium analysis
in a housing market model is able to show: that if
whites' preference for whites is stronger than blacks'
preference for blacks,an integrated equilibrium is
unstable and a segregated equilibrium is stable, if
blacks prefer whites greater than whites prefer whites
no segregated equilibrium exists and a stable integrated equilibrium exists.
In the integrated equili-
brium all sites have idential racial composition and
therefore racial composition has no effect on equilibrium rent distance function.
In the segregated
equilibrium where whites prefer whites and blacks
prefer blacks equilibrium rents on the white's side
of the boarder may exceed, equal or fall short of that
on the black side.
In the-segregated equilibrium
where both races prefer white neighborhoods, rents
18
on the white side of boarder exceeds that on the black
side provided there are no discriminatory practices.
Farley and Bianchi in [73] report a survey that suggests that whites prefer whites while blacks prefer a
50-50 integrated neighborhood.
Thus in a segregated
equilibrium one expects in the absence of discriminatory practices, that segregated white residential
rents will exceed black rents.
Miesykowski and Syroy
in [7] summarize current economic housing market
theory and their findings show that income differences
are a small factor leading to segregation, pure
preference differences are a significant factor and
lead to whites paying a premium for segregation, and
that whites overtly discriminate against -blacks
resulting in both higher housing prices and limited
job opportunities for blacks.
in (I]
Follain and Malprezzi
attempts to empirically test whether blacks
pay a premium.
They estimate a hedonic model in which
race is a variable and using micro level survey data
show that blacks receive a discount of about 15% in
owner occupied units and 6% in renter markets.
This
supports a finding that pure preferences are a dominant factor in determining market segregation.
19
The question of whether segregation is a result
of pure preferences or discrimination is important in
a number of areas.
If segregation results from
preferences of self association and if residential
housing market segregation does not cause disadvantage in other markets, programs to force integration
may result in everyone being worse off.
evidence, see for example [7,?],
There is
that housing segre-
gation leads to blacks being at a disadvantage in the
labor market.
In this case, one must understand that
improvement in blacks' welfare in the labor market
resulting from integration of residence are traded
off against losses from not being able to self-associate.
If, however, segregation is the result of
discriminatory practices, programmatic relief might
include both legislative and compensatory programs.
Both policy goals and programmatic content are affected
by the identification of those barriers which generate
residential segregation.
Whatever the cause of neighborhood segregation,
programs which successfully reduce segregation have
the effect of aggregating several nearly independent
submarkets into a larger housing market.
In many
instances a close examination of the housing market
20
reveals that it is comprised of nearly independent
submarkets.
Some of the factors which lead to this
market segmentation include: strong ethnic self-association preferences, transportation networks which
makes interzonal commuting costly, physical barriers
such as rivers, parklands and large expressways, and
historically generated governmental structures.
Frequently programs which are directed to these factors have as a secondary effect changes in market
segmentation.
In markets in which search is characteristic
there are frictional costs associated with search;
that is costs required to obtain price information
and costs associated with making decisions without
full knowledge of price structures.
The aggregation
ob submarkets into a larger market will change these
frictional costs.
In the first essay, we perform a
partial equilibrium analysis for a specialized type
of market aggregation to determine the effect upon
frictional costs associated with market aggregate.
An equivalent formulation of the problem of
aggregating several identical markets is to consider
the replication of a single market.
This latter
approach is easier to deal with analytically and is
21
In the market
the approach taken in the first essay.
are
n
buyers and
m
and
xn
the
.th
j
ith
xm
times there are, of course,
x
replicated
When the market is
sellers.
sellers.
Each seller has one unit for sale,
Y.
buyer has a potential bid of
for the
ir
In the replicated market the
unit.
Y. .
has a potential bid of
jt
for the
13.th
buyer
unit where
seller has a
jth
The
1 < t < x.
and
1 < r < x
buyers
reservation price or minimum acceptance price of
Xi;
in the replicated market the reservation price held
jt
by the
All analysis for a single
X..
seller is
In this time period each buyer
fixed time period.
independent of other buyers selects from a uniform
probability distribution exactly one seller to visit.
Buyer
will purchase the unit owned by seller
I.
'i
if buyer
Y.
1)
. > Y.
1,3
,
2) Y.
. > Y.
=1
1,3
visit
I.
where
seller
Y.
J.
3
and either:
I
J.
J
for all buyers
.
,3
I.,, that
J.
wins the toss of a fair
is the number of buyers
s
> X.
.
for all other buyers
.
that visit
and buyer
J.,
visits seller
J.
with bid
Y....
1
.
,J
=
Y.
1,3
Ii,
s
sided die,
that visit
The same trans-
22
action rules apply to the replicated market.
Since we are interested in the effect upon frictional costs associated with market segmentation and
aggregation, bid and reservation prices are held fixed.
In this model, the frictional cost in the replicated
.Ii,
market to the buyer
can be measured by
The frictional
the probability of making a purchase.
cost in the non-replicated market is given by
The frictional costs to the seller
P(l,i).
in the repli-
J.
the proba-
Q(x,j),
cated market has two measures:
P(x,i),
E(x,j), the expected
bility of making a sale, and
value of a sale given that one is made.
The first step in analyzing the impact an frictional costs associated with aggregation is to compute
and
Q(x,j),
P(x,i),
E(x,j).
Before giving their
values, it is necessary to introduce some additional
For an arbitrary set
notation.
whose bid for the
G.(r) = I {I.:Y.
that is
Let
for the
G.(r)
J
Jth/unit
let
its cardinality,
.<
S,
I[SI
denote
be the fraction of buyer
does not exceed
r}
r,
/m.
B.(r) be the fraction of buyer whose bid
J
jth unit equals r, and let H (r) be the
fraction of buyers whose bid for the
least
let
r.
Let
F(r)
j th/unit is at
be the fraction of sellers
23
whose reservation price for this unit does not excee d
F(r) =||{ J.: X. < r||}/n. Let C. be
3 J
J
the set of buyers whose b id for the unit owned by J
r,
that is
.
is at least as great as i ts reservation price.
Finally, let a = m/n
be the ratio of buyers to
sellers and let
=n
u(x)
x1.
It is important to
note that those measures that are ratios do not
change when the market is replicated.
Using the above nota tion it is shown that
1
=
P(x,i)
'x xm
[H.(Y)
3 l'
B (Y
]
i']
j FA 1e mB(Y.
1,3
3
[H (Y. .)]
-ixm 'l
Q(xj)
P(x)
-
1
-
y(x)
XJj
E (x ,j)
iEC.
x
[H.
I(Yi j) -B.(Y.
3 1,3 .) ]
Y. .
[(x) xm 31
mB. Y. -)
H. (X.)
J
1, J3
3 3
xm
(x)
1 _
While the number of times
x
[H.(Y. .)]
m3
1,3
that a market may
be replicated is integer valued (x>O), the expression
P(x,i),
Q(x,j)
and E(x,j)
have for each
natural extension to smooth function in
i
and
j
the x variable.
To determine the effects of frictional costs associated with aggregation one can examine the value of
24
3P(x,i)
3Q(x,i)
3x
3xX
'
and 3E(x,j)
for
x>l.
We see
'x
that for a fixed
j
if there are at least two buyers
with different bids for the
j th/ unit at least as
large as
then both
and
X
and if
3E(x,j) < 0.
DX
n >2,
'Q(x,) < 0
Thus in almost every case the
sellers frictional costs are increased by aggregating
equivalent submarkets.
Ii,
For the buyer
the results
are not as conclusive, however, the following can be
shown:
1)
sufficient conditions for
3P
(x,i)
0 are
that nx>2, A./0, and a>sup[H.(Y.1,3.)-B.(Y.
3 1,J .)
3
2)
a sufficient condition for
that
Ai
3)
if
cP-(x,i)<0
are
nx>2.
0,
and
nx>2 and a<5/7,
As a consequence of 2 above
-l
<nxin(nx-1)
then
P(x.i)<0
in markets with a large
number of sellers and in which sellers outnumber
buyers it will in almost every case be to the benefit
of the buyer not to have market aggregation.
In
general it appears that maintaining segmented markets
25
helps both the buyers and sellers by restricting the
competition that buyers feel from other buyers and
sellers from other sellers.
The exception occurs
for the buyer who tends to be a low bidder for each
unit he sees and who is in a market with many more
buyers than sellers, in this case the low bid buyer
would prefer aggregation.
In this case, it appears
that-aggregation gives the low bidder more opportunities for finding a unit for which he is the only
bidder.
The above analysis has clear limitations.
In
addition to assuming that the segmented markets are
identical, it is a single period model without dynamic features.
Nevertheless, it provides a framework
for illustrating the importance of frictional costs
that arise wherever search is a consideration.
In
particular it alerts the planner and program analyst
to welfare factors that should be considered whenever
programs directly or indirectly affect the parameter of search including market segmentation.
3.
Estimation of Housing Related Elasticities.
An
essential part of the planning process, especially in
housing and manpower planning, is the identification
26
of the current status of various market parameters.
In the field of housing analysis and planning the
important parameter includes price and income elasticities of demand and price elasticity of.supply.
f
Recall that if
0
X*=(X*,X
n
1' 2 ,...,X*)
Rn
is a point in
and
then the elasticity of
the point
by
Y X(X*)
J
Y
=
R
,
R ,
Y=f(x),
X. at
J
is given
with respect to
is denoted by
X*
into
is a function from
n Y.(X*)
The elasticity is a
-j (X*)
J
measure of the percentage change in
output for a per-
centage change in an input.
Much of the current debate among housing policy
analysts center on whether programs for low income
housing should feature supply or demand side subsidies.
Central to this debate are questions concerning
supply and demand elasticities.
If the income elas-
ticity of demand is high and the price elasticity of
supply is near zero, then income transfer programs
that raise the incomes of low income families will
tend to just raise housing prices.
If on the other
hand, the price elasticity of supply is very large,
such programs will then result in low income families
consuming more housing without large increases in the
price of housing.
If the price elasticity of demand
27
is near zero, supply subsidies designed to lower the
price of housing will not result in substantial
increases in housing consumption.
Knowledge of these elasticities is also important
in designing programs addressed to residential segregation.
Suppose higher quality, higher priced housing
is found in predominately white neighborhoods and
lower quality lower priced housing is found in
racially mixed neighborhoods.
If low income whites
have a higher income elasticity of demand than low
income non-whites, an income transfer program that
raises the income of low income families will result
in white families leaving the integrated neighborhoods
to move to the substantially wh.ite neighborhoods.
If
the non-white income elasticity of demand is the
larger, then the same program will result in non-whites
moving into segregated neighborhoods and thus increase
the amount of residential integration.
The usual procedure for estimating part or all
of an unknown parameter victor
6
in
RK
is to speci-
fy (assume) a linear relation
Y = XS+E
vector
Y
and data matrix
X
is observable.
vector
6
is estimated by
b0 1 s
(XT
where the
-1 XTY
The
and
28
is estimated by
the variance of b
T -1
ols
T
2-(Y-Xb o)s
(Y-Xb
)~s(X X)
w erIe
n-K
i'-I
a
is
Y
nxl
(n is the number of observations).
vector
Among the assumptions which are implicitly made
in using this procedure are that the error term
has zero mean and follows a homoskedastic
a
I,
that is
E(s eT)
scalar.
Under the usual
=
where
a
is a positive
assumption, it is not
OLS
b
difficult to show that the estimation
s
2
1)
b ols
2)
in the event that the data matrix
and
s
and
are consistent
non-stochastic,
b ols
X
6(bols
in BLUE).
is asymptotically efficient.
has zero mean, but
c
If, however, the error term
follows a heteroskedastic distribution,
E(E c )
a non-scalar diagonal matrix, then while
bols
still consistent, efficiency is lost and
s2
a consistent estimator of
a2
is
is
is not
so, in particular, the
usual hypothesis tests based on
performed.
is
is a best linear
b ols
unbiased estimator of
3)
S
of
have certain desirable properties including:
a2
of
distribution,
s2
should not be
29
Heteroskedasticity may be introduced into a model
because of both theoretical considerations, and as a
The most obvious
result of measurement procedures.
way in which heteroskedasticity is introduced is
through data grouping.
Y.. = X.
1
13J
+ e..
.
where
13
(1 < i < n,
If the model is
1 < j
e. .
are homoskedastic,
13
< mi.)
and the estimated model
uses group averages, that is the estimated model is
m.
Y.
X 6 + 6.
where Y. L .1Y
i1i1
X.
-
in.
m.
E.
=-
te
mi
c.
then the
X
j=l 13
scedastic distribution.
1
j=;
13
follows a hetero-
1
If the numbers
m.
are known,
one can perform weighted least squares by weighting
the vector
(Y.,X.)
by
v'm
the optional propertiesof
and thus regain some of
OLS.
If
m.
are now known,
one should perform a weighted least squares procedure
which first estimates the
m.
1
and then use these
estimates to weight the observations.
Heteroscedas-
ticity can also be present for purely theoretical considerations.
The best example is the problems associ-
ated with estimating consumption or demand functions.
In the simplest case, see [/S,/B)
consumption is
assumed to be a linear function of income and the
model
Ct
Ao + B Y+
consumption and
Y
t
is estimated where
is income.
C
It is observed that
is
30
the variance of
ct
varies with
Yt,
and it is easy
to develop a theoretical explanation for this phenomenom.
For an understanding of the consumption income
relationship as well as for prediction purposes it is
important to analyze and estimate this variance-income
relationship (see U2] for an introduction to the
relevance of the heteroskedastic structure to the
original model).
Heteroskedasticity is likely to be a problem in
the estimation of housing demand elasticities both
because of grouping and because of the theoretical
structure of the error component.
There are in the
literature a number of attempts to estimate price and
income elasticities of demand and own price elasticities of supply with wide ranging results.
example,
[4],
['/],
[70],
[3fj],
[72],
[/0/],
[[o], [R],
[/o,] and [/7].
[g/),
(See, for
[7], [F7],
The following
table gives an overview of the wise range of estimates that have been presented.
It is difficult to
analytically compare the estimates of different
authors since they use different specifications and
different measures for the various-variables.
To the
extent, however, that the concepts of income elasticity and price elasticity has developed into policy
31
and program planning variables, it is interesting
to see the variance among these estimates.
Dusenberg and
Kristin, 1953, [70]
Lee, 1968,
-. 078
.8 owners
.58 renters
[f'q]
Winger, 1968,
n
p
n
y
.15 (6)
[/o]
1.03
Maisel, Burnham and
Austin, 1971,
[R]
de Leeuw, 1971,
Carliner,
1973,
[g3-1
.7 -1.5 owners
.8 -1.0 renters
[]
.50 renters
King, 1976, [8',]2
Polinsky, 1977,
-. 89
.45
.64
[4/]
.75
Stegman and Sumka
1978 ,
[/0O]
Polinsky and Elwood,
1979 L3C)
McRae and Tuner,
1981 [oi-)
1)
.251
.337
.195
current income
-.400
(permanent income)
(Black Families)
.57
- .67 micro
-.72 grouped
.25
-
.39
.89
Their reported elasticity is the coefficient of
a linear demand equation.
The imputed elasticity
would be 0.6.
2)
King estimates a Lancastrian demand model and the
value 0.6 reported in the table is the imputed
32
elasticity of demand for space with respect to
income.
A central question in the estimation of income
demand elasticity is how to measure income.
In
general micro level household income demand is not
collected.
Furthermore it is not clear what concept
of income, permanent or current should be used.
Polinsky and Ellwood [35] attempt an analysis of
the specification error associated with various
choices of measures of income and with using micro
verses grouped data.
This paper critiques the
earlier work of Carliner
Burnham,
[8] , Lee [9/] , Maisel,
and Austin [9] , and Winger
[/17-]
and shows
that much of the divergence of income and price
elasticities can be explained by mispecification of
the income variable.
While Polinsky and Ellwood
observe that heteroskedasicity will occur due to
grouping, they fail to adjust for its presence in
the micro model estimation and assume in the grouped
estimation model that it arises
only from grouping.
In the last two technical essays two procedures
for estimation and hypothesis testing in the presence
of heteroskedasicity are presented and analyzed.
the first of these the variance covariance matrix
In
33
is assumed to be block scalar with a fixed number
of blocks.
the form
The model to be estimated could be of
Y..
= X. .
+ c..
1)J1
1 <
j
< m.
1
where
1)
and variance of
c.
1 < i < m,
22
=a2
1)
i
(a. >0).
The estimation scheme presented is an interactive
weighted least square procedure.
a.
2
1t
is estimated by
1
B
and
weighted least squares (weights
stage
and
a.
Br+l
is estimated by .
is computed using
=
1).
n. . 3=1(Y
r+1s_
In the
13
-
-
3
2
is computed using the weights derived
2
a. .
from the estimated
In the ensuing analysis it
is shown that the sequence
sure as
In the first stage
max(ri.)-+oo,
Sr
converges almost
it is also shown that the esti-
1
mates for the variance also converge.
estimation of
6
and
a- 2
The resulting
are shown to have the
usual desirable asymptotic properties, the estimator
for B being asymptotically equivalent to weighted
least squares with known true weights.
It is also
shown that the usual hypothesis tests using the
estimates in the weighted model using the weights
2
derived from the estimator a.
has the usual known
asymptotic distribution.
This procedure of inter-
active weighted least squares is appropriate when
34
using group data or when the variance of the error
term is thought to be related to the values of a
discrete variable.
In the last essay we present a procedure for
estimation and hypothesis testing in the model
Y
=
X +c
where the variance covariance model is
diagonal with diagonal vector
and where
Z
is observed and
2
a,
r
a
where
unknown.
2
=
ZF,
Starting
with the works of Glejser and White we develop an
easy to apply multiple step estimation procedure
that not only permits hypothesis testing on the
estimated
r
but also yields an estimator for
B
which is asymptotically equivalent to weighted
squares with known weights and for which the usual
In the first
standard hypothesis tests are valid.
stage
6
is estimated by
squares estimator and
y
=
Z e ,
(Z Z)
jth2 element is
r
where
bols,
the ordinary least
is estimated by
is the vector whose
e2
(Y -X. b
)2
Asymptotically, valid hypothesis tests can be
performed on
W1 2
1
nl1=1
:L
r
T Z.1
and using
y
using the estimated
as an estimator of the variance
covariance matrix where
W.2
l
2
-
Z.y)2
1
The
35
estimated variances are computed by
a2
=
Zy
weighed least squares are used to reestimate
and
S.
The
statistics reported from the ordinary least square
regression of the weighted model has the expected
known asymptotic distribution.
Heteroskedasticity has been an often ignored
problem in estimation and hypothesis testing associated with urban economic analysis.
In particular
it will be present in the estimation of demand
elasticities, which are themselves important variables in the formulation of urban housing policies.
This dissertation presents two procedures for handling
the heteroskedasticity problem.
The first of these
is appropriate when the variance covariance matrix
of the error term can be put in a block scalar form,
for example when the variance is related to the values
of a variable with finite discrete range.
The second
procedure is appropriate whenever the variance of the
error term is a linear function of observable variables.
This latter procedure is particularly appro-
priate for use in estimating demand elasticities
where the variance of the error term is likely to be
a function of income.
36
There are a number of research directions suggested by the works in the dissertation.
In
the areas
of search theory, one should begin to examine the
impact of aggregating non-identical markets and
should begin to develop multiple period search
models in which there is intertemporal dynamics.
In particular one wants to investigate how search
affects long run housing patterns.
In the area of
estimation and hypothesis testing in the presence
of heteroskedasticity, both the estimators presented
here should be compared to other procedures to
determine their relative powers.
In particular the
interactive scheme presented here can be compared
to the simple one interation weighted least squares
procedure in a Monte Carlo study.
Clearly, the
next obvious step in the estimation of housing
demand elasticity is to redo the study of Polinsky
and Ellwood using the heteroskedastic correct procedure developed in the last essay.
It will be
interesting to learn whether the divergence among
elasticity estimations can in part be the failure to
correct for heteroskedasticity.
CHAPTER II
MARKET SEARCH
1.
Introduction.
In markets with search, either
consumers or producers are making decisions with
less than full information about commodities and
prices.
Buyers or sellers are making decisions
based upon expected prices and not upon a commonly
observed market price.
Thus, a priori, one need not
expect that a commodity in a search model would have
a well defined price.
Therefore consumers and
producers are faced with not only the decision of
how much to buy and sell, but also if, when, and
where to make these transactions.
Both housing and labor markets are classical
examples of search markets.
The prospective house
buyer usually is unaware of the available stock, its
quality and its price while the prospective seller
is unaware of the potential market demand.
In labor
markets, the employer is uncertain of prospective
skills of an applicant and of the minimal wage
acceptable to the applicant.
The applicant is
unaware of both job openings and of their potential
wage rates. While
much of the work done in search
37 ~
38
theory has drawn from labor market oQfservations for
its motivation, many of the results are immediately
transferable to analysis of housing markets.
The remainder of the chapter is divided into
four sections.
In the next section is a brief review
of search theory literature with most of the articles
concentrating on search in labor markets.
The third
section contains a discussion of equilibria in search
module together with a tentative definition of equilibrium and an existance proof.
The fourth section
contains a simple one period housing search model
which is used to analysize the effects upon buyers
and sellers resulting from duplicating the market.
The last section contains a discussion of optimal
bidding in a search framework.
2.
Review of Search Theory Literature.
a feature in every market.
Search is
In almost every market
where the cost of search~is sufficiently high, transactions are made with the participants possessing
less than full information.
In some markets, the
marginal cost of search is so sufficiently small that
behavior in this market is perturbed but slightly
39
from that in a deterministic market with perfect
information.
In deterministic full information
models, however, it is difficult to support nondegenerate price distribution for homogeneous goods,
support un- and underemployment of resources, and
support the existence of advertising, while in a
search model these phenomenon are natural consequences.
Despite the obvious importance of search
in economic analysis it has only recently been
developed in the literature.
Economic search litera-
ture seems to owe its origin to the two papers by
Stigler [Nir].
In these papers, Stigler argues that
non-degenerate price distribution might be supportable if the cost of obtaining price information is
high.
In the later paper, he presents a job search
model that will support the job seekers accepting a
wage less than the maximum available in the economy.
It has been shown, however, that the search strategy
presented in this model is suboptimal.
Much of the literature subsequent to Stigler's
premier articles can be divided into two classes;
optimal search strategy, and existence of non-degenerate equilibrium price and wage distributions.
The
optimal search or optimal stopping time theory that
40
appears in economics also appears in sequential
analysis, in statistics, and in the study of stopping
times and smartingales in probability.
It should
probably be a meta theorem that any theory that
appears semi independently in three fields has relevance.
In any event, optimal search theories are
making a contribution in explaining labor market
behavior.
Much of the optimal search literature has
regarded the job seeker's and employer's problem as
distinct and separate.*
The job seeker's problem is taken to be some
variation of the following.
The job seeker samples
sequentially from the distribution of wage offers
incurring a cost for each sample and seeks a rule to
tell him when to stop searching and start working.
The employer's problem is taken to be a variation of
the following.
The employer sequentially observes
job seekers with a particular marginal product from
a distribution incurring a cost for sampling.
The
employer is seeking a rule telling him when to stop
sampling and make a wage offer
w.
It is surprising
*
For the purpose of simplicity, the language of
labor market analysis is adopted throught this section.
41
It is
that these problems are not consistent.
assumed that if the employer makes offer, it will be
accepted while the job seeker's problem is when to
accept an offer.
Furthermore in most of the litera-
ture, the employer is assumed to have a fixed wage w,
so that his decision is only to offer or not offer a
job.
The complexities of the analysis depends upon
the assumptions that are made on the objectives
being optimized and the learning process.
Some of
the variations that have been analyzed in the optimal
search strategy literature include assumptions of
infinite time horizons no discounting, finite time
horizon positive time discount, random number of job
offers at each period, underlying wage distribution
known, underlying wage distribution learned through
a Bayesian process, risk adversion, and wealth constraints with bankruptcy.
The literature has also
addressed the question of search strategies when one
can choose between distributions and search strategy
when one can, at a cost, affect the distributions one
faces.
The former model is used by Wohlstetter and
Coleman [10],
Kosters and Welch [20),
and McCall
[u]
to explain observed discriminatory behavior in the
work place.
The latter model has been used by many
42
to explain advertising.
Kohn and Shavell [/71, made a substantial contribution by reformulating the optimal search problems in
sufficient abstraction and showing that the same
analysis applies to both the employer's and job
They are then able to show that in
seeker's problem.
the majority of the variations, the optimal stopping
rule is a switch point rule.
d,
optimal stopping rule
n
one has not stopped after
samples, then there is a number
stops at time
the
That is, if under the
(n+l) th/
n+1
s
such that one
if the utility associated with
observation exceeds
s
and continues
*
if it is less than
s.
They are then able to determine what happens to
the switchpoint
They show that
s
s
under a variety of conditions.
falls with an increase in time
preference, and with an increase in next period's
expected search costs.
They are also able for special
cases to determine the effect upon
s
of increased
risk in the sense of Rothschild-Stiglitz [37] and in
the sense of Diamond-Stiglitz [//]
While their analysis holds for other economic
problems involving optimal search, Kohn and Shavell
have chosen to use the language of the expected
utility maximizer.
43
For the reader interested in the mathematics of
optimal stopping, Chow, Robbins, and Siegmund [8 ] is
an excellent, though difficult reference.
Good
sources for the probability theory necessary to understand the optimal stopping literature are Ash [3],
Chung
[1] , and Feller [/3].
Perhaps the greatest motivation for search theory
research has been in the analysis of non-degenerate
wage distributions and in- the analysis of unemployment.
The optimal search strategy has attacked these problems
from a partial equilibrium analysis, that is regarding
either the employer's or job seeker's behavior as
exogenous.
Early equilibrium wage models were
generally unsuccessful in supporting non-degenerate
wage distributions.
Indeed, in many of the early
models, the wage distribution collapsed to the single
monopsony wage.
assuming that:
This has been shown to be a result of
there is a single market, the number
of employers in the market is large, the cost of
search in positive, employers maximize profits,
employees maximize discounted net wages, and the
equilibrium distribution is known by all.
In the
early 1970 1 s a number
models in
of authors presented
which some of the above
assumptions were relaxed
and
44
which sustained non-degenerate wage distributions.
Mirman and Porter [28],
Lucas and Prescott [23],
and Telser [4fy] have
Mortensen [30], Diamond [/0],
each presented equilibrium models explaining wage
distributions.
More recently Varian [5L] has shown that the
search structure can explain the existence of sales,
[7],
Butters
uses a search structure for analyzing
advertising, and numerous authors have used the
search structure for analyzing effects of government
policy on unemployment.
3.
Equilibrium in Search Models.
In this section a
definition of equilibrium in search models is presented, and for certain elementary models this
equilibrium is shown to exist.
Much of the notation
and many of the concepts in this section are taken
from Arrow and Hahn [2 ].
In an elementary general equilibrium model there
n
might be
firm
f
a utility
h
F
firms, with
possessing a set of feasible production
allocation
holds
distinguished goods,
Y- in IR
,
and
H
households with house-
having an initial endowment
function
Uh:Rn
-+
,
xh
in
and a share
JR
d(h,f)
45
> 0
Here d(hf)
f.
of firm
and for each
f,
a consumption
A price vector p*,
H
allocation x* c G Rn , and a production allocation
F
h=l
y* C (DYf constitute a general equilibrium if
Sh d(h,f)
1.
=
f=1
(a)
p* > 0,
where
p* c IRn,
j
1,2,...,n
p*(j) > 0
=
p* > 0,
and
and for some
that is
j,
p*(j) > 0.
*
(b)
Zh
*
h
f yf
h
h
subject to
(c)
Yf
maximizes
p*yf
(d)
x
maximizes
Uh(Xh)
subject to
*
*
*
p Xh < p Xh
+
yf c Yf.
*
-E f d(h,f) p yf
In the general equilibrium framework households
and firms have full market information.
No
ntility
maximizing household will make a purchase of good
from firm
f
price for good
if firm
i.
f
i
does not post the lowest
Since any firm would capture
the entire market demand by any undercutting of the
market, it is easy to show that in a full information
competition market model all firms and households
face the same prices.
In a search model, price
information is not universally distributed.
House-
holds or firms act upon their expectation of prices.
46
Even though a household's decision of how much to
buy might be based upon an observed price, its
decision of where to shop is usually based upon price
expectation and not upon a full set of observed
prices.
A single
n
commonent price vector, since
it need not exist in-a search market economy,
cannot be expected to erradicate excess demand as
it does in a full information general equilibrium
model.
hold
vector
In the search model we have for each househ
and firm
p(f),
household
h
f
where
a price vector
p(h,f)
expects firm
is the price that firm
f
p(h,f)
and a
represents the price
f
to change and
posts.
only households are searchers.
p(f)
In this model,
In equilibrium it
is reasonable to expect that a household shops where
it expects to maximize utility and that for this
firm the expected and posted price should agree.
This leads to the following definition of equilibrium
for a search model.
DEFINITION:
A price profile
consumption vector
a choice function
equilibrium if:
x*
p*(hf), p*(f),
allocation vector
C:H F
y*,
a
and
is a competitive search
47
> 0,
(a)
p*(hf)
(b)
y*
(c)
x*
over all
n maximizes Uh(x)
hx
there exists f c F with
maximizes p*(f) yf subject to
xp(f)
(d)
n p(f)
<
c*(h)
f
Z
Z
+
x
yf
Y*
such that
d(h,k) p*(k) yj
kEf
implies that
p*(f)
xh p*(f) < in
(e)
> 0
p*(f)
(x
d(h,k) p*(k) y*
+
kEF
<<y
-
h
c* (h) =f
(f)
p*(h,c*(h))
= p*(c*(h)).
Conditions
a-e
have obvious interpretations.
Condition a is that expected and posted prices
satisfy the standard notions of a price, that is
they are non-negative and that the price of some
good is positive.
Condition b is that firms are
profit maximizers while conditions c and d are the
conditions that individuals are expected utility
maximizers.
Condition e is that in equilibrium
there is not excess demand felt by any firm.
Condi-
tion f is that the expected price held by a household agrees with the posted price at the firm where
48
the household has transactions.
The obvious question is whether such an equilibrium exists for a search model.
It will be shown
below that the answer is affirmative if we have
sufficient continuity conditions on the household
demand and production supply functions and if we
have a Walras' law type assumption on each demand.
We begin by letting
F
c
be a function from
H
into
and listing our assumptions.
For each
Assumption 1.
firm
f,
that
pyf(p)
max
p
into
p > 0,
p e Rn
there is a choice of
> py
yf(p)
+
all
y(p)
y 6 Yf.
in
and each
Yf
such
Further more the
is a continuous map from
{p:p>0}
Yf.
F
Assumption 2.
p(f) > 0
of
all
For each
O
p s
f=1
h,
and each household
xh(p)
in
Rn
such that
R
there is a choice
Uh(xh(p) > max{Uh(x):
such that
x
p(c(h))(x--Th) < d(h,k) p(k) Yk(
keF
(xn-i ) < -E
kEF
p(c(h))
further the map
{p:p s
with
0
f
p
-+
k))
d(h,k) p(k) yk(p(k)); and
xh(p)
Rn , p(f) > 01 into
is continuous from
Rn
49
For each firm
O
let
IR ,
Zf(p)
f
f
=
and price function
n)
E h xh (P)~
h
p
in
f(P)
c (h) =f
Walras' law states that
ZfeF p(f) zf(p)
0,
=
we
need a somewhat stronger assumption which is as
follows.
Assumption 3.
For no
p
in
&9 Rn
f
with
(the unit simplex) is it the case that
implies that
p(f)sSn
Zf (P)(i) > 0
p(f)(i) = 1.
This is a condition on the function c,
In
essence it states that if for some price function
p
if there is excess demand in the system then there is
some firm
good
f
i
f
experiencing excess demand for some
where the price of good
i
charged by firm
is not the highest price the firm could charge.
In a general equilibrium model assumption 3 is a
consequence of Walras' law that
pz(p) = 0.
We prove in Chapter IV the following theorem:
Theorem:
Under assumptions 1, 2, and 3 a competitive
search equilibrium exists with
C* = C.
The proof of this theorem is a direct application of a Browner fixed point theorem.
It is
similar
to the proof for the existence of a competitive
50
general equilibrium appearing in Chapter
and Hahn [2
2 of Arrow
J.
An interesting corollary on price distributions
can be obtained by appending two search models
together.
Corollary:
In a search equilibrium two different
firms may post different prices for identical
commodities.
4.
A Housing Search Model.
In the next section I
present a simple one period search model which has
significance in housing analysis.
In the hosing
market we find that potential buyers visit (according to some process) sellers to gain information
about the characteristics of the unit the seller
is offering.
The potential buyer, without full
information of the housing market and with knowledge
of the seller's asking price but not of his
reservation price makes a bid on the unit.
The
potential seller must await bids from buyers and
must decide the level of his asking price as well
as when to accept a bid which might be below the
asking price.
51
It should be clear that the individual search
processes in the housing market do not follow any
simple model.
The housing market is a dynamic
market with buyers and sellers learning as they
sample.
Buyers do not sample at random but
rather develop a search strategy.
Sellers need not
wait for buyers but may and do advertise.
Further-
more, market brokers (Realtors) exist to facilitate
the exchange of price and quantity information.
However, the data transmitted by realtors need not
always be accurate.
A search model which attempts
to incorporate each of these factors will be
intractible to mathematical analysis.
The obvious
hope is that as with labor market analysis, a
simplified model will capture enough of the behavior
to yeild valid analysis.
The seller's problem is much the same as that
of the job seeker's in labor market models.
The
seller is faced with a sequence of offers and must
decide when to stop sampling and sell.
This problem
is well researched and the optimal strategy under
a wide range of assumptions concerning the seller's
objectives is known.
The potential buyer's problem
is not well developed in the literature.
The buyer
52
does not know whether or not a given bid will be
accepted and thus this problem is not covered by the
optimal search literature for employer's strategies.
A simple model of the buyer's problem can be stated
as follows:
m
the buyers samples from among
n,
At time
Associated with each unit is an
classes of units.
unobserved reservation price below which it will
Within class
not be sold.
j,
the reservation price
F.(-).
J
For each sample, the buyer incurs a fixed cost
is a random variable with distribution
If he purchases a unit from class
j
c.
at period
n
he then enjoys the payoff,
P.(n),
J
U(X.,Y-P.(n)-nc), where Y = individual income.
J
J
We assume that the probability of drawing a unit
for price
j
from class
for all
n
n.
P.(n)
Let
for unit
j
and let
p.(n).
Let
sampled at the
ni-
values
n th
at the
i(n)
draw is
pj
constant
be the buyers bid at time
P
be the function with
be the class of unit
period and let
Z(n)
be the
actual reservation price for the unit sampled at
the
nih
If the buyer has income
draw.
bid structure
P,
he will enjoy pay off.
Y
and
53
B(YP) = U(Xi(n)
Y
- nc)
(n)
P
-
and
if
only if
P
i (n)
(n)
> Z(n)
-1
and
P.(j)
j
< Z(j)
(j)
for
P
The buyer's problem is to find bid structure
EB(Y,P).
which optimizes
Let
if
LP*(n)
W(Y,P)
optimizes
P*
=
j<n.
P*(n+l)
J
=
E[B(Y,P)],
W(Y,P),
optimizes
then it follows that
then
LP*
defined by
W(Y-C,P).
It is important to note that
P
define a stopping rule for the process
does not
i(n)
but
does define a stopping rule for the process.
[i(n), Z(n)].
It is the unobservability of the
random variables
Z(n),
that distinquishes this
problem from that solved in the literature.
To
the best of my knowledge this problem, even under
simplified assumptions, has yet to be solved.
In
the model presented below this problem is finessed
by simply assuming the buyer has a bid structure.
54
A number of factors will affect the welfare of
One factor which
individuals in the housing market.
is either affected directly by housing market policies
or is indirectly affected by transportation policies
is the size of the market.
In the next section, I
analyze how expanding the market affects through the
In
search process the welfare of buyers and sellers.
particular I show that sellers are made worse off by
expansion, and whether or not buyers are made better
off depends upon the ratio of buyers to sellers.
The basis for the analysis is a one period market model with
buyer
I
where
P..
m
buyers and
Each
sellers.
n
is assumed to have a bid structure
,
for sellers
i
is the bid of individual
P
13
j's
unit.
Each seller
J. is assumed to have a
J
for his unit. At the beginning
X.
J
of the period, the buyers are distributed indepenreservation price
dently of each other among the sellers such that proba-
I.
buys unit
if
I.
1
visits
1i
J.
J
J
visits seller
I.
1
bility that indivudal
1/k
with probability
P.
.
1,3
> X., P.
_
J
.
1,3
> P
-
s,j
J.
3
is
1/n.
if and only
for all
See appendix to this essay for further results
on this problem.
55
individuals
I
iisiting
sJ
individuals going to seller
is
exactly
k.
J.
and the number of
J.
P
with bid
3
s, )
.
=
P.
1,3
This one period model can be thought
of as an equilibrium model where the market process
is such that buyers and sellers are rpelaced as they
are successful in the market and in which bids.and
reservation prices are independent of experience. *
Before continuing with the analysis it is
necessary to introduce additional terminology.
BASIC Model.
I = {I
| i = 1,2,...,m}
set of Buyers.
J = {J
I j
set of units.
=
1,2,...,n}
For an arbitrary set
cardinality.
Y. .
1J1
X.
J
(n)
x
S,
nx 1 with
be the bid of buyer
let
y
I.
||S||
=
for unit
be the reservation price for unit
G (r): =|I{ I
Y.
< r} II/m
denote its
J.
J
J.
this is the frac-
tion of buyers whose bid for the unit
falls below
B. (r): =11 {I
J.
J
r
I Yi
= r}|| /m
In general, this assumption will not be consistent with optimal search with positive search costs.
.
56
H. (r):
J
F(r)
C.:
J
I
=
=
=
{ I
== r}II
Y..3
|[{J. I X. < rl1
J
{
Y.
> X.}.
J
1J1-
A.: = {j
1
Y
|
ual
I
/n
>
j
if
X };
then individ-
c A
has a bid for unit
J.
at
least
as large as its reservation price.
PROPOSITION 4.1.
I.
will make a successful bid is given by:
m[H.(Y..)
3 13
1
1y
mB (Y..
jEA
j iJ
J1A
Proof.t
ful and
ders
is
The probability that individual
t
The probability that
visits unit
1
mH.(Yi.)
J(Y..)
L
1)
I.
- B.
for unit
J.
3
J.,
is success-
and the number of bid-
3
with
I.
Y
ti
=Y..
13
equal to
k
given by
1 1
mB
(Y
.)-
1)
1 k-l
n
n-1 mB
(-)
k-1
Provided
Let
Y.
P(j,k)
13
>
-
3
,
)- k
n
(n-1 m[H-3 (Y.
1 J
n
(*)
(Y
it is
-
B.(Y..)]
J
13j
0
otherwise.
denote the expression given in
then the probability that
I
i
(*),
is successful is given
tDetailed proofs appear
in Chapter IV.
57
by
mB.(Y..)
B (Y i
P(jk)
which equals
jeA. k=l
~ m[H.i(Y..
J3
1 1
j J1A
.
mB.
J
(Y..
iJ
mH(Y..)
- B.i(Y..)]
1J3
3
1J
)
The probability that J.
mH. (X.)
sold is given by 1 - p 3
is
.
If unit J.
3
the expected value of the sale is given by
is
PROPOSITION 4.2.
Y..
m[H. (Y..)
3
13
13
- B.(Y.
3
MB.(Y..)
.
13.
3
.)]
mH.(Y..)
3 13
1J3
mH. (X.)
13
1-py
Proof.
sold,
1)
3
Unit
J. will not be sold only if
3
each bidder with bid for J. at least as great as
3
X. visits a unit other than J.. There are mH. (X.)
3
J33
J
bidders with bids for unit J. at least as big as
3
reservation price X..
its
The probability of going
3
to a unit other than J. is given by
3
2) If Y.. > X., then the unit J. will be
133
3
Y. .
sold for
visits
J
Y . > Y..
tj
13
13
if some individuals with bid
and all individuals It
go elsewhere.
is sold for exactly
Y
.=
s3
1J
with bid
Hence probability that
Y. -) (where
Y..
Y. . > X)
is
J.
given
58
m[H.(Y..)
mB.(Y..)
by
i
[1-
J1j
1
1
J
(Y)
-
J
1
1J
which
equals
mB.(Y..)
3
3
mB.(Y. .)
m[H.(Y. .)-B.(Y. .)]
k~lik1 mB (Y)
The probability that it is sold for
3)
(Y
X.)
Y..
1J
given that it is sold is simply the proba-
bility it is sold for
Y..
1J
divided by the probability
that it is sold.
4)
To obtain the result stated, we need only
observe that the set of individual with bids for
at least as great as
J.
J
is the disjoint union of
X.
J
classes of individuals whose bid for
over all
J.
J
equals
r
r > X..
In a full information deterministic market model
a seller can affect the share of the market captured
by varying his prices relative to that of other
sellers.
Proposition 2.
tells us that it is a con-
struct of this model that the welfare of any particular seller is independent of the reservation prices
of other sellers.
Proposition 1.
states that buyers
in this market compete with each other and that the
buyer's problem is a game theory problem involving
59
the action of both the other buyers and the sellers.
Suppose now each of the individuals in the model
is replicated
x
times to give an expanded market.
Our goal is to determine the effects upon the buyer's
and seller's welfare from such an expansion.
This is
a partial equilibrium analysis in that we assume that
price structures are not affected.
If buyers and
sellers determine their bids and reservation prices
upon the distribution of bids and reservation prices
and independently of the market size then the price
structures will not change.
Behavior of buyers and
sellers independent of market size is suboptimal, however, since changing the size affects the probabilities of being visited and of having competition in
a
bid.
The expanded market is the
x
time disjoint
union of the market in the original model.
new set of sellers can be denoted by
i = 1,2,...,m,
s = 1,...,x}.
variables in the new model by a
I: = {I.
If we denote the state
^,
we observe the
following relations:
J = {J.
m
= xm
Thus the
| j = 1,...,n s = 1,.
,x}
60
n
= xn
y
=
G
B
H
y(n)
,s)
nx-1
=
(r)
nx
G.(r)
=
j
= B.(r)
(53,s) (r)
J
.
= H.(r)
(r)
3
(3,s)
F (r) = F(r)
C
i e C.
= -{(i,s)
(3 ,s)
3
j
: = {(j,s)
A
Proof.
indivudal
xm[H.(Y..)
13
J
1
mB.(Y..)
13
3
=
Ux~
A.
1,...,x}
will make a
I(is)
given by
successful bid is
JeA
C.
1,)...,)x}
=
s
c A
=
In the replicated market model,
PROPOSITION 4.3.
the probability that
s
- B.(Y..)]
J
1J()
)
Use the fact that
A(i,s)
xmH.(Y..)
11 (x)
s=1
A
and
then use Proposition 4.3. replacing all the variables
with their values in the replicated market model.
PROPOSITION 4.4.
will be sold is
of the sale of
The probability that
1 - y(x)
J( 5 5 )
xmH. (X.)
33
.
(3,s)
The expected value
given that a sale occurs is
61
Yi
1
p'(x)
3
xm[H. (Y. S)-B
31l
mB. (Y. .)
.
3 1,3
i3C
Proof.
xmH.(Y..)
1(Y,3)]
p3x)
-
xmH. (X.)
J 3
(X)
1 -
replacing variables
Use Proposition 2.
by their values in the replicated market model and
C
use the fact that
USl
(jIS)
C.
We are now able to determine how expending the
market through replication affects the welfare of
buyers and sellers.
Let
THEOREM 4.5.
1
P (x,i)
I
]EA.
xm[H.(Y. .)
1, 3
-P x) 3
B. (Y. .
(the probability that buyer
the market replicated
A.
a
>
[H. (Y.
31
1,)
buys a unit
I (i,1)
times.
in
a = m/n.
Let
- B.(Y. .)]
3
1)3
)
P(Xi)- > 0
-1
nx > 2
are
and
0.
P(x<i)
A sufficient condition for
b)
that
sup
]
xmH .(Y.
Sufficient conditions for
a)
that
x
mB. (Y. .)
a
1nx-l
nx ln
ticul ar if
nx > 2
and
A.
nx
a < 5/7,
3P(x,i) < 0 ( nx>2).
0.
0 are
In par-
62
a
One can reasonably interpret
as a measure of
congestion among buyers or equivalently for a fixed
market size,
is a rough measure of the competition
a
Theorem 4.5
between buyers.
states that when this
competition is low, when there are proportionately
more sellers than buyers, buyers are made worse off
states that if a
Theorem 4.5
by market expansion.
buyer tends to be a low bidder on each unit whose
reservation price doesn't exceed his bid, then market
expansion makes the buyer better off.
Market expan-
sion affects the buyers by increasing the number of
competitiors and by increasing the number of opportunities.
gives sufficient conditions
Theorem 4.5
for one of these effects to dominate.
THEOREM
the probability that
is replicated
x
Q(x,j)
Let
4.6.
J .
() ,s)
times.
Let
=
1
xmH. (X.)
(
- pi(x) x
is sold when the market
E(x,j)
Y. .
.
=m
1E.J
p(x)
xm[H.(Y. .)
3 l'
-
B.(Y. .)]xmH
- p(x)
'
3
xmH. (X.)
1 - p(x) x
be
.(Y.
3
1i,)
.)
'
(
be the expe.cted value of the sale of unit
J(j
s
63
given that a sale takes place in the market replicated
x
a)
Then if
times.
Necessary and sufficient conditions for
< 0
b)
nx > 2.
are that
C. / 0
and
Necessary and sufficient conditions for
(xj) < 0
YDX
Y
Y.
.
> Y.
1,
Proof.
.
.
,J
are that there are
x
i
i'
with
> X..
-
3
See Chapter IV.
The interpretation of Theorem 4.6 is straightforward.
Expanding the market never is beneficial
to the seller.
In particular, if there are at least
two buyers with different bids exceeding the reservation price, the seller is made worse off by expans ion.
64
5.
An Optimal Bidding Problem in a Search Model.
In some markets in which search is a prevalent
feature, there is sufficient flow of information that
buyers and sellers have full knowledge of price
distribution.
buyers,
Search still occurs since prospective
although they know price distributions,
do not know which seller is posting the lower prices.
An example of this is the residential housing market
in urban areas where realtors maintain extensive
records of past transactions and make these available
to prospective home buyers.
The potential buyer
visits a unit and then may tender a bid for that
unit.
The decision of how much to bid is in part
determined by the bidders expectation of the seller's
reservation price and upon the bidder's wealth.
In this section it is shown that an optimal bidding
strategy exists and that if search is costly this
strategy need not result in a bid pattern that is
monotonic with respect to time.
The bidder samples sequentially and at time
n
(Xi(n),
samples the pair
i(n)
E {l,2,.
equivalently
. .
,m}
i(n)
and
is
Z(n)
Z(n))
e ]R.
where
Xi(n)
observed but Z(n)
however the distribution of
Z(n)
given
or
is not,
i(n)
65
is known and denoted by
i(n)
that
=
j
F i(n
A.
J
The bidder starts with income
c {l,2,...,m}).
Y
and incurs a fixed cost
at time
n
for
P i(n)(n) > Z(n)
Let
P
for
X.
c > 0
Xi(n),
and
profile
n,
Y - nc - Pi(n)(n))*
P
(j)
< Z(j)
provided
for
is the bid
i(n)
then associated with the bid
W(Y,c,P).
bidder's problem is to find a bid profile
maximizes
j < n.
P.
is the expected payoff
P
Pi(n)(n)
he will enjoy one time
be a bid profile, that is
at time
per each draw
If the individual bids
U(X i(n),
payoff of
The probability
is fixed and denoted by
(j
of the sample.
).
p*
The
that
W(Y,c,P).
The bidder's problem is not too difficult to
understand.
If he bids too low, he will fail to
make a buy and then must incur the cost of additional
search.
If, on the other hand, his bid is in excess
of that needed to make a buy, then the difference
of his bid and the minimum needed to secure the buy
is lost opportunity.
In the model described above,
it is assumed that search requires little time and
To study the
so utility is not time discounted.
bid profile, the shift operation
where if
P
is a bid profile,
L
LP
is introduced
is a bid profile
66
with
(LP),
= P (n+l).
(n)
PROPOSITIC N S.1.
with
(L)
L et
(n) = P (n+j).
p 1
W(Y,c,P)
Li
p
Then
be the bid profile
U (X , .Y - c - P (l)) X F (P. (1).)
+
i=l
m
+
[1
i=l1
Proof.
{=
1=1
Pi (1))]
F
-
W(Y
c, c, LP).
1
1)
m
Let
h(P,n)
Prob Pi(n)(n) > Z(n)
> Z(n)}
xF (Pi(n))
H(Y,P,n)
x U(X , Y - nc - Pi(n)(n) ) F (P (n))/h(P,n)
= {expected payoff given
P (j)
Assume
P i(n)(n)
> Z(n)
and
< Z(j)}
U(X
,Z)
argument for each
i.
is nondecreasing in the second
67
Then W(Y,c,P) =
+
h(P,1)H(Y,P,1)
[i-h(Pl]h(P,2)H(Y,P,2]+
=
[l-h(P,1)][1-h(P,2))h(P,3)H(Y,P,3)+...
j=l
(1-h(Pi)
i=0
]h(Pj)H(Y,P,j)
[h(P,0)
E 0]
n-1
2)
Note.
H
and
3)
4)
[1-h(P,i)]h(P,n) = probiPi(n)
_=0
Pi(j) (j)
< Z(j)
H (Y,P ,n+1)
= H(Y-c,LP,n)
W(Y,c,P)
h(P,1)H(Y,P,1)
-
h(P,1)H(YP,1) +
~ i-i
-
j=3
i=1
.
(by def h(LP,0)
+
lh(P,2)H(YP,2)
[1-h(P,)]
[1-h(P,1)] {h(LP,1)H(Y-cLPl)
h(LP,i-l)]h(LP,j-l)H(Y-c,LP,j-1)I =
(1
h(P,1)H(Y,P, 1)
+
[l-h(Prl)
j-i
_
H1_
m
W(Y,c,P) = Z
+
[
0)
=
n=l, 2 , 3...
[i-h(LP,i)]h(LP,j)H(Y-c,LPj)
5)
> Z(n)
,i))h(P,j)H(Y,Prj)
j=3 i=2 (1-h(
+
< n
= h(LPn) for.n=l,2,...
h(P,n+l)
-
j
(n)
Xi U (X irY-c-Pi(l)
i=i
X iF i(P i(l)
F i(Pi(l))
)]W(Y-c IcLP).
+
68
PROPOSITION 5.2.
P*
Suppose
Fi (P* (1))
I.
if for some i
optimizes W(Y,c,P),
1 l, then LP* optimizes W(Y-c,c,P).
Proof.
1)
It suffices to show any bid profile, Q.
W(Y-c,c,LP*) > W(Y-cc,Q).
2)
Suppose W(Y-c,c,Q) > W(Y-c,c,LP*)
Let P be the bid profile with Pi(l)
= Pi*(l)
Pi(n) = Qi(n-1) for
n
>
2
Then
W(Y,c,P)
=
[1-X.
i-x Xi
m
U(Xi,Y-c-Pi*(l)Fi(Pi*(l))+
n
Fi(Pi*(l))]W(Y-ccQ)
and
3)
W(Yc,P)
-m
[l-n
> Im
U(Xi,Y-c-Pi*
(1) )Fi (Pi* (1))
+
Xi Fi(Pi*(1))]W(Y-c,c,LP*)
and
4)
W(Y,c,P)
> W(Yc,P*)
contradiction
Proposition 2 states that if we know the optimal
structure
from time n+1 onward or
W(Y- (n+1)c,c,LnP*), we can compute
particular Pi*(n) must be
if we
P
bid
know
...
P*n).
the bids that maximizes.
In
69
m
(A)
{
i=1
U(X., Y - nc - P.
.F.(P.)
11
m
+ [
.F. (P)] W(Y - nc,
The maximum value that
(A)
W(Y-(n-1)cc,L (n-1)P*)
and the value
mize (A)
P*(n)
become
Pi*(n).
(and hence for
c,
takes on is then
P.
P*(l),...,P*(n))
perhaps helpful to note that if
F.
W(Y,c)
is increasing in
as expected.
1
reduces to
p
that maxi-
[U(X ,Y-nc-p) = W(Y-nc,c,PnP*)]F.(p).
mizes
c
that maxi-
The problem of solving for
solving the simpler problem of finding
then
L P*)
W(Y,c)
Y
=
It is
sup W(Y,c,P):
P
and decreasing in
In the event that the distribution
is associated with a probability measure with
finitely many atoms, the search for optimal
be restricted to the atoms.
P.
In the case that
can
F.
are continuously differentiable we can develop further
results.
Let us now assume that each
F.
1
tinuously differentiable in interval
lim F (p)
=
0.
is
twice con-
(0,o)
and that
We further assume that for each
i
p- o
U(X ,Z)
is twice continuously differentiable in
3U(X ,Z)
3Z
2U
>
Z>
z) 2 (X.,Z) < 0 lz>O
Z.
70
lim U(X.,Z) = 0.
Z+1
THEOREM 5.3.
Under the conditions above an
optimum bid profile exists.
Proof.
1)
W(Y-nc,c) = 0
be the least
let
for all
To find
p
pute
To find
n > N,
and for
> 2c,
we can find
and of course
=
M(p).
p. > c m(p.)
P *(N-1)
< 0;
p
=- Zn
11
and we can com-
. U(X.,Y-(N-1)c
(Pi *(N-1))
p.*(N-2)
).
For
m(P ) < 0;
0 < p. < 2c,
thus can compute
-
i
D *(N-2)
pi < 0,
since
-
m(p)
m(Pi)
that maximizes
and
that
pi
we need to find
m(p ) = [U(XiY-(N-2)c-p)
W(Y-(N-2)c,c)]F
p
Then we can
U(X.,Y-(N-l)c-pi)Fi(p.).
W(Y-(N-2)c,c)
maximizes
N
we need to find a solu-
0
=
Thus we can find
4)
and
-Let
is continuous, so there exists
- Pi* (N-l') )F
for
Y-nc < 0.
i
Z < 0.
nc > Y.
U(X.,Y-(N-l)c-pi)F(p.)
m(p )
that maximizes
3)
such that
Pi*(N-1)
< 0 m(p.)
furthermore
for
= 0.
tion to maximize
Now for
n
such that
= 0
W(Y-(N-l)c,c)
U(X.,Z) < 0
for all
n
P (n)
2)
Since
=
0
and
is continuous
m(P )
W(Y-(N-3)c,c).
and
71
5)
W(n) = W(Y-nc,c);
In general, let
pi *(n)
we need only find
[U(X ,Y-nc-p )
W(n)]Fi(p.).
-
m(p ) = 0
ous with
p
Now
is continu-
m(p.)
m(p ) < 0
and
=
m(p.)
that maximizes
pi < 0
for
to find
for
so a solution is always possible.
pi > Y-nc,
In general we seek solutions to the problem of
maximize
F.(p:)
=
M(p ) = [U(X ,Y-nc-p ) - W(n)]Fi(p ).
0
for all
p.
such that
we can choose for our optimal
m(0)
= 0.
m(p )
If
> 0
pi,
for some
pi < Y-nc
pi = 0
If
then
and
then our
pi,
optimal solution must satisfy the differential condition.
0 = m'(p) =
[U(X,Y-nc-p)-W(n)]F'(p) -
F(p) U 2 (X,Y-nc-p).
Thus at our optimal solution
p,
we must satisfy,
F'(p)
F (p)
_
U 2 (X,Y-nc-p)
U(X,Y-nc-p)
**
- W(n)
It is somewhat surprising that the optimal bid
profile,
Pi*(n),
need not be monotonic in
n.
*
We have dropped the subscript
ence of notation.
**=
U2(XZ)
=
(9U
Z (XZ).
i
for conveni-
72
Pi*(n)
maximizes the generic function
+ (1-F(p))W = M(p)
As
H(Y-p)F(p)
n
goes from
n
both
Y
M t (P)
= 0 = [H(Y-P)-W]F'(P)-F(P)H'(Y-P).
and
then m" (P)
W
fall.
F'(P)H'(Y-P) + F(P)H"(Y-P)
is a unique
P = P(Y,W)
P
solve
9M' (P(Y,W))
3Y
9P
9Y
-
such that
g
90
3Y
90
-
MW
If
F"(-)<0,
F' (P)H' (Y-P)
M"(P) < 0
so
M(P)
= 0.
M'(P)
n+1
to
p = P,
At the optimal
= [H(Y-P) -W]F"(P)
+
= 0.
and'there
Now let
Taking
M'(P(YW))
W
=
we find that
0,
F(P)H"(Y-P) - H'(Y-P)F'(P)
[H(Y-P)-W]F"(P), - 2 F'(P) H'(Y-P) + F (P) H t (Y-P)
= F(P)H"(Y-P)
- H'(Y-P)F'(P)
(P)
> 0
F'(P) <
0
M"(P)
Since
as
n
increases both
Y
and
W fall
there is no conclusive determination of what happens
to the optimal
bid
P *(n).
One might suspect and it
is easy to construct examples where
F. (P)
discrete probability distribution and
falling as
n
increases.
is a
pi*(n)
is
The following example
reveals that it is possible for
pi*(n)
< p1*(n+1).
73
Example.
M = 1
and hence
Y = 14
F(-)
X
Q
C = 5
= 1
= 3
P = 2-1/2
is discrete with
F(t)
0
t<2.5
.571 2.5 t<3
.6
1
3<t<l00
100<t
= 1
U(Xl)
U(X,1.5) = 1.05
U(X,6)
U(X,6.)
Note
1.49
=
1.538
=
is chosen such that the above values
U(X,-)
could be generated from a function
U(Xt) > 0
D 2U
(X
,t)
with
U(Xt)
< 0.
t
P*(n) = 0
F(-),
Y-3c = 14-15 = -1,
Since
1)
n > 3.
for
p*(n) c {2.5, 31
2)
follows that
=
U(X,
4)
for
n = 1,2.
P(2) =
1.5 )(.571 = U(Y-2c-P)F(P)
Q
F(P)U(X,Y-c-P)
- F(Q)U(X,Y-c-Q)
1.134
Furthermore, by the nature of
U(Y-2c-Q)F(Q) = U(X,1)(.6) = .6 > .59955 =
(1.05)(.571)
3)
it follows that
P(1)
< P(2).
= 3
and
W(l)
+ (1-F(P))W(l)
= .6
it
.
= 1.135598
+ (1-F(Q)]W(l). So P(L)=P=2.5.
74
Search is naturally a part of the urban economy
and particularly of the housing market.
There are
numerous directions that search theory research
might take with respect to urban economics.
One
area of research which seems especially fruitful,
is to develop rational bid rent search models.
A
rational bid rent search model is one in which
buyer's bids and seller's reservation prices are
consistent with optimal search strategies given
some rule for determining how buyers and sellers are
brought together.*
It is interesting to note that
if in a given bid rent model,** the equilibrium bids
satisfy the following conditions:
1)
For each
I.
there is a unique' J.
such
J0
that B.
-1,J0
j(i)
=
B
s,30
i / s.
(denote
j 0 ).
Examples of such roles are:
1) Naive rule each buyer independent of the
action of other buyers visits a given seller at
random with equi probabilities of visiting one
seller.
2) Maximal expected utility: Buyer I.
chooses from random with equi probability among
those sellers J. that maximize expected utility.
3
Sell Alonzo [1] and Wheaton [5-6].
75
2)
For each
j(i)
Then if let
=
there is an
J.
I.
such that
jo.
X. = sup. B.
J
reservation prices
1
X.
,
.,
the bids
B.
.
and
1,3
are consistent with the
search rule that each buyer visits the seller which
maximizes his expected utility.
A natural consequence of the full information
bid rent model is that individuals with the same
income and tastes will end up enjoying the same
level of utility.
This fact is often exploited in
empirical studies to estimate parameters of individuals utility functions and to estimate marginal
rates of substitution between various housing
characteristics.
In a stochastic search model
the hypothesis of constant utility for individuals
with some preferences and initial income is not
supported and hence parameter estimates based upon
the assumption of indifference need not be consistent.
It remains to determine, however, the degree of inconsistency that this introduces.
See Wheaton [57-] .
CHAPTER III
ESTIMATION AND HYPOTHESIS TESTING IN THE
PRESENCE OF HETEROSKEDASTICITY.
Introduction.
1.
The often encountered one equation
liner model can be written as either:
K
n=
and
Y
where
B
matrix,
B
c
and
a
c
.Z
i=0
xn,i
i + 6n
Y = X3
+ 6
Nxl
are
Kxl
involves the estimation of
on
B,
or predicting
E(ce
and
c
)
B,
X
is an
NxK
are observed,
is a vector of
The usual analysis
0.
=
yn+1,
are given or predicted.
that
Y
are unobservable, and
E(c)
X
vectors,
vector,
random variables with
or
testing of hypothesis
when xn+,l 1'' 'Xn+l,k
The usual naive assumption
is a scalar diagonal matrix cannot
usually be supported.
When the observations are
generated by time series data one would expect
serial correlation,
E(c s t)
/
0
when
s /
t,
while cross sectional and grouped data frequently
2
imply problems of heteroskedasticity,
when
i /
j.
76
E(2 )/E(e
2
)
77
E(E T)
The naive assumption of
to give up.
) 01I
is hard
=G2
Under this assumption, together with
assumptions on the data matrix
X,
one has the well
known Gauss Markov theorem which states that the
ordinary least squares estimator
^
T -l T
=
is a minimum variance estimator.
X Y
(X X)
given by
,
Furthermore, the predictor
y, = xB
linear unbiased predictor of
y
is a best
given
x,.
additional assumptions on the data matrix
asymptotic distribution of
Q
B
Under
X,
the
can be computed.
If
is given by the
lim N (X X),
then VR(f- )
N
has a normal limiting distribution with mean 0 and
coaracemtrx
covariance
matrix
E = Y
where
a2
- XB,
2
a 2Q
S
-1
Furthermore if
S
2=
TC
N-
is a consistent estimator of
Under the naive assumption of
E(cE
)
scalar,
one can easily compute the limiting distributions for
63,
compute asymptotic confidence intervals for
3,
and test, at least using asymptotic theory, linear
hypothesis on
B.
Indeed, most regression packager
automatically report all these statistics which of
course are valid when
E(eT)
=
E(sE
)
is scalar.
If
Q is not a scalar multiple of the identity,
then the statistics reported by the usual regression
78
packages do not have their usual meaning.
To
examine the loss resulting from using ordinary least
2
When
squares.
is not a scalar multiple of the
identity, we consider the case where the data matrix
XTX exists and
is non-stochastic and the lim N
N
equals some KxK matrix Q. The ordinary least
T -1 T
XY _
squares estimator 1 is given by 1=s(XTX)
X
(XTX)~
mator
=
X T(XS+-)
(XT X)~
+
XT.
The OLS esti-
is un unbiased estimator of
1
1
1 +
p lim 3 = p lim
T
X X)
N
N
is a consistent estimator of 1
D
diagonal operator
such that Q = U 1 D 2 U;
SY = SX8+Sc.
Now,
X
T
If
.
=
Q
and
1,
so)
so
is positive
is symmetric, we can find a
Q
definite then, since
1
-1
1
and unitary operator
let
S = D -1 U,
E(SET S )T
D
U
then
T
UU-T
D2 D- = 1, so by the Gauss Markov theorem,
T -1
-l T (XT
T
-1 TT
-1 Y
X
X S SY = (X (- 1)X)
(X S SX)
13
D
the least linear unbiased estimator for
1
is not efficient.
ticity,
(S)i=
given by
(XT X)-
Q
(a
is
3 and thus
In the case of heteroskedas-
is diagonal and
S
is diagonal with
2 -1^
The covariance matrix for 1 is
-1
E[(X]T
T
X
Tc
X(XT -1=
E[(-)(-)T] = E[(X X)
2)
2
X IX(XTX) -1
and not by
a 0 2(XT
-1
79
Therefore
/N(3-8)
is not asymptotically distri-
buted with mean zero and covariance matrix
2
G0
-liT
-l
2
=
X X)
(N
0
Q.
printed statistic of
Furthermore the commonly
-
N-K
(N
X TX)
is not a con-
sistent estimator of the covariance matrix for
/N(6-3).
It is clear that the computation of a con-
sistent estimator for this covariance matrix requires
a consistent estimator for
XT QX.
In the case of heteroskedasticity,
gonal the operator S
(a.
=
squares.
If
and the model
model
Q
is simply weighted least
is known then
Y = X3+c
SY = SXB+Sc.
is dia-
is diagonal with
and 8
2) 2,
Q
S
can be computed
can be transformed to the
Not only does
OLS
on the
transformed model give optimal linear estimator for
3,
but the usual test statistics computed by the
standard regression packages for this transformed
model can be correctly interpreted.
The identification of the heteroskedastic structure has importance beyond that of statistical consideration.
In many instances the data has a
natural grouping such that within each group the
variances are constant.
This may suggest to the
80
investigator that while the group share the same
B,
structural parameters
the proc.esses generating
the structure are not the same across groups.
The
idea that the heteroskedastic structure conveys
theoretical information is explored in [/2]
In general,
Q
is not known and
difficult to estimate.
special forms of
Q,
1 XTX
.s
The solution, at least for
is to either develop suffi-
ciently good estimators for
Q
and for
S,
so that
the transofrmed model using these estimators has
desirable asymptotic properties or to develop consistent estimator for
N
XT X
so that hypothesis
tests can be done using the OLS' estimator
S
on the
original model.
In the case of heteroskedasticity, HalbertWhite has proposed a consistent estimator for
N~ X TX
so that asymptotically valid confidence
intervals and tests can be developed using
5.
This
procedure is explored in the next section.
2.
A Theorem of Halbert White.
One approach to the
problem of hypothesis testing when
Q is diagonal
has been to look for a simply computable consistent
estimator of the convariance matrix associated with
81
the OLS estimator
[5q ],
White in
.
proposes such
an estimator and as a corollary develops asymptotically valid confidence intervals and hypothesis
tests based upon
B.
Before stating White's results,
it is useful to motivate his approach.
purpose, assume that
X
For this
is non-stochastic, although
this assumption is not necessary to obtain White's
results.
As we have already seen, the covariance matrix
X(3)
for
is given by
and that for
-l1TT
(N
IN( -)
X TX) (N
-
(XTX)(X X)l
(XTX)
is given by
1
X X) 1 .
Since
X
(N
X X)
is observable, the
problem reduces to developing an estimator for
(N
1
XTX).
The matrix
X
can be written as
X1
X
2
where
X
n
X
is a now vector with
XT
can be written as
is a column vector with
(X.)
=
X
.
(XT, X ,...XT)
T
(X. ).= X..
taJ
that
X
is a real matrix so
.
The matrix
where
XT
[Please note
J1
XT = X*.]
In the case
82
2
of heteroskedasticity,
is a diagonal matrix of
the form
a1 1
2
2
aY
22
0
0
2
NN
The critical matrix
N
-1
N
E
a..
2
N~
reduces to
X TX
X. T X
i=l11
Under fairly general restrictions on
X
and
c,
the strong law of large numbers can be evoked to show
that
lim|l N
IN
1
N
I
-
(
2 -
E.
) X2
X.
= 0,
where
|| - |1
1=1
RKxK
can be taken to be any norm on
This suggests
1 *N
2 T
that N 1 2 N
XT X. would be a good estimator
of
N -1 XT GX,
However,
E.
but
s. 2
like
a..
2
11
is not observable.
which equals y. -X.6 is observable,
1
1
1
which leads one to speculate that since
strongly consisted estimator of
B,
B
is
83
N
N-1
^ 2
X.
T
X.
i=1
might be a good estimator for
N~1XTQX.
It seems
likely that White followed a similar line of reasoning to obtain the following results.
Before stating White's results it is necessary
to introduce several new definitions and notation
and to enumerate the formal assumptions of his
theorems.
Al)
This we now do.
The model is known to be
Y
i
(Xi ,E)
where
0 +
i=2
is a sequence of independent (not
necessarily identically) distributed random vectors,
such that
X.
(a lxK
T
E(XiE.)
satisfy
=
0.
unobservable while
Y.
vector)
and
Ei
The scalar valued
and
1
The parameter vector
0
X.
1
(a scalar)
c
are
are observable.
is a finite unknown
Kxl
vector to be estimated.
A2)
(a)
and
A
E(IX
There exists positive finite constants
such that for all
XikI
)+s< A
,
i,
E(E:2 1+s)
j,k = 1,2,...,K
< A and
S
84
all
n
-l
-
(b)
=
1 =) E(
XT
is nonsingular
,1X
sufficiently large and for
large let
A3)
(a)
and
A
n
for
sufficiently
M n > S > 0.
There exist positive finite constants
such that for all
E(fc. 2 X
(b)
S
i
Xiki+s) < A
j,k
,K
1,2,...
=
The average covariance matrix is
V n:
E(E 2. TX)
and for
n
sufficiently
i=1
large
6 ,
V
n
is nonsingular with
Let
Bn
let
i
(XTX) -1 XTY
=
Y.
=
Finally let
R
i
det V > S > 0.
n
be the OLS estimator of
- X.6
and let
i n
be a known fixed
V
qxk
let
r
A4)
There exists positive constants
be a fixed known
that for all
j,k,k
=
i
E(IX.
i
qxl
X
ik
}X..
X
= -1 Zn
X .
n
i=1in
n
matrix and
vector.
X.
iA
li+S
S
and
A
such
< A
1,2,...,K.
With the notation developed above, White then. proves
the following:
LEMMA 2.1.
surely for
n
Given Al and A2,
Bn
sufficiently large and
exists almost
6an aS.
n
+0'
85
LEMMA 2.2.
Under Al
V
V
A3,
-
M n(n-0)
A
0
N( 0)IK)
THEOREM 2.3.
(i)
IVn
n
-
a.s.
A
(iii)
n(R n_
2
Xq
T
T
Vn Mn
-IT-
T
1l
)T[R(XTX/n)~
Vn (XTX/
RT
given the null hypothesis
and under Al
3.
- Mn
Vn (X X/n)~
|(XTX/n)
(ii)
under Al, A2, A3(a) and A4.
0
in
0.
1
-l R~n-r)
H 0 :R
A
r
r
- A4.
Block Scalar Covariance Matrix.
stances
a.s.
which heteroskedasticity
observations can be grouped
In some circumis
present,
the
into a small number of
groups such that variances are constant within each
group.
If
the data is
matrix
Q
will be of a block scalar from where the
so grouped,
blocks may have unequal sizes.
equivalent
Y..
E(
=
general
this
is
to having a model of the form
X.j + .,
2) =
In
the covariance
2.
(j=1,2,.. .,J,
i=1,2,... ,N)
with
In this case one tries to estimate the
a. 2 JJand then to use these estimates as weights to
3
transform the data.
If
the
c..
1J
are independently
86
distributed with normal distribution of zero mean and
variance
a 2,
one can use a maximum likelihood
procedure to jointly estimate
and
0.
However,
even in this case the computations requires the solution of a nonlinear equation.
An alternative pro-
cedure, the one proposed in this essay, is to iteraS
tively estimate
estimator of
In the case
S
E
and
and
Q
2, and then to take as our
the limit of their iterations.
is normally distributed this becomes
the procedure proposed in Oberhofer and Kmenta [33].
They however, use an erroneous argument to show that
such a limit exists.
In this section, it is shown
that this iterative procedure, regardless of the form
of the likelihood function does converge and the
resulting estimator have the usual desirable asymptotic properties.
Proofs for the theorems in this
section are given in Chapter IV.
87
Consider th e linear model
(j
=
written as
Y = X 0
3
s
+
XjO0
Y. =
+
X -
E. -
+
which can also b e
i = 1,2,... ,m.)
1,2,...,;
.=
Y
' ''
C.
or
where:
1)
Y..
and
2)
X.
is a
3)
30
4)
Y.,
3
1J
1J
13
m.xl vectors with real entries
J
elements are Y.. and
C..
th1
whose
fixed vector with real entries;
Kxl
are
J3
vector with real entries;
lxK
is a
C.
are real valued;
E. .
i
1'
1J
respectively;
5)
X.
J3
is a
i
whose
6)
Y
ing
and
3
N = E
X
th
c
Y.
is a
matrix with real entries
m.xK
3
row is given by
are
vectors formed by stack-
Nxl
j
and
1J
where
'and
1,...,J,
=
m3;
and
NxK
matrix with real entries formed
by stacking
X.
j
The problem is to estimate
=
J.
1,.1...,
0
where
and
Y
X
are
observed, and where
0
E
If
3(E..)= 0
the
a2
J
and
E(c
1J
r,s
aY2
J
(i~) /(r,s)
(i,j)
=
(r,s)
are all known, then the appropriate linear
estimator is weighted least squares, where
(Y
X., )
X..)
88
are weighted by
If there is no knowledge
1//aV.
J
about {a jj=l,... ,J}
but the likelihood function of
is known, then one may use a maximum likeli-
(YIX)
hood estimator which in general requires the solution
of nonlinear equations.
An alternative to the maxi-
mum likelihood estimator which is also often used
when the likelihood function is unknown is a weighted
least squares procedure.
the true values of
a?
Estimates of
a
J
replace
in the first mentioned
J
weighted least squares procedure above.
In
this paper we discuss an iterative weighted
least square procedure that can be described as
follows.
Step 1.
Select any
them by
a 2 ,...
2
J
positive numbers 'and denote
E
Let
be the block scalar
matrix where
2!
20
2
aT
)
o
2
-
1,J
1 ,J 1,
89
2.
Let
T 1
(XI ZX)
-1 T
X.
Step
B (a
1
Y
Let
Step 2n+1.
' a,2'j' xY)
1 ,)G 2
1
1
m.
G2
n+1 ,j
(Y. -X.B )T (Y
J
3 n
J
let
+1
-XB
and
N)
m1
1
2
cn+1 ,1
m.
G2
n+lJ
m
a2
n+l1,J
Step 2n+2.
Let
B
n+1
(a,2
.,
11'**.
2
lJ
XY)
-(X-
n+l
X) I
- 1 y.
XT En+l
any limit point
Choose as the estimate of
{B (a 1 1'''
&1J2 jX, .,Y)
of
n > 1}}
_
In this paper we shall show that under fairly
general assumption:
(1)
point
for all pairs
(2)
N > N
=>
I
2
cjYX~ln
a.
>
{B n(a21
(Y,X)
For any c > 0
prob{{Bn (011'
satisfying the assumptions.
Ltere
'..
' '
exists -N
< aosuch that
XY)1 n>l
converges (to
a unique limit point) is greater than
N =
m.
has a limit
1-c.
Where
90
(3)
If an estimator
,
a1 1 ,...
fixed
{Bn (a1 1 ,... , 1
is a limit point of
O(X,Y)
X, Y)n>1
satisfies that for some
0
then
p lim Q(X,Y)
=0
Nn+o
is a consistent estimator of
0
i.e.,
(4)
0,
and
Under additional assumption on X and s.,
the estimator described above has the properties that
p lim (0-0) = 0
a)
and
N+co
b)
)
N(0,Q^
is asumptotically distributed
A(-0)
Q
where
is a fixed positive definite
matrix.
c)
l
2
If we let
be given by
a.
J
(Y.-X.0)T(Y.-X.0)
and let
I
be given by
ml
2
m
1
a2
m2
2
a2
Q
then
and
Q
(Q)
"
=
1
X
T -1
X
2
aj
is a consistent estimator of
is a consistent estimator of
Q~
Q
91
Definition and Conventions.
The model:
.
Y..X.. j 3 +C..
Yij =
=
0 +:
(j
=
Yu
(j
= 1,...,J)
Y
Y..,
= X
0+ g
u
0
=X0+
or
or
where
Y., )X.
e..,
X.j,
i = 1 ,...,m.)
J,
1,...,
Y, X,
.,
and
C,
are as
described in the previous section.
Assumption I:
The data matrix
X
is nonstochas-
tic.
Assumption II:
j
such that for each
Z XX
X < inf
ZC
k
There
Z <
|1Y.-X.B |
, T,
m.
3
sup
ZTXIX J
Z cIR k
M.
Z < T
j
For all
and all
is
NxK
m.
The observed values taken by the
dependent variable are realizations of an
X
<
0.
Assumption IV:
random vector
0 < X < T
II 11=1
Assumption III:
inf
and any
m
ZI
11=1
exists
Y
N
element
Y = X O +F.
which can be written
matrix with real entries satisfying
assumptions I and II and
60
unknown real numbers.
E
is an
N
IX)
= E(c)
= 0
bance vector with
E(E
V(£jX) = V(e|X) = V(c)
is a
Kx1
vector of
element disturand with
being a diagonal matrix
o
92
comprised of
blocks, and being of the form:
J
2
a1
0
0
2
V(C)
-.
2----------
o
0
2
0I
I~J
identity matrix and
I. is the m.xm.
J121
J1
a positive unknown real number.
where
For each
1)
Assumption V:
j2,
either
2
a.
c..
121
is
are
iid random variables, or
j
for all
2)
m (N)
and
and all
ITAR
r-
M < o
N,
such that
2)
are independently distributed.
{s. . }
1J1
1
(Y-X.B)T(Y.-X.B).
2121
m
J
21
When there is no danger of confusion z. (m . . . ,m
z.(m
DEFINITION:
X,Y,B)
where
N
ZN(B).
,j
X,Y,B)
will simply be denoted by
J
=
DEFINITION:
Z - (B)),
..
1
-
z (B)
or
z N (B)
21
21l
Z (m ,Y. . .,Ym
XYB) =
(Z I(B),YZ2 (B)
this will frequently be denoted by
(Z.(B): = z-.(B)).
21
J1
,..,
Z(B)
or
93
DEFINITION: -Z(m ,...
Z (B)I
X,Y,B) =
,m
0
1
0
Z2(B)I
1
0
2
0
21
where
I.
is the
m xm.
identity
matrix.
0
0
X(m,... ,m
X,
Y, B)
Z (B) I
will also be denoted by -Z(B)
or E (B).
In the following section, we state and prove our
results concerning the iterative estimator.
The
iterative estimator is constructed by choosing a limit
point from the sequence generated by the iterative
process.
In the following sequence of propositions
and theorems, it is proved that the iterative process
generate sequences with limit points, that as the
sample size increases, the set of limit points
degenerate to a singleton almost surely, and the
resulting estimator is strongly consistent.
PROPOSITION 3.1.
Let
for observed data matrix
(a1
X
satisfying assumption I - IV:
,.G..
)
]R
and dependent vector
,
Y
94
I
1)
Z
Let
1I1
a2
=
0
0
al
0
2
2
0
-I -
a
0
0
--
-
- -
2I
I
Let
B
3)
Let
En+1 = Z(B n)
4)
Let
Bn+1
A)
For each
B)
The sequence
=
-
XT E1
2)
XT
T-1X)
n+1
1
XT- n+l
Then
n > 1
Bn
exists;
has at least one limit
{Bn}
point;
If
C)
{B }
then
If the
part
is a limit point of the sequence
B*
B*
=
c..
(XT
(B
the tuple
B*,
Y.
are normally distributed, then from
of Proposition 3.1.
c
XTI(B*)
)
it is easy to see that
Z 1 (B*),... ,Z (B*)
satisfies the first
order conditions for maximizing the likelihood function.
To see this, we need only observe that part c
of Proposition 3.1.implies that XTE(B*) Y T
- X B* = 0 and that (if we consider B
X Z(B*)
to
95
dh
dh = kH(B)
be a column vector)
X TE(B) IXB].
T
[X T(B)
e..
vi
O(X,Y)
0
with the
2
( 211,.. .0G3aj)
property that there exists
B).
are normally dis-
tributed we have that any estimator
22
Bn (1 '' ''J,
Y -
is a constant independent of
(K
Therefore in the case that
such that
-1
£
R +J
is a limit point of
,mi, X, Y)
in,...
will be consistent.
The next result shows that the normality assumption
is superfluous.
PROPOSITION 3.2.
g(B)
g:IRk
be defined by
= (XTE(B)'X)~' XT (B)'Y [=(l XTE B)X)-
X7Z(B)
of
Let
g,
1
Y)].
and let
YO > 0,
F
Let
d = Sup{|| B-B 0 ||:
d < y
Let
almost
N t o.
COROLLARY 3.3.
Let
property that for each
ail
,... lalj
,
such that for each pair
2
point of
B c Fl.
then under assumptions I-V,
surely as
number
be the set of fixed points
{B n a 1,1~
Assumptions I-V,
'''l,
0
o
N,
be an estimator with the
there exists positive
(perhaps depending upon
(X,Y),
2
O(X,Y)
,XY)}.
N)
is a limit
Then under
is strongly consistent.
96
PROPOSITION 3.4.
the property for each
bers
2
2 ,...a
a1
Let
0
N,
there exists positive num-
such that for each pair
is a limit point of
G(X,Y)
be an estimator with
(X,Y),
J2
{Bn (al 1 2 '''''
X,Y)}.
Bn
Then under Assumptions I-V, the sequence
2
2
is convergent almost
X, Y)
1,1 '''''1,,
N + c.
surely as
The next result is a minor improving of the last
proposition.
It will however pave the way for showing
that almost surely as
N
is independent of the
(a1
1 ...
2,
..
PROPOSITION 3. 5. Let
g:IR k
by
g(B)
K(N):=
=
XTE(B)
X)
gets large, our estimator
1
T (B)
,
2)
k
Y).
g(B - g(B2 )11
71B1
B2 11
Sup
-
B2
|
E
Rk
B1 - B 0 l
<1
11 B2 - B0 l <1
0 <
Then
K(N)
11B
< 1/2
1
- B2 1
'
almost surely as
N t-
selected.
be defined
Let
97
g(B)
=
implies
be defined as before by
Il B-
g(B)
=
JIB, - B0
< 1
and
points we have
l
11 g(B 1 )
B1 - B2 11
and thus
1[B
PROPOSITION 3. 6.
as in Proposition 3.5.
-
B2 1
=
are fixed
B2
and
B1
Since, if both
-
< 1/2
has a unique
g
then it follows that
fized point.
g(B-2 ) 1 < 1/2
0.
IRk
Let
g:IR k
Let
F = {B E mkg(B) =
be defined
[(Y.-XiB) (Y.-X.B) mj
Let
h(B)
(2-re)
=
1 =
B
Suppose
j[g(Bl) - g(B 2)
implies that
< 1
B 1 - B 2 1|
and that
B0 l <1
|J B1 - B 2 jI,
(B) ~ Y).
X
(XT (B)
B 2 - B 0j1
H|
IRk
g:R k
Let
p.
-1/2
m
then:
1)
F
2)
There exists a unique
is a singleton almost surely as
almost surely as
3)
R k
F
Sup
BEF
that maximizes
h
N + o.
B*
where
that maximizes
Proof.
1)
=
B
N t C.
is the unique element of
almost surely as
h
Proposition
IJB - B 0 |C1
B*
N + *.
3.2 gives us that
almost surely as
N t 0.
98
Proposition 3,5 yields that:
g(BI)
g(B21
Sup
2)
B1 -B
1
<l
H B2 -B 0
-
< 1/2
B2 1
almost
surely as
l l
N t
F
Proposition 3.1 shows that
o.
cannot be empty,
thus we have;
3)
-F
a singleton almost surely as
is
N + co.
Proposition 3.1 gives us that there exists
B E Rk
B
is
that maximize
h
a fixed point of
g.
and furthermore any such
The rest follows immedi-
ately.
COROLLARY 3.7.
Let
the property that for each
a
2(N),...
E(X,Y)
2(N)
,j
1)
E(X,Y)
h(B) =
N
(2 7 re)
there exist
positive numbers such that
is a limit point of
{Bn (a 1 ,1 2 (N),...,a1
be an estimator with
O(XY)
Bn(
2 (N),X,Y))}
1
2(N),...,
J
then:
maximizes the function
(y-1/2
).T
mj~/
(jY.X.B).T(Y--X.B)J
-N2
2
almost surely as
.
mN/
N t
o.
99
2)
a
®(X,Y)
2
(N)
is independent of the choices for
...
2
a
Proof.
a1
2
j=
g.
B
If
almost surely as
a
For any choice of
limit points of
points of
(N)
2
Bn (a
,.,a
.2
a
2
Z (B1 ),
= B
,XY)
of
g,
where
then it
a
2,...,
2
{Bn a 1 1 '''
,XY)
are fixed
g,
and if
F
for all
n.
is the set of fixed points
the sequence
2
'0 1 ,J X,Y)} must converge to
F
is a singleton,
pendent of the choice of
a
®
is always a singleton.
F
EXAMPLE:
Let
Ytl
Xtl
Yt2
Xt2
t =1
1
0
1
0
t =2
2
1
2
-1
B = 0, then
g(0) = (5/4)0 = 0.
Z 1 (0) = Z 2 (0)
is inde-
5/2
of fixed
The following
example shows that this is not the case.
Let
B*.
a2...,a 2
One might conjecture that the set
g
Therefore
follows that for any choice of
Therefore whenever
points of
the
then we have
Bna1,''
F = {B*},
2,
Part 1 of the corollary now follows.
is a fixed point of
if
N to.
so
100
2 (5/2) 2
- 2.
= (2-re) -2 (17/4)
h(0) =
h(2)
(27re)
h(2) > h(0)
g
so
0
does not maximize
B
/
=
hence
h,
Maximizing
has at least one other fixed point.
we find that
h,
are also fixed points of
g.
In this section we analyze the asymptotic distriSince
butional properties of our proposed estimator.
this estimator equals the maximum likelihood estimator
(when the
are normally distributed) almost
c.
surely as the sample size grows large, it is not
surprising that it has optimal asymptotic properties.
Before proceeding to this analysis we need to
introduce the following assumptions-.
Assumption VI:
1
N
N
j=l
lim
T
X. X
and -N
±2 -
appropriate diagonal matrix.
a
EN
1 Nl
0
0
0
0
0
-1
m.-(N)
3N
The lim R X - N X
Assumption VII:
1 XTN XN)
j,
For each
0
a
INj
exists
exists where
is the
101
where
I
is the
m.(N) x M (N)
identity matrix.
Assumption VII (Replaces Assumption V).
N,
| j
2
{s..[
J
IJ
=j
1,2,...,J
For all
i = 1,2,... ,m.(n)}
is a collection of independent and identically distributed random variables.
LEMMA 3. 8.
A)
all
B)
C)
lim a
Let
m. (N)
aN
j=1
N
2
a. 2
3
exists and max a. 2> a
N
3 j
= a
N
trace
1
lm
N-*m
-1
X (aN
N)
-1
min
j
a.
3
2
N
all
N = N
aN
T
Then:
exists and is positive
X
definite.
PROPOSITION 3.9.
with the property that
g(B)
=
(XT
Let
Q(X,Y)
g(O(X,Y)
X)-(XTZ(B)~ Y)
(B)
be an estimator
O(X,Y)
=
where
then under the
hypothesis above:
A. 0 is asymptotically equivalent to the weighted
least squares estimator with known variances in the
sense that if
W
is the latter,
p lim N(O-W) = 0.
N
+ co
B. 0 is asymptotically normally distributed with
mean vector
(XT
-1
-1
B0
1
and variance covariance matrix
XT. E-1X -1
102
C.
(.
X
surely as
-I
1
-
X)
(
XT()lX)l
-+
0
almost
N + *.
It is impossible numerically to identify the
limit of a sequence.
In paractice, one calls a term
of a computed sequence a limit if it differs from the
previous term by an amount less in norm than some
preassigned tolerance level.
The above theorems
state that if this procedure does pick a limit of
the iterative procedure, then the statistics computed
in this last stage have their usual expected asymptotic distribution, that is, one can do asymptotic
hypothesis tests and construct asymptotic confidence
intervals using this output.
4.
Linear Variance Model.
Block scalar covariances
arise when the variances are structurally related to
discrete valued variables.
In the case that the
variances are structurally related to at least one
continuous valued variable, the likely candidate for
the variance structure is that it also follows a
linear model.
component
the vector
Let
2
(f2)
E(n 2.-
2
is
be the vector whose
2
given by
F-
and let
ith
2
2
The linear- variance model for
be
103
heteroskedasticity assumes that the variance covariance matrix
Q.= a2
2
is diagonal with diagonal elements
and furthermore that
is an observed matrix and
matrices
X
unrelated.
1)
and
F
1
The
may share columns or may be
r
sufficiently well, that the esti-
yields estimator of
well used in estimating
tests on
r is unknown.
where
The usual problems are to:
Estimate
mate of
Z
a 2 = Zr,
6,
a2
that can be
, performing hypothesis
and constructing confidence intervals
on
S.
2)
Test hypothesis on
3)
Construct confidence intervals for
r.
1.
Glejser [/q] and Park [3f] have suggested two
similar procedures for estimating and performing
hypothesis tests on
F.
Both of these procedures
are supported by heuristic arguments but both are
known to lead to inconsistent hypothesis tests.
In
this section a new procedure which uses White's
theorem is proposed which has the advantages of
being easy to construct and understand, and which is
shown to yield, consistent test statistics for
testing hypothesis on
P.
This procedure yields
104
F
estimates for
and thus for
a2
that can be used
in a weighted least squares procedures to estimate
6
13. If
is estimated by this multiple stage
weighted least squares procedures, the final stage
6
estimate for
together with the usual generated
statistics have the expected known asymptotic distributions.
In this section we give an heuristic description of Glejser's procedure and describe our analysis.
In the following section we present and prove
our two main theorems.
We end the section with some
comments about the validity of our assumption and
some comments for extending this work.
Let the model be given by:
(1)
i = 1,2,...,n
Y.. = X.0 + Ei
10
i
i
where:
(2)
E(E.)
=
0
1o-.
E(EiE ) =
2
ij=3
We also assume that:
(3)
and
a. 2 = z.r
31
0
i = 1,...,n
Y.
is an observable
real valued variable,
1i
Z.
are vectors of real valued variables,
has dimensions
1 x k
and
Z.
1 x m.
It is
X.
X.
105
possible that
80
Z
share common components.
0 F Rk
are real valued unob-
i (i=1,2,,,.,n)
-
r0
and
are unknown parameter vectors
r0
and
X
servable random variables,
a. 2
numbers also unobservable.
We assume that the
(E
vectors
i = 1, ...
,Xi, Z.)
are positive real
,n
are independently
distributed.
Following Glejser's suggestion we let
^
=
T
(X X)
-l T
X Y,
in
we let
=
Yi - XOn'
squares residual.
r ^T=
.th
i
in
where
and then
is the ordinary least
r0
Now we estimate
2
(Z Z) -1 zT^
Tn,
vector whose
0,
the OLS estimate of
^ 2
cn
is
the column
(C.
component is
by
)
2
.
Glejser
in
then suggests we perform our hypothesis tests on
r0
by using
.n*If our observed rn "1supports"
-2
the model a2
= Z r0,
we then use a weighted least
square procedure to reestimate
our estimate of the variance
0
using
a. 2
Z.
F
as
In Glejser's
paper, he proposes the model
6
=
vP 9(Z
g
where
E(v.=
v.
0
3
is known,
= v {m 0 + mIf (Z.) +
+ m f(
)]},
are independent random variables,
E(v v.)
2
=&--a ,The
13
in.
13
Z.
and function
3
is unknown and each term
is assumed positive.
Thus he has
a, 2 =
3
f
kk
mkfZ.
2 [P
(Z
g J
2
106
and by taking absolute values and then expectations
EI
.
J
=
estimating
m
gets
E(Iv.I) -P (f(Z.)).
g
J
He then suggests
J
by regressing
on the values
Jin
R. E. Parks [34] suggests a similar pro[f(Z.)]
J
w.
cedure where he assumes a. 2==S a 2 Zj
e. j,, taking
J
2
2
He
+ yanZ- + w..
= Zna
logarithms he gets Ana.
J
J
J
then uses
^~
s.
Jn
2
^'
knil
regression on
a.
to replace
2
2
and performs a
£na
to estimate
2
and y
We find that for our analysis it is more convenient
that is
to use the model resulting in (3),
1
.
= v.
(Z. F)
2,
We then get (3),
with
2
E(v. )
=
a
2
=
1.
E(v.)
=
0.
a.2 = Z
zi3 0 . '
The apparent problem with all of these procedures is,
as Glejser observes, that the estimated
coefficients are biased.
Glejser optimistically
states that we should ignore the bias effect in the
hope that it will generally be unimportant compared
to other contributing terms.
Returning to our model for the variances, we
observe that:
2 -a 2
2
2
+
2 =
) and that E(e. -a. ) = 0.
= Z.r + (E. -_
E.
(4)
3
J
3 0
3
2
Therefore, if we could only observe e. , the OLS
2
would yield a consistent
regression of Z. on e.
J
3
107
estimator for
2
E2
-
.
2
r%.
Of course, the error term
while having' zero expectation, need not
have constant variance.
So it would appear that we
have again returned to the problem of hypothesis
Now,
testing in the presence of heteroskedasticity.
however, we are in the position of using White.ts
procedure to generate a consistent estimator of the
variance covariance matrix and are able to correctly
perform asymptotic
x2
tests on hypothesis concerning
restrictions onthe parameter vector FO. Unfortu2
it is the substance
nately we cannot observe E. ,
of this paper that we can replace
e.
2
E.
by
J1
2
and
JFI
that in so doing we will get estimation and statistics
that are asymptotically equivalent to those when we
used
E. .
In the next section we formally state our principle result, proofs are given in Chapter IV.
Before
doing this, however, it is necessary to introduce
additional notation.
We also state without proof
some elementary propositions on convergence in probability and almost sure convergence.
In the next sec-
tion we shall make frequent reference to convergence
in norms.
Thus for real valued, variables
is the absolute value:
if
x
is a
JZxl
xi,
x.
vector in
108
or a
JR
lxX
vector in
standard Euclidean norm.
the norm is the
IR
Since
norm is the same as the dual norm.
we use the norm in
R nxk
in
X
yeIR
For
R
X
,
this
a matrix
L (R k, JRn),
that is
Since on a finite
n.
11 XY
Sup k
n=
L (JR k
J)*=
(R
iYfl =1
dimensional Banach Space all norms induce the same
topology, if
then
|jxnx
0
is a sequence of matrices in
Xn
| -- 0,
R kxj
if'and only if
L(]Rk YRj )
- X0
| Xn
(ij)
in
-|| 0.
-
That is we get convergence
(ij)
the operator norm of
jIR)
iff
L(IR k
we have con-
vergence for each matrix entry and hence if we have
convergence in the Euclidean (Hilbert-Schmidt) norm.
We are now in a position to list our assumptions and to prove our results.
For the convenience
of the reader, we have followed much of the notation
of White [59].
We have also borrowed liberally from
him on the wording of our assumption.
Al)
The model is known to be
Y=
+
)
=E(c
2
E(e. )
=
0
a=
2
Z r0
i
=
i,2,...,n
i
=
1,2,...,n
109
Where
and
Ei
is a
X.
1.
Y
is a
lxk
vector of random variables,
kxl
Y.
vector of real numbers.
observable,
e.
Z.
80
is unobservable and
is a
1xm
and
1
estimated or hypothesis concerning
tested.
%0
are real valued random variables,
0
X.
1
are
is to be
are to be
vector of real valued random
variables which may contain some or all of the variables in the vector
r
X..
is a
mxl
unknown vec-
tor of real numbers which is to be estimated or hypo-
r0
thesis concerning
Let
W.
are to be tested.
be the vector of length
variables whose first entry
W i
p
of random
is the scalar
1
and whose other entries are exactly those random
variables that appear in
E(W. .W.r.)
1)
= 0
lrT
We let
y
1 <
X.1
j,k1
.
denote
or
< p
Z..
1
and
- a1 2
We assume that
E(W.T 1(.
The vectors
.2))
1i
= 0.
(W.,e.)
1
1
are assumed to be a sequence of independent though
not necessarily identically distributed random vectors.
A2)
I)
There exists
such that for all
0 < 6 < 1
A < co
i:
a)
E(Ier W. W.is W.W.
it lv
b)
E(E.W.kW.W. WitW.
c)
Wikir isW it Wiiv
E(1W ij ikirs
1
and
lr
) < A
1 < r.,s,t,v < p
p
)+ < A l<k,r,s,t,v < p
11+6 )<
_
1<5 ,k,r,s,t.,v <p
110
d)
E(1p.
e)
E(JE
II)
Let
i
1+6) < A
-
3W
W Wi
'urisit1 1+6 )<
< ~
A
Ma =n 1E(XTX.i)
nn~
T
n
M b -1I
1=
<
p
and let
E(Z.TZ.)
ii
We assume that there exists
NO < O
0 < X
and
n > No, the minimum eigenvalues of Ma
n
b
exceeds X and the minimum eigenValue of M exceeds A;
n
such that for
(Note by the first part of A2) this is equivalent to
the property that for n sufficiently large det Ma and
n
det Mb is bounded away from zero. Also, observe
n
that we can choose
A3)
6
and
A
Let
Va = n1 En
E(x
2X)
n
i=1ii1
let
Vb
n = n1 Eni=l E(y i ZZ.)
i
We assume that there exists
that for
so that they are equal.
n > N ,
0'
and
N0 < O and
minimum eigenvalues of
A > 0
such
Va exceed A
and the minimum eigenvalue of V exceeds A.
n
n
(There
is no loss in generality in assuming that NOA is A2
and NOA in A3 are the same).
In the presence of A2,
A3 is the equivalent of the assumption that for
n
111
sufficiently large minimum
(det Va, det V )
.R
%n
is
bounded away from and above zero.
The first theorem is a restatement of a result
] and its proof can be found therein.
found in White [
Before stating Theorem 3.
we introduce additional
notations.
(XTX)
Let 6
Let
if
(XTX)
is nonsingular
0
if
(XTX)
is singular
T-l T 2
(ZT Z) ZT
if
(Z Z) is nonsingular
if
(ZTZ)
lXTY
=
an
2 is the
(Ci )
T
n
0
C2
.
nx
column vector whose
is singular
i th entry is
.
Let
i= -
E .
in
Xin
1
2
Let
pin
=
- Zan
E
Let
^a =n
n
Let
Vb
n
n
Let
Ra
i=1 in
n
n
=
i=
nin
be a
qxk
full row rank and let
.
T
^2
y2
'
i
i
T
i i
matrix of real numbers with
ra
be a
qxl
vector of real
numbers.
Let R
be a
row rank and let
qxm
r
matrix of real numbers of full
be a
qxl
vector of real numbers.
112
THEOREM 4.1.
(White)
Under Assumptions Al,
A2, and A3, we have the following:
i)
ii)
0
n
an
-
11)/ni[
a
n
XTX -1
n
a
n
-12
n
n
)1
E. 2
^b
Z
-
ra
1(a,
a
an
A
n r
T-
are not observable,
statistics associated with
N(0,I k)
R aSO
H0
X TX)aT
0) ,
[R
(
)
under
hypothesis
RaO
R 1] Ho: (R-r
n the V
n
n(
n
T
Since
X
n
under the hypothesis
ra T[a
n(ra -vi)
r)
n1
a.e.
XX - 1 ^ a
n
Vn
...
v)
r0
(a.e,.
almost everywhere
r
r
A
an
2
q
q
2
and the
are not computable.
It is a principle result of this paper, that we can
^2
E
replace
by Lin
and obtain asymptotically
equivalent results.
This notion is more carefully
stated and then proved in the next theorem.
113
Before stating this theorem it is once again
necessary to introduce additional notations.
2
Let
entry
column vector whose
nxl
be the
n
ith
2
is
i
Let
r
T
(Z Z)
=
n
-1
T^ 2
Z
T
if
Z Z
is
nonsingu lar.
otherwise.
0
A
Let
W.
in
=
A
2-
-
Sc
i n
n
n
2
i=l
Theorem 4.2.
and
z.r
.
in
T
in
i
i
Under assumption Al, A2, and A3, the
following hold:
i)
n
+
0
a.s.
Tc -1
iii)
T
_1
vn
[n
ii)
^
0
2(n
under the hypothesis
A
m N(0,Im)
H0: RI'0 = r
(where
R,r
are as in Theorem 3. .
n(Rn -r)
(Note:
T
[R(
ZTZ )-1
c
n
ZTZ -1 T -l
Zn)
R]
2
(Rnr),Xq
A part of the statement of this theorem is
that matrices whose inverses must be taken well as
n + w
be nonsingular almost everywhere].
-
114
Thus we have shown that we have a valid asymptotic test for testing linear restrictions on the
parameters in the variance model.
Our next and last
result is to show that under additional assumptions,
we can use our estimates from the variance model to
reestimate the original model and obtain a estimator
that is asymptotically equivalent to weighted least
squares with variances known.
A4)
There exists
x > 0
such that for all
i
a.2 >
1
AS)
For all
2
i,
E(.I|W. ,..., 2
i
E(EsIW .,... ,W.)
=
W. ,...
0,E(s
=
)
W
1P
zi 0r
.
A6)
There exists
M <
>,
such that for all
i
|| Z.|| < M.
Theorem 4.3.
Let
matrix whose
(i,i)
denote the
nxn
Z.r
.
i n
Bn
Let
(XTQ
n
0n
denote the
entry is
1
matrix whose
nxn
2
(i,i)
diagonal
and let
entry is
denote the Aitken estimator given by
X)1 XTl Y where
n
XTQ n X is nonsingular
B0
n
0
n
if X TQG
n
X is singular.
115
Let
B
be the weighted least squares estimator
n
given by
(T"0Q 11
B
(XT n 19
.
Tf
1
is nonsingular
X
n
0
X Qn
if
X
is singular.
Under Assumption Al-A6,
i)
ii)
I|
Bn ~
0
+1
p lim v(Bn - B)
V/i(n~ XTQ
iii)
a.s.
0
If
R
is a
1X)
qxk
full row rank and
0
=
n ~
and
0
N(0,Ik)
matrix of real numbers with
r
is a
real numbers, then under
qxl
H 0: R 0
vector of
r
XT^
T
XQn -lX -1 T-1
A 2
n(RBn - r)[R(
n
- R]
(RB n-r)
Xq
CHAPTER IV
MATHEMATICAL PROOFS
1.
Introduction.
This chapter contains the complete
statement and proofs of the theorems developed in
Chapters II and III.
Because of its mathematical
nature, the chapter is designed to stand apart from
the rest of the dissertation and may be passed over
by those with low mathemtical inclination without
loosing the content of the rest of the dissertation.
The proofs in this chapter, especially those of
theorems appearing in Chapter III may be of interest
not only because they yield further insight into
the contents of the theorems, but also because they
are examples of the application of elementary functional analytic and Banach algebraic techniques to
statistical analysis.
The norms used in showing
convergence in the lemmas and theorems in Chapter
III are the operator norms.
Since any two Hausedorff
topological vector spaces over the same scalar field
and of the same finite dimension are isomorphic as
topological vector spaces, convergence of a sequence
in one norm implies convergence in all norms.
Definitions and notation, where not explicitly
116
117
restated in this chapter, are taken from Chapters II
or III where they are first introduced.
2.
Market Search.
Al.
For each
in
y
F,
p
in
Rn,
there is a choice
in
> py.
Yfpyf(p)
each
of
For each
f,
p
IRN
in
X (p)
p
-*
fF Rn
and for each
f
p(f) (Xh(p)
and
in
yf(p)
and for each
h
yf(p)
yf(p)
H,.
can be
is continuous.
p(f) > 0
with
in
f
such that for all
Furthermore
choosen such that the max
A2.
p > 0
for
there is a choice
such that
Xh) <
-
[ d(h,k) p(k) yk(p(k))
kcF
Uh(X (p)) > sup{Uh(x):
for all
x c
IRn
where
Z
p(f)(x-n) <
d(hk) p(k) yk(p(k)). Furthermore,
f
kEFf
xh(p) can be choosen such that the map
p -+ xh(p)
is continuous for each h in H and f in F.
Let
F.
C
be an arbitrary function from
For each firm
p(f) > 0
f
for each
Zf(P) =
and each
f,
p
in
H
G0 ]Rn
f
(X(p)-h
p > 0 Ei=
p(i) = 1}.
with
let
-
Yf(p).
S
Let
be
n
h:c(h)=f
the unit Simplex in
into
Rn,
For
that is
p
0
f F
Sn: =
Sn,
p
R n
we'have:
118
Ef PEf) Zf(P) =Ef
p(f) (Xf
~ n)
h:c(h)=f
Ef p(f)
If
Yf(p).
= f,
c(h)
p(f) = p(c(h)),
then
so by A2,
h:c(h) =f p(f) (Xh(P)-Xn
d(h,k) p(k) Yk(p(k))
:c(h)=f
+
I
r
kEF
d(h,k) p(k) Yk(p(k))
p(k) Yk (P(x))
hEH
kcF
kEF
fEF p (f)
Yf(P(f)).
f EF
A3.
For no
Zf(p) (i)
A4.
> 0
For each
p
in
Q
Sn is it the case that
f
implies that p(f)(i) = 1.
h
in
Uh : R n'
H,
R
is
a continu-
ous function.
THEOREM 2.1.
Under assumptions Al, A2, A 3 and
A4, there exists a search equilibrium with
Proof.
Let
K =
(D
hEH,fsF
S
(G
S ,
C* = C.
then
K
n feF
is a compact convex subset of the finite Cartesian
product of copies of
JR .
We identify an element
119
of
K
and
by the pair
has
in
for each
Sn
(p*,q*)
p(h,f)
h
in
is
H
function
We seek a continuous
F.
in
is
q(j)
where
(p,q),
as a fixed point only if
in
and
S
n
f,j
ip:K+K
th at
(p*,q*)
is
a price profile for a competitive search equilibrium.
Let
1
n +R n
4:Sn x
2
n)
where
be given by
xRn
p3:S
IR
is
defined as follows:
1p1 ) (a 1\J)
$ (pi,p' Y ''' - 1n, a ,a2,...,a n)
J+$
Here
and
a)
=(-$VO),
(p,a) = (P
,...,pn,
aVb = max(a,b).
-Pj+1)(a +\10)
a1 ,...,an),
The function
$p =
j (pa),
has the
following properties:
a)
$ is continuous.
b)
For.all
c)
$(fa) > 0
with
d)
pi
(p,a)
in
Sn x Rn
, (p,a) >
0.
if and only if there is an index
1
and
a. > 0.
There exists at most one index
j (p,a) / 0.
j
with
i
120
X be the function from
Let
defined by
= (aVO)/(1
, (a)
into
IR
IR
Then >, is
+ (aVO)).
[0,1)
and only if
if
0
and
p c
a < 0.
Sn,
0
For
let
and
h
f (p)
X(a)
in
equals
f
H,
is
X(a)
continuous function with the property that
in the half open interval
a
in
F
be defined by
hEH,f F
(
Xh (p)
=f
$
The function
$(p,q)
and
=
(
$2 :K
Uh[X
X{U h[Xh(P (h,-))]
X
9
-*
S .
(p(h, -))]
.
can now be defin ed as follows
2 (pq)) where
p,q)
(h)
)1:K
-+
hD
Sn
he-H .fcF
is defined by
The function
fEF
}
1 (p,q)(h,f)
p(h,f)
+
q
if
f = c(h)
hf(p)- p(hc(h))+[1 -A (fp(h,
if
q(f)
f)
f A c(h)
+ $(q(f),Zf(q))
2
1 + i
Clearly
K
*
is
$(q(f)
Zf(q)) [i]
a continuous function from
and so by Brouwer's fixed point
vex subsets of
IRn
4
K
into
for compact con-
has a fixed point
(p*,q*).
121
Let
and let
C* = C.
Since
= p*,
$ 1(p*,q*)
p*(h,c(h)) = q*(c(h)).
p*(h,f)
=
p*(h,f)
=
A2,
<
p*(h,c(h)).
Since
i
In this case
j
or
*
=
by assumption
with
Zf.(q*)(i)
> 0
$ (g* (f), Z f(q*) ) > 0,
ej(q*(f),Zf(q))
with
+
$
> 0
(q*(f) ,Zf (q*))
> q* (f)(j)
1 + $d (q*(f),
Ef(q*))
This contradicts the fact that
for
= 0
q*(f)(j) / 1,
q*(f) (j)
point
(p *)
$ 2(p*,q*)
Otherwise,
and an
so there is an index
and since
Xh
either
f.
f
q(f)(i) / 1.
and
then since
In either event,
for each
we can find a
follows directly that
f / c(h),
p*(p*,q*)(h,f),
0
it
If
U(xh(p*)) < U(X c(h) (p*)).
Zf(q*)
y* = yf(g*(f)),
let
c(h) p
x* =
11,
and so
Zf(q*)
(p*,q*)
< 0.
consumption allocation vector
by
X*(h)
=
y*
defined by
defined by
equilibrium.
a fixed
The price profile
(p*,q*),
XA,
is
defined
X*,
the production allocation vector
y*(f)
=
C
= y*
and the choice
C*
is then a competitive search
122
COROLLARY 2.2
In a search equilibrium two
different firms may post different prices for identical commodities.
In a simple general equilibrium model
PROOF.
such as that described in Chapter 2 of Arrow and Hahn
[
], we see that the equilibrium price is not indexh
pendent of initial endowments
Consider two
simple general equilibrium models identical except
and the resulting equi-
xn
for initial endowments
These models are identified
1*
1 1 (H,F,Xnp*)
and
(H ,F ,Xh'p )'
librium price vectors.
by the parameters
where
and
F
F
are both singletons.
H U H1
the search model with
firms and choice function
if
h
is in
H
and
y*, y*,
householders,
defined by
c(h) = f1
The consumption vectors
vectors
C
Now consider
x*
Xht
Xh x2
if
h
F U F
c(h)
is in
=
f
H
and production
from the simple model will also
be the desired equilibrium commodities in the search
model.
We assign ownership of the two firms as
follows:
d(h,f)
=
S(h,f)
.
0
if
h e H, f c F
elsewhere.
or
h e H1 f e F
123
Here
is the assignment of ownership in the
S(h,f)
we have an ordering on
p > q
defined by
Sn
H C H1
in
h
For each household
original model.
if
and only if
Sup{U(X):
X e IRn
p(X - Xh) <
such that
h
*
< d(hf)p*yf
+
1
1*
d(hf )p ly
*
}
f
Sup{U(X):X e IRn
n
is greater that
dh 1~
< d(hf)p*y* + d(hf 1)p
q(X - Xh)
For
h
and for
p
in
h
such that
choose
H
in
in
choose
H
Ph
Sn
1* *
y}
f
such that
in
such that
S
No
p h'
Now define a price profile by
p*
if
h e H,
f c F
if
h c H,
f c F1
if
h E H1
if
h C HI, f
if
f 6 F
if
f 6 F
p(h,f) =
p
and
p*
g(f)
=
p
1*
p* > ph
1
f e F
£
F
124
(p,q) xh, Yf C
Then it is easy to see that
a competitive search equilibrium, but
3.
q* (f)
comprise
/q*(fi).
The Housing Search Model.
Lemma 3. 1.
For
ln(fX-1) +
nx>l
nx
1
> 0
nx-1
Proof.
1)
Let g (z)
2)
g' (z)
=
ln( z+J. ) + 1z
_ z 23 -z (z+l)
z (z+j)
1
z2
1
(z+1)2
_ z 1
z
_
-1
z2 (z+1)
< 0 (z > 0)
3) lim
g(z) = 0
z+=4)
therefore for z > 0 g(z) > 0
(g decreased down to zero)
for nx>l 1n nx-1)
nx
+
> 0
nx-1- = g (nx-1)
Lemma 3. 2
For nx>2,1 <nx in nx-1< < 1.4
Proof.
1)
Let g (z)
=
z in
2)
g' (z)
n(
z)z1
=
z
z-1
1
- z-1
,
therefore
125
3) g"(z)
=
1
1
z-l)2 ~ (z) (z-1)
is
increasing
g' (z)
4)
= 0, g'(z)
im g'(z)
9" (z) > 0
< 0
for
5) g is a decreasing function for
z > 2
6)
7)
g(z)
z
-z-l
z in
lim
z-+CO
= lim
-
-
lim i ln
C--o eC
-
ln(1
L'Hopital's
> 1
z
> 1 and for
= in 4 < 1.4
nx in nx-1 = g(nx)
nx>2
8) By
< g(2)
z
< g(2)
1
1
=
< 1.4
lim iln
C-4' e
lim g(z)
rule, lim
-
ln(1-c)
--1
=lim -
1~E 1
c:*o
9)
+1
g' (z) < 0 so g is decreasing for
z
> 2,g(z) > 1
Lemma 3.3
Let Yi,Y
2
.-- Yp be a nondecreasing sequence of
nonnegative numbers.
Let Ml(-),M2(-),-..Mp(-)
be a
sequence of positive differentiable functions satisfying
p
Z p YrMr (x).
aLet f(x)
= r=1
r=1 Mr (x)
If
xl
> xo
and
d Mr(x)
Mr-1
dx
(X)
Xo < x < xi
< 0
--
r=2,3,...p
126
then f(xi) < f(xo).
Yp > Yj and
The inequality is strict if
d M 2 (X)
ddM
X0
<
x
< xi
(x)
1
< 0
dx
Proof.
Mr(xl) < Mr(xo) for r < p-l,
1) If
<
If
Mr+l(xo) -
contradicting
not, then
2)
M1(x
Mr+1(Xl) > Mr+1(Xo)
Mr(X1)
Mr (xo)
d Mr+1 x)W
Mr (X)
< 0
dx
1
then Mr+l(Xl)
xo < x < x1
-
iot Mr (xl) < qr (xo) for
) > M 1 (xO); If
r = 1,2,...p
p
p
Zr=l Mr(xo) = l contradicts
Zr=1 Mr(Xl) <
p
Zr=1 Mr (Xl)
3)
If
=
Mr(Xl)
holds trivially.
for some r,
1
r = 1,2,...p
= Mr(Xo)
then lemma 3
Therefore we may assume Mr(Xl)
and for some r Mr (xl) < Mr (Xo)-
d Mr(Xo)
Let j be
the least integer such that Mj(xi) < Mj(xo)
p
4)
5)
r=l[Mr (Xl)
-
Mr(xo)
1-1 = 0
=
j-1
p
Lr=1 Mr(xl)-Mr(xo) + Zr=j Mr(Xl)-Mr(xo) = 0
j-l~
z r=1 Mr(xl)-Mr(xo)
p
~
r=j Mr(Xo)
Mr(xi).
(all terms in both summands are positive)
127
j-1
6)
Z r=1 Yj-1
[Mr (xl) -Mr (xo)
p
r=j .Yj
<
[Mr (xo) -Mr (Xl)
j-1
7) Z r=1
since for r <
Yr
8)
p
Zr=j Yr
r [Mr(xl)-Mr(xo)1 i
> Yj;
j
[Mr(xo)-Mr(xl)]
1 Yr < Yj-1 and for r > j
-
to get
rearrange
p
p
zr=1 YrMr(Xl) .< r=1 YrMr(Xo)-
9) If
d M 2 (X)
Mi(x)
MI()<< 0
<
Xo
x
<
Kl
QX
> Ml(xo).
then Ml(xl)
(If Mi(x)
Mi(xo)
=
M 2 (xo)
then since M 2 (x)
M(Xo)
MJ(x)
M 2 (xl) < M 2 (xo) and by
(1) Mr(xl) <-.Mr(xo)
r=2,3,...p. This contradicts 1 =r=
r=1 Mr(xl)
rXi
p
r=l' Mr (xo))
10)
dM2( W
If d Mi(x)
dx
then
X0 < x < xi
Mg(xj) < Mg(xo).
Therefore examining
(7) we see
M 2 x)
if
d MI(X)
dx
<
0
Xo : x
~~Xl
p
p
zr=1 YrMr (Xl) < Zr=l YrMr (Xo)
and
Yp
> Yi
128
PROPOSITION 3.4.
Proposition 4.1.)
(Chapter II,
mBj (Yi
'jP(j,K)
lk=1
=
mBj (Yi , j)
I
k=1
n k (
) -1
mBj (Yi
I
mBj(Yij)-k.
1 k-l n( n)V
n
k-1
n-1 m [Hj(Yi,j)-Bj(Yi , j) I]
(-) n
K
(
mBj (Yi , j)
k
k
)
mBj (Yi
k=1
.
/n
1
m(Hj(Yij)-Bj(Yi,
11
mBj (Yi, j)
2)
Binomial
n-1 mBj(Yij)-k
n
N N
theorem states Zko k
4)
Z N
k=1
N
pk (1 -p) N-k
N-k
1
[(P+(1-P)] N=
3)
pk (1-p)
)
-1-N
k
mBj-(Yij)
Pjk
4)k=1
1
mBj (Y,
1
mBj(Yi,
SmB
(Yi,j)
m[Hj(Yi,j)-Bj(Yi,j)]
P-
j)
[m[Hj(Yij)-Bj(Yij)]
)
(1 = nn )
mHj(Yi,j)I
I
129
Propositions
above are proved in the body of the
text.
Theorem
(x,e)
(Chapter II, Theorem 4.5.)
3.5.
1)
Let f
2)
P(x,i) = ZjEAi
U(x) xm
=
then
,
1
rnBj (Y , j)
[ f (XH
(Yi ,j)
-Bj (Yi,
j))
f(X,Hj (Yi,j) I.
3)
If
Ai
/d .9,
Then
af
< 0
if
(x, i) > 0
x, 6)
6E[Hj (Yij)-Bj (Yi,j)
and
aaxaf e
(x,i) < 0 if
Hj (Yij)]
> 0
(x,6)
6C [Hj (Yi
j)-Bj (Yi,)
,
Hj (Yi rj)]
4)
ax
a2 f
axa e
u(x)xme me[ln(n nx
[in nll)
nx
+
1
+
1nxlm
5)
a f
6)
By lemma 1, sign a f
axa e
nx
1)
+
nx-1
[
(x) 40+
iy(x)
xm ln u (x)].
y1) xme [1 + exm in yI(x)]
2
7 i= =
7) ignaxae
sign [1 + 6xm lny(x)].
gf
sign [1 + e~a nxnxln(nx-1
nCnx
130
8)
9)
for
0 < 6 < 1 (6
1
Oanx
+
= Hj(Yi,j)-Bj(Yi,j)
e
= Hi(Yi
> 1 + a nx In(nx-1
nnx-1
1 + anx ln( nx-1
nx
or
> 0 if
a < -
nx-1
nx In (nxnx
1
nx In( nx
nx-l
10)
nx
for nx>2, nx In( nx-1
By lemma 2,
> 0
2 e
aa1)
5/7
if
nx>2
aP (X, i)
< -nx-1
nx In (
if a< 5/7 < -
< 0
ax
< 1.4,
1 + eanx in nx-1
nx
(13)
If
(14)
If
< a
sup
=
1 -6
anx In
1 + 6 a nx In nx-1
n1
nx-1
< 0,
{[Hj(Yi,j)-Bj(Yi,j)] -1
[1 + 6anx In
nx-1
nx
> 0.
-
there if
< a
< 0
(6E:[Hj(Yi,j)-Bj(Yi,j),Hj(Yi,j)]
ap (xti)
x
1nx-1
JcAi
sign
and
nx In (nx
nx
nx>2
Ai /
(12)
therefore
and
ea
))
131
THEOREM 3.6.
(Chapter II, Theorem 4.6)
1)
Q(x,j) = 1-p()x)mHj (xj)
2)
2
(x,j)
= - p(X)xmHj(xj)
By lemma
1,
3)
If
and
4)
Let
nx>l
if
i
m
nx
n(nx-1
nx
nx>l,
Yi
+
3Q
ax
Hj(xj)
= E(x,j)
f(x)
mHj(xj) [ln( nx-1)
1
0;
nx-1
+
1
nx-1
1
so
< 0.
(xj)
=
)
[mHj(Yi,j)-mBj(Yi,j)]
xmHj (Yi, j)I
u x)
xmHj (xj)
1 -
5)
Partition Cj
11x)
1
2
into k disjoint subsets C.,
C9,
-J
J
k
J
such that:
6)
a)
i E C
b)
i E C
t e C
J
Y.
J
1,]
t C Cs+ 1
J
J
. =
Y
y.
.
< y
iJ
Let is be chosen such that i E C
5
t,j
J
t,j
s=1 ,2,...k,
then
1lCs li = mBj(Yi 5 j)
k
7)
E(x,j)
=
f(x)
s=1
III
Y.
1j
(Yis, j)-mBj (Yi5
x) [mH
xmfHj (xj)
1
- 11 (x)
j)_
j
(X)
xmHj(Yi5 ,j)
132
rnHj (xj)-mHj(Yi Sj)+mBj(Yisj)
8)
E
e=
mHj (xj) -m~j (Yi sj)
(X)
EIli
xmHj
+ 1
(x)-e
xmHi-j (x)
-
Wi (x
-e+iJ
xmHj (xj)
2.-p
X
x[mHj(Yisj)-MBj(Yis
pj Wx
, j) I
xmHj(Yisj)
-v()W
xmHj (xj)
)
(3.-PW
9)
m~~j-~(~sj+~.Yi~)Mjx)MjYsj
(s=1,2, .. .k-1)
mHj(xj)
mjYlj
=
mHjCYik,j)
10)
mBj (Yik, j)
=
mHj (xj)
E(x,j)-= f(x)
r=l
x[mHj (xj)-r
(x)
x[mHj (xj)-r+l1)]
-Vi Wx
xmhHj (xj)
1
where
Yr
=
-
p W)
yi srj
if m~~j-~(if)lrmj~j-~
11) f (X) nHj (xj)
r=l
(i~)mjYsj
Yr mr(XW
x [mHj (xj) -r])
where Mr(x)
J
x[mHj(xj)-r+l
- li Wx
=-W(X)
1
xmHj (xj)
(X)
133
x[mHj (xj)-r+1]
x[mHj(xj)-rl
12)
v (x)
Mr (x)
Mr-1(x)
1(x)
-W (x)
x[mHj(xj)-r+2)
x [mHj (xj) -r+1]
-i (x)
[r=2,...mHj(xj)]
(x)
d Mr (x)
Mr-l(x)
13)
1
_
xl
(x)2x
dx,
nx-1
nx,
+
1
]
nx-1
d Mr (x)
Mr-
(x)
< 0
for 2<nx and r=2,3.. .mHj(xj)
dx
14)
mHj (xj)
Mr(x) = 1, so apply
Ir=1
if
x 1 > xo
(xo
>
2/n)
then
and then exist i and i'
f(xj)
1emma
3.3 to get
f(xi <
f(xo)
with Yi,j
>
if xo > 2/n
Yi, r
xj
then
< f(xo).
rX Theorem 2
4.
Block Scalar Variance Covariance Matrix.
PROPOSITION 4.1.
observed data matrix
Let
X
(a1 1P*.
1J)
and dependent vector
ing Assumptions I-IV:
1)
Let
E
=
a2
101 1
0
0
0
2
0
0
2
01,2 2
0
2
Gi
Ij
FR
Y
$for
satisfy-
134
=
T
X)-1 XT
2)
Let
B
3)
Let
En+1 = E(B n)
4)
Let
Bn 1
1
1Y.
( T -1 X) ~ XT
1
Y.
Then
A)
For each
B)
The sequence
C)
If
B*
n > 1
exists;
B
{Bn I
has at least one limit point;
is a limit point of the sequence
B* = (XT E(B*)1 X)~
PROOF.
{B i
then
XTI(B*)~1Y.
Our proof is similar to that done by Oberhofer
and Kmenta.
1)
Let
f:IRk
"
-+R
f (B, Z 2 ..
be defined by
N/2
Z.
Z 2) = ( 2 7)
exp - 1/2 E 1
J=
[HjJ -1
(Z i
2)mj
-1/2
(Y.-X.B) T (Y.-X.B)
3 3
z.
2
J
(N =.
f
mi )
is, of course, the likelihood function should the
ii
be normally distributed.
2)
The concentrated likelihood function is defined by
h(N)
3)
Since
=
(2Tre)-N/2
lim
1B I +
3
sup
M.-
[
(Y,-X.B) (Y.-X .B)
3
3
3J 3
m.
J
2
f(B,2
I =J
1/2
3
2
3..z
J
lim
B1
h(B) = 0.
-|oo
It follows that for any 65 > 0
135
{Be]Rk 13
Z
(Z2...
f(B, Z 2 ,.
c ]+Ji
2
Z. 2) > 6
.
J)
is
either bounded or empty and hence has compact closure.
4)
It also follows from 3 that the function
5)
It is immediate that for any
f(B,Z 1 (B),. .. ,Z (B))
(Z 2,. .. ,Z )
for all
It is also immediate that the unique
,Z 12
Z1 2
where
2)
numbers is
given b)y B
Z1I1
0
o
222
Z$I
0
0
0
Z I
J 3
2
Let
. a
2
2
I.
is
XT -1lX
{B n}
be any
=
2 all
j.
J
that maximizes
are f ixed positive
m. X m.
where
identity
matrix.
exists by induction on
J
2
n.
positive numbers.
T
X TX
The matrix
B
T -l
(XT X-1X) -1
'(X E Y)
=
We show that the sequence
7)
IRk
2) => Z.(B) =
J
J
f(B,Z 2
1J
f(
is bounded.
and that
f(B,Z 1 (B),. .. ,Z.(B))
6)
£
> f(B,Z 2,. .. ,Z )
c IR+
J
B
f
=
jl
a
m.
1
3
XT
.
M.X
For each
is a positive definite symmetric matrix
J
m.
and
32
1 ,j
is
a positive number.
is
Therefore,
a
positive definite symmetric matrix and, in particular, is
nonsingular.
XT-lY
Therefore,
exists.
B
which equals
(XT -1 X)-1
136
Assume
infk
BeJ
||
B1 ,... ,Bn
so that for each
J J
In the above argument which shows
1
8)
By assumption
Y.-X.B| > 0,
2
0
exist.
by
to see that
Z (Bn)
j,
Z.(B ) > 0.
B1
exists, replace
exists.
[X]
A.
{B n}
is
It follows from 5) and 6) that
f (Bn'
1
f(Bn+1,
(Bn)'... Z(B n)) < f (B n+1'Z1 (B n+1
'l''' .. Zj (Bn+l))
hence from 3)
bounded and therefore has a convergent subsequence.
[X]
9)
Let
B*
be one limit point of
B
f(Bn
.
and let
{B i
+B*
, Z 1 (Bn ) , . . . , Z
(Bn k))
f (Bnk
), ..
Z (Bn
k+1 1
k+1
B.
<
f(Bn+1 Z
(Bnk
),.
. . ,z
(Bnk ))
,Z (Bn
< f(Bnk+1, Z1 (Bnk+l
, ..
),(Bnk+l
Since:
i)
ii)
f
is a bounded function.
B
= (XT (Bn )
XT E(Bnk
X)
so
k
Bnk +1
converges to
which we denote by
Letting
function
k
increase to
f
and
Z
o
(XT (B*)
~ x.
yT E(*
B*.
and using the fact that the
a
are
continuous, we have
137
f(B*,Z
(B*),...,Z
(B*))
< f(B*,Z
the unique element of
f(,Z 1 (B*),.. .,Z
IRk
(B*)),
(B*),...,Zj(B*))
is
B*
that maximizes
s* = B*.
therefore
[X) C.
Let
PROPOSITION 4.2.
g(B)
=
Let
F
g:IRk
-1
XTE(
-1 -1iT
(XTZ(B)
X)
X (B) Y
be defined by
IRk
1 T=
-1 (1 XT
-1Y]
XTE(B) Y)].
XT Z(B)X)
be the set of fixed points of
d = Sup{l B-B 0 jfl
d < y0
tions I-V,
B e F}.
g,
y0 > 0,
Let
almost surely as
N + oo.
appear explicitly in the definition of
fore, for each
remembered that
Recall that
N =
vector.
X
j=l
m.
j
(X,Y)
is an
X, Y, and
N
all
and that,thereIt should be
and that, in effect, m.=m.(N).
3
let
B
be a fixed pointof
real matrix and
N x K
3
Y
is an
Where no confusion is likely to arise, we
drop the superscript.
g(B) =
1)
g,
is a random variable.
N,d
For a fixed pair
N x 1
then under Assump-
It should be understood that
Proof.
g.
and let
Thus:
X
(B)~ Y)
N X
B)Z
B.
Recall that:
Y = XB 0
2)
and rearrange 1)
3)
+
E
to get:
T
1 [(X T E(B) 1l
X) (B 0 -B) + X
N
Al
-(B)E]
= 0
SpT
Now premultiply both sides of 3) by (Bo- B)
get:
and expand to
138
.-
T
XXA
- 3
y
4)
Let
=
{
I
m
S 2 = {j
bounded } .
is
$
M.
-
-i-v
N + =}
is unbounded as
M (N)
(N)
o
Z. ($)
S
is the empty set.
cannot be empty
S2
Z'J = 0
We therefore get from equatioi
4):
r
r
(B
M.
5)
jeS
o -B)
jES 2
m.
A
^rX-..
(B -B)
+
o
(B -B)
om
Z. (B)
L
-7
M.
4X
A
A
N
(B-_B)-0
X2
Mn.
-B
(
(B-_B)
o
X.E
+
I..
(B-sB)
Mn.
o
N
We now analyze the
RES
of equation 5) and examine each
term in the summand to get:
XX
6)
M.
n
[(B 0
M
.
.X
(B
A
and let
By convention, we take
however, may be empty.
where
o
L
S
(B
+
(B -B)
M.
f (B -9)
m
AT X.e.
-3
1
-B)
-$)
+
A
nr (B--)
(B)
J
(B O-$)
]
I0
-
139
X .E.
N
(^
Z6.B)
Li4 XZji(
j CS
and
e.
,
and for
N
then
are independent of
Z ($ )
N
.
Hence,
for
N
inf
BF]RK
m (N)
Z.(B)
3
4a
,
X
is
sufficiently large, and so
is bounded away from zero.
For
lim
N+)00
N
ax 2 -bx >-
sufficiently large,
3
also independent of
b>O
A
4
N
For
(From elementary
calculus if a>0
12
1
m. (N)
N
Let
j C S2
,
m (N)
is
bounded,
therefore,
for j ES2
=0.
y > 0
it follows immediately from the above that:
,
that:
,r
(B
m. (N)
es2
8
j F-S
+
-
(B -B")
0
-a-M.
>-
N
almost surely as N + =
By 5) and 7) we now have that
8)
o
mN)
3N)
3N3
(
T
for arbitrary y > 0
(B-B
0
)
+ (B 0 -B
(N)
J
z . (B N
Y
140
almost surely as
N + w .
Before proceeding with our proof of consistency, we
needrteoestablish
9)
Lemma 4.3.
on
t'vo elementary numericail lemmas.
Let a, b, c, be positive real numbers, then
{x I x > 0},
{i
the function
f(x)
functiobx+c
ax
=
is an
increasing function.
Proof.
10)
Lemma 4.4.
ac
(b x4-c)
> 0
Let a,b,c,d, be positive real numbers.
f(x) =
Let
=
f'(x)
ax-bx
cx2+2bx+d
x >
Then for
-
a--
f (x)
2ab 2
4 (c+a)b 2 +a 2 d
a
b>
Proof
.a
Now use lemma
-
2b
a
2
2
cx 2 +2bx+d
2
2_
-bx
-
__
(., +a) x
2
+d
4.3 to get:
2
cx
ax
bX
+2bx+d
2
2ab 2
.
4b 2
c+a)a2
+
d
4(c+a)b 2 +a 2 d
>
141
We now analyse the
2
LHS
JIB 0 -B"1t>
For
of equation 5)
m
+
J (B 0 -B)
J
)
(B 0 -B
mn
-T
Xme
+
(B -B)
mn.
ll)
($")
Z
.
XI B -B
2K
0.
IB O-BI
-
I
I
E.
X.T C.
2 + 21B
o -B
I~-
T| B -B
Z. (B
(Recall that
= (B
+ 2(B.-B
0
For
+
(BO-B)
3
use 10)
+
mn
to get:
3
IIB 0 -B II
A,
4
T
>2
J.
X.
(B 0 -B)
12)
-B)
-7-
XJ
m
m
-
m.
M.
-r
(
(BO-B
+ (B 0 -B )
x
2X
2
M.
J
x7 E.
4(T+>)
2
3
+
X2
T
S3
mnj
(B )
Z
x.
.
I
C.
142
if
<
3
and if
,
< V
then using 10),
12)
J
becomes:
|1
JIB -B
For
> 6
13)
(N T
(B0 -B)
X X.
3 3 (B 0 -B ) + (B 0 -BN T
XI
3
3
"N
Z . (B)
J
N
8 (T+-X)
6 2 + 2V
m. (N)
T
For all
( 2
1, therefore for N sufficiently
j=l
m. (N)
large
N
jE S
it follows
V > a.2
y=1/2
k62
> 1/2
.
Let
a.2
, then
above,
let
V=1+
j=1
all
j
.
In
equation 8)
8 (T+X) 6 2 +2V
By the Strong Law of Large Numbers we have
T
(14)
almost surely
max
je S(
N + W and
15)
almost surely
<2
max
j CS
N + =
|M.(N
J
<
V
143
8) gives us that
T
A(
zjS
(B 0 B
(Bo-B
5
m. (N)
ND
N
1
)+
T
A
)
(B0-B
(N
Z. (t")
1/2 X6
2
8 (,T+X) 62 + 2V
almost surely
N +
.
Now observe that if :
M (N)
Ij5
X.
> 1/2,
$N
max
E.
M N
j (N)
j.S
2
max
(N
, ES
xTr
7-T
M. (N)
and
Zj ES
(B -B
N
)
Xix.
I< v
(BO-B)
+
(B 0-B
)-
AJ)
Z. (B
1/2 X6 2
8(T+X)6 2 .+ 2V
Then
|1B
A
0 -B
II <
We can now conclude that for
almost surely as
N + = .
6 > 0
IBo- B I
< 6
144
COROLLARY 4.5.
Let
0
property that for each
2)
number
1
,..
such that for each pair
2
{B n a 1 ,...
point of
Assumptions I-V,
0
the property for each
(X,Y),
2
a1 , 1 ,..a
E(X,Y)
(X,Y),
2
,a1
,
I-V, the sequence
Let
N,
2
0
be an estiator with
there exists positive
such that for each pair
Then under Assumptions
XY)}.
{Bn (1,1...,a
convergent almost surely as
Proof.
is a limit
Then under
2,X,Y)}.
1
2
...
O(X,Y)
N)
is a limit point of
2
{Bn (a
perhaps depending upon
is strongly consistent.
PROPOSITION 4.6.,
numbers
there exists positive
N,
2
.,
be an estimator with the
XY)}
2
is
N t o.
Consider the mapping
g:
IRk
+Rk
defined by:
1)
It
g(B)
=
(XT
(B)~ X)
1
XT Z
Y(B)
Y
follows immediately from the definition of
the proof of Proposition 4.2. that
that
0(X,Y)
is a fixed point of
g
Bn+1 = g(B n)
g.
and
and
As in the proof
of Proposition 4.2,
let
B = 0(X,Y).
exists a subsequence
B.
converging to
Since there
B,
-the
145
{B (
n1
g is
original sequence
converge to
if
B
1
. .a
2
g(B)
H1
2)
< 1/2
11B - B
only show that
we need
{B n
implies that
1/2
<
- g(B)l
will
a contractionmapping near
Therefore to show convergence of
B.
XY)}
Y~
,J 2
1 B
-
BI)
[Here as elsewhere in this proof, all norms refer to
To see that this suffices, observe
the operate norm.
that
B.
1 Bn
g (B)
-
-+
B
j<
B
imDlies that there exists
1/2.
(B),
= g
B = g(B)
Since
an d since
G
such that
= g(g(B)):
=
+r
Pn
p
g (B
P ),
we
have that:
|| Bn +r
P
3)
| Bn
(1)2
gr(B)H
g ( n )
=
~2)
( 1
r+l
p
4)
g(B)
-
g(B)
XTE(B)
=
N XTE (B)
X)
( XN
TE ()X
)-lX)
Y)
-1. 1 XT
Z (B) - Y)
Use the following facts:
Y
5)
6)
11AB
CD 11 < | Al l
=
XB
0
|[ B - D
+A
[
- Cl
||1 D 1|
146
1
11A
7)
- B_1 1 < 11A -
to get from
1
A - BI1
1 B_ 1 1
4):
8)
11g (B)
-g(B)
ell
x (B)
- XT
1 X)
B
XT
1(
-1 -1
XxT E (B)
1
-
11
XTE (B)
II
- g(B)
|| B - B I
1/2
<
1
1X
We wish to show that for |1 B - BII
Jg(B)
XT Z(B)
El) X
-
X Z (B
1 T
+
1 X E(B)
T
N X-Z(B)
B,
about
-1
XI
< 1/2,
almost surely as
since we have shown
I B - BO l1 < 1/2
N + o.
almost surely as
Let
S1 = {jM (N)
is unbounded as
Let
S2 = {j M (N)
T
is bounded as
N + o}.
N ±
T
(Be-B)T - 1 -- (BA -B)
m - U
-u
J
Z. (B )=
therefore for
B0 - B|1
9)
2
T
<
_
1,
< Z.(B)
-- 1
m -
m=i min
J
0
(a 2 )2
m < M <
+3
Jn
we have
T
T
m.
T
_
J
T
F_j
co}.
+ 2(B 0 -B)T xm.
-
then
1X)'hI1
We can confine our analysis to the unit ball
N + o.
Let
-I
< T + 2
and let
and for
3 M3
m
+
. 3S
m.
M = T + 2 +-
jC
S 1and
llB-B0 1
147
we have:
almost surely as
< M
M < Z. (B)
10)
- J
For
B
such that
(
X
(B)
11 B - B0
Since
X)l.
,
<
N' t 0
We wish to find
N XT
(B)X
is positive
definite symmetric, to find a upper bound on the norm
of its inverse, we need only take an inverse of a
The
positive lower bound of its minimum eigenvalue.
XT (B)
minimum eigenvalue of
11)
For
ZT(1 x T(B)
inf
eIRk
H Zfl
Z_ E R k,
N
X)Z
Therefore by 10)
N + 0.
(
$(XT
X)
particular, since
N + co,
we have:
=
and for
Hence, for
X E(B)
lx)z
Z
.XTE (B) - X)Z
eigenvalue of
is given by
=1
M.
zT(
12)
X)
(B)
|
X)
zT x-T X
Z X. X. Z
3
m
N=1
Z (B)
B - B0
<
< *
M.
3
CS 1
Z .3
the minimum
almost surely as
JIB - B 0jjz1
almost surely as
JIB - B 0 l < 1/2
N t
n.
almost surely as
In
148
1
()
N (XT (XzB
13)
2M
X) -
N t m.
almost surely as
T
1 T
Ngx- E(B)
14)
1
T
RX, E(B)
- Z (B))
+
eS2
Now
m.
Z (B)
=
3 (B0 -B) + 2(B0-B)T
M.
J3
(B 0 -B)
-T
x.Tx
Jm.3
(B0-B) T
(B 0 -B)
-
3
Z ( )
XTX
Z (B)
j2
Z..B
M.
Z. (B) Z.i(B)
X.
jcS N
T
m
m.
21
N Z. (B) Z.(B)
21
21
2
2(B0-B)T
B
X.T
M3
X. TE
M.
15)
< T11 B0 -B I
Z. (B)
31
Z(B)
|1 B-Bli
+ T11 B0 -BI1
B-BI|
T
+ 21
< T
B -B
+
|1B 0
T
B0 -BII
T
+
+2 x ,. E
2
J
Let
0,
B - B
-
_
j c S1
then for
and
I B - B1
<
1,
it follows that:
X T
16)
-. B)
J1
Z. (B)
j
e S2, X , m (N),
independent of
(Zj(B)
- Z (B))
1 <
11B-B
N t o.
almost surely as
For
m.
N.
21 are all eventually
By assumption
Z (B) > 0
all
H
149
B
£
j
and for
IRk
e S2,
m. (N)
0,
lN
N
=
2N to
N
therefore
fJ B - Bol . :
we have for
T
17)
Ej
.1j 11
S2
Z.(B) Z.(B)
J1
J
YI
11B - B1
Combining 15)
-
^
j
almost surely as
- Z. (B)) I1
21
N + 00.
and 16), we get:
E(B) I
18)
(Z. (B)
21
m 3
-
p
TE B)
Sup
0 <
B 6 IR k
|B-Bo 1
11
B - B0 11
N t o.
almost surely as
m. M.XT
X.TeE
1XT(B) 1
jES
+
m.
N
jES
2
m. X.T
3
3 3
N
m.
Z. (B)
and
1 Z . (BI)
therefore for
lIB - B 0 l
T ,(-B 1
X'
(B)
£1
19)
N
<
XTZ (B)
and
-<1
y2 > 0
Hence,
almost surely as
2
Y2
E
almost surely as
N t o.
ST
20)
X-TE(B)
-x
XT Z(B)
Xj ES
N
M.
21 Z (B) Z j (B)-
T
m. X. X
(Z.(B)
21
- Z.(B)) +
9L
jeS 2
Zj(B) Z. (B)
(Z. (B)-Z. (B))
1
21
150
By 15 and 20) and for
21)
I
|1 B - B0 1
- 1
SIT
-lX
XTE(B) 1 X SI XT (IB)
XI I
N
almost surely as
<
m
y
22)
2
[2T+3]
B-BI|
o.
N
Combining 8), 13) , 18) , 19) , 21)
choices of
we get
Y2
and
and by the proper
we get:
g(B)
Sup k
| B
0C1| B-Bj < 1/2
g (B)I
-
< 1/2
-B
N t c
almost surely as
[X]
The next result is a minor improving of the last
proposition.
It will however pave the way for showing
that almost surely as
independent of the
by
g(B)
= (1 XTZ(B)
Let
K(N):
2
(1
PROPOSITION 4.7..
gets large,
N
2
Let
X)
1
X1
T
( X- Z(B)
Sup
B
||
E
R
B 2 c ]Rk
|| B - B0
|IIB 2 -B0
~z1
K
0 < |1 B 1 -B
2
1
selected.
-*1R1,
g:IRk
g (B
=
our estimator is
)
B
be defined
1 Y).
g(B 2)
- B2 1
151
Then
K(N) < 1/2
Proof.
almost surely as
N t
m.
As in the previous proposition all norms
refer to the operator norm.
We use the same notation
as in the proof of the last proposition.
1)
-
g(B
1
)-g(B
2
2
)yX -
X)1
T (B 2 )
111XTE (B 2 -1
X
)
X
XT(B
1)
1
1
1f T(E(B2
-XTX(B ) 1 c
1
XTE(B ) l6 e
x(B
X
.1
_
X)
.
From the proof of the last proposition, we have already
shown that:
2)
Sup
almost surely as
(XT Z(B)
1
B - B f0
1<
2M
N + o.
Here as elsewhere in this proof, let
elements of the unit ball about
B1 , B2
be
B 0.
X.X.
Z (B1 )
-
Z (B 2 ) =
+
2(B 0
-
(B0 - B
T
m.3
(B0 - B1 ) +
X.T C
M.T
B1) T
JJ
-
(B 0
-
B 2 )T
T
___
inM.J(B
0
-
B 2)
23
-
2(B 0
-
B2 ) T
T
.
3 M. J~
23
152
and so:
3)
IZ (B)
< 2 1B 2 - B1 |
Z (B2)
-
T
x.E.
T
x. e.
+ 2 |
4)
2 - B
B|
> 0,
y
Let
Z
j
N
5) S 2 11N
X.,
N
and
NX E(B 2 )
=j
J
j
.
2)
Z (B ) Zj
-1
B
-
E
m, 3(
(B()
j(B
--
N t o.
1
-1
T
B )
1
(B2) Z(B )
and therefore:
B-B
are all eventually
B2 1
E
1
1
that:
almost surely as
1T
c.
X.
2
(B2
it immediately follows from the
m.
5)
- B
N +
t.
m (N),
independent of
2
it follows that
(B1 )-Z
e S2 ,
facts that
]B
c Sl,
2)
almost surely as
For
_
<B
[3
j
then for
)
T +
X
E.:
m
3
Z (.Z
i(B1 )
(
(B
Z
2)
B
153
6)
III
Sup
SIB 1 -B0 1I<_
XTI(B
E -
l
<
JIB 12 -B_oi<
HIB
B1
1 x T( B 2
Y
B 2 11
1
d B2
almost surely as N + c.
Let Y2 > 0, we showed in the proof of the last prop that:
7)
III XTr(B)~ 'i
Sup
IIB-B
almost surely as N + w.
-2
<1-T
o0
m
XT (B2)
j(B
I<
~IX
-
))- Z (B 2)
XTE(B )
+
J Ss
2
X.
I
2
X.X.iT
N
M.
3
m. X. X.
N
3
3
1
1
WW
( 2)zW(B 1
Z (B
(Z. (B1 )
and so by 3) we get:
8)
Sup
IIB
XTE (B)
-
I jB 2
B0 1
< 1
X -
i B2
x (B1
1-
2T + 3
B1
B 0 11 <.1
B
B2
Now by the appropriate choices of y
and Y2 we get our desired
result that:
9)
Sup
IB1-B o
IB 2 -B_Oi
B
# B2
IJg(B 1 )
< 1
< 1
JIB
- g(B 2 ) 1
- B 2 11
l
almost surely
as
NT
-
154
Let
= (XT
g(B)
implies
11 B2
(B)~
-
be defined as before by
X) 1 (XTE(B)~1Y).
|| B - B0 |
- B0 l
B
-R
k
g:IR
< 1
< 1
implies that
Since, if
points we have
B
both
| B1 - B2 11
-
|| g(Bl)
g
< 1
g(B 2)
and
- 1/2
has a unique
and
g(Bl)
1
=
= B
g(B)
B1 - B0!|
and that
then it follows that
B2 |!
fixed point.
B2
are fixed
- g(B
2)
|
< 1/2
and thus || B 1 - B 11 = 0.
2
11 B1 - B2 11
PROPOSITION 4.8.
Let
as in Proposition 4.7.
Let
Suppose
Let
J
1
(2Tre)- N/2
h(B) =
k IR
g:iIR
F
Y.
j=1e
-
Rk |g(B) = B).
T
- m. -1/2
X.B) T(Y. - X.B)
3
3
3
mM.
=
{B
be defined
£
Then:
1)
F
2)
There exists a unique
is a singleton almost surely as
F = {B*}
where
that maximizes
Proof.
1)
that maximizes
h
N + o.
almost surely as
3)
B
N + o.
B*
h
is
the unique element of
almost surely as
N + o.
Proposition 4.2. gives us that:
supj| B - B0!! < 1
almost surely as
Bsf
Proposition 4.7.
yields that:
N +
Rk
155
Sup
BI-B0 H< I
2)
B 2 -B 0
g(Bl)
B1
-
g
-
B2
1/2
2
1
almost surely as
N + o.
Proposition 4.1. shows that
F
cannot be empty,
thus we have:
3)
F
is a singleton almost surely as
N + *.
Proposition 4.1. gives us that there exists
B E IRk
B
is
that maximize
h
a fixed point of
g.
and furthermore any such
The rest follows immedi-
ately,
COROLLARY 4.9. Let
the property that for each
a1
2 (N),.. . ,a
G(X,Y)
h(B)
N
there exist
2 (N) positive numbers such that
is a limit point of
2 (N),XY))}
{B n(a1,12(N),...,
1)
be an estimator with
O(X,Y)
O(X,Y)
then:
maximizes the function
(2re)-N/
n=1
almost surely as
[
m
(Y. -X.IB)
N t c
(Y- -X . B)
m
-1/2
156
almost surely as
2)
E(X,Y)
2
0,
(N)
almost surely as
limit points of
g.
2
,
the
are fixed
{Bn(
2
2
{n 01,1 '''' 0 1,J ,X,Y)}
Part 1 of the corollary now follows.
is a fixed point
B1
N +--
2
a1 ,1 ,-.aig
For any choice of
Proof.
If
2
+
is independent of the choices for
2
ca1,1 (N) ,...,j
points of
N
of
g,
and if
Z(B ),then we have
if
2
2
'' 1,J ,X,Y) = B 1 for all n. Therefore
,1 ''
G = {B*}, where F is the set of fixed points
of
g,
a
{B
then it follows that for any choice of
2
,1,. ..
1,1
2
0
,
, J2
the sequence
1
1
Therefore whenever
F
B*.
must converge to
,X,Y)
is a singleton,
pendent-of the choice o T
2
,
,J
0
2
is
inde-
157
LEMMA 4.10.
jm,.(N)
Let aN =X
j=
N j=l
NN
Then:
Cj 2
aN: = a exists and max aj 2 > aN>min
-a
A)
lim
N+ -o
B)
trace a N -1ZN = N all N
C)
1 T
lim
K X. (aN
N + co
1)
Part A follows immediately from assumption 6 and from the
2)
2
exists and is positive definite.
N)~X
fact that
aN
is a convex combination of the
Trace a
N
N
N
N
aN.
trace ZN
N.= aN
1 --N
J
aN
1
3)
+). co
-lX
XT
lim
N
N-+o N
4) lim
N +
aN
1T
N
2
2
= N
N
N IX
N
aN exists and by Assumption VII
1 X (a
IZ )~'X exists.
N
N
(X (aN
-~
1
X_
X)N
aN
Thus
I
(N)
N
J
j=1
,
m (N)
a~~T
max a.2
[I[
(X (aN
1
J2
min a. 2
N)X)
j
and hence:
N aa
a.
exists; thus
1 (XT
.
j=1 m (N)a
31
N
j=l
T
N
5)
mj(N)
L.a2=--2
= a
(X N(a(rN)'X
NN
By part A) lim
alN
1 XT (aN
lim
N
N)
max a.
j
min a.
j
j
X is invertible~with the norm
2
+-
of the inverse boun ded by
1
j
*1
158
LEMMA 4.11.
T
E {}
2
< KTa.
Proof T
2
/Ek
1
i=1 X
2Am1 jik ji
k=1
S0
Since E (x(Xik
.-.-. i1Xask
i
/
as)
-1.
V-
s=1
7-
2
x
E
jsk Eis
.
2
i
=
ik
we have:
2
X.T
k
21 m m. j
2
m
i. 1 Xjik
k=1j
=
1)Ej
(where
x.
jik
is the ik element of the matrix .X.).
X TX
<
T,
and
since
is positive semi
m
~m.
definite symmetric, each element on the diagonal must have values
X.T X.
J
By assumption
less than or equal to
2)
I
m
j=l 1 m.
T.
i2
uk
<
-
Therefore:
T
2
a 1.
and since xjik are non
Since E(z j2
stochastic, we have from 1):
T
E
2
Xj
=kE
k=1
a
j
2
i
-L
i=1 m
x.2
< K a 2 T
jik -
The next lemma is an immediate consequence of
Chebychev's inequality.
j
159
LEMMA 4,12.
EYn2 < M < m,
Xn
+
0
in probability and suppose
XNYn
+
0
is probability.
Let
then
6 > 0,
C > 0,
For any
Proof.
we have:
IYn
and
Prob{IXn n < 6 ) > Prob {IXn! <
1)
c}.
I
By Chebychev's inequality we have:
EBY 2
Prob { 1 n
2)
y > 0
Let
ently large that
can find
Hence for' n > No,
-2
C} >
Prob {Yn >
such that
N
2
>
C2C
by 2) we can choose
6 > 0;
and let
I--
< C} >
Prob {IXn
<
< y/2 + y/2 = y and so Prob{IXn
<
By hypothesis we
c+} >
Prob {IXnI
we have
suffici-
C
-
and
-
for
or
n
<
n>No
'YnI
C}
>
>
C}
-Y
[X] Lemma 4.
PROPOSITION 4.13.
g(o(X,Y))
property that
g(B)
=
=
e(X,Y)
Y),
-1-1Tr(B)
)
(XT(
be an estimator with the
e(X,Y)
Let
where
g(B) =
(XT(B)
X)
then under the hypothesis a
above:
A: e is asymptotically equivalent to the weighted least
squares estimator with known variances in the sense that if W
p lim /N(O-W) = 0.
N+ o
is asymptotically normally distributed with mean vec-
is the latter,
B:
tor
e
B0
(XT -1
C:
and variance covariance matrix
-1
(1XTE - 1
11
-1 _
X
lT
.
1
-1
-+ 0
-1l
almost surely as
N + o.
160
Proof.
=
(XT (aN
X
is a
Let
VN
N) -1N
N x K
-1
aN
=
-1
T
W
then
N,
N
(X
=
1 X)
(XT V
Y)
N
X)
T
wYre
-lXT yN- 1Y
where
matrix.
m.(N)
Let
e(XY)
cN
(X
=
zj(e) and let VN
N
= j=1
VN
X
A
and
B
-+
V)X
N
XT ( VN_
plim
ii)
Principles of Econometrics,
it suffices to show
N1
plim NT XT vN
1)
0 and
=
VN- 1 ) e: = 0
O4N
N)X 1
1 T ^
NX (VN
T -
N X
N_
)-
NZ
NZ)X
-
Therefore:
N
1)
then
*
T" N-
By theorem 8.4 of Thiel's
to prove
(6),
=N
XTN-1
VN)X
m (N) X
=l
=
N
T
X.
a
a
6(
)
Z.
m.
ai3
m.(N)
= Z
Recall that
m. (N)
Z (0)
N
N
(e)
j
and that for any J,
*
m. (N)
aj2 + 0
- N
J
almost surely
N
+
oo
Hence:
2)
0 almost surely as N + co
aN
T ^
1
and we have N X (VN
V N~)X = 0
(1
(I)
) XT(
N 1
-
N~)e =
N
IT
almost surely as
T
X (
- TaNZ
)& and thus
we have:
3) (X
(VN 1
V-N
Im.N
E. .
XT
iN
(
N
Z (6)
aN
161
x TE
By lemma 2
E I
2
< KTa. 2 independent of N.
'---|j
/m.(N)
j
For
e S ,
1_
aN
aN
31
j
2
0 almost surely and since
+
a_()__
m.N
a
< 1,
0 <
NN
it follows that
a
2
t.
almost surely for N +
If
j
M < w. such that
C S 2 , then since 9
aN
aN
Z()
3 e
m.(N)
N
N
and since
M. (N)
aN
Z(-z )
N.
N.
0 as N + c,
+
aN1
-
almost surely as N +
.<M
j.I
)
0 almost
.
we have
surely as N + m.
We are now in position to use lemma 3. to conclude that,
T
j =1
m
m(N
-
Z.6
N
(N)
converger to zero in probability and thus:
3) plim
1 xT(VN
N + c VN
TExT
-
XT E ~ X -
-
VN
Nm.(N)
Xx
N N
XT ( e - x =
so part C follows immediately.
)s = 0.
NN
X.TX.
N
Z 6)1
3~
162
5.
Linear. Variance Model
(Strong Law of large numbers)
PROPOSITION 5.1.
Let
$
IR
such that as
Let
{x n
be a positive even and continuous function on
increases
[xl
and
tt(x)
x
+
be a sequence of independent random vari-
ables with
0 < a n t 0.
E(x n) = 0. for each
E($(X ))
n
Proof.
x
n
.EJ=1X
1
n
almost every and
and let
< o then
(a
a n)
if -n
n
+ 0
n
converges
na
a.e.
Chung [7 ].
Let
PROPOSITION 5.2.
{Xn }n>1
be a sequence of
independent random variables and suppose for some
1 < p < 2
for all
m, < 0
there exists
n.
1.1,
lim sup
Then
with
EIXnIP
X |I < m+2
almost
everywhere.
Proof.
1)
Yn = Xn - E(Xn'
Let
- 1+
IE(Xn)I < EIX_ i
2)
(E JYnIP)
EJY nP =fIXi/p
= (fIX
m
E(Xn)
- E(X n)
+ (fIE(Xnp)
1
By Minkowski
p)
lip
< Cf IXn
1
/p
/2
(E[Yn Ip1/ 2 < m
= 2m + .1
+
P,
163
3)
let
In Proposition
an = n.
5.1.
let
= |x|P
$(x)
and
We then conclude that since
EYn
np
n
n
2m+l
P
n np
Y. +0.
n
in
4)
lim sup
5)
lim sup
6)
lim sup in
n -1 <
< 2m+1-
n np
a.e.
Yj
n
n=1
n
n 1 X.
n=1
3
<
-
1n
and
a.e.
j=l
E(X.)j
3
<
I ae.
in.
n+OO
PROPOSITION 5.3.
=1 X.i < 1 + m + 1 = m + 2
Let
{Xn n>l
a.e.
be a sequence
of random variables taking values in the same fixed
E
space
{Xnln>l
Let
(where
R
or
]Rjxk ).
Let
have a limiting asymptotic distribution
{Y n>1
then
{Yn n 1
tion
D.
Thiel
D.
be another sequence of random variables
taking values in
Note:
E = RI ,
E
and if
p lim
||Xn
-
n1|
= 0,
has the limiting asymptotic distribu-
[z19]
Convergence almost everywhere implies conver-
gence in probability.
164
PROPOSITION 5.4.
Let H denote an arbitrary Hilbert
Space and P(H) c L(H) be the set of bound-d positive linear
operators.
maps
P
Then the positive square root function
onto
(A'* A2 )
P
that
is uniformly continuous.
[AcL(H) is positive if A=A* and (Ax,x)>0 for all x in H.]
(A=A* is implied by (Ax,x) real valued for all x in H).
Proof.
1)
(Sketch)
The positive square root function is a monotone function when restricted to
2)
P(H).
(0,co)
The square root function defined on
is uni-
formly continuous.
3)
Therefore if
{An I
and
{B'}
n
are sequences of posi-
tive operator such that || An-Bn|+
then there is a N0 < o
n > N 0 An < Bn + cI
4)
/X~
v/B
n
and if
e > 0,
such that for
and
B
< A
+ n
n + EnI = /n~
n
B (6)
n + nA (6)
n
0
/~
+ EI.
n < /A n +
where lim"|B (E)||
E:-* 0
n
=
0 independent of Bn*
We are now in a position to list our assumptions and to prove
our results.
For the convenience of the reader, we have
followed much of the notation of White [51).
We have also
borrowed liberally from him on the wording of our assumption.
Al)
The model is
(6)
Yi = X i0
(7)
E(c.)
(8)
E(e. 2
=
known to be
+ L.
0
= a. 2 = Z
r0
i
=
1,2,...,n
i
=
1,2,...,n
165
Where X. is a lxk vector of random variables, s. and Y- are
%0 is
real valued random variables,
bers.
Y
a kxl vector of real num-
and X. are observable, Es is unobservable and 60 is
to be estimated or hypothesis concerning
0 are to be tested.
Zi is a lxm vector of real valued random variables which may
contain some or all of the variables in the vector X-.
10 is a
mxl unknown vector of real numbers which is to be estimated or
hypothesis concerning F0 are to be tested.
Let W
be the vector of length p of random variables
whose first entry W
is the scalar 1 and whose other entries
are exactly those random variables that appear in X. or Z. . We
1
assume that E(W *Wir£i) = 0 1 < jk < p and E(W T(
We let y
s 2_ a 2
denote
The vectors (W.,E)
1
2
2 )=0.
are assumed
to be a sequence of independent though not necessarily identically distributed random vectors.
A2)
I) There exists 0 < 6 <
a)
qnd A
a)EJSW2
W . 1 +6 ) '<
E(I 2 Wir w
Wis WitWiv
1
<
suc
i-
h
tA
at
for all i:
r, ,t,v < p
b)
E(ISEWikWirWisWitWiv 1+6)
1 < k,r,s,t,v
c)
E(JW
1 < j,k,r,s,t,v ( p
d)
E( yj2!')
e)
E(
f)
Ejc
II)
Let
WikWir W isW itWiv1+)
E?
f
<
WWit1)
< a
1 ( r,s,t
W.irW.
is
) <A
Ra = n 1
n
n E(XX.)
i
i=l
Mb
n
=
< p
n 1
Z.
i=1
1
E(ZTZ.)
i
i
and let
a
< p
r,s < p
166
We assume that there exists NO < o and 0 < X such that
for n > N0 minimum eigenvalue of Fa > X and minimum eigenvalue
> X > 0 (Note by the first part of A2) this is equiva-
of R
lent to the property that for n sufficiently large det Ra and
n
det M is bounded away from zero. Also, observe that we can
6
choose
A3)
and
X
so that they are equal.
Let Vn = n -- E
E(e 2X
let Vb = n 1
ni=1
2 ZT
1
X.)
and
1
We assume that there exists N0 < c and X > 0 such that for
minimum eigenvalues of V>
n > N
X > 0 and minimum eigenvalue
0-n
> 0.
of Vnb >
that NO,%
(There is no loss in generality in assuming
is A2 and N0 ,X
in A3 are the same.
In
the presence
of A2, A3 is the equivalent of the assumption that for n sufficiently large minimum (det
a , det V ) is bounded away from
n
n
and above zero.
The first theorem is a restatement of a result found in
White [
] and its proof can be found therein.
Before stating
Theorem 1 we introduce additional notation.
(X TX)
Let s
XTY
if
CXTX) is nonsingular
n =
0
if (X X) is singular
(ZT Z)
ZT2 if (ZTZ) is nonsingular
Let an
0
E
is the nxl column vector whose i
Let Ein
Let
if (Z Z) is singular
=
i
- X .n
-
Z ia
entry is
( i) 2
167
Let ga
1
n =
n ^2
T
i=l in
i
Let Vb = nl z.n
n
2
i
Z.
i=1lin i i
Let Ra be a qxk matrix of real numbers with full row
.a
rank and let r
be a qxl vector of real numbers.
Let R be a qxm matrix of real numbers of full row rank
and let
r
be a
ax1
THEOREM 5.5
vector of real numbers.
(White)
Under assumptions Al, A2, and A3, we have the following:
i)
Sn
o
ii)
an
r0
a.e.
T
1
iii)
r
iv)
V)
[ (n
almost everywhere
a
T
An~$o)
Yb
(TZ)l
Z
-1
4
under n the hypothesis
n
n H o:
n
n(Ra nr a T [R a
vi)
TA
)Xn-l
; [(n)
-1
under the hypothesis H :
n(Ra
T
-r)
(a.e.)
an-a )
rQ
N(0,Ik)
A
N(OIM)
Rao =0 rna
X
RI'
q
~
)R aT 1-l(Raan-r a
A X2
= r
a
1 ^b Z Z -lRT
(Rn ()
n ( n
I
(Rn-R
-r)
A
2
X
x
168
Before stating this theorem it
is
once again necessary to
introduce additional notation.
S2
th
be the nxl column vector whose i
entry is
n
Let En
Letn
2
=(ZTZ~ ZT
0
Let
N.in
Let
9cn
'
2
=Ci
in -
E.
2
in
.
if ZTZ is nonsingular
otherwise
Z3.n
and
n~1 Z n
2Z. T Z.
i=1in i
i
THEO.RFM 5.6.
Under assumption Al, A2, and A3, the following hold:
i)
rn
r
a.s.
T
n
ii)
iii)
vri
[(
n
)-l vc(Z
n n
-1J- 2
n~Io) At N(OIm)
M
F-o
under the hypothesis H0 :
R' 0 =r (where R,r are as in
theorem 1
ZZ-c
(Rrn r) T [R(-)n.nRn
n
Vn
n
2
ZZ-l
-TR T-l
1- (Rr nrr)
n
n-r ^V Xq
(Note: A part of the statement of this theorem is
whose inverses must be taken 4
almost everywhere).
Proof.
1)
as n + oe-
that matrices
nonsingular
See Chapter IV.
The idea of the proof is quite simple.
We show that
the difference is norm between the statistics stated in Theorem
2 and the associated (noncomputable) statistics in Theorem 1
converges to zero as n + w almost everywhere.
Hence by
169
proposition 3 and theorem 1, the desired results hold. Unfortunately, the only proof I know involves a large amount of
computation.
0l
2) 11 F r
| rn n||n,
it suffices to show
3)
+
TA^2
0 a.e.
ZT(eFn2_E2
-1
Zy
^^
r n-a n
1 xn-rO 11 and so by theorem I,
+
rn
<
n
2
n
11 (ZTZ)1
11
T
n-an
ZT ^ 2_ 2
Z
n
E )
(n
n
By SLLN (Prop. 1)11 Z
By A2 for n > N0.'
- nl
E(ZTZ.)I|
0 a.e.
n
i
-1l1
[n1 E.=1 E(Z Z )]
-_ *
It follows from standard Banach Algebra techniques that if
l TZ)|
E(Z
)-n2
ZTZ -1
(1n )
2
<T.
ZT ^ 2_ ZT 2
n
n1 ZT
n
1
n2
Z TX0
Zn
n-1lZn
n Xi ~
n1-1IEn
il
Therefore, to show i) it is enough to show
0 a.e.
-+
2)
is invertible and
<
=
+
E-
Xi2]
ZT
A
2-
)2
[(Y
Z
n
n 2
+
A~
[Xi"(%-n)Xi(%-n) + 2s
0
Z
[X (6
n0
0
n)
+
X.02
Tm
m
i r=
0ir nrX is0
n s
)
c
1 ir
0
n r
Z (E:-E2 ) is a mxl vector and m is a fixed finite number; thus to prove convergence to zero, it suffices to prove
component wise convergence to zero. Furthermore, since the
sums indexed by r and s are of a fixed finite length, we are
Now n
reduced to showing.
4)
n-
(E 1ZirX
Xis)
0
nr(
n
+2
Zir(iir
0
n r
170
(1 < k 1 r,s < m).
converge to zero a.e.
This is an immediate consequence of A2, proposition
5)
ylim fl
If
li
I XY
CnDnII
AnB n
=
0, it suffices to show:
-
Cn
a) plim IAn
=
DnII=O
b) plim IBn
c)
JI, then to prove
X!Y
>
for n
such that
wand N(s)
< m
N (e)
< M(C)}
prob {IIAn
<
M(E)
there exists
for any s > 0,
>
1-c
and
prob {JJ
Bn 1 < M(e)} > 1- s
This is true since
AnBn
< IIAnD l IIBn - DnI1 +
An-Cn11
< 11An
I1 Bn
6)
- DnI
=
above, to show plim V'F (n
AnDn + AnDn - CnDn
Bn1 + JB n1
-
n 1
Z
,.n,Z-l
nn
AnBn
n 1
+ IAn - Cn 1(I1 Dn
( na n)
/n
CnDni
-
an
ZT (
n 2_2
as w e
saw in 3)
0 we are reduced to showing for
1 < k,r,s < m
a)plim
o- Rn r)
Zir XirXis )
No~
n) s
n
b)
plim (n 1
1
(So
VM
(Va
n
o
Zikir
b)
M~a)
n
n
o)
(M-
on
ns
nr
i= ioki
n r
) )
Va
0o=
1
o N(0,I).
-1
.<a-
u
Assumption A2,I.a. ensures us that Cnis uniformly bounded.
0
171
for any
c < 0,
n > N(E)
implies probability { iIvl(6 n
)
n
0--
there is
a M(c)
< o and an N(c)
-
Since (n-60)-+O
E
a.e., n
< M(e) } > 1-E.
-
0 a.e. and since
-ZikXir+
there exists M < o such that
(n
fl
T
-1c
Vcn _
8)
n
n[
n
n
n
b 1 (Z_
-1
b-n~
zn
=1
n
0.
=
TT
TC
n
a.e.
n n
n
n
n
n
Ij< M
n.E il ZikXi X.
6a and 6b hold, and plim V/i (Fn-a)
co),
< o such that
in
2
in
2 )ZTZ.
3.
It is necessary to show that plim 1|^ n _
n
n
|
=0
Unfortunately this requires the following long computation.
9-AC
is
a mxm matrix and m is
nn
show for 1 < j,Z < m plim n
Z zi
Z
Since
2
win
-
v
2Win
~in2) ='0
2
^
2 ^
^
I - Z an + Z (an ~rn
^
Zin
a fixed finite number
+
in
2 + [ei
2
n
{[Z i (an
Z
n - z Zi-1zjziz
n rn )
i.n
22
I)
C in
2
i
2
A
Vin
9)
2
in'
Ci
n
)
^
n
Z
z
33,
in
2
in
^
(c.--Z. a1ni.)Z
2)
2
^
(a - d )+2 (E ii -_Zian)(E. n.
1-E.)+2Z.
a -i(
ia
i
n n.
2
2
. - i:. }
in
It is now convenient to proceed term by term
10)
n
-l
m
n
Z Z1
m
r=l s=l(n
^
Z
^
2
[z(an)J 2n=
m
_1
zi
ijZiZ'ZirZi-))(ann
(s -nnr
172
Since
-
a
-inn
0 a.e.
-n||l
E Z Z [Z
(an
(n
rn)] 2
-
0.
+
it
o),
-+
follows that
(n
a.e.
+
o)
(11)
i.
1
=i
in
i2= S
Xin=
ian
-
2+
2
2s.
2
+
n
X.6+s.
i
i+
X
-
-
X.Sn
ian ==*iX.(G-Un)+s.
na- d+
+
N-
2
n
2 + 4
=
-
[X.e
n
]4
kr=i=14(ZZ
k
k
1
nr+
XXins
n
n
i
ir
s
n
S Z ijZitXirXisXitXi)
k
*k
k
k
r=1 s= 1 t=1 v=l(n
nt
a-an
+)-
0
a.e.
(n
assumption A2, n 1
+
o),
Z
n
ns
n r
so again by
22iE
[
Z
i=1 ij ii[Ein
0
a.e.
(12)
n2
n
i Z .Zi 2(c.-
n
m
22(n1 Z. i1
r=1
2Z
m
m
r=1 s=l(n-
^
Zcan)ZC(an
2
Z Zi
i ijiZir
2Z
-
n
-
rn
n(cn)r
n r+ +
an-
n
zi=iZijZiZirzis)ans
(an~
n r
173
Now a
so
= F Os + (a nl
n-POs)
a an o< I r0 + 1 a.e. (n
Thus,
n
nls
-1
n
z
Z. Z
i=1 ij
o).
2E
iz
-
1
Z1 a f
ii
nn
n)
+a
0
a.e.
(13)
n2
n =1
n
-
I j Z Z 2(
n
i1 Z..Z.
k
n
r=1
(4n
k
sE
- Z
ifn
2
I
n
2
) (2Ei X (-
Z Z Xir
3
)($-n )r
n) +
[X.(-n
2
+
2
e Z
I
ZXirXis
n r +
n s
n
Z. s: Z.Z. Z. X. )a~ ($-p
+
1= i ij
iZ ir is
nr
n s
4(
r=1 s=1
n
2
2(C
k
k
r=1 s=1 2(n
m
2
-
-
(
m
k
k
r=1 s=l t=1
n
=2(
Z Z Zir is it
iz
So we conclude nE
z
-Z
2(c
n)
nr
i
n t
n s
C ) +0
a.e.
(14)
-1
n
Z
.Z
2Z
n
I
2
i
2
(an
rn
in
-1 nA
n~
i1 Z Z.2Z.(an -r) (2e.X.-$n)
m
k
r=1 s=1 4(n
m
k
)
nn
Z
e
k
Zir is
n
n r
+
n s
n
r=1 s=1 t=1 2(ns nZ
+ [X (-$n)
so n
and thus
1
2(
Z
Z
Z
~
ZirXisXit) (an~
2
-T (.-EQ
0
(s
r )($
nr
(n
a.e. (n
n
n)
-
174
15)
16)
0 |
-
D
-
+7 0
|IV
=
0cn
hence
a.e.
(n
n-
EZ
-
nI
+
0
-*-
c).
Z )-|
E1(
a.e.
(n
-+
1
For n sufficiently large, (ViY)nn
<
($ )
Therefore
a.e.
implies the existence of
T
17)
z
nB
iT
nZ
-
|
-
)
(Z
a.e.
(n
+)
) .
exists and
-
| (V'II<1
)
which of course
w) ,
(V )
a. e.
(n
CO) .
-
By assumption A2.c) there exists M < w such that 11Mb
all n, hence I
18)
(u
:S
By Proposition 1,
i
0
(n
0
)j
a.e. as n
< M+1
j <M
-.
Furthermore, by assumption, for n sufficiently large,
(
) -1
a4
.
T
T
YZb
(Z ZZ)
19)
We conclude then that
2 2
1
n
<
(M+l)
a.e.
(n
-+-
cc)
and that
20)
21)
)
[(
nn
By 20)
[(
n )
Z b -l
Z 1 -c(
vn n
n )-
1
n
oZ~l--L,
a. e.
and Proposition 4).
nb n )
-1 [(
n
-v c
n
n ZP)
-1--]||
0
a.e.
175
22)
It
follows from White
<
n
Mn (a -r
there exists M(E) <.m,
such that for n > N(E)
prob {'a
24)
--b ~
that
[ J,
N(0,I ) and hence for any c > 0,
N(s)
-b-
llan~r o
M(E)}
>l-E
Recalling 5) we see that
pli
1
b Z
[
-(r
[rn
)-
nn
-~
(Z
-0-
n
n
where we need only note|| v/ (no
r. o)
n
r
nn
Thus part ii follows.
R(
n
)-
T
T
T
R 2 1
(Zc _bl (ZZ) -ylTn
n
n
b
n
c
n
-l2
n
hence
R Z
c
Z-lRT -
n
n
n
)
n
RT
b Z
n
n
(n + o)
28)
Let P be a positive mxm matrix and jP-
II
< S
Now R is a qxm matrix of full row rank; hence there is
a
6 > 0
inf 11RTX I > S.
Xe
X11
||=1
<RPRTX,X>
=
<PRTX'RTX>
=
hence RPRT is invertible and
R TX 112
11(RPRT-l
_
6
P
6
176
S
29
-
zT
V n(-n
-)
Z
n
-l
Vc Z
n n
T
where
i
is a positive operator, such that
(
- -l
<
)
M+1
(M+l) 2 2 a. e.
a.e.
(n -+
).
(n -+ co)
Hence we conclude
[R(
n
) 1
Z
c
n
- lRT
l
-
n
[R(
n ) Yb
n ) -lRT
n
-lD+0
a.e.
30)
|v| (Rn-r)
--
(Rrn-r)
< l R
plim
IVrn(R
n-r) -
V
Under H0 ;V(Ra n-r)
(
vln-(an
r) T-
(RPn
n)O
(R n-r)
n
plim 11v(R nr)T
-n-
31)
II = i
Rr n-r)TI|
=
0
VW(R& n-R 0) = Rv(Q -^ro);
thus under H0 , for c > 0, there is a M(e) <
N(c)
< o such that n > N(c)
prob {fn(Ra n
and
implies
*
Observing that our norm structures are such that
X II| < || XII | Yl||
11ABC-DEFH|
1A
-DII
and hence that
<| Al] 11B|| 1| CF | + | Al
FH IIB-El
Ell 1 F| , we have just shown that:
+
177
32)
Under
H0 ,
|
T
(R n-r)
V
r- (Rrn-)(RT
and so part iii
[R (
T
)
z z
Z
(
R
-l
T
-(RTn
T
^CT
-1 T -1
n- r)T=
follows.
[X]
Thus we have shown that we have a valid asymptotic test
for testing linear restrictions
ance model.
additional
Our next and last
assumptions,
on the parameters
result
is
in
the vari-
to show that under
we can use our estimates
from the vari-
ance model to reestimate the original model and obtain an
estimator that
is
asymptotically equivalent to weighted least
squares with variances
There exists
A4)
known.
X > 0 such that
for all
i
a3 > X.
A5) For all i, E( 1Wis, ...W p) = 0,E( i2 wil, ... Wip) = Z
A6) There exists M < c, such that for all i
Z
J<
M.
THEOREM 5.7.
Let
2
nn denote the nxn diagonal matrix whose
is a. and letI
Z rn.
n denote the n n matrix whose
(i,i) entry
(i,i) entry is
Let Bn denote the Aitken estimator given by
(XTn -lXIQ
nn
~1 Y
where XTh n -lX is nonsingular
Bn
0
if XTnn~1X is singular.
Let Bn be the weighted least squares estimator given by
178
-lY)
1
-l (T-g
(T-1
Sn
B
n
if
Xg
- lX is
nonsingular
=
iX X
if
0
singular. -
QT
n1 X is
Under assumption Al-A6.
i)
-S 0
a.e.
0
+
plim v(B n-B ) = 0 and
ii)
A
(Bn~o) A N(0,Ik0
lXT^nl
n
If R is a qxk matrix of real numbers with full row
iii)
a qxl vector of real numbers, then under
Xn ^ -l -lT-l
A 2
) R ] (RBn r %Xqn
n(RB -r) [R(
lq
nn
rank and r is
H0 R 00n = r,
Proof:
1)
Consider the transformed model
= X
+ E
a.
So
a)
E(
i=l,...n and a.
dP
=
J1)
a2
=
G2
1
x.
.sC.
1]
z.r
1 dP
1 0
X. .
E (E
w
.Z
dP = 0
ip
jillii-
b)
c)
b)
E.
E(-)
E(
=
E(.
S. .
o
1-2
- )
10
aZ
i
zi
1 < j
1
iro
E(Ew
..
=
f1
< k
o
wip)dP = 0
Z.r
i
o
Z
r
dP
179
2)
The OLS estimator on the transformed model is
B
Since assumptions A4 and A6 insure that
0 < X < a 12 < M|il
0<
c> for all
i, so all the assumption of
White's theorem are satisfied and we conclude
a)
B n~o+
b)
rn(n~ iXTQ
c)
under the hypothesis H :
0
0
n(RBn-r)
3)
T
a.e.
1 X)k(Bn~o
A N(0,Ik
T -1X)1T-R
[R(XT n
T%
T -1
the case X £n~ X and X 2n
In
Bn-Bn =
(XT
_
(XI
-lX)
-1-T
n
-X
4)n~X
1
n -n
X
n
A
2
X are both nonsingular,
-l (X+E)
T n X)
-lXT
T1(n
1 -1 XT
nn)(
X
Q X -l
1
1=T
XT
n r
XT Q n
Tl
l -1l
nXQE
whe
wherei~
a 2 in
=
rn
zir
n in
nin
ni 1
5)XT n-1i
n
n
iti
T
tin
o 2i2
so
ar am
2
ft
T S1-ixis akxk matrix where kis
n
finite number.
To show lixT0
§lx
a
- XTQn- 11-) 0
fixed
a.e.,
it suff ices to show convergence for each matrix entry.
180
6)
nn1
X. X
(
1 )
I ] ik
in
CF . Cin
n
max
ai
l<i<n
7)
1+1
a.
-
1
a.
in
Zi (ro -n)
Z
i
>
I
in
=
1
Zirn
I > Dziroli
a.e.
ZinI > I
o)
i
and
and hence
(n
)
-
and
0
Mpfn-Ioi
supi 1
l i~n a.
a.
so
i orI - lZi(ro-fn)
HZ in
X - Miro-fnH
Z (Tn~
Thus
1
T
io )
A2(Z
I
IXinXi
in
Z i'r'ni +
Z in
ri=2.
n
A2
1
a.
e.
a.
e.
(n
+ o ).
(n
+co).
a.
in
Therefore by proposition 2 and assumption A2', for 1<j,
in
1
Z Xn
(1- --)+I
i
n1
T(X
n
X)
-
-
0
a.e.
(n
0
a.e.
-+
w)
and hence
in
n
(X
n~-X)
(n
Z<m,
+
c)
181
(8)T
Q)ZZ>
Let Z E 3R; <n 1 (XTn
n
= n~-<4n~
XFZ
<2n XZ,XZ>
2 > n'
> n 1XZ
m ||T
<n
1
<XZIXZ>
I
m Tlo
XTxzz>
1
<n lxTxzz>
M Broil
IZ 112
1
(XTx)1H
jj[n-
M 10 H
(providing these exist).
For n sufficiently large-I] [E(n~ XTX)] 111
<n ~1(X T Sn~ X)Z,Z> >
a. e.
1 Z 12
M 1
ff1r
n~~
(n +) co ) .
and thus
[ n~
(XT0n
1
In[n~(XT
and
l
X)]
M2
2
X)]
E - XT
n
(n
a.e.
[ n~l(X
-
)
-
1|1
nX)
0
a.e.
(n
+
X
X.
s
c)
(9)
n
[(XT
n
n -n
-
E)]
XT(
ii
2
-
1
n~
[ XTT
XT1 -
E]
22
in
is a kxl vector so again we
need only show coordinate convergence to zero.
In_
.C)I<(
-
a.
I
.
ai
in
MAX
in
n 1
n-
I
a.
I
a.n
in
.1
182
The-refore~ we conclude
.
X1
1nE =nX
21 ZT
n 1T
1-1
|
- n
P 1
0 a.e.(n- o).
n
ij
therefore
e i,
X
there exists C < a such that
1|n~XTQn
10)
Hence
$n-Bnn
A
|I
and
|
{Recall
11)
0
+I
0
|UJv-xy
T[n~
XT
plim plmvr-
a.e.
< C
eC
-1
(n +) w) .
a.e.
(n
+
c)
a.e.
(n
+
0)
iIV-YH + llu-X l(HV
U
) e]
-
(part i)
[n~X
Tn~
( n
n-XMn-
|
+ I|y-V I|)).
is a kxl vector, to show
n)s)
=
n
0, it suffices to show
that this holds by coordinate.
v n~ [n ~
1
CFi
X
X...a.
En
n
1
ain
1
2 2 v~-Z 1 [ nnor
iin
m
_-
r=1(n
n X. e Zr
Ei= 1
2^ 2
Siin
n-o r
12) We have already seen in the proof of theorem 2, that for
any E < 0,
n > N(c)
there exists M(s)
implies
prob.
< c and N(c)
{v/ ||l rn-
O
< o such that
-ol
M(e-)}
> 1-6.
183
We now must show that
i=
13)
Zir 1+6
E
) <
1 1
41
i
A2.
j
0 a.e. 1 4
2-i2
1 < r 4 m
k
a. a.
i.X
E
r)
n X..e.Z.
n1j i ir
(where 6, A are as in assumption
Therefore by proposition 1,
-1 n
c.X.i .E
in
(n
n
XZ
E:.X. .jZ.i
- n
j=JE(
4' 1
+
0
a.e.
and hence
+u)
n
2
-2 0
ir
a.
e.
(n
+) co).
ai ai
n e X- .Zir
2
n -1 Zi=1
14)
Xn EX.Z
L113
n
<
1r
2-2
i=1
+
i. in
i
n113 nZir
2
11
+
lz
c
n
X.Zir
MAX
+
1
a.
So
-1
vrn[n~ 13
nX
X)
I1v (n~ X1 n
VlV (Bn
-)I
-
-
n
(Bn
(n~ X
a.e.
(n
~10
- n
T
)
-
Qn~X)k
-
-+
a.
=)
)] ^
n
iXT
n
X)2(Bn
(n~ XT
n
X)
(n
v
sX. .Zi
in2
2
Cin
0
-+.
2- 2
aiain
n
1
l~~
X Z ir
n
n Ei=1
and plim
15)
aiin
I
~<o
+
184
(I
V(Bn-B n)
-
(n~1XTn -lX)
|
+
1
~^XT^
(nlXT
|| v/-n- (R%n-r ) - v n(RB n-r ) || <
so plim
I vj
(R n-r)
follows.
En (B^n-B n)
and
(RBn-r) 11= 0
V
-
R
'
-1X)'1)
therefore plim | V(n~1 X T
X) k(B
0
nl
n-+
v'ii(n~ XT n 1
= 0 and part ii
Bn-60)
16 )
X)
n +)-
plim ||/ (R n-r) T
6(RBn)
-
n +*c
Under the hypothesis Ho:
exist M(s-)
prob
17)
< o N(-)
{|fi
11 (nXT
n
X)
f|
(RB n-r)
-
1
-
for an e > 0 there
R$ 0=r,
-
(n~
(
implies
M(E)}
R(n~lXT n
-
XTn n 1 X)
1
0
=
< o such that n > N(6)
R(nXT n-lX) -lRT
R112 11(n
T
T1
X)~lRT
-
(n-X Qn-X
XTQ n~X)H
+ 0
a.e.
(n
-+
co)
(from 8) , Therefore
(n~XT
-[R
n1
-1 R ]
-[R
(n-1
T-£ -l
[Rn
x
-l Rnl
T
0
(n
18)
R T is a mxq matrix of full column rank,
+
a.e.
)
3
therefore there
185
exist 6 > 0 such that IR TZ j> 61ZI for all
<R(n1 X TnnX)RT Z,Z> = <(n
>n- iT
The
n-1
nn
62
XT n
X
RTZ,RTz>
n
X
(s,t) component of n
X)
ZE3Rm
i =
ln
t
il
n
so there exists C < w such that
In 1 XTn~1 X
C C
a.e.
(n
-+
oo).
<
a.e.
(n
X Qn
X)~nRT]
+-
1RT -l
Therefore I [R(n~1XT n~X)
+)
and
X)~RTV-
[R (n~ XTn
-
[R(n
1
0
a.e.
(n
no
(n
+~
cc).
We can therefore conclude that under H :
0
19)
plim
|ln(RBn
T [R (n~
n
X T n-X)1RT
-
n
-
n(RB n-r) [R(n ~XQ
n-r)
n
n~ X) ~R] (RBn-r) j = 0
and so by proposition 3, part iii holds.
[Xj
CHAPTER V
SUMMARY AND CONCLUSIONS
The state of the nation's housing program, particularly those for the urban poor, should convince
even the skeptic that we do not have a firm understanding of the housing market.
unique among economic markets.
This market is
Patterns of resi-
dential development have a profound effect upon the
social and economic development of the family, the
municipality, and the nation as a whole.
Urban
economics and planning professions must seek to
develop new theoretical and empirical procedures to
help us analyze the housing market.
In the preceding chapters, I have presented new
theoretical procedures that should be of interest to
the urban economist and policy developer.
meant to be nor is it a finished tome.
It is not
Rather it
points to new directions for future work.
In this
last chapter, I'll briefly discuss directions that
future research might take.
The search model presented in the first essay
is not completely analyzed.
It would be of great
interest to build into it a mechanism for replacing
186
187
buyers and sellers and then examining the implications over tine for an individual buyer and seller
resulting from market aggregation.
Ultimately one
desires to have a search model where bid structure,
sellers behavior, and search strategy are all endogenous.
One would then be able to examine the time
paths of the buyers and sellers behavior as well as
that of the market as a whole.
It would be advan-
tageous to be able to determine the effects on this
model from altering the distribution of incomes of
buyers and from adding more buyers and sellers from
different income groups without having to replicate
each agent.
If we had this type of model, we would
be in a much better position for understanding
effects of discrimination on market prices of housing
as well as in neighborhood residential patterns.
While White's work and that in the third essay
may seem to answer the questions of estimation and
hypothesis testing in the presence of heteroskedasticity, this is far from the fact.
The results that
we have derived, and those in the second essay also,
yield asymptotic properties.
What is clearly needed
are small sample properties of the various estimates
and statistics that are designed to deal with
188
heteroskedasticity.
As computers become more power-
ful and more available, maximum likelihood procedures
become more accessible.
It would be of great value
to know the circumstances under which each of these
estimators dominates in the small
(finite) sample
case.
It is clear that urban policies and programs
have little chance of success until their designers
gain a better grasp on the behavior of the urban
housing market.
While much theoretical and empiri-
cal research remains to be done, there have been
large gains in developing the theoretical and technical tools necessary for effective analysis of the
housing market.
However, the gains made -in the
development of these theories and techniques will be
insubstantial unless the policy and program planners
develop their technical skills and mathematical
maturity sufficient to understand and utilize the
theories.
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