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PLACE IN REIURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
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2105 WWW-p15

 

 

 

 

 

MULTINOMIAL LOGIT MODEL

BY

CHI FAI CHAN

A PLAN B PAPER

Submitted to
Michigan State University
in partial fufillment of the requirement

for the degree of

MASTER OF URBAN PLANNING

School of Urban Planning and Landscape Architecture

1986

133515
, .

TABLE OF CONTENTS

PAGE
INTRODUCTION ...... . ................. 2
THE MODEL ........................ 3
DATA ...... .................. 7
THE RESULT ................... ..... 9
CONCLUSIONS ........................ 10
NOTES ........................ 12
APPENDIX A ......... ........ ....... 15
(MAIN PROGRAM
AND EXPLAINATION)
APPENDIX B ........................ 23
(DATA FILE A)
APPENDIX C .............. ...... .... 29

(DATA FILE B)

BIBLIOGRAPHY ........ . ......... . ..... 35

ABSTRACT

MULTUINOMIQL LDBIT MODEL

By

Chi Fai Chan

The principle of multinomial logit model is to
model the choice behavior of individuals. Knowing the
diITiculty and great expense in obtaining survey data, the
author attempts to use the widely available but more
aggregated census tract data in U.S.A. for estimating the
choice patLern of housing consumer in Chicago SMSQ with
special reference to its relation to transportation to the
Central Business District (CED) in Chicago.

The results review that the mode ”drive along” has
the largest eifect on the people's choice of rental
housing market location in the area as compared to the
other {our modes; carpool, bus, train and walk. The rent
and travel cost show even more proTound effect on the

utility level of the choice of rental housing location.

INTRDDUCTIUN

The widely used Multinomial Logit (MNL) Model in
modeling housing market and transportation mode choice
(with special purpose such as shopping, work, recreation,
business) considers choice of housing as independent to
the choice oi mode with special travel purpose. These
models assume sequential choice pattern, which is
operationally more eTTicient than simultaneous choice
pattern. In fact, not many research has been done on the
difierence between sequential choice model and
simultaneous choice model. Manheim has shown that when
certain conditions are satisfied, the two approaches are
equivalent. He ponited out that

"Ideally a single simultaneous~choice model

incorporating all relevant choice dimension would

be utilized. This is impractical because of the
large number of dimensions in real situations. In
practice, some sequential models must be

utilized."(Manheim,M p.421)

Alex Anas in his Residential Location Market And Urban
Transportation, attempts to consider both the choice of
mode to working in CBD and the choice o4 housing location
in Chicago SMSA in his model.

He employed 1970 census tracts data in the Chicago

SMSR to estimate the utility coefficients in his model.
Zonal aggregated attributes as well as disaggregate
attributes are used, and are compared in his work. He
comes up with the probable conclusion that

” errors in the estimated coefficients arising
from the aggregation of certain attributes
(such as rent) to the zonal level appear to
be smaller or comparable in magnitude to
errors arising purely from sampling”
(Anas,e p.159)

Based on his observation, this paper attempts to
use 1980 census data in the Chicago SMSA to estimate the
residential location markets and urban transportation

choice behavior in Chicago SHEA.

THE MODEL

The sequential choice pattern of the individual is

I
u"
Ln

. 'umed to take the following form
Step 1: Choice of the itfl residential neighborhood

from the existing residential market (in

this case, the itfl census tract)

Step 2: Choice of the ktfl dwelling unit given the
choice of itfi_neighborhood
Step 3: Choice of the mtg mode for work trip to CEO

given the choice of itfl neighborhood and
the ktfl dwelling unit

Two utility functions appear in the choice model

U1 3 E xiu ' xv;

Ll

a measure of the utility level for each individual

5.;
if!

choosing the igfl neighborhood. This utility depends on the

socio-economic status of the individual which is
d

incorporated into the vector attributes X, . Examples of

these attributes include income, rent of housing, family

size, housing quality etc. O('s are the coefficients to be

U1": = Z Ximv ' Pv

V

estimated.

is a measure of the utility level for each individual
choosing the ith neighborhood, mtg mode of travel to the CED.
The vector Elm consists of zonal attributes that depend on
mode of travel. Most commonly used attributes of this
kind are travel cost, travel time and dummy variables for
each mode.

The estimated probabilities that associate with

the above three—step choice model are given by

r:
-_J

. A I: A '
respectively, PinH1 ,and Pfl1k(1)-The magnitude of these
probabilities depend on the utility which the individual
can derive from his choice.

With the log-likelihood function defined as

J

a“ A d
log L '-= Z 2N1m1og P...“ {Nilog $1+constant
i m 1

one could solve for the coefficients F,’s and oL's by the
maeimimun likelihood method. This is equivalent to finding
the solutions for the following set of maximization

P." \

eguations(ei

 

 

e 0 for all v
9h
21:49 L:
for all u

§D
R
K

II

where L: is the marginal part of the likelihood function
and is given by the second term of in the log-likelihood
function, whereas L1 is the conditional part of the function,
and is given by the first term of the function.

By using two step methodiE), the equations can be
solved by using numerical method.

Newton~Raphson method is used for solving the above

n on---l i near equations . The it 2‘!" at i r. n equation f or

finding @s is as follows

A

.4 2 -\
($r-H-13P” _ (V109 L1 WV”? L1

where n stands for the nth_iteration.‘7log L1 is the gradient
of the log-likelihood function ,and Vflog L1 is the variance
covariance matrix of the estimated coefficients.(4)

For testing the significance of the estimated
coefficients, one can use the t~statistic to test the
hypothesis

Ha : =0 for all v by using t, = PVs,
where
s; is the square root of the itfl‘ diagonal elements in the
variance~covariance matrix and t1 is the standard t—

statistic.

DQTR

Table 1 summarizes the attributes that are being

used in this paper.

 

 

Attributes for zone Attributes for zone—
specific utility u; mode specific utility utm
(X1u) (Ximv)
Number of rooms in Dummy variable for each
the census tract mode of travel
(room), (Dim)
Per capita income Travel cost to CED
for family (cost);m
(incomeii Travel time to CED

{time} 1m
Rent

(rent);

 

Five modes are generally available to residents

living in the Chicago SMSR: car(mode l), car-poo1(mode 2),
bus(mode T3) , train(mode 4) , wal|<:(mode '5).

For those residents who are not living in the CED,
it is assumed that moie 5(walk) is not in their choice

set.

The rent for census tract i'is calculated from

H1 =3 {gr}, + {1“+1)V3 /10

where 41 is the proportion oi zone i’s occupied dwellings
that are renter occupied; r1 , the average annual rent o4
these renter occupied dewllings; and V. ,the average annual
market value oi the owner occupied dwellings in sane i.
This formula is a rule of thumb for estimating zunal rent.
The attributes, number of rooms, per capita income
and rent are available in census. However, travel cost and
time to CED are not available. These data have to he
collected {rem Chicago Transportation Authority
.Study(CeT3). The CATS does not provide zonal travel cost
and time. It provides a more aggregated sevenmsector
division 0% the Chicago SMSR, and from which the traval
cost and time for each census are estimated. The seven

$5

(r-
. a
I

'.tors are. CED North Chicano South Chica o Northwwest
. 9 3 5
Chicago, Western sector, South-west sector and South

-'~ H"-r
(CSP‘DIA I

sector o4 Chica o

9
It has been studied extensively the a sample size

of about 300 can give accurate estimation for the result.
A sample size oi 293 censLs tracts is chosen randomly from
the existing census tracts(about 4920) with the number of
work trips to CED from each tract as the weights of
choosing them.

The utility Tunction U1 and Uim read

U1

' drlncomei + OlanJomi

Uim

P,~l.og(F(ent1 + Costim)+ €2'(Ti‘7'91m)+F3'D13-'= +

F4'E'13 + @3‘D14 4" @"Difi

THE RESULT

Since t e fortran program for this paperisee
appendix) uses the Newton—Raphson apuronimation method to
solve -hm* 9’s” and the numerical method used does not
Igurantee feasibility of solution, the computation ends up
with an approximation {cm €‘sXED values as follows

(~ssa, 14.2, ~o.05, —l7.8, ~1a.4, ~48.5)

Checking the sign and relative magnitude of the
coefficients for the {our dummy variables, ~é.QS(car
pool), —17.8{bus), -lb.4(rail), —4B.5(walk), one can {ind

that this is consis;ent with one's intuition in that all

10

these modes give negative utility as compared to the
reference mode, drive along(mode 1), which takes up eero
for comparison purpose. The relative magniture for each
mode reveals the individuals' preferences(measured in
terms of utility) towards different modes. The smaller the
coefficient is, the less satisfaction one can derive from
choosing the mode.

The coefficient for log(Rent+Travel Cost) is very
small<~256), this means that people are very sensitive to
the change in rent and travel cost. 9 small change in
these values would d:cr:css(or increase) their utility
level. by a great. amount.

Since the iteration process in the numerical
method does not converage to a unique solution, further
inference on the values and the significance of these

estimated coefficients are not possible.

CDNCLUSIDNE

Although the computation ends up with an
estimation of the coefficients for the sons—mode specific
utility Uh ,the result agrees with what most studies have
identified by using similar model.

Because of the scone of this waver the demand
8

side of the model has been investigated while the supply

11

side of the model was left aside. Thus care must be taken
when one tries to compare the results of this paper to
those obtained by A.enas because he applied the
equilibrium model on the 1970 data, which take care of
both the demand side and the supply side of the model.

The aggregated data used in tflis paper, the census
tract data, give reasonably well estimation. The use of
aggregated data at the census tract level can be widely
adopted if some of the necessary travel attributes such as
zonal travel time for work trips to GED and zonal travel

v ilaole in the

if:
I) .

21a

cost to BBB for each modes, are al o

census tract.

12

NOTES

The exact forms for these probabilities are

A

P S" U+1-6’I ZSr U+l- I
i ‘ ieXP( i ( ) i)/ jexP( j ( 0) j)

j

A
Pkfi' 1/31

A
Pmtik= exP(Uim)/ e:p(Uim)
where I. = log Zexpw. )

J m 3m

with I = (l-T)
6 = 1-(1-1) (1'6)

where6'is a measure of correlation among unobserved
travel mode related attributes within a zone

tiis a measure of similarity of dwelling units
in unobserved random attributes

6 is a measure of mode similarity in unobserved
random utilities.

Notice that the P . is aggregated probabilities
because within-zogg variation of dwelling units cannot
be observed with the census data

13
The exact forms of the equation read

_®Q_{ 2. Zfiimlog smu"§i; 51 21:“(Emfi'smi) ximv

9v 1 m

=0 for all u

9 ~ A _ ~ ~
-—9—3—( :1 N.log P.) —i:(P.-Pi)log Si

1

where 3 . represents the estimation of the probability
by usinaucensus data; P . and Pi are given by

m
Pi =§Nimm

~

Pm‘ Nim/ 21le

and
'fii = total work trip to C30 from ith census tract
fiimé total work trip to CEO from ith ecnsus tract

through mode m

Forcing X to equal 1 and forcing 6 to equal 0, one
needs only to solve the first two sets of the above
equations. This is justified on the ground that‘l and 6'
ususlly take up 1 and 0 respectively in most studies,
and this could facilitate the calculation in this paper.
The reader can find similar examples in A.Anas's book
listed in the Bibliography.

One step method involves solving the @‘s and.dfls at the
same time whereas the two step method solve for 9's and
substitute these values into the other set of equations
for solving the rest of the coefficients. The one step
approach always give superior result to the two step
approach, yet the computation is more complicated. This
paper uses the two step approach for simplicity. Notes 3
indicates the two step approach that is used in this
paper.

1h

The exact form of the matrix is
I» A A
=2?“ (1?:- ximvpmti) ( ; ximupm) -
i

Z ximvximqufi.)

for all u,v for F's

o L.
Fv

(-0.0391, 0.0399, -0.0722, -0.1442, -0.1329, -0.0406)

The corresponding values for are

which is reasonably close to zero vector.

APPENDIX A

15

EXPLAINATION OF SYMBOLS USED IN THE PROGRAM

= ith census tract ,i= 1 to 293

= mtg mode ,m= 1 to 5

DA(I,1)= per capita income

DA(I,2)= number of rooms in census tract

DA(I,3)= number of dwelling unit in census tract(Si)
DA(I,4)= 511

DA(I,5)= N12

DA(I,6)= $13

DA(I,7)= Ni4

DA(I,8)— fiis

DATA(I,1)= annual rent
DATA(I,2)= travel mode

DLIKFU(U)=first derivative of log-likelihood with respect
to P

D2(U,V)= varignce-covariance matrix for solvingck's

DZIN(U,V)= inverse of D2(U,V)

TIME(I,M)= travel time to CBD from zone i through mode m
COST(I,M)= travel cost(annual) to CBD from zone i through

mode m
PI(I)= E1
PIM(I,M)= P .
ESTI(I)= 9?“
ESTPIM(I,M)= ma
EI(I)= Ii

ALPHA(U)= oUs
BETA(U)= p's

FUNCTIONS:

UTILI(I)= Ui

UTILIM(I,M)= Uim

DIz= Di2’ D13= 013, DI4= 014, DIS= Dis

Others are variables necessary for the computation process,
and they carry no specific meaning.

101
102
103
104

899

111

898

16

MAIN PROGRAM

CHAN,PN1315683,rgz,jc499.
attach,mnlda1,mnlA2.
attach,MNLdat,mnle.
ftnS.

lgo.

listty,i=tape6.

PROGRAM MNLB

real da(293,8),data(293,2),dlikfu(6),beta(6),residue(6)
real delta(2),d1mafun(2),time(293,5),cost(293,5),PIM(293,5)
real estpim(293,5),sum(293,6),d2(6,12),d2in(6,11),PI(293)
real WTMASUM(2),W2MASUM(2,2),D2MAFUN(2,2),ALPHA(2)
REAL EI(293),ESTPI(293)

common/coml/beta,data,time,cost

common/COMZ/ALPHA,DA

open(7,file='mn1da1')

open(10,fi1e='mnldat')

OPEN(6,FILE='OUTPUT')

do 101 i=l,293

read(7,102) DA(I,1),DA(I,2),DA(I,4),DA(I,5),DA(I,6),DA(I,7),
DA(I,8),DA(I,3)

format(F6.0,F3.1,6F5.0)

do 103 I=1,293

read(10,104) DATA(I,1),DATA(I,2)

format(F6.0,9X,F2.0)

DO 899 I=1,293

DA(I,1)=DA(I,1)

DATA(I,1)=DATA(I,1)

alpha(1)=0

alpha(2)=-0.05

do 111 i=1,6

residue(i)=0.0

DO 898 I=1,293

DA(I,1)=DA(I,1)/365

DATA(I,1)=DATA(I,1)/365

BETA(1)=-0.05

BETA(2)=-0.05

BETA(3)=0.0

BETA(4)=0.0

BETA(5)=0.0

BETA(6)=0.0

do 105 I=1,293

if(I.lt.10)then

TIME(I,1)=O.82

TIME(I,2)=0.82

TIME(I,3)=0.77

TIME(I,4)=O.63
COST(I,1)=2.62
COST(I,2)=2.92
COST(I,3)=1.59
COST(I,4)=1.35
else if(I.lt.134)then
TIME(I,1)=O.47
TIME(I,2)=0.47
TIME(I,3)=O.77
TIME(I,4)=O.37
TIME(I,5)=O.25
COST(I,1)=2.12
COST(I,2)=1.79
COST(I,3)=1.59
COST(I,4)=1.35
COST(I,5)=O
else if(I.lt.235)then
TIME(I,1)=O.98
TIME(I,2)=O.98
TIME(I,3)=1.05
TIME(I,4)=0.9
COST(I,1)=2.65
cosm(1,2)=2.41
cosm(1,3)=1.59
COST(I,4)=1.5
else if(I.lt.272)then
TIME(I,1)=1.67
TIME(I,2)=1.67
TIME(I,3)=1.25
TIME(I,4)=0.9
COST(I,1)=3.3
COST(I,2)=3.00
COST(I,3)=1.59
COST(I,4)=1.5
else if(I.lt.276)then
TIME(I,1)=2.5
TIME(I,2)=2.5
TIME(I,3)=1.7
TIME(I,4)=1.37
COST(I,1)=4.13
COST(I,2)=4.16
COST(I,3)=1.59
COST(I,4)=1.89
else
TIME(I,1)=1.57
TIME(I,2)=1.57
TIME(I,3)=1.2
TIME(I,4)=1.28
COST(I,1)=3.42
COST(I,2)=3.16
COST(I,3)=1.59
COST(I,4)=1.81
end if

105 continue
TOTAL:0.0
do 210 I=1,293
PI(I)=0.0
do 211 J=1,5

211 PI(I)=PI(I)+DA(I,J+3)

210

310
311
999

213

214
212

778
777

215
216

217
218

555

389
388

TOTAL:TOTAL+PI(I)
do 311 I=1,293 18
do 310 J=1,5
PIM(I,J)=DA(I,J+3)/PI(I)
PI(I)=PI(I)/TOTAL
do 212 I=1,293
if(I.LT.10)THEN
K=4
else IF(I.GT.133)THEN
K84
ELSE
K=5
end if
A=O
do 213 M=1,K
A=A+EXP(UTILIM(I,M))
do 214 M=1,K
ESTPIM(I,M)=EXP(UTILIM(I,M))/A
continue
do 777 I=1,293
do 778 L31,6
SUM(I,L)=0.0
continue
do 216 I=1,293
if(I.Lm.10)THEN
=4
else IF(I.GT.133)THEN
K84
ELSE
K=5
end if
do 215 M=1,K
SUM(I,1)=(PIM(I,M)-ESTPIM(I,M))*LOG(DATA(I,1)+COST(I,M))+SUM(I,1)
SUM(I,2)=SUM(I,2)+(PIM(I,M)-ESTPIM(I,M))*TIME(I,M)
SUM(I,3)=SUM(I,3)+(PIM(I,M)-ESTPIM(I,M))*DIZ(M)
SUM(I,4)=SUM(I,4)+(PIM(I,M)-ESTPIM(I,M))*D13(M)
SUM(I,5)=SUM(I,5)+(PIM(I,M)-ESTPIM(I,M))*DI4(M)
SUM(I,6)=SUM(I,6)+(PIM(I,M)-ESTPIM(I,M))*DIS(M)
continue
do 218 L=1,6
DLIKFU(L)=0.0
do 217 I=1,293
DLIKFU(L)=SUM(I,L)*PI(I)+DLIKFU(L)
continue
WRITE(6,555)DLIKFU(1),DLIKFU(2),DLIKFU(3),DLIKFU(4),DLIKFU(5),
DLIKFU(G)
FORMAT(6F8.4)
do 388 I=1,293
do 389 L=1,6
SUM(I,L)=0
continue
do 401 I=1,293
if(I.LT.10)THEN
K=4
else IF(I.GT.133)THEN
K=4
ELSE
K=5
end if

402
401

702

701

610

601
602
603

404
403

406
405

443
444

000000

1
do 402 M=1,K 9
SUM(I,1)=SUM(I,1)+ESTPIM(I,M)*LOG(DATA(I,1)+COST(I,M))
SUM(I,2)=SUM(I,2)+ESTPIM(I,M)*TIME(I,M)
SUM(I,3)=SUM(I,3)+ESTPIM(I,M)*DI2(M)
SUM(I,4)=SUM(I,4)+ESTPIM(I,M)*DI3(M)
SUM(I,5)=SUM(I,5)+ESTPIM(I,M)*DI4(M)
SUM(I,6)=SUM(I,6)+ESTPIM(I,M)*DIS(M)
continue
do 701 L=1,6
do 702 N=1,12
D2(L,N)=0
continue
do 610 I=1,6
DZ(I,I+6)=1.0
do 603 L=1,6
do 602 N=1,6
do 601 I=1,293
D2(L,N)=D2(L,N)+SUM(I,L)*SUM(I,N)*PI(I)
continue
continue
do 403 I=1,293
if(I.LI.10)THEN
K34
else IF(I.GT.133)THEN
K=4
ELSE
K=5
end if
do 404 M=1,K
D2(1,1)=D2(1,1)-LOG(DATA(I,1)+COST(I,M))*LOG(DATA(I,1)+
COST(I,M))*PI(I)*ESTPIM(I,M)
D2(1,2)=D2(1,2)-PI(I)*LOG(DATA(I,1)+COST(I,M))*TIME(I,M)*
ESTPIM(I,M)
D2(1,3)=D2(1,3)-PI(I)*LOG(DATA(I,1)+COST(I,M))*DIZ(M)*
ESTPIM(I,M)
DZ(1,4)-D2(1,4)-PI(I)*LOG(DATA(I,1)+COST(I,M))*DI3(M)*
ESTPIM(I,M)
D2(1,5)=D2(1,5)-PI(I)*LOG(DATA(I,1)+COST(I,M))*DI4(M)*
ESTPIM(I,M)
DZ(1,6)=DZ(1,6)-PI(I)*LOG(DATA(I,1)+COST(I,M))*DI5(M)*
ESTPIM(I,M)
D2(2,2)=D2(2,2)-TIME(I,M)*TIME(I,M)*PI(I)*ESTPIM(I,M)
DZ(2,3)=D2(2,3)-TIME(I,M)*DI2(M)*PI(I)*ESTPIM(I,M)
D2(2,4)=DZ(2,4)-TIME(I,M)*DI3(M)*PI(I)*ESTPIM(I,M)
DZ(2,5)=D2(2,5)-TIME(I,M)*DI4(M)*PI(I)*ESTPIM(I,M)
DZ(2,6)=DZ(2,6)-TIME(I,M)*DI5(M)*PI(I)*ESTPIM(I,M)
D2(3,3)=D2(3,3)-DI2(M)*PI(I)*ESTPIM(I,M)
DZ(4,4)=D2(4,4)-DI3(M)*PI(I)*ESTPIM(I,M)
DZ(5,5)=D2(5,5)-DI4(M)*PI(I)*ESTPIM(I,M)
D2(6,6)=D2(6,6)-DI5(M)*PI(I)*ESTPIM(I,M)
continue
do 405 I=1,2
K=I+1
do 406 J=K,6
D2(J,I)=D2(I,J)
continue
DO 443 I=1,6
WRITE(6,444)D2(I,1) ,D2(I,2) ,D2(I,3) ,D2(I,4) ,D2(I,5) ,D2(I,6)
FORMAT(6F10.4)

704

706
705

708
707
709

713

714

888

802

803

804

998

711

710

801

822
824

833

832
834

do 709 M=1,6

J=12-M 20

do 704 L=1,J
D21N(6,L)-D2(1,Ld1)/D2(1,1)

do 705 I=1,5

do 706 L31,J
D2IN(I,L)=D2(I+1,L*1)-D2(1,L+1)*D2(I+1,1)/D2(1,1)
continue

do 707 K=1,6

do 708 N=1,J

DZ(K,N)=DZIN(K,N)

continue

continue

EPSILON=0

do 714 J=1,6

RESIDUE(J)=0.0

do 713 I=1,6
RESIDUE(J)=D2(J,I)*DLIKFU(I)+RESIDUE(J)
BETA(J)=BETA(J)-RESIDUE(J)
EPSILON=EPSILON+RESIDUE(J)*RESIDUE(J)/(BETA(J)*BETA(J))
EPSILON=SQRT(EPSILON/6.0)
WRITE(6,888)BETA(1),BETA(2),BETA(3),BETA(4),BETA(5),BETA(6),
EPSILON

FORMAT(7F8.4)

IF(EPSILON.GT.0.001)GO TO 999

do 802 I=1,293

EI(I)-o.o

do 804 I=1,293

if(I.LT.10)THEN

K=4

else IF(I.GT.133)THEN

K=4

ELSE

K=5

end if

do 803 M=1,K
EI(I)=EI(I)+EXP(UTILIM(I,M))
EI(I)=LOG(EI(I))

continue

ASUM=0.0

do 710 I=1,2

do 711 J=1,2

W2MASUM(I,J)=0.0

WTMASUM(I)=0.0

do 801 I=1,293
ASUM=ASUM+DA(I,3)*EXP(UTILI(I)+Ei(I))
DO 824 K=1,2

do 822 I=1,293
WTMASUM(K)=WTMASUM(K)+DA(I,K)*DA(I,3)*EXP(UTILI(I)+EI(I))
continue

DO 834 L#1,2

DO 832 K=1,2

do 833 I=1,293
W2MASUM(K,L)=W2MASUM(K,L)+DA(I,K)*DA(I,L)*DA(I,3)*
EXP(UTILI(I)+EI(I))

continue

continue

do 806 I=1,2

do 805 J=1,2

805
806

809

810
811

812

813

815
814

816
825

818

819

21

D2MAFUN(I,J)=(WTMASUM(I)*WTMASUM(J)-W2MASUM(I,J))/ASUM
continue
DET=D2MAFUN(1,1)*DZMAFUN(2,2)-D2MAFUN(1,2)*D2MAFUN(2,1)
DZIN(1,1)=DZMAFUN(2,2)/DET
D21N(2,2)=D2MAFUN(1,1)/DET
DZIN(1,2)=(0.0-D2MAFUN(1,2))/DET
DZIN(2,1)=(0.0-D2MAFUN(2,1))/DET

do 809 I=1,293
ESTPI(I)=DA(I,3)*EXP(UTILI(I)+EI(I))/ASUM

do 811 J=1,2

D1MAFUN(J)=0.0

do 810 I=1,293
D1MAFUN(J)=D1MAFUN(J)+(PI(I)-ESTPI(I))*DA(I,J)
continue

DIFF=0.0

do 813 I=1,2

DELTA(I)=0.0

do 812 J=1,2
DELTA(I)=DZIN(I,J)*DlMAFUN(J)+DELTA(I)
ALPHA(I)=ALPHA(I)-DELTA(I)
DIFF=DIFF+DELTA(I)*DELTA(I)/(ALPHA(I)*ALPHA(I))
DIFF=SQRT(DIFF/2.0)

if(DIFF.gt.0.001)GO TO 998

XLIK=0.0

do 814 I=1,293

do 815 M=1,5
XLIK=XLIK+DA(I,I+3)*LOG(DA(I,M+3)/TOTAL)
continue

ESTLIK=0.0

do 825 I=1,293

if(I.LT.10)THEN

K=4

ELSE IF(I.GT.133)THEN

K=4

else

K=5

end if

do 816 M=l,K
ESTLIK=ESTLIK+DA(I,M+3)*LOG(PI(I)*ESTPIM(I,M))
continue

CHISQ=2.0*(ESTLIK-XLIK)
write(6,818)ALPHA(1),ALPHA(2),XLIK,ESTLIK,CHISQ
format(2F8.4,3F10.4)
write(6,819)BETA(1),BETA(2),BETA(3),BETA(4),BETA(5),BETA(6)
format(8F10.4)

stop

end

function UTILIM(I,M)

real DATA(293,2),BETA(6),COST(293,5),TIME(293,5)
COMMON/COMl/BETA,DATA,TIME,COST
UTILIM=BETA(1)*LOG(DATA(I,1)+COST(I,M))+BETA(2)*TIME(I,M)+
BETA(3)*DI2(M)+BETA(4)*DI3(M)+BETA(5)*DI4(M)+
BETA(6)*DI5(M)

return

end

function DIZ(M)
DI2=(M-1)*(M-3)*(M-4)*(M-5)/(-6.0)

return

ENTRY DIB(M)

22

DI3=(M-1)*(M-2)*(M-4)*(M-5)/(4.0)
return

ENTRY DI4(M)
DI4=(M-1)*(M-Z)*(M-3)*(M-5)/(—6.0)
return

ENTRY DIS(M)
DIS=(M-1)*(M-2)*(M-3)*(M-4)/(24.0)
return

end

function UTILI(I)

real ALPHA(2),DA(293,8)
common/COMZ/ALPHA,DA
UTILI=ALPHA(1)*DA(I,1)+ALPHA(2)*DA(I,2)
return

end

APPENDIX B

DATA FILE 1

COLUMN 7
COLUMN 1
COLUMN 2
COLUMN 3
COLUMN 4
COLUMN 5
COLUMN 6
COLUMN 8
4.9 413
5.0 1334
6.7 1750
5.0 1229
5.2 791
4.3 1205
5.3 1867
5.6 3568
5.0 2155
4.2 1780
3.6 691
4.0 1022
4.9 293
5.1 2116
5.1 2093
5.1 2492
3.1 1783
4.1 568
2.8 1368
5.0 912
3.0 1158
2.8 841
2.9 746
5.4 866
3.9 400
5.2 1160
5.0 682
3.4 902
4.6 200
3.0 1067
4.9 244
3.8 516
2.9 1146
2.7 1270
4.3 529
5.0 496
3.2 824
5.1 406
3.4 1235
2.1 376
3.4 974

= DA(I,3)
= ROOM

= DA(I,4)
= DA(I,5)
= DA(I,6)
= DA(I,7)
= DA(I,8)
= INCOME
82 9
273 13
315 17
399 134
195 79
314 111
394 127
822 154
598 170
638 544
263 522
408 351
85 108
779 353
830 605
994 860
745 1630
153 304
528 1402
490 446
430 1022
560 1012
710 990
203 209
162 244
402 454
270 225
494 1687
141 106
554 2702
51 118
177 279
500 3020
559 3029
300 331
226 123
284 1287
369 278
434 1605
100 658
320 1052

43
296
224
313
251
395
289
91
66
1038
754
1295
151
467
385
464
1552
150
1064
504
441
432
484
160
138
350
180
163
95
212
134
647
72
141
573
765
52
55
149
168
337

24
81
63
110
115
210
147
80
122
400
378
260
83
124
229
330
547
174
196
281
191
574
251
71
142
275
50
115
100
173
91
472
394
623
279
246
238
37
852
1009
1341

23

451

1912
1630
2310
1373
2187
2298
3141
2854
3970
2363
3269
486

3765
3647
4373
7566
1523
6360
2441
4248
4775
4589
1129
1202
2388
1334
3644
612

5458
599

1955
6272
6329
1775
1565
3335
927

5249
2943
5551

9282
11042
11131
9870
8932
7591
9181
7717
9170
8636
8382
7805
8180
11592
9281
8772
11023
7747
10268
8098
9351
5068
5615
10114
6852
7838
8044
14779
7132
14962
7426
10415
17190
17877
10884
12747
18014
17621
30320
8398
28636

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1164
176
1028
1309
418
560
315
888
871
2005
510
117
360
600
371
353
409
426
422
74
340
162
286
469
402
2470
804
361
1682
1749
1305
96
709
836
292
2379
526
1037
1291
427
487

12

103
296
420
326
351
368
629
510
761
148
610
153
489
299
168
63

403
586
205
446
403
393
212
195
480
337
675
301
32

122
484
176
248
959
113
137
83

175
106
171
226
85

761
290
201
614
792
435
81

420
344
100
691
181
342
524
120
265

367
50
113
218
152
386
242
665
506
507
110
543
186
443
351
128
140
809
1016
430
716
806
494
504
363
572
582
1158
232
177
209
382
170
225
462
343
670
154
261
148
227
353
352
1633
859
383
1617
1534
483
52
385
555
113
628
180
374
392
50
256

50
99
152
308
253
212
251
555
535
292
147
307
98
130
135
24
43
347
941
336
27
82
182
100
89
125
252
137
43
36
51
119
96
37
32
43
76
15
54
130
101
310
203
361
243
83
431
552
539
31
415
167
44
279
240
331
218
131
86

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178
87
74
172
359
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171
61
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220
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394
88

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246
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67

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26
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105
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97
29

166
266

324
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44

169'

180
365
171
174
156
305
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225
250
411
93

149
168
155
146
244
318
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324
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514
149
688
145
62

156
345
231
457
259
316
209
651
209
130
409

47
147
24
165
200
121
127
139
35
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195
69
143
127
140
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97
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71
257
184
34
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111
91
34
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102
50
177
99
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136
69
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96
573
126
238
67
286
11

43
27
118
67
81
142
190
85
61
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156

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2142
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3498
1633
1952
1175
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1098
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1186
1549
3358
1627
1389
1323
2747
2938
1381
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4509

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7882
8896
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6950
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7526
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4103
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6503
9021
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10158
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38
123
112
138
325
98
239
89
182
64
126
65
59
64
94

23
90
33
120
94
27

27

3532
2728
4112
2133
5778
5522
588

1586
1865
2625
2311
2990
1611
2785
2272
4775
2421
2085
2835
1882
8440
1476
2029
2350
2510
2229
2891
1162
691

4101
910

2009
836

2588
1851
909

2498
3622
5822
7815
2994
4289
3012
3817
2267
2870
3444
3638
4424
1875
2254
2149
945

1635
2415
320

1266
1947
1361

7608
7715
8246
8420
10038
8749
7219
7761
6541
9481
11231
9467
10893
10900
17486
11303
8309
10030
11072
9995
11281
13456
9498
9630
8801
10782
10350
8819
9531
10598
9426
11071
14390
11730
11642
10032
10713
9571
9504
9388
10910
10506
11062
11685
14669
11101
11049
10340
11229
11478
10807
10024
11237
7253
9784
10365
6718
12945
26688

UlmmmmmmmOSflmUIO‘UIQO‘

WHuomeUHufiUO‘UIl-‘m

1027
2251
1557
1303
2650
457

2136
3363
2532
6888
5530
4364
2736
5767
787

908

213
610
310
274
873
62
467
1019
651
1916
1803
933
496
1917
497
278

SOOhOQO

17
12
33
52
11
17
48
28
0

291
206
10
293
21
15
356
62
291
878
465
165
411
574
11
253

376
64
35
112
50
29
38
346
182
343
113
55
48
109

24

28

1346
2214
1460
1672
2627
442

2052
3797
2767
7416
5639
3955
2734
6228
1428
1182

16829
11201
9187
14845
8065
10621
17679
14428
9420
9416
8045
8656
8394
8149
5262
9857

APPENDIX C

DATA FILE 2

COLUMN 2
COLUMN 1

.ounmcnunuraa-ucnu:brauautuhnhcspcduhh4sh1npaH1»hamraenhhaprac-Haouawrd

6524
6005
9526
4857
4977
3950
5426
5563
3704
3511
3884
3512
3715
6427
4977
4398
3763
3450
3522
4116
4786
2513
2809
6813
2830
3338
2877
5336
2673
4987
2654
4321
6526
7471
4595
7405
6196
7425

10779

3054

11171

3809
6418

ANNUAL RENT
TRAVEL MODE

29

HthIl-‘HNHNHUNU'IHHHHHNU‘UIHuuuuNNPb)NbuwNUUIHNHNUHHHUIUIUHUDHHUQHHUIHH

7189
5507
8728
5421
5008
3006
4904
4963
4165
3187
3604
5943
4074
4097
2759
2612
2568
2266
2450
2225
1598
2186
1573
3702
2659
2731
2218
2517
2223
1677
2544
1830
2417
4428
9423
1822
7490
2124
4327
3929
2233
3060
2585
3163
2746
2630
3747
2608
2379
3115
1372
4383
3459
3412
3983
2062
1789
2737
3273

30

NoahHHHHHHHHfiHHHHHNHHhuHNHH-hwwuhHI-‘NHHHHHNUNKJNHmmmuMHmeHwi-le-DN

3715
5159
3854
3273
2508
1983
2133
2264
1762
4593
3060
4700
4160
4799
3652
3160
2142
1965
2222
2598
2575
5331
4848
3229
2858
3480
6387
3409
3761
3822
3983
3958
3425‘
4620
6173
6531
4894
4323
8673
6732
10512
4958
5628
3608
14116
7028
2891
6009
7256
5931
7604
7893
9479
6161
4612
8199
6984
8516
11351

bidNHNHhHl—‘HHNNHHHHNHHhHHHNHHHHHHhHéHPD—‘NHHHHHHHhNHhHNHNbHHbI-‘H

9945
6543
6522
5465
8519
4939
4996
6248
5962
7393
3254
9199
8094
3574
7524
6715
8145
8739
6412
7611
6570
6292
9103
9919
9948
17075
19175
18998
6197
10084
5236
7263
4842
5129
5704
6282
11130
7099
14638
14557
11305
5768
7386
9177
8706
7341
6306
5550
5507
6814
6876
4923
8563
8366
3970
5584
9979
5253
3084

32

 

HHUHHHHNH-hNl-hbHHHHHHHHHHNHHNHHHHHHHHHHHHNHl-‘HHH#NHHNHHHHHHHHH

4685
5128
8948
6400
5887
5392
6541
5362
7611
4847
8042
5431
9498
7105
5636
9102
5939
6086
8171
8676
6198
5493
6150
8667
7343
5936
4088
6682
7803
7570
13226
5296
5812
9169
7881
5190
7228
7195
7400
6695
7927
6790
6451
5945
5373
6079
7569
5374
8681
7856
7516
5007
6792
7535
3504
9667
15168
10871
9521

HHNHHHHHHHNHHH

6359
10731
5369 34
8783
11171
5727
7228
6531
5734
8342
4836
5729
2690
4278

BI BLIOGRAPHY

35

M.Ben-Akiva and Lerman, DISCRETE CHOICE MODEL, MIT Press,
1985

A.Anas, RESIDENTIAL LOCATION MARKET AND URBAN
TRANSPORTATION, Academic Press, 1982

Chicago Area Transportation Study, PERSONAL ENERGY
CONSUMPTION, 1980

G.Ingram(ed), RESIDENTIAL LOCATION AND URBAN HOUSING
MARKETS, Cambridge,Mass: Ballinger, 1977

M.Manheim, FUNDAMENTAL OF TRANSPORTATION SYSTEMS ANALYSIS
Vol 1:Basic Concept, MIT Press, 1979

D.Sega1(ed), THE ECONOMICS OF NEIGHBORHOOD, Academic
Press, 1979

U.S.Census Bureau, CHICAGO SMSA CENSUS TRACT, 1980

 

. llHIHWIHI|||||l|l|||l|IllIlllllllllllHlHHlllmIIIIIH\l

 

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