[; T T il..'r t: O lin.', !i I i : I LD
rI Dl :L i.1 ii lii.Lf PGDEfI liLif CiOIC' S
,P".SOUT OUllIJG OR Lro !TIiG
Pobert P. Trost
A DISSCRT/,TIO[tJ PF''SFrJTLD TO TIIE GRADUATE COUNCIL OF
TI! U;!I\L.ESIT'I OF FLOr'iDA
Ill PARTIAL. FULFI LL t1JT OF THE PEQ,.U 1 REILTS FOP THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OFT FLORIDA
1.977
UNIVERSITY OF FLORIDA
3li 1i262 08552 5276 11111I
3 1262 08552 5276
To my parents
AC FI PNILEDGE1 I El iTS
The author '.'ishes to thank his Chairman, Professor
G. S. Maddala, for his guidance throughout the course
of this invest igation. Special thanks also is given to
Professor P.. B. Roberts for his help and friendship.
The author also .*:ishes to express his sincere apprccia
tion to Professors R. D. Emerson, J. W. tilliman, and
D. G. Tay.lor for their contributions to his education
at tie Unl"ersity of Florida.
Appreciation is also extended to Hs. Candy Caputo,
v.lo demonstrated not only expertise as a typist, but
infinite patience and cooperation.
Financial support from the National Science Founda
tion under cirant 5OC7604356 to the University of Florida
is gratefully acknow..ledged.
Finally, the author's parents ha.e been an important
influence on his life, and they are deeply thanked for
their love and urnderstandinci.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . iii
LIST OF TABLES . . . . . . . ... vi
ABSTRACT . .. . . . . . . . vii
CHAPTER I INTRODUCTION .. . . . . 1
CHAPTER II A CRITIQUE OF PREVIOUS STUDIES . 3
CHAPTER III A MODEL OF INTERDEPENDENT
CHOICES . . . . . . . . . 7
1. An Interdependent Choice Model Based
on Utility Maximization . . . . 7
2. SpcciEcatiiocn of the Model . . . 19
CHAPTER I'.' TIO0 STAGE ESTIMATION . . . 24
1. Estimation Problem? . . . . . 24
2. Discussion of the Results . . . 30
CHAPTER "' A::iA IflUlM LIKELIHOOD ESTItlATION . 39
Int1rod auction . . . . . . . 39
1. Amremicya 's T'..'o Step Ilaximum Likelihood
Procedure . . . . . . . 40
2. A Comlpi. iion of OLS and Ila,: iIum
Li l:e1 lihood Ltimates and a Test
for Simultaneity . . . . . 16
CI: ',PTE' R VI CO CLUSI S . . . . . . 56
PPE'i' JDI.X 1 DISCUSSIONJ, OF THE DATA . . . 60
APPENrDI:: 2 E:iTEIJSIOi OF THE MODEL TO THE
11ULTIPERIOD CASE . . . . . . . 65
TABLE OF C01TlSEN'['TS CO1TIIUED
Pa e
PEFEPENCES .
BI OGPJRPHl TCA L S ETCH
. . 74
LIST OF TABLES
CHAPTER IV
TABLE 1 Probit Coefficients for Owners
TABLE 2 Comparison of Two Stage and OLS
Estimates for Renters . . .
TABLE 3 Comparison of Two Stage and OLS
Estimates for Owners ... . ..
CHAPTER V
TABLE 1 Probit Estimates of the Decision
Function . . . . . .
TABLE 2 '..o Step fla::imum Lifelihcood sti
mntes of the Decision Function .
TABLE 3
 iiCuising E:penditute Equation of
the O .ner . . . . . .
TABLE 4 lio.iusinr.g E::end iture Equation of
the P enters . .
TABLE 5 
TABLE 6( 
Iousin q Ex:penditLure Equation of
the OwC'.n eer . . . . . .
i ising L:: end ture Equation cf.
the F]entcrs . . . . .
.PFENDI:T 1
TABLE t1 Comrnpar Lson of Cetnsus and .F'C Data:
Toal Famil, Income by Pace, for
all rain i e es . . . . . .
T,.LLF 2 Coipara1so of cenri us. and S:.' Data:
race b '.'ar io',s D remiogq graphic '.'ari
ables, fo. r Fariliesv '.'ith T'..'o r
lore tlembers . . . . . .
Page
31
S 47
Abstract of Dissertation Prosented to the
Graduate. Council of the Unive.rsity of Florida in
Partial l'ulfillment of the Pequirements [or the
Degree of Doctor of Philosophy
ESTE"] ATIOIl OF A, HOUSING[ DlEfl.'D
riODEL 1WITH 1 IITERDEPCFrDIT.IJT CHOICES
/,BOUT OI'[IIJG OR RPE[TING
F.'y
Pobert P. Trost
flarch, 1977
Cha i rmn : G. S. a addala
Major Department: Economics
The first limited dependent variable model was proposed
by Tobin. His approach is now known as Tobit analysis.
Tobit analysis uses a maximiumn likelihood procedure to esti
mate models with a truncated  that is, a limited  depend
ent variable ".'hen a large number of observations take on the
truncating threshold. Tobin's model has recently been ex
tended to handle more complicated situations. One example
of these e::tensions is a switching regression model where
the s'..witchinc function is endogeneous to the model. In this
thesis the switching regression model is used to study hous
ing demand equations.
The thesis argues that previous studies on housing only
emphasize one part of a complete model. Some of these past
studies only consider demand equations for owners and renters,
vi'
while others only consider the consumer's rent versus own
decision. The thesis proposes and estimates a complete
model which allows for the simultaneous determination of
whether or not to own and how much to spend. In this
model estimation of the housing demand equations by OLS
will be biased if there is simultaneity between these
equations and the switching function. To avoid this bias
the parameters are estimated by a two step technique and
a two step maximum likelihood procedure. Both these pro
cedures are asymptotically consistent and the two step
procedure also is asymptotically efficient. In order to
test for simultaneity the demand equations also are esti
mated by ordinary least squares (OLS). A maximum likeli
hood ratio test then is used to compare the two step and
OLS estimates. The results of this test show that simul
taneity is present. Therefore, the thesis concludes that
'OL cstimrrates of the demand ecquations are bi.aed .nd the
ti..'o stage and tL'...c step prLocedures are more appropriate.
CHAPTER I
INTRODUCTION I
lany models of housing demand have been presented in
the literature. The earlier models ..ere estimated with
aggregate tjme series or crosssectional data. For example,
see Muth (60), Reid (62) or Winger (68). More recent studies
such as Carliner (73) and Fenton (74) used micro data to
estimate demand for housing equations. In general all these
past studies use a least squares technique and estimate t..'o
housing demand equations  one for owners and another for
renters. One purpose of this thesis is to sho'.. that under
certain conditions these least squares estimates will be
biased. This bias becomes apparent once the owner and renter
equations are v.iew.ed as limited dependent .ariables that
are determined simultaneously ..ith a rent v.ersus ow..'n choice
function. It is this simultaneity which causes the OLS
estimates to be biased. Therefore, other unbiased estima
tion procedures have to be used. One such method is a "t.wo
stage" technique proposed by Lee (76). Another is the t..wo
step maximum likelihood procedure described in Amemiya (74).
Both procedures are used in this thesis and the results are
compared to the usual OLS approach.
I limited dependent .'ariable is a dependent variable that
has a lower or upper limit, and takes on the limiting value
for a substantial number of respondents. Models of this
type have attracted great attention recently. For examples,
see Cragg (71), Maddala and Nelson (74) and Heckman (74).
A second purpose of thi: thesis is to derive a model of
housing demand with interdependent choices about owning or
renting. To .accomplish this task housing is viewed as a
bundle of several goods such as location in a given area,
type of structure, size'and quality of neighbors. In general
any given bundle will have one price if it is owned and
another price if rented. Following this line of thought,
some families will be better off renting while others will
prefer to buy their own home. The model derived in this
thesis assumes that this rent versus own choice function and
the expenditure equations depend on certain independent
variables as well as stochastic disturbances that are not
necessarily iidependent. Thcse '.ariables include a measure
of permanent income; personal characteristics of the family
head such as race a r and se:: family background 'ariables
such as mobility and family size: various price indices; and
regional variables to capture the effect of varying land
val'ues.in different parts of the U.S.
In Chapter II past housinQ studies are critiqued.
Chapter III presents the formal. model. In Chapter IV con
sistent estimates of the rent and Cown expenditure equations
are ob:tainedr by the t'.o stage method. The choice function
is estimatedl by Frobit analysis. Chapter '.' estimates the
m,.,iiIel '..ith .a tl.'o step maximum likelihood procedure and tests
[or simultancitY '..'ith a maximum likelihood ratio test.
Cliapitr ''I coniailns the conclusions. Finally, the appendices
contain a discussion of the data and an extension of the
model tc the multiperiod case.
CHAPTER II
A CRITIQUE OF PREVIOUS STUDIES
In the past fifteen years numerous studies of the house
hold's demand and choice of housing have been presented in
the literature. The earlier papers studied macro data such
as aggregate U.S. time series data or the average income and
housing expenses of given SMSA's and cities. The main con
cern of these studies was to obtain estimates of the income
elasticity demand for housing. In general these elasticity
estimates '.'ere greater than one. For example, see Muth (60),
Reid (62) or Winger (68). luth's (60j study uses time series
data and, in a separate analysis, he also uses crosssectional
data. He concludes that the income elasticity for desired
housing stock is at least 1.0. In a review. article, de Leeiu
(71) makes adjustments in the Muth (60' reid (62).and Winger
(68) estimates. lie concludes that the income elasticity of
rental housing in the United States is between .8 and 1.0.
A second conclusion is that the income elasticity for ow'.ner
occupied housing is probably higher than that for rental
housing. llowever, even de Leeu'.:'s results are quite high
'..hen compared to the estimates obtained with micro data.
For example, using micro data Carliner's (73; estimates .were
]These "adjustments" stem from the fact that past studies did
not use the same data base or the same definition of variables
when estimating the elasticities. For example, some did not
include imputed rent in the definition of income for homeowners.
3
.631 for owners and .520 for renters. Also using microdata,
Fenton (74) estimated income elasticities of .41, .46 and .46
for all renter households, and high and low income renter
households, respectively. Polinsky (75) concludes that these
past macro results are biased upward and the micro results
biased downward. 2 He then states that the true income elas
ticity is around .75 and the true price elasticity is approx
imately .75. While Polinsky's criticisms deal with the
specification of the price variable, the critique in this
clha ipter is i me1d it: the technique used to estimate these pist
model]p. Since i typical Timple is comprised of families '.:ho
either o'.:.n their home or rent, these models cin be divided
into or e oF thi re .roups.
;, nai e approach is to diide: the simple data into
o,..ner s .nnmd renters and estimate t each subg rroup sepr tely.
Thi method rei'iults in t'.:o unique e~:pe.' di t:t e equations:3
oniE for homeo'.:ners, estimated withoutu t .usin the information
in the renter subsample: e ncother for renters, estimated
..itliout ,using the in form tion contin ed in the o.'ner sub
sample. For e::amples see Lee (68), Oh]s (C), C Irliner (73)
or Fenton (741 .
l'o l 1. insi ;y L'.:u s tIiC t Lhoe omission or misspecif ication of the
price term Lbas'.? the u.1n rol ups (i.e. micro) estimates do'..'nwa.rd
and: the rocups (i.e. micrc) estimates up'.:ard.
P' !tlher thln cli'.'ie the iriiple into o'..ners *and renterFs one cn
assume 1that all or 1,part of the parameters in the t'.:o expendi
tuLr eCquations are the sime for o..wn.ers and renters. A single
equationb can then be estimated] from the entire smple. Dunmmy
*rir .bles are uscd on the subset of para meters assumed differ
zent for o..:wners jnd renters.
The attractiveness of tie above approach is that an
ordinary least squares estimation technique (OLS) can be
used. A second alternative, the Tobit model, requires
maximum likelihood estimation. This method, first des
cribed Iby Tobin (58), results in two unique demand equa
tions. It differs from the first approach in that each
equation is estimated using all the sample information.
This is done by assigning a zero value to the renters
ownersr) when estimating the demand equation for owners
(renters). In his paper Tobin examines buyers and non
buyers of a durable good in anygiven year. He shows
that OLS on the subset of buyers is both biased and in
efficient. He then suggests a maximum likelihood techni
que (Tobit) that is both unbiased and efficient.
In these. first t'.o approaches either housing demand
or e::penditure equations are estimated. The dependent
variable is either the homogeneous commodity called "housing
services" discussed in Olsen (69) or various Lancastrian
"housing characteristics" described in King (76).
A third approach is to only model the choice of whether
to buy or rent. This can be accomplished with a linear pro
bability model, a Probit model or a Logit model. With this
method one does not obtain demand equations but rather an
equation to predict the probability that a given family
will own their ow.n home. Examples of the Probit model can
be found in Ohls (71) and Poirier (74). Quiqley (76),
in a similar analysis, uses the Logit model to estimate the
probability that a given family will choose among 18 types
of residential housing. The coefficients obtained by these
models are estimates of the true coefficients in the choice
equation divided by a common scale factor. The coeffici
ents can only be estimated up to a scale factor because the
dependent variable is never observed and therefore the
variance of the disturbance term cannot be estimated. For
example, if the dependent variable is the utility obtain
able '.when a family o'w.ns less the utility obtainable w..hen
thit sme family' rents, this difference of utility is never
obscr',,,d. Rat her, ..'e only obse'rv.e a one (o'.'n.. inqg) when this
difference is positive and a zero (renting) %'..'hen this differ
ence is negat'.'ie. The Probit and Log it modIels use a ma:
likelihood technique to estimate the cor fficients. These
crefficients are then used to assign each family '..'ith a
number bet.'een ;ero and one. This number can be interpreted
as the probab ]lit tllh t a gi'.'en family '..'1ll o..'n their own home.
For Probit Analysis this probability is derived by first
multiplying the independent variables by their respecti'.e
coefficients. The scalar obtained from this multiplication
is a standard normal '.ariate. The probability of owning is
simply the Standlard Normal Cumulati'.e function evaluated
at this scalic.
While numerous e::amples of the abo'.'e alternatives can
be cited in the literature, no one has studied models where
the decision on how much to spend is determined simultaneously
'.*:ih the rento.':n decision. An approach designed to fill this
gap in the literature is deriv'.ed in the following chapter.
CHAPTER III
A MODCL OF IiTEIRDEPEIiDEINT CHOICES
1. An Interdeplendent Choice Model Based on Utility
Mlax: imiza t.Lon
The major focal point of this section is to sort out
the factors thnt determine '.hy some families ow..n their ow.n
home and others rent. While many specific reasonssuch
as investment criteria or simple Tobit analysis for buying
i home can be i.'en, the choice mechanism presented here is
based on the simple economic principles of utility analysis.
The hope is to h.'e a general and '.idely applicable model.
The model will ':ie..: the decision to procure shelter in
an imperfect world. First, the choice of a family to rent
or buy in a particular neighborhood is limited to an "either
or" situation. In w'.ords, a family must choose bet..,een a
number of discrete alternatives available. That is, given
price and nonprice constraints some alternatives may not be
feasible; for example, renting an apartment '..:ith a large
q rden. Second, individual families face a number of con
straints such as budget and time constraints. Financial
IIn Tobin's; (58) pionocrin i piper oil limited dependent vari
ables he presents an estiimtion technique for models .w.hen
the dependent '.',riable often takes on a lower or upper limit.
His method is now know.'n as Tobit analysis and has many appli
cations in economTics. For e::ample, 3 family's e::penditure on
a major durable such as house may be zero until the house
hold income exceeds a certain level.
7
institutions constrain families to loans of limited size
by requiring, for example, that housing expenditures not
exceed 2025 percent of income. Third, families purchase
both durables and nondurables, consuming durables over
time. Fourth, housing is not a homogeneous commodity.
Rather, what we call housing is really a bundle of several
goods: location, house size, yard size, type of structure,
central airconditioning and'even type of neighbors. To
get around this nonhomogeneity problem Olsen (69) defines
a homogeneous commodity which he calls'"housing services."
Olsen argues that whenever the housing market is out of
equilib: rium t:he e 1:istinq housing stock .'ouldl filter (i.e.
and increase or decrease in the qluantity' of housing
serv.icosi up or do..:n and net.' dwelling units '.*.ould be con
structed L'isequillbr ium is said to e:.:ist whenever r price
is greater tlhan long run average cost. This filtering and
construction continues until there 3re no profits to be
rad3e on bundles of housing of an size. This zero profit
equilibrium ,r's ition requi rre the price per unit of housingq
services for bundles of all sizes to: be the same. Follo'..'
ing this line o:f thou iht, if t..'o families in an equilibrium
market speijr different amounts on' housing, the implication
is that both families are colnsumiing different quantities
'f the same good" lousincT setr'.ices." Tl is quantity com
parison is possible because of tw..o crucial assumptions.
First, it assumes that the equilibrium price per unit of
"Ihousing services" has been reached. Second, it assumes
that families do not have preferences for particular types
of "hoIusi n servicCes" such a li'.'ing in do.wn to'...'n few' York
City versus living in flew Y'cI' I: City suburbs.
A Lancastrian approach to the nonhomogeneity problem
is taken b:y j:inq (76'). He 'postulates that housing is
really made up of several hon'ogeneous characteristics.
An individual has separate demands for each of these
characteristics but not a demand for*housing per se.
SIn order toc distinguish bet'.:een the demand for renting
..which .was primarily the demand for apirtments in 1971and
the demand for hou]Ises, a slightly diff rent ,approach to the
ronhomrrogcnei Lt issue is taken here. For simplicity this
approach is presented as a one period maximization problem.
Ho..e'ver, the model is general enough to be easily extended
to the nmultiperiod case. For example, one could assume
the fainily mr.xiimi zes ;:pected utility summed over the
entire occupancy of a dwelling. A multiperiod model taking
'this approach is presented in the Appendix.
Assumi'e for the moment that housing merely amounts to
the size of the dw..:elling. The larger the size, the greater
is the quantity of Ihousinq consumed. In this simple case
an owned home and a rented apartment are perfect substitutes.
Indeed, they are the same commodity and a family's decision
to o'..'n or renLt '.:i]1 l e based on price alone. From consump
tion theory w..:e kno'..' that in equilibrium the family will
equate the r:atjos of marginal utility to the ratios of
marginal cost for all qoods. That is, if the family owns
IlUm (( 0)
fIU (I) o
will hold, where MU is marginal utility, MCo is a marginal
cost function for owning, H is the size of the owned home,
X represents all other goods and P is the price of all
^x
other goods. Similarly, ifthe family rents H units of
housing
SICr (H)
MU(II) Cr(
MU(X) P
x
will hold, where MC is a marginal cost function for renting.
r
If MC and MCr are different functions', then given prices,
preferences and income, the family will rent or own depend
ing upon which is cheaper.2 For example, consider the partial
equilibrium ain.alysis dej:,icted in Figuire 1. In this case '.'e
would observe family 1 renting H1 and family 2 owning H2. 11
igenera 1 the shape of these marr,:ainal cost functions would de
pend on the location cf the structure. A family in t[ew York
City with higi h land values s would therefore face a different
mi:rgiqnal cos L pattern than a family in rural Alabama with
relatively, io. land '.alues.
,!liil, in qgneral the choice between t'o perfect substitutess
result in an cithlier'or situation there is one ec:eption. If
the ma.rgninal 1ite of sul:.stitution and: the p.:ice ratio ha'.'e
iden t ica slopes, then the consumer is indifferent between
ani con: bina t ion of the t,..o commodities. Unless the individual
could rent Lhhrce troo'ms of a five room house and o...n the other
tw.:o, I1iis ''LouIl not be the case foi housing. For most con
sumiersG the implicit time cost of consuming t ,..io separated hous
int units constrains them to an cithcrori choice. Of course
thrre may I:b situations hereee a familIy would simultaneouslly
o( . ii reGnt. C'n o.al'e is thle businessman \ o ons a house
in tl!e suburLs but also rents an aparttrient near w'.or For him
thl opportunity cost of commuting ever,' day outw.'eiqhs the cost
of renting an apartment.
lo.' consider housingor if you like, housing services
as? being made up of several commodities. Let this set H be
represenEted by a vector of n different housing service com
modities such that II = [hi i=l . n, .here h. is one
element of the sectorr 1. lie::t, let the set of prices of
these n cominodities be sector r functions of h.. Call this
set P for renting and 0 for o.'ning such that
P(11) = (h ], i = 1 . n
0D(1) = [o (h.)], i = 1 . n
R' (il) = (r (h i = 1 . n
i i
(11) = [o '(h .)], i 1 . n
where P' (ll) and O' (H) are sectorss of marginal cost functions.
At any point in time a family can consume all the h.'s or
only a subset of them. Utility theory states that in equili
brium the family '.'ill equate the ratios of marginal utility
to marginal cost for all goods. Utility theory also tells us
that each family :.ill ha''e demand equations for all the hi
goods. As shown.n in the partial equilibrium diagrams of
Figure 1, wvhethcr they o:.'n or rent these qoods '.'ill depend
on the marginal cost and marginal utility functions. But
at least four problems arise when it comes to estimating
these separate demand functions. First, a family generally
does not o\.'n housing commodity h. and rent housing commodity
i. For e:.aminlc it v.'ould be very difficult cost wise for
a family to rent a lot with a nice location and then build a
dre'aml hlcuse :.'hichl they plan on taking with them when they
move. Second, most or all of the h. 's are either unobserv
able or difficult to measure. Indeed, many of the hi 's
MC (H)'
R
AC (H)
/ R
/ / MC (H)
I
/
I
/
/
 I
rIi (U it
II
House Size (lI)
Fiu.jre 1.PaLr al equilibrium housing demand
4
may be subjective qualities in ique to e.'ery family. Third,
because housing for any family may only. contain a subset
of the h.'s, ..e have a limited dependent variable problem.
That is, for any h. we willconly observe positive quanti
ties bcing consumed by some of the: families. For the rest
of the families w.e ..ill observe some limited quantity being
consumed. In general this limited quantity is zero. Fourth,
the price functions o. (h. a' d r. (h a are hard if not im
1 1 1 1
possible to observe.
Ideally of cours.:e w.,e should estimate demand equations
foi: all the h. 's, but qiv'en the four problems just discussed,
other approaches must be substituted. One method is to
reduce housing to a fewr. reasonable and measurable Lancas
train characteristics and estimate demand equations for each
of these char icter istics. Tiis is the approach taken by
i.ing (76). Another alterna tive is to simply define a homo
geneous good called "housinci services." As discussed in
Olsen (69. the problem no'..' reduces to the estimation of a
single demand equation. Uhile both of these approaches
have merit, they give little insight into the rento'.'n
decision. The analysis to follow combines these tw'o ap
:)roaches. The result is a model where the expenditure
decision is determined simultaneously '..'ith the rentov.n
decision.
Kecpingj in mind the fact that families face time,
financial, and budget constraints, the model can be stated
as follow.s. The family can choose to purchase or rent
some amount of the homogeneous housing commodity H.*.
Here 11.* is simply one of the many possible subsets of
H such that 11.* = F.(I) j=l . m. Here F.() is some
function. More precisely, ;.* should be viewed as a homo
geneous subset of H with the remaining mj commodities
held constant at some level.3 This fixed level can be zero
or any positive number. The idea here is to break housing
up into several types much like one'can break fruit up into
apples, oranges, and pears. For example, you often hear
a family say they are looking for a "big house.in the city,"
or "a ranch house in the suburbs," or "a split level just
outside the cit, '..ith a pool," or "a two'. bedroom apartment
near '..ork," or perhaps "just a one bedroom unit near a
shopping center." .After pricing these specific types of
rhousi ri the fla il rlI.:i\' or may eo t settle on their first
choice. They will of course settle for some unique com
bin action of the set H. The simplifying argument made in
thi' [ 1.'p" r is that '.:E can .'ie'.. this decision. as a choice
oL one t,'ye of housing from a set of several la 1ternati'.'es.
The per unit cost of buying H is represented by O.*H. )
suchI that
C0 (I i ) f Cf OlH) = 1 . m,
'..lh re f, is soni' function. Similarly, the per unit cost of
rental. I. 1 Ls hi'.'en bL such that
i .
I4ll ) = f [ (II)] i = 1 m ,
...'herc f is some function. If '..e ignore for the moment
"If I i = II, then this model reduces to the one discussed
in C'lIen (69).
the possible ta:: reductions From o'.ning, then if the family
were to purchase housing the maximum utility obtained would
be derived from
(i1) la:: (4 *ix, H )
Subject to [" X, H. ) 1 = 1 . m, .'where : is a
bundle of all other goods including sa'.'ing, F f (, HI. ) is
the "ow'. inci consumption possibility frontier." In this
simple case Fr (:, 11. ) is ni''en b:
F :' r ;, ) = 'P( ) 0. (H )H ,
1.I 1 k H:; H j .
v.'here 0 ((ll includes ani' financial payments or opport
unity costs and IH only includes that part of the dur
able qood bought for consumption purposes. That part of
11 + purchased as an investment is included in X. Y is
income in period t. The first order conditions to (1) are
U Fo
(2) i F
J 3
.here U Fi a nd F 1 are partial deri'vati"es. rl,
U F. and 1r are nothing more than the marginal
1 1
utility of X, the marginal utility of H.*, the marginal cost
of X and the marginal cost of buying 1i respectively. When'
(2) liolds let the utility obtained be represented by U (i ,*),
where 1 = 1 . m. Assume that when ow.ned the Kth subset
of H produces the highest le'.'el of utility. Let this utility
level be represented by .U o* (11 ).
A similar anial.'sis holds for renting. If the family
rents the subset II. the maximum utility would be derived
froi (3)
(3) Max U(X, Hi*)
Subject to: F (X, H.*),
where F (X, II.*) is the "renting consumption possibility
frontier" and is given by
FR(X, H.*) = YP(X)XR.*(H *)II*.
j ( J J J
The first order conditions to'(3) are
U FR
x X
(4) F
H .* F
] .*
RJ
where F is the marginal cost of renting Hj*. When (4)
holds let the utility obtained by represented, by TF (nI *)
\where j = m. Assume that the Lth subs.'Eh cf H pro
duces che highe.st le'.eel of utility by renting. Let this
Ltilit'y le'.'l. be represented by II* ( ) Figure 2 sh'..o s
R LL
Ht e eC]qu 1 iibr' i ,m po, c :ition ii'.'enll by" b I H *) and *0H ).
The f mily' ''ill buy, tlhe su:,set of housing ser'.'ices H *
if lU li(H.* is larger than UI ) Con.ersel the family
'...'ll r nL the subset cf housing ser'.'ices 1 L' if U l(HL i
greater than LU Itl *I ) If one includes the ta:.: ad'.'antaqes
from owning, then the "ou'ninli consumption p.ss ibilitt
frontier" constraint reduces to
r" IX, [ *) = C'rM.1) il t)PI ::) .'O .*(H .*)ll.*,
] j .i j
..'here r is the interest rate orc the mortqgae, I1 is the amount
of the mortgaLq"e ou stand inq and t is the tax: rate. In
general lhore o'.ninq '.'ill no'..' be more attracti'.'e the frontier
for o'..'ninq '...'ill slope, more to the right in Figure 2), but the
analysis w.iill bte the same as before.
t could, of course, be made a function of income.
S :: l )
U 4 4, 1
St A ( ,,l, I 4 1 = I. ll )
T B R L
FIGUR.C 2. Consumption possibility curves
Up till now I have proceeded as if the quantities ,Hk*
and II can be observed and measured. Indeed, I have pro
L
ceeded as if the whole set of housing types H.* can be ob
served and measured. In pr.~ctice, both observation and
measurement are difficult. To get around this difficulty,
housing can be measured in units of dollars. These dollar
units can then be deflated by regional price indices. The
dependent variable for bothowners And renters now reduces
to deflated dollar expenditures. This view of housing is
easily reconciled with consumption theory. The consumer
:..ill ima::imi utility by e.quating the mariaginal utility of
the lSt ldolla.r spent on housiln to the m.arjinal utility
of the last dcll.ar spent on all other gcds. More speci
fically, the model de'eloped: in this chapter can be written
03S
=5) 'In n' 1 + L n
15) In n + 2n
*71 I =' ,' 1i 1
hereee 'ln is LOal annual e:xpenditures on housing if the family
owns, is r' al anni.ial expenditures on housing if the family
rents, I is an unobser\ vablle inde::, : n: ::, nd Zn are "ectors
of incaeperenten v.ar tables and l Un ', U ) are triv.r iate
In _n 1
normally d t s tribl.u Id Also, the model assumes Yn and n
.rie rILItL.1ly l :I cusi'.'e and cannot be observed simultaneously
for any one indi.'jdual. .One either observes the family' own
111I ajnd sFpndinr' Y if I 0 or renting and spending 'IY
IInIl orall this is a s intc i e ssi
if I 1). Mrire formally, this is a 's%.itching regression
model '.with sample separation and can be writtenn as
(8) lIn = ln. 'l. + tn
(9) '2n = :2. + U
n 2n
but w.e only obser"e Y where
'Y = IY ,iff 1,, U i
n In Z n T n
(10)
In 2n i n Znn *U
This s'..itching regression model is nothing more than an
extension of Tobin's (58) paper. In Tobin's paper the
dependent variable is zero or some constant if a family does
not have expenditures on a durable .ood. In this paper the
dependent variablee is rent ex:pendituresa random "ariable
for families w..ho do not ow.n.
In this section I ha"e described a .'itchinq repression
model with sample separation where the expenditure equations
and choice function are simultaneously determined. Although
the model itself is simple, its estimation is not straight
forward. The estimation problems stem from the fact that U
is assumed to be ccLrelated with U1 and Un .
In n
2. SpecificaLion of the Nodel
The model was estimated on a sample of 3,028 families
from A Panel Study of Income Dynamics, "olume II (72),
(hencefor th tie "Panel") These data include voluminous
q]uesti.onnairec data for a sample of 3,4152 American families
over a period of fi.'e years. A detailed explanation of
this data base is presented in the Appendix.
For empirical purposes, a logarithmic functional form
20
was used in the demand equations and the choice equation.
This makes the estimated demand equations compatible with
previous studies.
The dependent variable .was housing expenditures
described as housing cost in the Paneldivided by a price
of housing index. This variable includes utility payments,
amount saved on additions and repairs.when the work was
done by a family member,5 property t'xes for homeowners,
6% of house value for homeowners and annual rent payments
for families who rent. That is, the department variable
is 1'in its of housing as measured in real annual dollar
e :: nd i tur c s.
The p[:rimary ind.1eperndent ariable '.'as a measure of
prrmanent income. The variable e used: '.:as a fi'.e ye.ar
a'.:C'rag of "fmrimily noine,'"as described in the Panelplus
inp,:ute.d rental income for .homeo'.'ners (i.e. 6l. of net equity),
all di '.1id1: by a general price inde>:. This '..r able in
cludes labor income of head and/:or w'.ife: asset inbo me from
fam or business; rental, interest and diidend income;
arid transfer pri, ments such as Aid to Dependeint Children.
Other iiid.iepc dcrt "ariables included city population
and distance from the center of the nearest city of 50,000
.r mount saved ion additions and repairs \'as included to make
hou sing c:*: pcrn.d it L es fo.r renl tle s andr o..lners comparable. In
_inCrCL ii 'L'c, c.: icnterls dc thicir c'.'n main tcnance '.'crl:.
Tiiis is beciusc t;hieir r,:nt]tal p:'aymen ts.and therefore housing
e::pend i turesi nclu.de a3n implicit maintenance cost.
6
population or more. These two variables '.'ere included to
capture the effect of higher land values as land becomes
relatively scarce in densely populated areas. Also, to
capture demographic differences, family size '.*.as included
as an independent variable. This variable is defined as
a number of people (children plus adults) living in the
family unit.
Dummy varliables for .age, sex:, and race of the family
head were included, as '..as a dummy variable if the family
moved more than once between 1963 and 1972. The reason
for including this lst variable in the expenditure equa
tions '.:as to capture the different search costs as '.ell as
different demographic factors of the mobile families. It
was included in the choice equation for other reasons.
The moving family, because of the transactions cost in
volved in buying a home, is more likely to rent than own.
Finally, six price indices '..ere included. Each house
hold has associated with it a relative (to all other goods)
price of renting, a relative (to all other goods) price of
owning, a price of renting, a price of owning, a price for
all goods other than housing if the family, rents and a
price for all goods other than housing if the family ow.ns.
These price indices were based on data taken from the
In the Panel these two variables were coded in groups. For
c:amIpl1e, c ty size W.'as class ficd as beinti in one of six
groups: rcatcr thD1n 500,000; 100,000 to 199,999; etc. To
make the variable cont iluous, a real iumber the midpoint,
when appropr iate',..as used for each gr [)oup. That is, for
the Z 'wo above examples the family was assigned a value of
1,000,000 or 300,000, respectively.
Bureau of Labor Statistics (DLS) Handbook of Labor Statistics,
1972. The three "renting" price indices were constructed
from Table 136 in the BLS "Annual Budgets at a Lower Level
of Living for a 4person Family, Autumn 1971." This table
breaks the budget into nine categories for fortyfour cities
and nonmetropolitan areas in the United States. The housing
expenditure data assumed all families were renters and was
used to construct the rent index. The price index for all
other goods was constructed by subtracting the housing expen
diturres fromL tLhe total bl:.udget. Similarly, the three "ow'.ning"
price indices ..'ere constructed from Table 138 in the BLS 
"Annlual Pudgets at a Higher Lc'.'el of Liv.inq for a 4person
Family', Autumn 19'71." In this table the housing expendi
ture data v'ere ..'eiqhted by the following proportions: 15
percent for rental cost and 85 percent for homeow.ners cost.
Using these t:.o BLS tables, it '..as threfore possible
to assign one price imnde:. for renting and another for ow'n
ing to each of the fortyfoour 3reas. Depending on where
they li'.'ed, each family in the Panel could then be qi.'en
three price indices for o.'wning and three for renting. For
e::ample, if a family li'.'ed in the Boston SMSA the price
indices for Boston ..ere used. If a family li'.ed in the
ilor'th :ist andj in a city of less than 50 thousand popula
tion, then tlhe .price indices for nonmetropol itan areas in
the lortheas:t .ere used. A similar procedure '..'as followed
for other parts of the country.
i.]thouigh the abov'.e price indices made careful use of
all available data, at least five cautions should be made.
First, section one of this chalipter argues that housing is
really a bundle of several goods. One price index is only
an approximation for several price indices and therefore
its coefficient is only a surrogate for several price
elasticities. Second, Polinsky (75) notes that even if
housing is a homocieneous good, using metro housing price
indices ignores price variations among observations from
a given metropolitan area. This can' cause do',.'nward biased
estimates of income elasticity and up'w.'ard (toward zero)
biased estimates of price elasticity. Third, the price
inrd:.: Fnr n.wninn waq hansrd on 1071 data. IUnless rPlativ'e
prices across cities remain constant over time, the rice
index for o..'ners is only valid for families '.ho bought
their homes in 1971. Fourth, since the rent index '..'as
based on lower income families, it may not reflect the
true price of renting for higher income families. Simi
larlv, the owniing prrice index may not reflect the true
price of owninq for lo'..'er income families. Fifth, families
may believe that the selling price of houses will rise
faster than the general price index. This may encourage
them to buy, a house larger than their consumption needs.
Their dollar expenditures on housing would now..' reflect
investment as .,well as consumption decisions.
CIIAPTER IV
TWO STAGE ESTIMATION
1. Estimation Problems
Because of the assumption of correlated disturbances,
estimation of equations (5) and (6) in Chapter III with
ordinary least'squares is biased and inconsistent. As in
the Tobit model, this bias and inconsistency is due to the
presence of a disturbLaince term \whose e:xpectation is nonzero
and not constant for all obser'.'ations. In the econometric
litertLjure the e::pectatioin of this disturbance tern is
o fteri referred to .as a "Hissincg Variable." lore specifi
cally, it can be sho.wn that
E ( I ) 0 E (U ) 0
In n1 n 2n n n
.aind lso tllat the error sterns U 1 will be correlated with
S1n' 2 n
their regCressCors. Thus, the direct least squares method can
not Ib:, applied and other consistent estimation methods must
be used.C While one possible procedure_ is nm.aximum. likelihood
estimation, Lee (76) hais proposed t'..o much simpler computa
tionll "t'.'o staqe" methods. These procedures can be applied
to eqij'ations 57 of Chapter III as follows.
Let the obser'.'ed sample separation be denoted by a
dichotomous '"'ri.alle I That is,
(1) I = 1 if0 f U
]n n n
If '..e assume U is normally distributed the above can be
w.'rittoen as the following model.
(2) I = 
n n n
but we only observe" I where:
n
(3) 1 = 1 iff I 0
n n 
n
= 0 iff I 0.
Finally, the Probit model y' ilds the relationship
(4 ) I = F ( ) + E 
n n n
where F is the cumulative distribution of a standard normal
variate.
The first procedure proposed by Lee (1976) uses the com
bined owner and rcnter sample and estimates B1 and 82 together.
To derive this procedure let i' be the observed sample of the
th
endogenous variable for the nt family. Combining the two
separate equatioiLs into one singIle equation, .'e have
S + (Il I 1 ''
n 11 n I L 
(5 = + (1 I : + I U + (11 )U
(5) = 1 ]lit + n n 212 n in n 2 "n
.Al1thloug:j (5) looks like a single regression equation, ordinary
least squares cannot be applied directly to it. First, since
U of equation (2) is correlated '..'ith UI and U,2 the error
terms do not have zero mean. In fact, Iwe have
(6) E(I Ui ) = Ol U f(U )dU
n In 1 1 n n n
Z 'y
(7) C(11)U2 = ~2. n U f( )dU
where n", 2' 1 L0, denote the standard deviations of Uln,
U and the correlation coefficients between U and U and
2n In n
between U2n and Ul respectively. Also, f(U ) is the standard
normal density function._ Thus, the mean error term of (5) is
(8) E[I U + (1 I )U ] = (P lp 2) U f(U )dU
Oo
Furthermore, the error terms of the equation are also
correlated with the regressors since
(9) E[InUnI Xln] = EmUln]Xln = POlXn/ Unf(U)d f 0,
OO
and
(10) E[(1I )U (1I )X ] = E[(II )U ]X
n 2n n 2n n 2n 2n
Z Y
= p OX21 U f (J )dU 0
P2 Z X2/ n Unf n)dn n 0
O
In order to estimate (5) we must first adjust the mean of the
error terms to zero and take care of the correlation between
Lth error trms a1nd the rec ressor We no'.' ha'.e
(11) Y = F(, ,)::ln + (1F C n ));:2n 2
n n In 1 /n
+ (I'1 r'n .' i n U (LI Id + W
...'hL,e U = I iU i (1I U.. ( .i. n, ,.. 1 L fW( )dUL
I'I I III I 0. I IIl "1 n .
'11 4 1 ^ ,n n'
It can be c sil' sl..n that Ef I = 0 .nd W .is uncorrela ted
l' rn
'.' l hI tI1, r.;i res or s.
Ho..'e"r, erjuation (11) is nonlinear in the parameters
1 id :2. But since s,imple separation information is
a'.'i lable, a simple technique to estimate the unkno'..'n co
*ffici .ntl s 1 1 : d i. ., is possible.
FiLst, .]uation i) crn l e estimated b.y th.: usual Probit
nal1sis. This is the first staqe of the estimation procedure.
ilir scon.i stage is to estimate the fol l'.'inl equation by
oud. inary 1' I nse t souar ,
(12) 1 r ( "):.:: 1 + (1F (Z ') :.: ".
II In 1 n
S7i n ),
+ ( i"' 0,0,) .1 U f (IU Id U + W
z1 I n1 n' n n
Since is a consistent estimate of y, U will have an
asymptotically zero mean and be asymptotically unrelated
with the rcqressors. Even though W does not have the
same variancec e for each n, the estimated ~1 and 8, by ordinary
least squares w.'ill be consistent.
To see ho..' P 1 and F '2 are identified, equation 12 can be
rewritten as
(12a) = :' + X2 + , n + . ,
(n in 1 2n 2 n n
w\.herIe X = F'. = F(Z ) 2 a ( P F0 )
in 12 I11 1 2 1
and 3 = i U f )d" When all the independent .variables
3 n n n n ,
in X: and are different, OLS on (12a) yields estimates of
., ', './ar (,) and '.'c r H( 2 directly. fiHowev.'er, if =
"" 2"2
2n, OLS on (12a) will only gi' estimates of ,, )
'.'ar (2) and '.ar ( 2,). This is because (12a) now
reduces to
(12b) = :: 2 ( [ ) + + n'
4. ^
1 2 1)1
e = = and : = F Z ,) Estimates of ..'ar
e n In "2n 1n n n1
(B6) can be obtai ined by using OLS on the follo'.'inq
(12c) : ;= + >; (, ) + a:e; + ,
S1 2 2 I 3n
where r = F(Z x) x. Since the three sets of independent
variables in equations 12b and 12c ..'will be highly correlated,
estimation of 1 and , \.'ill ha"e multicollinearity problems.
In order to av.oid these multicollinearity problems a
second two stage technique is available. This method esti
mates F1 using only the o,..ner data and 2 using only the
renter data. That is, consistent estimates of 81 are
1
The standard errors will be slightly biased. This is because
unbiased estimates of 01 and a, require a somewhat different
forimulation than the one used in OLS.
obtained by using OLS on
f(Z y)
(13) Y X lnB1 + T ]( + V n
Similarly, consistent estimates of B2 can be obtained by
using OLS on
f(Z y)
(14) Y = X 2 + o [ J + V .
2n X2n 2 Zu (Z) + 2n
In the two above expressions Cu and a 2uare the covari
A lu 2u
ances between Uln and Un; and U2n and Un respectively. Vln
and V2n are stochastic disturbance.
Equation (13) is der1.'ed by t,~kinq the expectation of
iln conditional on I = 1 arid adjusting the conditional mean
of the error term to zcro. That is,
(15) EC(Y In = 1) Xl n + E(U 1n In 1)
n + I In d
= n1 I + 1" Ulng (In n = l)dU n,
..'here 1l(U n I = 1) denotes the conditional density of Uln
conduit ioncd on In = 1. Before proceeding, rev..'rite q(U r, I = 1)
as the joint density divided by the marginal density,
q (U In = 1)
(1;J (( l II = 1) = I rI
In n qC(I = 1)
n
n f(U IU )cU
in' n n
f (11 ) d
n r
77 1 .rA f (II.
F(7 F)ln 171 n
n
1 .n
.I f (U ) f (U )dU
F(Z ) In n n n
=, 1 n (f ( lU U ) f (U )dUn
n ,
..'lher: f (U .n',I n and f('ln IUn ) are the joint and conditional
I.i'.ariate normal density functions, respecti'.'ely. By
substituting (16) into (15 it follow'.s that
(171 ) l i n I 1) = v I ..
in n n
+ n J (" U f(U ILl ) dL ) f (U ) dUl
F( Z1 ] 1 1 n In n n
1By substitutinI in i or f (U I U ) '.'hen i 2 = 1 and
i nte r a t i ri n '.'.e r e t
(19) CFl'inI = 1) = : n 1
nr r. U f (ndu
1n in 1
1 n'
n U f L )dLI
*I n I n n n
n 1 lu F (( )
Ln n
in 1"I
Finally to use OLS on Chapter III's equation (8) directly,
'..e need to adjust the conditional mean of the error term
to zcro. ['oinci this '.:c cet
in = in 1 lu F (Z )
n 7
+ n ilu L C.Fn)
S[ E(Z1 1
S"1'nl lu F(Z ) n'
.where .' = tL , and C' l I = 1) = 0.
Similarly, Equation (11) is deri'.'ed Ly taking the
c::pectation of Y2, conditioned on I = 0 and adiustinq
the conditional mean of the error term to zero.
As mentioned earlier, this two stage approach avoids
the multicollinearity problems that arise when Xln =2n
Since nine of the ten variables in Xn and X2n are the
same for the model specified in Chapter III, 81 and B2
were estimated from equations 13 and 14 respectively.
2. Discussion of the Results
'The model was estimated.,usinc 1971 data. The Probit
estimates of the choice equation are reported in Table 1.
The OLS and two stage estimates are reported and compared
in Tables 2 ard 3. A discussion of the results found in
these tables ollo'..'s.
Table 1 contains the Probit results w...ith el.een in
dependent .ariables. Both age dummies are significant
and inJndcate that families .'ith a head o'.'er G6 are the
most likely to o..wn their ow..n home and as expected, families
headed Lby an individual under 36 are the least likely to
own. Thie coef[icient for The Black durmmy is highly signi
ficant and neqativ'e, indicating that Blacks are more likely
to be renters than owners. This could be due to either
price discrimination or merely a stronger preference for
those housing goods that are cheaper to consume by renting.
A similar interpretation applies to the ngati'.e sign for
the female dummy.
'ihe ineg t0i'.'e coefficient for the mover variable sug
gests that transactions costs of buying and selling a house
effectively raise the price of a home for the family that
frequ,'nt 1y mov.es. These families are therefore more likely
to rent.
TA BL 1.
Probit CoefficienL:s for Owners'
Intercept
Age < 35
Age 3664
13 lack
Fem ale
flo'.'er
Log (City Size)
Log (Distance from Center of Cit'y)
Log (Family Sizel
Log (Ile.al Permnainent Income)
Log (Pelati.ve Price of Penting Housing)
Log ( lciati.ve Price of Owning lorme)
Lumber of Owners
Number of Renters
'Standard errors in parentheses
4.6586
. 5350)
1.2001
(.109 4)
. 6567
(.099 1)
. 3256
(.0728)
(. 0632)
.9591
(.0721)
.1060
(.0199)
.1389
. 0276)
S1328
(.0498)
.7586
(.0535)
2.1735
(.4015)
.3684
(.3793)
1803
1225
TABLE 2
Comparison of Two Stage and OLS Estimates for Renters*
Intercept
Age < 35
Age 3664
B a c:
F em,: I e
Lo;i (['lct.incc froni Cen ter of Cily)
Log (Fariu1l; Siz,)
Lo (. (F.1e PcFi:ranent Income)
Two State
Estimates
2.0515
.1994
(.099.0)
.1320
(.0730)
OLS
2.0965
.1774
(.0580)
.1190
(.0558)
.1823 .1776
.04941 (.04641
.1061 .1107
(.0329) (.0282)
.12 6 1407
(.0593) (.0286)
.056 4 .05 81
(. 100) ( 007 6
. 0 n005 .0017
(.0310 (.0299)
.i 0 2 .0863
S0215 (.02041
.50231 .490:
(.0506) (.0236)
Loi (Pe la t "i r'Lice of PFenting Ilousin )
1.2:372
(.22 65)
.0329
(. 1192)
1.3216
(. 1 3 5)
[ iussini '.', i ,:a b 1 e
n2
[lumber of Obscr'a t ions
*St.andard errors in parent theses
1225
.4 2r 9
1225
TABLI: ]
Coniparison ofL Two' Stage and OLS Estimates for Ow'ners*
T',iwo SLta
 Estimates
2.0392
Intercept
Ace : 35
Ag le 36641
Blacl:
Female
lover
OLS
2.0909
.154 .1343
(.0693) (.04071
.1563 .1505
(.0440) (.0330)
.2534 .2502
(.0585) (.0561)
.0772 .0006
(.0352) (.0308)
.1018
(.0660)
.0909
(.0367)
Log (City Size)
Log (Distance from the Center of City)
Loci (Family Size)
Logr (Real Permanent Income)
Log (Relative Price of rOwning Home)
flissinci Variables
.0282 .0295
(.0088) (.0060)
.0193 .0182
(.0363) (.0357)
.029P .0285
(.0234) (.0224)
.5712 .5644
(.0396) (.0206)
.1784 .1768
(.1914) (.1912)
.0195 
(.0979)
.441 77
Number of Observ'.ations
1803
.4490
1803
*Standard errors in parentheses.
The next two variables, city size and distance from the
center of the city, indicate that in rural areas the family
is more likely to own their own home. This could be caused
by higher land values in thi more densely populated areas.
In populated areas where land is scarce and therefore
expensive relative to land in rural areas one would expect
to find more people living in land economizing apartments.
Conversely, in rural areas one would expect to find a rela
tively limited number of people living in apartments. For
S:;amp le, few fami lies could d af ford to Gjuy land and build
single: family lhous es in :do.'nto:'n He'. York City. Instead of
houses :.'e see high risc apartments' that economize on the
scarce inpuit land. Ia sed on mcre availahi] ity then, it
foillo.'s that city' size should have a negative sign and
distance from the city a positive sign in the choice equation.
ThI coefficient for family size is positi'.e. This
i.lndicat.cs that larger families prefer the more spacious
living conii it ions pro'.'jded by home ow.'nership. The positive
coefficiECrit for the income variable suqqests that lo\.'er
income families are unable to obtain a mortgage for the
size of house th'v desire and therefore rent rather than
o'.'n. Finally, the t.wo relati'.'e price 'ariables ha'.'e reason
tile sigq ns. ,fs the price of renting goes up or the price of
owning i J goes .io'..;n, a fariily is more likely to own their o'.'n
home.
Tables 2 and 3 contain the t,..'o stage estimates of the
demand equations (equations 13 and 141). For comparison pur
poses these tables also contain OLS estimates of the renter
and owner demand equations. These OLS estimates were ob
tained by splitting the sample into two groups, renters
and owners. Demand equations were separately estimated
for each group. A discussionj of the two stage estimates
follows.
The coefficients for theage dummies are negative
and significant for both owners and renters. For owners
the coefficients give a Ushaped pattern of housing ex
penditures versus age of head at given income levels.
For renters, expenditures on housing increase as age in
creases. The coefficients for owners are consistent with a
theoretical result derived in Muth. (74). Using what he
calls the "income effect, price effect and length of stay
effect," AMuth presents a justification for this Ushaped
patt rn.
The black and female dummies indicate that blacks
spend less on housing than nonblacks, while females spend
more on housing than males. While this may suggest the
existence of some discrimination against blacks in 1971,
it does not prove it. In general the outcome for both
blacks and females largely depends on their preferences
relative to others. Ceteris paribus, a stronger (weaker)
preference for housing relative to all other goods would
result in more (less) housing expenditures.
The coefficient for the mover variable in the renter
group is positive. Therefore, moving renters spend more
than nonmovers. Mobile families may spend more for rental
units because oC search costs. That is, since they only
36
spend a relatively short time in each location, the marginal
benefit from an additional search in terms of a lower price 
does not outweigh the marginal costs. For the nonmoving
renter the optimal searching rule in terms of the number
of searches or stopping price probably results in a rela
tively lower expenditure. For a review of search theory see
McCall and Lippman (75). In the owner group the mover co
efficient '.'as negqati.v This ma; be due to three factors.
First, transaction costs are lo..wer for less e::pensi:.'e houses
rincc reactor fees and the like are a percentage of the cost.
Hoi.ile families can economize on these costs by buying less
e::Lensiv.e Ji..wellings. Second, it may be easier to resell a
lo'w.,er priced hocusc. Third, mobile families ha:.'e little
incenti.'e to "buiy hlat dream house" and maintain it. Rather,
they aLre more liLely to settle for a simple place to li.'e.
Thie ; positi.'C coefficients for the city size .ariables
indicaLc thlt: bctlh osv'ners and renters spend m,re on housing
Ias city size increases. Since the dependent variablee is real
dollar e::penditure the expected a prior result depends on
the '.arlious price elasticities of housing goods. Since the
demand for "housing scrv.'ices" are generally .'iewed as price
inrelastic, then the c::pected effect on expenditures would be
p:ositiv.e. That is, a larger city implies higher land values
.which in turn simply higher housing prices. Given inelastic
demarid, hiihe r prices would lead to more expenditures on
housing. 7ilso, if w.e assume that all families require some
nILnimuni size house, and therefore some minimum size lot,
another explanation is possible. .Once this minimum size is
reached families can no longcr economize on lot size by
buying smaller lots. Higher land values s therefore imply
higher expenditures once this. lot size constraint is
ffectliie. The two stage estimates of the distance co
efficients are positive for both o''ners and renters but
are never significant.
The coefficients for family size show that as family
size increases renters spend'more oi housing. For owners
hov.'e'.er, family size has no significant effect on housing
expenditures.
As expected, if income increases both o',ners and
renters spend more on housing. These coefficients are well
below one on similar to those found in Lee (68), Carliner
(73) and Fenton (71).
The coefficient on the relati'.'e price '.ariable is
negative for renters but positive for owners. Gi.ven all
the problems of norhomogeneity and cautions about the
price indicies discussed in Chapter III, these results
are hard to interpret. However, the 1.2872 result for
.renters is similar to Fenton's (1.27, 1.35 and 1.09)
for all renter households, high and lower renter households,
respectively). Also, the positi'.e coefficient for owners
is not significant.
The coefficients for the "nfissin '.'ariable" in the
renter and owner equations are estimates of the covariances
clu and o2u, respectively. The negative sign, although in
significant, indicates the renters spend less as renters
than they would spend if they oi.ned. Similarly, oi.'ners
spend less as owners than they would spend if they were
forced to rent.
Finally, although the two stage estimates are theore
tically better than the OLS.estimates, both sets of results
are similar. The reason for this similarity is the in
significance of theMissing Variables. This implies that
olu and o2u are not. significantly different from zero.
Without using the two stage.iethod however, this zero
correlation conclusion could not have been drawn.
CIIAPTJP '. V
MAXIflUHl LIVELIHlOOD ESTIfU.TIOf[
Iii trod uc tion
In Chapters III and I'V' 7 housing expenditure model with
interdependent choices betw..een o'...ning or renting was studied.
Because of this interdependence the binary o'.r'n or rent choice
functions is determined simultaneous ly with the housing expen
diture equations. It is the simultaneity '.w.hich males that
model an extension of past studies on housing. While others
such as Poirier (74j and Quigley (76) have presented models
of consumer choice, they did not include an analysis of corn
sumer e:penciture equations. Similarly Lee (E), Carliner
(73), Fenton (74j and Polinsky (75) ha'e studied and esti
mated housing expenditure equations without considering the
implied choice equation. To the casual observer these over
sights are minor. A closer examination, ho..'ever, showed
that estimation of the expenditure equations is not straight
forward. In general simple ordinary least squares (OLS) on
the expenditure equations w..'ill lead to biased estimated.
Other unbiased estimation techniques have to be used. One
such procedure is a two stage method proposed by Lee (76)
This is the procedure used in Chapter IV to estimate the
model. Ilow.ev'.er, the estimates of the correlation between
the disturbances in the expenditure equations and the dis
turbance in the decision function were not significantly
different from zero. This is an important result since it
implies that simultaneity does not exist and therefore OLS
estimates of the expenditure equations are unbiased.
In this chapter a further test of the simultaneity
issue is made. To do this I start with the two stage
estimates as initial consistent estimates and then apply
a two step maximum likelihood procedure (2SML) to. give the
final estimates. Since the 2SML approach does not ignore
the simultaneous effects and the usual OLS approach does
ignore these effects, a maximum likelihood ratio test'is
employed to compare the two methods. If simultaneity does
exist thnii this test should show. the 2S;1L estimates to be
signify icitly Lbtter than the OLS estimates.
1. '.iim'miyia' s T',o Sterp l:.in.ium Likelihood Procedu're
lThe iiimodel presenrtc here is the same as the one in
Chapter. III except for tl, he specificatiron of the prices.
In Clhapter III t';.o sets of indices .e.'ere used, one for
o'.wning a'd one for renting. This clhnpter only uses cne
set Cof Cg'ij):re,.ate price itddices for bo th iwners and renters.
A different specific:i tiocn '.as emnplo,'/ed because the positi'.'e
coef ficient for the .price :.'arL ible in the owner eqluatin
obtained in Chapterl I" is lIard to, justify. Each h household
h1ia associ'atdc with it ] housing price inde:: and a price
inIle:: for all gcqoods. The purpose of these indices is to
stanJardize housing e::penditures and incoine for different
cost of housiingr and cost of living across the U.S., res
pecti.ely. Gi.'en this price specification, the relative
price of housing services w'.ould not enter into the rentovwn
decision function. It o.'ould, of course, influence the
allocation of the budget between housing and all other
goods. iore specifically', the model studied has the follow
ing specification
(1) crn in + r n
(2) C2,1 2n L 2n
(3) = ,
n I n
,.'here Cn .ire annual expenditures on housing if the family
owns, C2 are annual expenditures on housing if the family
rents ard (r 'c are tria'.'riate normally distributed
H1 (0, ) Here Q is a 3 x 3 co"ariance matrix. Without
loss of generality the .variance of E is assumed to be one.
I is an unobLserv'.able index. Also, assume C and C, are
n In 2n
mutual l e'c;lusiv' e and cannot be observed simultaneously
for any one ind '.'iual. What is observed are oxogenous
.ariables , the binary inde:: I and e::penditures
Iln 1 n n L
on housing, C such that
(4) C = C ; I1 = 1 iff I 0,
(5) C C, ; I = 0 other..ise.
n 2.n n
The price indices were constructed from table 128 in
the Bureau of Labor Statistics (BLS) Handbook of Labor
Stattistics 1972 "Consumer Price Inde::, 23 Cities or
Standard metropolitann Statistical Areas, All Items and
liajor Groups, 19741971." This table gives six price
indices including ai housing index with a base of 1967
= 100. T'.entyone of these SMCSAS (Honolulu and Washington,
D.C. ..'ere excluded) were grouped into one of four regions:
Northeast, North Central, South and West. Two average
price indices and a relative price index were then cal
culated for each of the four regions and assigned to the
appropriate family. The rast of the data are from A Panel
Study of Income Dynamics (72) on a sample of 3,028 families.
In the expenditure equations the dependent variable
is logarit}hm of housing e:pendiltures, di" ided by' the regional
price of housing inde::. Th'" same tn explanatory '.ariables
are used in thll t.wo eX.pend iture equations. They include
the personal characteristics of the family' head (age, race,
se::) ; fa;mi 1'. bac.: round (mover, logarithm of family size);
loqgaithm of family "tcrmanent income"; regional variables
(logarithm of city size and logarithm of distance from the
conter of the city) and a relative price index of housing.
.'*11 these e;.'pl nnatory '.'ar able e..cept the relati'.'e price
of housing are include inue n the decision function. For a
more detailed description of these 'variables and their
e:.pected effects see Chapter III.
The model can be re..'ritten as the following switching
repression model
*I6) C = "'" E. + iff :'
n IIn In n I
7) C = + 4C ifF Z y F n
n an 11 1 
.A\ssulime that : L, and e are tri'.'ariate normally dis
tLibuLed wi th zero mean and co'.'arlance matrix ? such that
I 2
'01 12 2Ic
^ l a2 2
o7 1
L 
As noted by. Lee (76), all thi, parameters in the above model
are identifiable except n12 cov'c I' 2). The maximum like
lihood procedure described in Amemiya (74) is used to esti
mate this model. Followinrg'Amemiya 's proof it can be easily
shown that those estimates are asymptotically efficient.
Let f and f, be the jointly normal distributions of
(51 ,:) and (, ) respecti.',lv. Dropping the n subscripts
on C, i ], 2, Z and I, tile likelihood function for this
model is
(S) L( IB 2 'O 1 ', 2 ') 0
T Z',
T 2
= II [. l C 1 )d?] f 2 (Cj122' c)d ] 12I
S1 (pc:.; I )q1 (c'Y I d. ]I ,
n=n=l1
n 1 1 i ( 1 1 f C C J ( X. )d 1I
S '" i\2 27 2 2 2
i 1 (CX11) /f1 1Cx1)JcII[g2(Cx22) f~2(c CY22)1I
i = 1'.'. "
1 1
(C: ) = e::p (C: )2
TI I
1..n 2 2
!, 11'1
fl(c C::1 1) 1
.1 11
cxp (  l (CXlB )]2},
2(1Pl ) 1
44
f2(EICX2) = 1
2 /2 / 2
1pq
2
exp 2 [E P2 CX( 2
2(1pq ) 2 % 2
where'p1 and p2 are the correlation coefficients of (cEE)
and (E2,c) respectively.
The integrals in equation 8 can be simplified by trans
forminc the variables. Following Freund (62),. if
"y
F(Zy) = / f(E)dE,

then given a new variable t such that "
1
t = h(E); E = h (t),
where h(E) is increasing, one can transform variables and
obtain
h(Zy) 1
dh (t)
G[h(E)] = / f [H( t)] dt.
dt
Ey selectinrq li( ) as
t [> (C:'. ,l
I 1L
and sol'.'inq for G.[ () ], thL first integral in equaLtion 8
simplifies to
2
S I 
L (dt.
S2ii
A sinlil ar analysis holcs for the second integral in equation
P. Ilnial. ly afctern. making these t..'o transformations and
taking .loqarithms one gets
( ) L = In L(e ',l,2 ,, l1, 02 , 2P ,
T
I ln(q] (C:; 1. )) +
t=l
in 'I (t) dt)
(1I) []Jn(q2 (C .' C' ) + In ( .f.(t)dt).]}
wherc P'( t) is the standard iiormal density function,
1 . r
S2 01
('2 L2," '2'2o) __ [ , (C 2e2) ])
S2 2
In ordLe to calculate the maximum likelihood estimates.
a numerical optimization algorithm must be used. Since the
likelihood func lion dcri'.'d above is nonlinear these com
putations will be :quite complicated. However, a simplified
procedure discussed in A;memiya (74) is available if con
sistent estimates of ,L,':2,,',o l,,' 1 and 2 can be
obtained. Let (D0 be these consistent estimates. Amemiya's
"t'.wo step maximum likelihood estimates" (2S1iL), 0 M, are
calculated from
InL.(11) I. nL ( )
(10) ', = )[ L
l U O ,: , "
The first step of A.'memiya's procedure is obtaining the con
sistent estimates 0. These can be found using a two stage
method described in Lee (76). Lee's procedure uses probit
analysis in the first stage and OLS in the second. The
details of this procedure are found in Chapters 2 and 3
r
of Lee (76). The second step of Amemiya's procedure is
substituting 0 into.(10) and solving for 0M.
2. A Comparison of OLS and Maximum Likelihood Estimates
and a Test for Simultaneity
Tables 1 and 2 present the probit and two step maximum
likelihood (2SML) estimates of the decision function res
pectively. Both estimation'procedules yield similar results,
but the 2SML estimates have slightly smaller standard errors
for all the estimated. parameters. Allthe coefficients are
significantt and ha'.'e the e::pected signs discussed in Chapter
III. This implies that all the included variables have a
significant impact on the buy .'ersus rent decision.
Table 3 presents the t'.'o stage and 2SHL estimates of
the e::l:pernd:it ure equat ions tor ,'..'ners. Table 4 contains
similar estimates for renters. is e::pected the asympotically
efficient 2Silj. estimates generally .have much smaller stand
ard error s tha t the t.'o stage estimates. Hlo'..'ever, the two
sets _of estimates are quite similar with three exceptions.
First, thie m .'ver coefficient in the o:.'ner equation changes
from the tw'o staqe estimate ,of .04407 to the 2Sill estimate
of .1 61.99" and becomes significant Second, the city size
coeFficient in the ''l ner e]ua tion decreases from the t'.'o
tage estimte of .065C2 to, the 2SnL estimate of .04237.
In terms of the 2SIML standard errors, these represent more
than a three standard error change. Third, the distance
arablee in the renter equation changes from a positive
TABLI 1
Probit Estimates of th., Decision Function
\' I.. L E S CSTI ATTS STArIDARD ERROP
Constan t
Aqe 35
36 .: aqge : 6;
Black,
Female
In (City Sizo)
In ( Distnce from
center of city)
in (Family Size)
1n lati"e
permanent income)
3.2r6292
1.17Q45
0.6 .3 96
0.30715
0. 32102
0. 9.1359
0. 14969
0. 12779
0. 13134
0.69437
0.47282
0.1075,1
0.09768
0. 07131
0.06729
0.07093
0.01763
0.02721
0.04929
0.05168
TABLE 2
Two Step Maximum Likelihood Estimates
of the Decisicn Function
VARIABLES ESTIMATES STANDARD ERROR
Cons tant
Age < 35
36 < age < 64
Black
remale
 rri, m I e
flo'.'e
I.n CI ty Size7
1n (Lr i t .rin c e ) o m
center', of citLy)
In i[aniily Size)
in P i co
r imnc n>ti L income)
3. 24123
1.16230
0.63151
0.30971
0. 31895
0. 94527
0. 14991
0. 12 864 .
0.13381
0. 6 3159
0.47183
0.10681
0.09711
0.07076
0.06625
0.07053
0.01755
0.02700
0. 01904
0.05155
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si nll to t i"e. sia.n nd ':...comes s iqni ficant. CGi'.'en
Chapter. III's is us ioi.i on l .:rid .al, es .aid cl ca t .ion this
neigati .'a sn scenes more appropsri, te. i itli the ex::eption
of price in the renter jequrtion,', all the .riL c e fficier, ts
ar,. siq r i if ri nt. I s th, : a ve h .: te s s i s 
.cuissecl in Chaptr III .
ithl u.: li i [ :st [.:.a .LE suci aus .ch i':, Carliru 3 ,
Frni to n i 4i} ani rolinsk. 7. sp d iC si ii t] di fcr
,nt .xI iCl iditur: euations F f ..' c Mlpa sons are pc'ssib'l '.
ITe er ,riii t inc,,m, ci 'L ic i t L i rites. *f t.':" ? fo r
o'..'I ..rS ]and .lOJi. for r n tr arc .:u re C lose t' Ca ir l ir r' s
1
Lst nats of 31 .:r o..rneirs .:inc:] '. 0 moi : ,EVncr. S. rcn to'n
estlimrated perranenrit Ii u e plastic ities *f 1i, 16 arrid 16
[cIc all re, tu househlcds and high an i. isc,"TI r1ent er
hlciusehol'ds, r [cti'. Le' i ",.'.: r ?[''L '1 se'c.'iL l s
of .income el sticity c l;.ri atus. His *.:= ir.at s of p rPf nrr i,'
i.n.:,oinr1 e l a:s ti ci L t r oC o C.n:. I rt s ,nrigqc f iom ...0:. to '.2 '. L
renicIrs hi [.rma 13 1,2.it income elasticity: estimates ranged
from 387 to 6 2. Pol insk; c nci'lud s tliat tI2 true irnc .
clastici t,' is appr :: irna t l,' 5 f .c b'tl r n teirs a iid C'...'rier s .
Finally', the co efficient ts for the re lativ.'e r:.rice :ariab es
are .n2qaLi'.'e. l'] se coeffici'en s :re riot reliable e.tinrates
of trad i io ai 'ri, c lastLicit l'rC r ntl:age clhaiin je in c1uL nti tv
[)er pe rcen ta c n qe ii [ '' : This is tri.r": for ti.'c ru s sr s .
C r 'rline"r U.1 S tie sanec data h.base a.= 'ii C]c. ilis s tc Limate s of
. 631 arnd 51'0 aire based oin a re ;riess '1in usiing fIour year
'.'eraiCi .i.nico'nie, re' la iv 'e p1r ice o:f Iou s i rn. :nId di i r s f, 'r
Llack, f:iiale ind ,a e is c::planato'y '.' r ii ] r l o. l'i
cocff icic' ts 1 for the BlacIJ :, female and anci cluiiic ::, are
si. r, la r L o, I'l' e s ti lna .c s .
First, the dependent variable is deflated dollar expendi
tures on housing. The price coefficients are therefore
percentage change in deflated dollar expenditures per
percentage change in price. Second, the poor quality of
the available price data makes any estimate of price elas
ticity equally poor. This would be the case even if the
dependent variable were measured in units of quantity.
The negative signs of the two price coefficients do indi
cate that families 1. ing in those regions .. here hoi_is in
is relati'.elv i.to al 1 other g ods) more e.:pensive tend to
spend less o:n housincq.
It '...ould be useful to compare the S'lfL estimates of
the e>:[pe id iture equations with O LS estimates. The OLS
estimates .ere oltained b.' di.i ng the sample into t.wo'
grouf'.s ow..rers and renters and appli..'ng OLS to each
grCoup separately. These results are presented in Tables
5 a'ni 6.
cc:mpar iscn of Tables 3 and 5 shco.'s the 2SI1L and
OLS estimates to le: sligihtl '.' different but co nm a tile.
One e::ception to this cormpr: tibility. is the distance V ari
able coefficient. The '.alue obtained w.;ith 2SI1, is five
L .
times greater than the OLS estimrlate. Also, the 2 S1,
estimate is igiifican t a=id the OLS estimriate is insignifi
cart.
comparisorn of Tables 4 and C aqain shows a difference
in the distance coeff icierit. Thle 2Sl1L ,estimate is negative
and significant. The OLS estimate is .ositiv.e and inrsinificant
All the other coefficients lc nearly the same.
IWhile these casual comparisons do reveal some differ
ences in the t'.wo approaches, a more detailed investigation
on w.'hetlier or not simultanra.t occurs is possible. This
investigation in'.ol'.'es the use of a maximum likelihood ratio
test. Consider the null hypothesis that there is no correla
tio! b t..'een the disturbances in the.housin.g expenditure
equations and the decision function. That is, the null
hypothesis contains rl, and 2 to zero. The maximum
likelihood estimates of the expenditure equations no,., reduce
to simple OLS estimates, '...hile the maximum likelihood esti
mates of the decision function are profit estimates. Denote
these estimates as 0 Tne alternative hypothesis does not
constrain 1, and nL to zero. The maximum likelihood esti
mates are no'...' the 2Si!L estimates presented in Tables 24.
Denote these estimates as Given *' and C.I the ratio 2
In h< as a ,. distribution asymptotically. For this
model the ratio is 6.771 and significant at the 0.05 level.
This implies that simultaneity does occur, albeit the e"i
dence is weak. It then follows that OLS yields biased esti
mates of the e:.xpenditure equations, eren if this bias is not
very strong. Therefore estimation procedures such as the
two staqe and 2SrIL approaches, v.hich account for the simul
tarn eituy, are more appropriate.
TABLE 5
Housing Expenditure Equation of the Owners
COS
VARIABLES ESTIMATES STANDARD ERROR
Constant 1.49477
/1qe 35 0.15 45 0. 0413
3n .ae 64 0 i 520 0 .0359
Black 260 0 i: 057 : 9
F.i .J e 0 I. i: 3 2 0312 9
ro'.'er 0.09916 0.03732
SIn (C it Seze) 0.05.922 0. 00.572
In i c)istance roi n
cen er of cit ') 0.:001 563 0.0 3646
In (Fainir,' Size) 0. 03752' 0.02284
In ( Pel l : .'
pi.ice of Lousini) 2.62209 1.085
In r, n.' ..
p trm ian ieni t income) 0.5c399 0.02079
TABLE 6
Housing E:penditure Equiition of the Renters
'.'/.IABLES EST I PATES STAllDAPD FPPOR
Constant I: 1.77267
Ag 5 0. 13423 0. 05946
36 C : age 64 0. 12172 05634
Black 0.19701 0.0465
Female 0.11751 0.02936
l.o'.'e 0. 13378 0. 02877
In (City Size) 0.06964 0.00778
In I [Distance from
center of city) 0.00698 0.02998
In (Family Size) 0.08309) 0.02050
I.n i Re 1 In t 1'.'
price of housin.:j) 1.81576 1. 36963
In ,n1
pc.rian.ment income) 0.50212 0.02368
Estimated slandalrd error of disturbance , = 0.4348
CHAPTER VI
CONCLUSIONS
The.model derived in chapter three of this thesis begins
with the premise that "housing services" are not a homogeneous
good. Father, hoIsing is ,'iew'.'d .As a bundle of several commod
jties such as location, size of house, tLpe of structure,
neighbors ard other humanities. This "ie..' of housing is some
.hat different t han the Olsen (69) approach '.here homogenitL,
is assumed and the Fing (7i6 approach '.hich estimated demand
equations for Lancastrian characteristics.
It is then aLcqued that some bundles are cheaper to con
sume if the family buys their ov.n home. Conversely, other
bundles may' be cheaper to consume if rented. A family is
.assumed to. consume that bundle which maximizes utility.
Follo.in. this line of thought a general model of consumer
choice bet'..een o'ninq and renting and ho.' much to spend 
is developed. The model is a sitching regression model
'..'ith a choice function that is not assumed to be independent
of the expenditure equations. This differs from past studies
of housing '.'hCLec independence is tacitly assumed and simple
OLS is applied directly on the o''ner and renter e:xpenditure
equations. Gi'cn that some simultaneity is present, these
OLS estimates will be biascd. Other consistent estimation
procedures must be used. In this thesis consistent estimates
of the model are found wi ith F' "two stage" technique and
a two stop maximum li ke li hood procedure.
Both the "twoo stage" and "t..wo step" procedures perform
quite well in terms of standard errors and e::plained 'ari
ation. For example, all but one of the coefficients esti
mated Li the tw'o step procedure] e are significant at the 57
level. Only the estimate of price elasticity in the renter
equation is insignificant. .'Aso, w~lth the e:.:ception of the
price elasticity estimates the empirical results are com
patible :.'ith economic theory and other recent studies. The
two step estimates of permanent income elasticity of .60678
for owners and .50046 'or renters are significantly less one.
These estimates support the results found in Polinsky (75),
Fe:nton (74i Carliner 173) and Lee (69i but differ with
nutlh's ('30' results. ILuth f( 60 uses a l rej ate time series
data and concludes that the income elasticity for desired
housing stock is at least 1.0. The two steps estimates for
owners indicate that families living in rural areas, families
w.ho frequently mo1''e, and families headed by a black tend to
consume smaller units of housing. Con'ersely, large families
and families headed by either a female or by someone over 64
tend to consume lari2er units of housing. The results are sim
ilar for renters with one except ion. Families who frequently
nmo.ve tend to consume larger units of housing. The two step
estimates of the decision function indicate that families
living in rural areas, families who frequently mo,'e, and
families headed by a black, female or someone under 36 are
the least likely to own:. Conversely, the larger the family
size or the higher the permanent income, the higher is the
probability of owning.
The only discouraging results were the estimates of
price elasticity. Two different price specifications were
tried and both yielded unsatisfactory results. The specifi
cation used in Chapter IV resulted in a positive estimate
cf price elasticity focr ow,,ners. The. specification u sed in
chapter f iv'.e resulted in a nlega ive' but elastic (2. 59 for
ow.ners and' 2.11 for renters) estimate of price elasticity.
This ela.stic estimate is not supported b recent studies.
Fo r e..am ple, Pol ins.y .75) concludes that housing demand is
price inelastic. These poor price, elasticity estimates
sugglcest that price data should be collected by the Sur'.'ey
Research Center's ne:t "Panel Stud'.."
ito test the hypothesis that simultaneity bet'.'een the
e::penditure and choice equations does e::ist, the two step
ma::imum likeli hood estimates are statistically compared to
the u.uail COLS estimates. By using a maximum likelihood ratio
test to make this comparison, evidence ..,as found that simul
taneity doe. e_:ist. Therefore, the procedures used in this
thesis to e.=stimate the e::penditure equations are more appro
priate than 'the usual OLS approach.
Finally, a few suqgge. t ions for future research are
'.:orth making. Firstly, a housing demand model using the
entire f'.e year ?ur'.'ey Pesearch Center sample could be
deri'.'ed and estimated. Thi s '..could not be a simple chore
since the Lrquired estimation technique needs to be workede d
cout. In general, the procedures used in this thesis would
not be appropriate to estimatl:e a model based on pooled
crosssection and time series data. Secondly, a model that
described li'.'h families move '..'ithin a qi'.'en area could be
foLmulated and estimated. 'The underpinnings for such a
model are presented in the Appendi:x of this thesis. Lastl.y,
tlie computer proq ram used to obtain the t'...'o step estimates
has many applications. For example,'it could be used to
estimate a model that explai ns electric utilities' demand
for oil and coal and ''hy these utilities "switch" from coal
to oil and '.ice v.orsa.
APPENDIX 1
DISCUSSION OF THE DATA
The majority of the data were constructed from the
Survey Research Center (SRC) .sample. The only exception
was the construction of the sundry price indicies. These
indicies were constructed from the Bureau of Labor Statis
t1cs' IlI lnd: o.Il ofl L.ibor Statistics 1972. For an e:plana
tion of lhi'.' th,?se indicies '.*'ere formulated see Chapter III,
cctiron 2 Dnd Chl.pter '.', Section 2.
lThe ori. i1~l SPC simple cacm from t.o sources. Abo.t
10 percentt of the f:mrilies first inter'.ie'.'ed in 1969 had
been intcr'.'io*.'cd pF e'.'iously in 1966 and 1967 Iby the Bureau
of tllh Ccunsus .is pirtt of the S'.ilrey of Econcmic Opportunity
iSE 'l lhe repal.in in 60% con listed or a crosssection
smr'pi[plE rof ']'E IlinigCs of the coterrmiinouI United States. The
19).69 tr, 1972 sacrimples consisted of all panel members living:
in tamI ilies f.h.at '.*.ere inter'".'i:'.'ed the pre'.,ious yeir and
nc'.,ly foimcd famijilie containing ..ny adult panel mcibbeLr
'.'hoc had mn'.'crd since 1E968 f rom a sample fmnlily. The res
pondcnt '..'.is usual]' thc head of the family.
,'11 of tlhe origlinal SECO families selected in 1968 had
income s in 1966 equal to cr belo'..' twice the. federal poverty
line at that time. The selection formula w.'as $2000 + r (51000)
'...h1ere I is the number of indi'.'iiduial s in the family. E::cluded
were families where the head .'as o'.ver 60. Also, except in
the South the original SEO subsample only included families
w'ho live'cd in SlISA's. In summary, the 1968 interviews were
taken with 4802 families by'SPC, 1872 from the SEC selection
frame and: 2930 from its crosssection samples.
Li.en though the SPC sample seems hiased tow..ard poor
families in 1968, by 1970 the data 'w.'re a fairly represent
ative cuoszssectionh of families. This can be seen in Tables
1 and 2, w.lich were taken from The Panel Study of Income
Dynamics (1972 '.'olume 1. These tables compare the SPC data
for 1970 to i corresponding national sample taken at approx
imately the same time, the 1970 Current Population Survey.
Variables used for the comparison were family money income,
race, sex of head, size of family, number of children under
ei llteren in family aqe of family head, and size of place
of residence. For e:ample, both distributtions of family
income shio'. about the same percent below and above S5,000,
though SFC does have a slight underrepresentation of very
poor families (Income under 51,000). The comparisons according
to the demographic variables are also close. There are a
few notable exceptions. For black families SPC does have
a slight loss among two person families and those with no
children under aic eighteen. For white families there .is a
slight undcrepresentation of those with heads of families
fortyfive or older.
Finally, this thesis only uses the 1971 data. The
sample consisted of 3,452 families. E::cluded from this
TABLE .
Comparison of Census and SRC Data:
Total Family Income by Race, for all Families
family Census S
income White Black White
$1000 3.0, 6.9' 1.5
1999 6.4 13.0 5.5
2999 5.9 9.9 6.1
3999 5.8 10.4 5. 5
4999 5.4 9.0 5.9
32.0 33.2
24.0
17.4
16.5
15.4
23. 4
20.1
12.1
5.8
$50007499
$75009999
$10,00014,999
$15,000 or more
99.9 100.3
Total
money
Under
$1000
$2000
$3000
$4000
RC
Black
S2.7
13.0
14.5
10.0
9.0
18.1
12.4
14.9
5.4
99.9
100.0
TABLF .?
Comparison of Cenr u.s and SP.C Data:
Pace by .'Various Demoqgraphic a.'ariables,
for Families with Tw.o or More Members
C Th IN u SRI
Sexc: of family head White Black White Black
MIale
Fema.i i
Size of family
89.7
10. 3
90.0 71. 7
9. 1 28.3
00 .0 100. 0
69. 1
3 0. 9
100.0 100.0
2 [personsS
3 persons
4 persons
5 persons
6 person s
7 persons or more
l.imber of children
One
Three
Fouilr or more
[lot ascertained
.'. of f aimi y hI e, C ad
under 25
35. 2
20.9
19. 7
12 .6
6.5
5. 2
28.7
19.9
16.1
11.7
14. 1
100. 1 100.0
142. 9
18.2
10. :
9.1
30. 5
10 .8
17.1
11. 1
21.
99.9 100.1
100. 1 99.9
Under 25
252 c
30341
35 1
3I5 5 .1
*15 5.1
5561 ,
65 7.1
75 and over
99.9 100. 1
Si.ae of place of
r es i dence
Sonmetropolitan
[ionmet ropol i tan
63.7
36 .3
99.9 99.9
63.3 7.5.9
36.6 24.1
99.9 100.0
72. .
27.4
100.0 100.0
35. 3
20. 3
19. 5
12.0
6.7
6.3
22.4
21.3
16. G
1 4. 7
: .3
16.6
29 .
19. 3
19. 5
10.6
10. 7
0. 2
100. 1
2 1 5
1 9. 6
12.3
1.6
99. 9
6. 6
10. 5
.9
21.2
21. 3
16. 3
9 4
4 7
9.3
11. 6
11.
21.9
19.6
1 C
15.3
i.
3.E
7.7
11.0
9.o 2
23.2
20. 2
15. 5
9 .4
3.7
7.7
10. 2
12.0
23.2
21.2
12. 0
9. 9
3. 7
sample were families who at any time in the five year inter
view period lived outside the continental U.S. Also excluded
were those families who at any ,time during the study either
both owned and rented or di not own or rent. This second
group included families who received housing free from a
friend or relative, or received housing as part of their
wage. 'The final sample used for estimation consisted of
3,028 families, 1,225 of these rented in 1971 and 1,803 owned
their o'...rn homes.
APPECDBIN: 2
L:;E:TL'SIOI OF TIL' fICDEL TO T1CIE HULTIPERIOD CASE
In this secti.on I will assume there are only two differ
ent bundles of housing goods. One of the bundles can be
owned, the otler rented. Ii .ight of chapter three, this
assumption rmay' seem quite restrictive. It is not. The re
sults derived in this section apply with equal force to the
more genectal case of several housing bundles. The fl.lowing
notation will be used.
U = U(::,Ih is the household's utility function, where
h is the .ow.' of housing services received per unit
of time when the house is owned and < is dollars of
e:.:pendituire .per unit of time on all other goods.
U = U(::,hr) is the same as abo.e, except the housing is
rented.
l, = h l. (t) is the flow of housing services per unit of
time by an owned dw.'elling of initial size IB. The
depreciation function B(t) has the following pro
perties: (0) = 1 6 (t)0 as t",, and < 0.
hl = h .(t) is the same as the above, but the initial
size d,.,wellinlc, hI is rented.
pB(t,ll) = per unit price of owned housing.at time L,
p (t, IU)
'..,here   = PB (') 0.
icLre B n
PR(t,hp) = per unit price of rented housing at time t,
where p' () 0.
P. >
Y(t) = family income receipts at time t.
CB(hB) = transaction costs of buying a home of initial
size h These costs do not include the time and
money costs of moving. DCB(hB)
_h  Ct(h ) > 0.
Th B B
B
mb(lhB [t]) = transaction costs of selling a home in
period t. Again, mb(') does not include the time
and money costs of moving.
3m (h 6 [t])
h 6t = 6B(t)m (hb [t). > 0.
k = time and money costs of moving. For any family, k
is the same for all types of moves, That is, it
makes no difference if the family; mo'.es from a
rented house into an o(..ned house, or vice versa.
f(t) = a probability density function of li'.ing in a
gi'.en area for e::actly t years. Ef t) is defined
o'er tlhe int:erv.al 0 : t .: t w.'here t is the
m m
household's time to retirement. Ech family is
assulIcd to possess such a densi t:y function.
For s ir ii ci ty, I first assLue that the family can
occupy only one d'..elling during their stay in any given area.
This as sumpt ion w'.ill be relaxed later.
Follo'..'inq lith (71) if %'e assume no time preference
o'n the part of tie household, its utility aggregated for t
years over the occupancy of an o.'ned d'ellitn of initial size
h is
13S
nnL) = L1 :.:(.v) 1,h ]
1: B B
Apart from its initial '.ealth, the present discounted
'.alue of the stream of wealth up to time t for the family
who buys h1 i .
i i
:(t) y( h BPBI. ,h)6 B(')e~ 'd. C (hl
It
mb(h 13B1 [t]"e
where i is the household's discount rate.
In selecting an owned dw.'llin characterized by initial
size hp, assume the family maximizes
(1 C (I) = i'm U (t) f (t)dt.
The maximization is subject to the constraint that expected
v.'ealth when the family moves from the area is equal to 1:, or,
t
(2) C('..') = I m ...t)f(t)dt = Y h X T = k,
0 D D
where:
t t 1
y = m I y ,., e d.]f (t).dt,
0o o
BP = l'm [ tpB V 'hB )"B( ie "' "]f(tdt,
X = f m [/ xt:('7)e de f (t).dt,
0 0
T = C (h ) fm [m (hBC (t))e i lf(tdt.
Let L = EfU) + E (W) where is an undetermined multiplier.
The household selects the function x(t) and h so as to
maximize L. The function x(t) must satisfy
f(t) t U L.'))d.' nf(t) t e df' = 0
o A 0'
and upon differentitiatinq with respect to t
it
(3) U (t) = ne
:\
Differentiating L with respect to hg yields
t
(4) / [f 0 Uh B (v)dv]f(t)dt
(4)m [ft (pB(Vl1) 6B(V)
0 0 i dfc b
+ hgp (v,h )(p (v))e dv]f(t)dt
tit
{C (hB [r (i) + B6 (t)) 6B(t)e ]f(t)dt} = 0
0 B I B
Dividing by p we get
Ii ,
m [.i t b
0 0 .: ('') ]f (t)d t
/ m It (p ('.',h ) (.) + h p (vh )6 ('.'))e d'.']f(t)dt
t it
C' C h ) + m (mr n h (t) ) (t)e ]f t)dt,
or, substi uting u = and combining terms,
..t h %. 
P,B
(5) .m [  IC (.) ] ('.v)e 'dv )f (t)dt
L t
It
S ,, U _t L.
= C (h ) + 1 (hp [m (t)), (t)e ]f(t)dt,
',here I1C ('.') = P ('' h ) + h P ('.',h ), the marginal cost at
time '.' of buying a larger house of initial size h. The
cerni insideL braces in (5) is the '..'eighted average ('weighted
by ('.'.e.. ) )of the difference between the maLinal rate
P 
of cubstitution of housing for other consumption and the
relati'.'e (to other goods) marginal cost of ov.ned housing
ser'.'ices. The right hand side of (5) is the e::pected pre
sent discounted marginal transaction cost of buying (and
selling) h . The higher the probability that a family
will ino'.'e in the near future, the higher are these trans
action costs. These transaction costs do nothing more than
shift to the left Chapter IIl's "consumption possibilities
frontier for o'.ning." When (3) and (5) hold, the family
maximi :es (1).
Let the optimal solution to (3) and (5) be E(U*) where
t
m
( ) ) = (I*) I'* (t) f t)dt.
b n b
If the family chooses to rent, they max:imize (1) subject
to
(7) E U ) = Y P =
S m t i
pR = [..t p '. (v I .")e ]f(t)dt
n r, P.
Again, the function :(t) must satisfy (3). The optimal hR
must satisfy
t h
(8) / I [ I C (_.'I 1. ('.'e d'.'d f(t)dt = 0,
0 0 U R R
where 1CR(') = Pp('',hp) + h pN ( ',hR ), the marginal cost at
time :. of renting a larger apartment of initial size hp.
Let the optimal solution to (3) and (8) be C(l~') where
r
(9) CI(U* = i m U* (t)f(t)dt.
r ,1 r
*The farnil y '..'ll choose to own if (6) is larger than (9)
Simi larly, thle' will rent if (9) is larger than (6). In
qcneoral this choice will depend on U' (x,hb) U(x,h ) 6B(t), 6R(t)
P (), Pp(), y(tl, C (hi ), m (h (t) ), f(t) and the families
discount rate i. Ceteris paribus, a higher T or a higher
f(t) in early pcriods will slant a family's choice to,.ard
renting. This is consistent with the "mover" variable
discussion in Chapter IV. Prices and preferences enter
the rentown decision in a manner similar to Figure 1
in Chapter III.
Up to now I have assumed that the family must occupy
the initial dwelling until they move from the area or retire.
More generally, there will be some optimal number of moves.
That is, it may be optimal f6r the family to first occupy a
rented dwelling of initial size hR(0), and at some time, .tl,
move to ah owned home of initial size h (1). Conversely, they
may want to own first and rent second. If we assume that there
are only a finite nuLmblr cf moves possible, then the family
must choose mcv.in periods t t, . t (t t m and h
so as to rnma.;inmize
Li t L. t
I 1
t, t t,
+ I. U (:(v),h('.'v j_.' f(t .dt + . .
t t t.t
1 1 2
SI [r lm L (v ) h 0' Jdv I] f ( t)dt
t t .'t.t
In nl m
.'herl e l(vi, is either h, or h .r The nmax;imi nation is subject
to the cons'taint that expected w.'ealth when the family moves
from the area is again a constant I], or
(11; [EU = :. (h(1)P(1 + . h(n P(n)] TTC = K
t t t _
hl(llP( 1 = r p (v,h ( ) )6 (vie d'. If t )dt,
t . t t ,
(2 PI = i p(v,h(2 (v)e dv f (t dt, etc.
tl tl 2
TTC = total transaction costs of all moves. These
costs depend on w.lhen the moves take place and
the type of move.
Also, hi, p, 6 are either h' p 3, 6 or hp, pp or ( i. )Whi le
the solution to (10) and (11) is not straightforward, in
general the optimal strategy will depend on prices, pre
ferences and the density function f(t). For example, if
the optiimal strategy is t tm (and therefore, L,. t2,
S. all equal zero), then the solution reduces to (3)
and (5) or (3) and (8).
In this section I have presented a multiperiod model
of housing demand. It extends :;uth's model in four important
ways. First, I hnve allowed for t.w.'o homogeneous bundles of
housing services rather than cone. Second, iluth assumes the
price of housing is constant. In my model price is a func
tion of quantity. Third, I e::plicitly distinguish bet..ween
mov incr costs for owners and renters. :1uth does not. Fourth,
I allo'' the fara.ily to plan mo'.'es while still liv.inq in a
gCiven area. .luth treats moves "within the area" the same as
moves "away from the area." Therefore, in his paper each
time the household mo'.'es prior to its retirement, it repeats
the decision process under the conditions then prevailing.
In other %.orcs, flulth does not allow for the optimal moving
strategy implied in (10) and (11).
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BIOGRAPHICAL SKETCH
.Robert P. Trost was born on June 7, 1946, in Rochester,
S'..' York. le g. rld uated fro:nm .'.uinas Institute Hi. h School
in June, 19,I' lie received a Bchelor degree in Mlechianical
iEni ineer inii firom the UniveLrsity of Detroit in May, 1969.
. ipon ..ra iatingq, le ser"ed in the United States Marine
Corp i :,s f[ i co n June, 19'9 t o A ril, I1971.
In MarL. ch, 972, Mr. Trost enrolled in the GradJuate
Scliool c:,! tlie LUni"ersity' of Flori.ida to pursue doctoral
stud ies in the Department of Economics. He has held research
and teachiniii assistantshi ips in the Depar tment of Economics
from Tune, 1973 to March, 1977.
I certiFy thlit I h.a"e r:1 l this study .and that in Imy
opinion it conforms to accep'Jable standards' of scholar 'r ,
resenl. nation and is fully a rd.quate, in scone and] quality,
as a dissertation for the rdeiree of Doctor of Philosophy.
G.S. S l dda la a, Clha irma n
Professor of Economics
I c: r t i f that I Iha.'" t.eir this s ttrlyd anl thiat in m,'
opinion it conforms to acct.ptable s tandrarlds of scholar) ly
uree. n tat ion and is full'," .,:'adequ1at, in scope and quality,
as a r di.sert.ation for the degree of Doctor of Philosohy.
Pobert D. Emerson
Assistant Professor of FooCd and
Resource Ec onomics, IFAS
I ccirt if. thlit: I h1 .'1 read thiis st ud', arid that in my
opinion it confo'rmr to arceptiblc r standards of scholarly
presentation and is fully. adequate, in scope and quality,
as a di sertation for the de.qee of Doctor of Philosophy.
S'.,., 1. '. I i i. l .1ii,,_
Je'iome WI. Ilill.iman
Professor of Economics
I certify thLnt I ha'.'e read this cturl, and that in my
oniniron it confcrnm to acceptable standards of cholarly
niesentation and is fully adequate, in sccoe and qualityt,
as a dissrta.tion For the degree of Doctor of Phil os;nrhy.
Sa','mnd B. Roberts
Associate ProfessorL ocf Economics
This dissertation was submitted to the Graduate Faculty
of the Department of Economics in the Collecge of Business
Administration and to the Graduate Council, and was accepted
as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
March, 1977
Dean, Graduate School
