Estimation of a housing demand model with interdependent choices about owning or renting /

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Estimation of a housing demand model with interdependent choices about owning or renting /
Trost, Robert Patrick, 1946-
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Analytical estimating ( jstor )
Consistent estimators ( jstor )
Estimation methods ( jstor )
Housing ( jstor )
Housing demand ( jstor )
Income estimates ( jstor )
Mathematical variables ( jstor )
Maximum likelihood estimations ( jstor )
Modeling ( jstor )
Price elasticity ( jstor )
Dissertations, Academic -- Economics -- UF
Economics thesis Ph. D
Home ownership -- Mathematical models ( lcsh )
House buying -- Mathematical models ( lcsh )
bibliography ( marcgt )
non-fiction ( marcgt )


Thesis--University of Florida.
Bibliography: leaves 72-73.
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Statement of Responsibility:
by Robert P. Trost.

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rI Dl :L i.1 ii lii.Lf PGDEfI liLif C-iOIC' S

Pobert P. Trost




3li 1i262 08552 5276 11111I
3 1262 08552 5276

To my parents


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-

t-ion under cirant 5OC-76-04356 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.



ACKNOWLEDGEMENTS . . . . . . . . iii

LIST OF TABLES . . . . . . . ... vi

ABSTRACT . .. . . . . . . . vii



CHOICES . . . . . . . . . 7

1. An Interdependent Choice Model Based
on Utility Maximization . . . . 7

2. SpcciEcatiiocn of the Model . . . 19


1. Estimation Problem? . . . . . 24

2. Discussion of the Results . . . 30


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


11ULTI-PERIOD CASE . . . . . . . 65


Pa e



. . 74



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 ... . ..


TABLE 1 Probit Estimates of the Decision
Function . . . . . .

TABLE 2 '..o Step fla:-:imum Lifelihcood sti-
mntes of the Decision Function .


- iiCuising E:-penditut-e Equation of
the O -.ner . . . . . .

TABLE 4 lio.iusinr.g E::end iture Equation of
th-e P enters . .


TABLE 6( -

Iousin q Ex:penditLure Equation of
the OwC'.n eer . . . . . .

i ising L:: end ture Equation cf.
the F]entcrs . . . . .


TABLE t1 Comrnpar Lson of Cetnsus and .F'C Data:
To-al Famil, Income by Pace, for
all rain i e es . . . . . .

T,.LLF 2 Coipar-a1so 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 . . . . . .



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



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,


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.



lany models of housing demand have been presented in

the literature. The earlier models ..ere estimated with

aggregate tjme series or cross-sectional 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 iidepen-dent. Thc-se '.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' 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.

Cliapi-tr ''I coniailns the conclusions. Finally, the appendices

contain a discussion of the data and an extension of the

model tc the multi-period case.



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 cross-sectional

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.


.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 T-i-mple 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 di-ide-: the simple data into

o,-..ner s .nnmd rente-rs and estimate t each subg rroup sep-r 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 cont-in ed in the o.-'ner sub-

sa-mple. For e:-:amples see Lee (68), Oh]s (C-), C Irliner (73)

or Fenton (741 .

l'o l 1. insi ;y L'.-:u s tI-iC 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 thl-n cli'.'ie the i-riiple into o'..-ners *and renterFs one c-n
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 s-mple. 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) w-hen estimating the demand equation for owners

(renters). In his paper Tobin examines buyers and non-

buyers of a durable good in any-given 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-

a-ble '.when a family o'w.ns less the utility obtainable w..hen

thit s-me 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

numb-er 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 rent-o.'-:n decision. An approach designed to fill this

gap in the literature is deriv'.ed in the following chapter.



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 reasons--such

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 estiim-tion 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.


institutions constrain families to loans of limited size

by requiring, for example, that housing expenditures not

exceed 20-25 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 air-conditioning 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 iti-on 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 ap--irtments in 1971--and

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 nmulti-period case. For example, one could assume

the fainily mr.xiimi zes ;:pected utility summed over the

entire occupancy of a dwelling. A multi-period 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
other goods. Similarly, if-the family rents H units of

SICr (H)
MU(II) Cr(
will hold, where MC is a marginal cost function for renting.
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

wo-uld 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 cithli-er'-or situation there is one ec:-eption. If
the ma.rgninal 1-ite 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, I-1iis '-'LouIl not be the case foi housing. For most con-
sumiersG the implicit time cost of consuming t , separated hous-
int units constrains them to an cithcr-ori choice. Of course
thrre may I:b situations hereee a familIy would simultaneouslly
o( -. ii reGnt. C'n'e is thle businessman \- o o-ns 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 housing--or 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 w-ise 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)'
AC (H)
/ R
/ / MC (H)


- I

rIi (U it

House Size (lI)

Fiu.-jre 1.PaLr al equilibrium housing demand


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 (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 rent-o'.'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 rent-ov.n


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 m-j 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 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.*) = Y-P(X)X-R.*(H *)II*.
j ( J J J
The first order conditions to'(3) are

x X
(4) F
H .* F
] .*

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 I-I* ( ) Figure 2 sh'..o s
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 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-
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 both-owners 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.-ar-jinal utility

of the last spent on all other gcds. More speci-

fically, the model de'eloped: in this chapter can be written


=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 tables -and l Un ', U ) are triv.-r iate
In _n 1
normally d t s tribl-.u I-d 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
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:penditures--a 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 ccL-related 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


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 Panel--divided 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

a'.:C'rag of "fmrimily noi-ne,'"--as described in the Panel--plus

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

fa-m or business; rental, interest and di-idend 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 bec-iusc t;hieir r,:-nt]tal p:'aymen ts--.and therefore housing
e::pend i tures--i a3n implicit maintenance cost.

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 ' a dummy variable if the family

moved more than once between 1963 and 1972. The reason

for including this ls-t 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 cla-ss ficd as beinti in one of six
groups: r-c-atcr thD1n 500,000; 100,000 to 199,999; etc. To
make the variable cont iluous, a real iumber-- the midpoint,
when appropr iate--', 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 4-person Family, Autumn 1971." This table

breaks the budget into nine categories for forty-four 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 4-person

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 ' th-refore possible

to assign one price imnde:-. for renting and another for ow'n-

ing to e-ach of the forty-foour 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 hansr-d 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 ow-ninq for lo'..'er income families. Fifth, families

may believe that the selling price of houses will rise

faster than the g-eneral 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.



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 \w-hose e:-xpectation is nonzero

and not constant for -all obser'.'ations. In the econometric

litert--Ljure 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-

tionl-l "t'-.'o staqe" methods. These procedures can be applied

to eqij'ations 5-7 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:
(3) 1 = 1 iff I 0
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


The first procedure proposed by Lee (1976) uses the com-

bined ow-ner and rcnter sample and estimates B1 and 82 together.

To derive this procedure let i' be the observed sample of the
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 + (1-1 )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(1-1)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 l-p 2) U f(U )dU
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,

(10) E[(1-I )U (1-I )X ] = E[(I-I )U ]X
n 2n n 2n n 2n 2n
= -p OX21 U f (J )dU 0
P2 Z X2/ n Unf n)dn n 0
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 t-rms a1nd the rec ressor We no'.' ha'.e

(11) Y = F(, ,)::ln + (1-F 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 (1-I 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'- s-l-..n that Ef I = 0 .nd W .is unco-rrela ted
l' rn
'.' l hI tI1, r.;i r-es or s.

Ho.-.'e"r, erjuation (11) is non-linear 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

nal-1sis. This is the first staqe of the estimation procedure.

il-ir- scon.i stage is to estimate the fol l'-.'inl equation by

oud. inary 1' I nse t souar- ,

(12) 1 r ( "):.:: 1 + (1-F (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

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 f(U IU )cU
in' n n

f (1-1 ) d
n r

77 1 .rA f (II.
F(7 F)ln 171 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 These families are therefore more likely

to rent.

TA BL 1.

Probit CoefficienL:s for Owners'


Age < 35

Age 36-64

13 lack

Fem ale


Log (City Size)

Log (Distance from Center of Cit'y)

Log (Family Sizel

Log ( Permnainent Income)

Log (Pelati.-ve Price of Penting Housing)

Log ( Price of Owning lorme)

Lumber of Owners

Number of Renters

'Standard errors in parentheses

. 5350)

(.109 4)

-. 6567
(.099 1)

-. 3256

(. 0632)



. 0276)








Comparison of Two Stage and OLS Estimates for Renters*


Age < 35

Age 36-64

B a c:

F em,:- I e

Lo;i (['lct.-incc froni Cen ter of Cily)

Log (Fariu1l; Siz,-)

Lo (. (F.1e PcF-i:ranent Income)

Two State








-.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 )

(.22 65)

(. 1192)

(. 1 3 5)

[ iussini '.', i -,:a b 1 e


[lumber of Obscr-'a t ions

*St.andard errors in parent theses


.4 2r 9



Coniparison ofL Two' Stage and OLS Estimates for Ow'ners*

T',iwo SLta
-- Estimates



Ace -: 35

Ag le 3-6641






-.1-54 -.1343
(.0693) (.04071

-.1563 -.1505
(.0440) (.0330)

-.2534 -.2502
(.0585) (.0561)

.0772 .0006
(.0352) (.0308)



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 -

.441 77

Number of Observ'.ations




*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 coni-i it ions pro'.'jded by home ow-.'nership. The positive

coefficiE-Crit 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


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


The coefficients for the-age dummies are negative

and significant for both owners and renters. For owners

the coefficients give a U-shaped 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 U-shaped

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


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 hl-at 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,-r-e 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 longc-r 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


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 p-ositi'.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 the-Missing 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.



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 th-is 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 ic-itly Lbtter than the OLS estimates.

1. '.iim'miyia' s T',o Sterp 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 it-ddices -for bo th i-wners and renters.

A different specific:i tiocn '.-as emnplo,'/ed because the positi'.'e

coef ficient f-or the .price :.'arL i-ble 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 rent-ovwn

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, 1974-1971." This table gives six price

indices including ai housing index with a base of 1967

= 100. T'-.-enty-one 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 i-ndex 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 t-n 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

c-onter 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 = + 4-C 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 1 1 i (- 1 1 f C C- J ( -X. )d 1-I

S- '-" i\2 27 2 2 2

i 1 (C-X11) /f1 1C-x1)JcII[g2(C-x22) f~2(c C-Y22)1-I
i = -1'.'. "

1- 1
(C-: ) = e::p (C-: )2

1..n 2 2

!-, 11'1
fl(c C-::1 1) 1

.1 11
cxp (- ---- l (C-XlB )]2},
2(1-Pl ) 1-


f2(EIC-X2) = 1
2 /2 /- 2

exp 2 [E P2 C-X( 2
2(1-pq ) 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
F(Zy) = / f(E)dE,

then given a new variable t such that "
t = h(E); E = h (t),

where h(E) is increasing, one can transform variables and

h(Zy) -1
dh (t)
G[h(E)] = / f [H( t)] dt.

Ey selectinrq li( ) as

t [> (C-:'. ,l
I 1L

and sol'.'inq for G.[ () ], thL first integral in equaLtion 8

simplifies to
S I -
L (dt.

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 ,

I ln(q] (C-:; 1. )) +

in 'I (t) dt)

(1-I) []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 s-tep 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
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. All-the 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 ne-r e]ua tion decreases from the t'.-'o

tage estim-te -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


Probit Estimates of th., Decision Function


Constan t

Aqe 35

36 .: aqge : 6-;



In (City Sizo)

In ( Dist-nce from
center of city)

in (Family Size)

1n lati"e
permanent income)



-0.6 .3 96


-0. 32102

-0. 9.1359

-0. 14969

0. 12779

0. 13-134





0. 07131








Two Step Maximum Likelihood Estimates
of the Decisicn Function


Cons tant

Age < 35

36 < age < 64


-- rri, m I 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 im-nc n>ti L income)

-3. 24123




-0. 31895

-0. 94527

-0. 14991

0. 12 864 .


0. 6 3159









0. 0-1904


C -1* L 4' *:o0 --

r I -- r -'i (- I '--4 '.
"4 '1 r"- r ', rC rC .
'-4 0 0 0 0 '0

'0 0 0 .0 0

,. CO

r-I r']
,--4 r0
0 0
-7 -7

r-, CO
r- oI

r- rI,

*- 0



L. '


1, 0

[ *





r 1

L 0

[- r'-

iLr 'i




--- C -C) -7 -r- -7
3 ,', ,- r, 7

C)0 D.D
'- -O'. 0 '- '- ',D
r'l r-; r'| ,-4 ,-4
c'. u-r i) .r 'j'. r-l
0 0 01 0 0

r- r', r'l r r rJ

r-1 r, r- r' I -I L.'

r- ,-4 r ,- C)


r- r]i
,-4 r--i
r- u-i
r'1 rC

'D r i
n n

'0 0

'-4 r-,
C) C)
0 0





i 1

I .,




r'l -: 0C r' c r'I 0
Si in -4 r rCi -" i
G) r- Or *r', -4 n | u-I u.I
r- 0 0, ", -' '.*D r-1 rl -
L", rj -i4 r,' --4 D0 ,- 0

-4 D C'

,i) -I
U [

S (1)
o -

0 0
G '0

0 I

C: 3
,-4 4

.*4 0

0 }

In () _1)
O -

-4 q
0 C

*-1 V
,4.J QJ

3 4*
) ( *
:2:, 1
rC rL4o- 1

Li i r' ,

71 S II
4 r,-4 0
.0 Li
--1 '-D r .]

i .. I 31) I

.0 - -I j

1 0

JI I'4
3 U 0 "0
14 CO I'
(0 ,-I U
4 --1 0

,- ru-i! .
O 1 -< C

IO *. c
LO 0 *-I

Li :*: )
L4 C" LI
1) 1 E *'
er -.l -

Si-. n
C L4- (II

m' II L) "
,n i11 u

< 0 <0. -.

















E 0

0 -
i fl) L I
S 4- 4 N 0.
*' N *^ L ) 1_}

-4 U >
II L C "., -4
') --4 44
_ (l :,, 4J -4 4
PT- Ln L' :-, E ,- *T'
70 0 .1 -4 -44 -) 0l C

|| ,-4 Qi rj "' [- -4
*V U "' > 3-

C. 'o 'l 0 C C 1-i C C '
ri r -- -,3 O C




U) Q

4 ,0
WM 0



,l I




4:* J

. C ,r r

'.CI i PI -- 1 r- *. -
.-., ,r- r"- -' --4 --4 0'

0C. C.

I -. rq .-C
r- ,-I , C, C

r-, i C. C Co C
o. o. .

:.: C C. "1 '.

"- .H -,--4- ,- ,
S .,C C C, C,
I Ir -J I
o I r r ^ ^ r- I

r^ ^-i ^-i O- r-i r-

(N in
N rn
(N C)

rCl C
o o

o o

co N

< m

m o
r-1 C
- C)

r-4 to

. I, I IC
l-1 C-
C o

--l ,-

r-, r i
o C-
o 0

'. t

, icT,

0 0






_J =

Ii C-'








, N ---i4 CO !.
C rl o o o oN oo
S N i- r- o o i
Co r- in oo r o 0
ri in in 1 n "tN N oC
cl C) CD C0 CD 0 C)


a -
E -

,---I D_

.' II ,' C : u
i '.) 4
il '24 C : *-I

U -13 .1i
ul 1i1 :s J .-- *-1 4

Sf, 1J E 1 :c c
C .' -- ,'I 0 i C '- C CO
, : ,"- 1' L ,D--CO -- -4 .C








Sr w

,4 1 -

r I IJ

E -- r
.L 0


I' L ') n

C) 'l G!
,E O
II 'i 41

0 -1 q>
.1-J I.)

C 0'- '2J

4 1 u'4 4-

X I-4 1J
r j nilll

S 4-1 0 C
E 4c

0-1 40 '

0 ui 'J .0

n IC, -

E 0 c
-- J ,.

il r .











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, es .a-id 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 -.L-E suci aus .c-h i':, Carliru 3 ,

Frni to n i 4i} ani rolinsk. 7. sp d iC si ii t] di fcr-

,nt .-xI iCl iditur: e-uations 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 t-r arc .:u re C lose t' Ca ir l ir r' s
Lst nat-s 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 [c--ti'. Le' i ",.'.: r ?['-'L '-1 se'c.'iL l s

of .income- el sticity c l;.ri atus. His *.-:= 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

reni--cIrs hi [-.-rma 13 1, income elasticity: estimates ranged

from 387 to 6 2. Pol insk; c nci'lud s tliat tI2- true irnc .

c-lastici t,' is appr :: irna t l-,' 5 f -.c b'tl- r n teirs a i-id 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 u-siing fIour year
'.'eraiCi'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 i-i ] r l o.- l'i-
cocff icic' ts 1 for the- BlacIJ :, female and anci cl-u-iiic ::, 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 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- ol-tained b.' di.i ng the sample into t.wo'

grouf'.s ow..rers and renters and appli..'ng OLS to each

grCoup separate-ly. 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 2SI-1, is five
L .
times greater than the OLS estimrlate. Also, the 2 S-1,

estimate is -igi-ifican t a=id the OLS estimriate is insignifi-


compar-isorn 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-

mat-es are no'...' the 2Si!L estimates presented in Tables 2-4.

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, e-ren 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.


Housing Expenditure Equation of the Owners


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 S-eze) 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


Housing E:-penditure Equiition of the Renters


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



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. 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 s-itching 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:xpenditur-e

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-

ma-ted 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 re-j 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 hea-ded 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 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 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

Rese-arch 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 io-ns 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

-cross-section and time series data. Secondly, a model that

described li'.'h families move '..'ithin a qi'.'en area could be

foL-mulated 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.



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 s-imple 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 in 60% con listed or a cro-ss-section

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 '.*.ere inter'".'i:'.-'ed the pre'.,ious ye-ir and

nc'.-,ly foi-mcd 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 o-riglinal 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 cross-section 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 cuoszs-sectionh 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 undcr-epresentation of those with heads of families

forty-five or older.

Finally, this thesis only uses the 1971 data. The

sample consisted of 3,452 families. E:-:cluded from this


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





23. 4







$15,000 or more

99.9 100.3




















Comparison of Cenr u.s and SP.C Data:
Pace by .'Various Demoqgraphic a.'ariables,
for Families with Tw.o or More Members

Sexc: of family head White Black White Black

Fema.i i

Size of family

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


Fouilr or more-
[lot ascertained

.'. of f aimi y hI e, C ad
under 25

35. 2
19. 7
12 .6
5. 2


14. 1

100. 1 100.0

142. 9

10. :

30. 5
10 .8
11. 1

99.9 100.1

100. 1 99.9

Under 25
25-2 c
35- 1
3I5 5 .1
*15- 5.1
55-61 ,
65- 7.1
75 and over

99.9 100. 1 of place of
r es i dence

[ion-met ropol i tan

36 .3

99.9 99.9

63.3 7.5.9
36.6 24.1

99.9 100.0

72. .

100.0 100.0

35. 3
20. 3
19. 5

16. G
1 4. 7
: .3

29 .
19. 3
19. 5
10. 7
0. 2

100. 1

2 -1 5

1 9. 6


99. 9

6. 6
10. 5
21. 3
16. 3
9 4
4 7

11. 6
1 C

9.o 2
20. 2
15. 5
9 .4

10. 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.



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 f-l.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


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 time L,

p (t, IU)
'..,here --- -- = PB (') 0.
icLr-e 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

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 0 : t .: t w.'here t is the
m m
household's time to retirement. Ech family is

assul-Icd 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
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

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,
(2) C('..') = I m ...t)f(t)dt = Y h X T = k,
0 D D

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
(3) U (t) = ne

Differentiating L with respect to hg yields
(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
{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 %. -

(5) .-m [-- - IC (.) ] ('.v)e 'dv )f (t)dt

L t
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 maL-inal 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
( ) ) = (I*) I'* (t) f t)dt.
b n b

If the family chooses to rent, they max:-imize (1) subject


(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

(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 rent-own 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 multi-period 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).


Amemiya, T., "Multivariate Regression and Simultaneous
Equation Models When the Dependent Variables
are Truncated Normal," Econometrica, 42, (1974).

Carliner, G., "Income Elasti'city ofHousing Demand,"
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( .o.' 1973).

Ciragci, J., "Some Statistical fl1oels for Limite d Dependent
'.'.ar1 i i].es .-.ith Appl ication to the Demand for
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Incrme and the Demand for Pental liousinq," in
iiL iC,.nt .er for r1.rban Studi s, ,',nal'.-sis co
Selccectd Cenisus Wleirre Pr.'e o i api Data to
Dcterriiinie I laticn of iiouselr_'l d Characteristics,
iiousinQ lirlLet Characteristics, and Adiiii.nistra-
Li'.'c t l.'c fa Policies to a Di.eCCt 1lousinq Assist-
ance [ (Draft -- Final Peport, mimeo-raphed,
Jul' 31, 1974) .

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Ileclian, J. l '..' Pr ices, flar. et Wa.Tes, and Labor Supply,"
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rrldels by' T'-.o Sta.ig e Ilcthlods," U npubl islied Ph.D.
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197 .

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an [hree-Y-ear Peinterv'ie..' Sur.'ey," The Revie'.. of
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rad:dala, G. S. and Ielson, [-. "Maximum Likelihood
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Sea rch: A Si.irr ." 'Discussion paper 55, UCLA,
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(Ed.) 'i'l. Demand for -Durable Goods, Chicago:
Uni-ersity of Chica.o Press, 1960.

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.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 B-chelor degree in Mlechianical

iEni ineer inii fi-rom 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 M-arL-. 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 ,
resen-l. 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 an-l 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 re-ad 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.q--ee 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
ni-esentation an-d is fully adequate, in sccoe and qualityt,
as a diss-rta.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