Multinomial and Conditional Logit Discrete-Choice Models in Demography
Author(s): Saul D. Hoffman and Greg J. Duncan
Source: Demography, Vol. 25, No. 3 (Aug., 1988), pp. 415-427
Published by: Population Association of America
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Demiiograplhy,
Vol. 25, No. 3, August
1988
Multinomial
and Conditional
LogitDiscrete-Choice
Modelsin Demography
Saul D. Hoffmnan
of Economics,University
of
Department
Delaware,Newark,Delaware 19716
GregJ.Duncan
of
Institute
forSocial Research,University
Ann Arbor,Michigan48106
Miclhigan,
Althoughdiscrete-choice
statisticalteclhniqueslhavebeen used with incrcasinig
regularity
in demographic
anialyses,
McFaddein'sconiditionial
logitmodelis less well
knownand seldomused. Coniditional
logitmodelsare appropriate
wlleiltllechoice
is modeledas a functioni
ofthecharacteristics
ofthealterniatives,
amongalterniatives
ratherthan(or in additionto) thecharacteristics
oftheindividLual
makingtllechoice.
We arguethatthisfeature
ofconiditional
forestimatinig
logitmakesitmoreappropriate
anid
behavioralmodels. In this article,the coniditional
logit model is-presenited
betweenl
comparedwiththe morefamiliarmultinomial
logitmodel.The differcnice
ofthechoiceofmaritalanidwelfare
thetwotechniquesis illustrated
withanlanialysis
statusbydivorcedor separatedwomeni.
Statistical
techniques forthe analysisof discretechoices have beeinused withincreasinlg
regularityin demographic analyses.I The best known are the binomial logit and probit
techniques,bothofwhichare suitableforbinarychoiceproblems.For problemsinvolving
thechoice amongthreeor morecategories,
themultinomial
logittechniqueis mostoften
littlebecause of itscomputaemployed;the corresponding
probitmodelis used relatively
tionaldifficulty.
Virtuallyunusedthusfaris a closelyrelatedtechniquecalled conditional
logit,a modelthatis wellsuitedforbehavioral
modelingofpolychotomous
choicesituations.
DevelopedbyMcFadden(1973),theconditional
logitmodelis widelyusedin transportation
demandstudies(see Ben-Akivaand Lerman,1985) but is seldomused in demographic
2
research.
Conditionallogit is not simplya different
and arguablypreferable
techniquefor
thekindof modelsforwhichmultinomial
used. Rather,it is
estimating
logitis currently
fora different
is treatedas
appropriate
classofmodelsin whicha choiceamongalternatives
a functionof the characteristics
of the alternatives,
ratherthan (or in additionto) the
characteristics
oftheindividualmakingthechoice.
We believethatmanyproblemsofinterest
to demographers
and othersocialscientists
can be modeledby usinga "characteristics
of the alternative"
approach.Thus theyare
estimatedwithconditionial
we suggestthatit is often
appropriately
logit. Furthermore,
totheresults
ofmodelsthatfocusexclusively
on
difficult
toattacha behavioralinterpretation
the"characteristics
of thechooser"-thatis, thoseestimated
multinomial
byconventional
logit.
The nextsectionofthisarticledescribes
thebasicstatistical
oftheconditional
properties
logit(CLGT) modeland comparesit withthebetterknownmultinomiallogit(MNLGT)
modelsofindividualbehaviorthatlead
model.3 It also considers
theformoftheunderlyin-g
(D 1988Populationi
Associationi
ofAmicrica
Copyright
415
Vol. 25,No. 3, August1988
Demography,
416
The thirdsectionpresents
a briefdiscussionof some
to MNLGT and CLGT estimation.
totheCLGT model.The finalsectionuses
and estimation
issuesrelating
practicalstatistical
thedifference
betweenthetwo
datafromthePanel StudyofIncomeDynamicsto illustrate
women'schoiceamonga set
techniquesin appliedwork.We examinedivorcedorseparated
byfirst
usinga pureMNLGT model,thena pure
ofmaritaland welfarestatusalternatives
ofboth.
features
CLGT model,and thena mixedversionthatincorporates
Statisticaland ModelingIssues
Both multinomiallogitand conditionallogitare used to analyzethe choice of an
betweenthetwo,canbe put
The centraldistinction
individualamonga setofJalternatives.
verysimply:MNLGT focuseson the individualas the unit of analysisand uses the
variables;in contrast,
as explanatory
CLGT focuseson thesetof
individual's
characteristics
variablesare characteristics
of those
foreach individualand the explanatory
alternatives
4
alternatives.
ofthejth
ofindividuali andZi,forthecharacteristics
LetXi standforthecharacteristics
vectorsdenotedby/3and a,
alternative
forindividuali, withthecorrespondinig
parameter
(forthe moment,assumed
Let Jbe the numberof unorderedalternatives
respectively.
thatindividuali choosesalternative
j. The
andPi,theprobability
forall individuals)
constant
in theMNLGT and CLGT modelsare
choiceprobabilities
J
MNLGT:
P11= exp(Xi/31) I
Jk=
CLGT:
Pjj = exp(Zija)/
exp(Xi/3k),
: exp(Zika).
(1)
(2)
k =j
ofthealternatives
and theinidividual,
the
In a mixedmodelthatincludesbothcharacteristics
as
canlbe written
corresponding
probability
J
Mixed:
Pj; =
I
k= I
+ Zika).
exp(Xifjl+ Z1jfl)/Cxp(Xi,Pk
(3)
in thenextsectionand estimatesucha model
We discussthemixedlogitmodel(3) further
in thelastsectionofthisarticle.5
in equations(1) and (2). In theMNLGT model,theexplanatory
Note thesymmetry
acrossthe
of the individual,are themselvesconstanit
variables(X), being characteristics
choiceprobabilities
is byhavinga
theonlywaytheycan affect
alternatives.
Consequently,
Thus in practice,MNLGT estimatesa set of
different
impacton thevariousalternatives.
J- 1 coefficients
showthe
variable.The estimatedcoefficients
((31)foreach explanatory
relativeto one
of choosingeach alternative
effectof the X variablesonithe probability
Thereare onlyJ- 1 coefficients,
because
alternative
thatservesas a commonbenclhmark.
is arbitrary.
Thus it is necessary
to normalizeon one set of
the scalingof the coefficients
the correspondinig
by settingit equal to zero. For thisalternative,
coefficients,
typically
is 1/Eexp(Xi,/31),
since,3 = 0 and exp(O)= 1.
probability
variables(Z) assumedifferent
In contrast,
in theCLGT model,theexplanatory
values
in each alternative
on Z butnotX), buttheimpactofa
(notethepresenceofa j subscript
In
acrossalternatives.
unitofZ is usually,althoughnotnecessarily,
assumedtobe constan-t
is estimated
foreachZ variable,so theimpactofa variable
thatcase, onlya singlecoefficient
417
Multinomialand ConditionalLogitModels
in its value acrossalternatives.
derivesfromthe difference
on the choice probabilities
a Z (or X) variablewithno variation
Consequently,in the standardCLGT formulation,
When such variablesare deemed
has no impacton choiceprobabi1ities.
acrossalternatives
to be important,
themixedmodelis required.
is clearerwhen
betweenthe MNLGT and CLGT formulations
The basic difference
bythenumerator:
bydividingthrough
equations(1) and (2) are rewritten
J
P1j= 1
MNLGT:
P1j= 1
CLGT:
I
exp[Xi(3k
/k=l
E
k=i
exp[(Zik -
-
Pi)],
(4)
Zj)aj.
(5)
in the coefficients
across
in equation(4) dependson the difference
Here, the probability
in thevalue
whereasin equation(5), theprobability
dependson thedifferences
alternatives,
ofthecharacteristics
acrossalternatives.
betweentheMNLGT and CLGT modelsis notmerelyone ofstatistical
The difference
modelsof
in equations(1) and (2) reflectthe underlying
form.The choice probabilities
hypotheses
aboutthebasison whichindividuals
reflect
individualbehaviorthatnecessarily
moveto
Oftenthisis notmadeexplicit,and researchers
makechoicesamongalternatives.
behavioralmodel.In fact,
theunderlying
theirempiricalestimation
withoutfirst
specifying
oftheempiricalresults.
however,it is a crucialstepfortheinterpretation
j to individuali, and assume,as a
Let Vi, standforthe value (utility)of alternative
behavioralrule, thatan individualchooses his or her most highlyvalued alternative.
someunspecified
ofthealternatives
(Z1)through
SupposethatVii dependson theattributes
bya pairofequationsas
functional
form(fi). Then thechoiceproblemcan be represented
follows:
vil
=
f(Ziv),
Pi, = Pr(Vi, > Vik) all k notequal to j.
(6)
(7)
Withtheadditionofan appropriately
definederrorterm,6equation(6) leadsto theCLGT
ofthealternatives
are the
modelratherthantheMNLGT model,sincethecharacteristics
notonlyabout
ofchoice. The estimated
offi provideinformation
parameters
determinants
in equation
thechoiceprobabilities
throughequation(2) butalso aboutthevalue function
(6).
The specificformofequation(6) will,ofcourse,varywiththenatureoftheproblem
alwaysregardutilityas a function
and the discipline.Economists,forexample,virtually
the
(definedbroadly)or, equivalently,
of an individual'slevel of consumption
primarily
exogenousincomeand thesetofpriceshe or she faces.Viewedin thisway,equation(6) is
ofthealternatives
a statement
aboutthefunctional
relationship
betweenthecharacteristics
ofeach alternative
to theindividual(theVq1's)-inshort,a utility
(theZi,'s) and theutility
maximization
function.Equation(7) represents
thewell-known
applied
principleofutility
to a discrete
a versionofequation(6) in thelastsectionofthis
choiceproblem.We estimate
article.
althoughwe
Noneconomicmodelsbased on equation(6) could also be formulated,
knowof no attempts
to do so. For example,a choice modelof becomingmarriedversus
oftheeconomic
as a function
remaining
singlemightviewthevalueofthesetwoalternatives
and otherattributes
thateach provides,withthe
security,
companionship,
independence,
Demography,
Vol. 25,No. 3, August1988
418
perceivedextentoftheseattributes
in each alternative
obtainedthrough
survey
questions.To
avoidproblemswithrespondents'
rationalizing
pastdecisions,a usefulresearch
designmight
is ascertained
in thefirst
waveand
be a two-wave
panelin whichtheattitudinal
information
the behavior(e.g., transition
to marriage)is measuredin the second. Thus reportsby
ofthevariousmaritalstatesin a first
wavecould
abouttheattributes
unmarried
respondents
interview.
be used to predicttheprobability
ofmarriagein a follow-up
Anotherexamplewouldbe a child-carechoice modelin whichthe value of a given
child-caremodel (e.g., day care, sitterin own home) is taken to be a functionof
on childdevelopment,
itscost,and itsreliability.
characteristics
such as itslikelyeffect
of each
Note thatin thesetwo examplesthe perceivedor objectivecharacteristics
or satisfaction
are used to explain an
alternative
ratherthan its subjectiveimportance
individual'schoice. The statisticalmodel would then provideinformation
(i.e., the
about the relativevalue that individualsplace on the various
estimatedcoefficients)
fromtheiractualbehavior.
characteristics,
inferred
In general,MNLGT will be
What behavioralmodel leads to MNLGT estimation?
appropriate
whenequation(6) is replacedwith
Vi
= f2(Xi)7
(8)
functionrelating
wheref2is some unspecified
Xi to Vii. In equation(8), the value of an
is regarded
as a functionofthecharacteristics
ofthe individual.Assumingthat
alternative
equation(7) stillholds,equation(8) thenleadsto MNLGT estimation.
a nonbehavioral,reduced-form
Multinomiallogitcan also be shownto represent
versionofequations(6) and (7). Ifequation(6) holdsbut
zi, = g(Xd,
(9)
then
Vij=
h(Xi).
(10)
of the ith
Equation (10) relatesthe value of the jth alternative
to the characteristics
ofthejthalternative.
individual,butwithoutincludingthecharacteristics
In a sense,the choice betweenMNLGT and CLGT is the choice betweena model
represented
by eitherequation(8) or equation(10) and one represented
by equation(6).
we thinkthata model
Although
thereis,ofcourse,no generalruleaboutwhichis preferable,
basedon equation(6) and utilizingequation(7) has muchto recommendit. The general
in favor
modelsis one argument
modelsarepreferable
toreduced-form
notionthatstructural
of equation(6) and CLGT estimation
insteadof equation(10) and MNLGT estimation.
about
Even thoughmodelssuch as equation(8) mayprovidedirectand usefulinformation
whichindividuals
makewhichchoices,theyare oftennotwellsuitedto testing
hypotheses
ofmodelssuchas equation(8)
aboutwhythosechoicesaremade. Indeed,theinterpretation
of alternatives
availableto particular
to the (untested)characteristics
oftenmakereference
individuals.
in
Models such as equation(6) are especiallywellsuitedfortheanalysisof situations
of an alternative
which government
the attractiveness
policyaffects
by changingsome
relevantcharacteristic.
Examplesincludethe Aid to FemalesWith DependentChildren
whichprovidesincometo female-headed
(AFDC) program,
familieswithchildren;scholand
arshipaid forhighereducation,whichmaymakecollegeattendancemoreattractive;
ofwomen.To assessthe
subsidiesfordaycare,whichmayincreasethelaborforceactivity
effect
ofgovernment
whenpossible,
policiesliketheseon individualchoices,itis necessary,
in the choice problem.Since theseparameters,
to includethe policyparameters
directly
in question,a conditionallogitmodel
ofthealternative
though,aretypically
a characteristic
such as eauation(6) is theaDDroDriatemodel.
Multinomialand ConditionalLogitModels
419
StatisticalProperties
and EstimationIssues
In thissection,we presenta briefsurveyof some of the practicalstatistical
issues
involvedin theestimation
oftheMNLGT and CLGT models.Amongotherthings,we call
attention
hereto one ofthepotentially
undesirable
restrictions
imposedbythelogisticform
used in eithermodel. We also discusssome estimation
issues.
LikelihoodFunction. Despite the differences
discussedearlier,the CLGT and
MNLGT modelssharea commonlikelihoodfunction:
log
L
=
yi,P
'
(ll)
j and equals 0 otherwise.The difference
wherey,1= 1 ifindividuali choosesalternative
as in equations(1) and
betweenthemodelsis in theformulation
ofthechoiceprobabilities,
As a practicalmatter,
(2), and in theunderlying
behavioralmodelsthattheyrepresent.
these
and in the
in thewaythedataareprepared
forestimation
differences
oftenlead todifferences
software
used to estimatethetwomodels.
programs
StatisticalSpecification.Both the CLGT and MNLGT modelsare based on the
assumptionthatthe errortermsin equations(6), (8), and (10) followan extremevalue
distribution
and are independent
acrossalternatives.
The assumptionof independenceis
difficulties
critical;any otherassumptionleads to substantial
computational
involvingthe
The "cost"of the independenceassumptionis the
computationof multivariate
integrals.
ofirrelevant
so-called"independence
alternatives"
(IIA) problem.As derivedfromequation
foranytwoofthej alternatives
(2), theratioofthechoiceprobabilities
dependsonlyon the
characteristics
of those two alternatives.If, for example, there is a change in the
in thechoiceset,thisproperty
characteristics
ofanyotheralternative
requiresthatthetwo
in orderto preserve
theirinitialratio.7This is equivalent
probabilities
mustadjustprecisely
is equal, a responsepattern
that
to assumingthatthepercentage
changein each probability
maybe an unwarranted
and inappropriate
restriction.
For example,thepossibility
thatone
excluded.
choiceprobability
mightbe moregreatly
affected
bysuch a changeis thereby
As a practicalmatter,the independenceassumptionis mostlikelyto be problematic
one
whenthe alternatives
are similarto one another,so thatunobservedfactors
affecting
can be tested(Hausman
alternative
anotheralternative.
The IIA assumption
maywellaffect
and McFadden, 1984). Ifit is notsupported,
thereare twogeneralalternatives.
One is the
normalcorrelated
errorterms.The
conditionalprobitmodel,whichallowsformultivariate
otheris thenestedlogitmodel(Hausmanand McFadden,1984;McFadden,1981)in which
the choice processis viewed as a set of nestedchoices. This approach retainsthe
computationaladvantagesof the logit formbut selectivelyrelaxesthe independence
and thereby
allowsa variety
ofresponse
toa changein thecharacteristics
assumption
patterns
ofone alternative.
statistical
softwarepackagescontain a
StatisticalSoftware. Most general-purpose
bivariate
toestimate
theCLGT model
logitprocedureand an MNLGT procedure.Software
is less common.Our CLGT estimation
uses the DiscreteChoice procedureavailablein
withLIMDEP's Logitprocedure.The Mlogit
LIMDEP; theMNLGT modelis estimated
procedurein SAS can be used to estimateboththeMNLGT and CLGT models.
EstimationDetails.8 The estimationof a CLGT model is somewhatunorthodox,
because the unitof analysisis, in some sense, not the individualbut, rather,the set of
availableto each individual.
alternatives
To estimatea CLGT model,
each ofwhomhasJalternatives.
ConsiderN individuals,
an
an individual'srecordis transformed
into Jdistinctrecords,each one representing
arerepresented
in thesamesequenceforeach
alternative
forthatindividual.The alternatives
individual;the firstrecordrepresents
alternative
1, the secondalternative
2, and so on to
420
Demography,
Vol. 25, No. 3, August1988
recordand alternative
7. The explanatory
variablesare similarly
to reflectthe
constructed
valueofeach variableforeach individualin each alternative.
An individual's
choiceamong
is indicatedbya 1 fortheappropriate
thealternatives
record;theotheralternatives
arecoded
0.
Table 1 illustrates
the typicaldata structure
forCLGT estimation.In thisexample,
foreach offourindividuals,
whochoosealternatives
therearethreealternatives
3, 2, 1, and
2, respectively.
There is one X variableand one Z variable;the inclusionof individual
characteristics
means thatthe model is reallya mixedmodel ratherthan a pure CLGT
model. Z11 is the value forindividual1 of some characteristic
in alternative
1, Z12 is the
forthatindividualin alternative
2, Z21 is the value of that
value of thatcharacteristic
characteristic
forindividual2 in alternative
1, and so on. Estimationofthismodelwould
yielda singlecoefficient
forZ.
The finaltwocolumnsshowhowan attribute
thatis invariant
acrossalternatives
can be
introducedto createa mixedlogitmodel. Let D2 be a dummyvariableequal to 1 for
2 and 0 fortheotheralternatives,
and letD3 be definedsimilarly
alternative
foralternative
3. The variablesin thefinaltwocolumnsareD2X and D3X; justas in MNLGT estimation,
theygive the effectof variableX relativeto an omittedcategory,here alternative1.
Estimation
ofthismodelwouldyieldthreecoefficients-one
each forZ, XD2, and XD3. If
could be constructed
desired,constanttermsfortwoalternatives
by usingD2 and D3.
ofCLGT models.In somechoice
Thereis an additionaland particularly
usefulfeature
is availableto everyindividual.For example,women
situations,not everyalternative
withoutchildrenare categorically
ineligibleto receiveAFDC, and onlywomenlivingin
can chooseto be bothmarriedand receiving
statesoffering
theAFDC-UP program
welfare;
onlywidowswithlivingchildrencan choose to live withtheirchildren;onlyindividuals
owningcarsor livingnearbus routescan driveortakethebus to work,respectively.
Taking
in the size and compositionof the choice set availableto
properaccountof differences
and oftenleadsto clumsy,ad
is troublesome
undermostcircumstances
specificindividuals
so thattheanalysisis no longergeneral,or 0
hoc solutions.The samplemaybe partitioned
valuesmightbe assignedfortheindependent
variablesin caseslikethatconcerning
AFDC
benefits
ofwomenineligibleto receiveAFDC.9 A morenaturalsolution,however,is simply
'0 Withthechoice
toeliminatean irrelevant
fromthechoicesetforan individual.
alternative
forCLGT
Table 1. TypicalData Structure
Estimation
Dependent
Individual
Alternativevariable Z XD2 XD3
1
2
3
4
1
2
3
0
0
1
Z,1 0
1
2
3
0
Z21 0
0
1
2
3
1
0
0
1
2
3
0
Z41 0
0
0
Z42
Z43
0
X4
1
1
Z12X1
Z13
0
Z22
X2
Z23
0
0
0
Xi
0
0
X2
Z31 0
0
Z32
0
X3
Z33
X3
0
X4
0
Multinomialand ConditionalLogitModels
421
set as the unit of observation,
tailoringthe choice set to individualcircumstances
is a
straightforward
matter.
An EmpiricalExample:Remarriage
and WelfareChoices of
Divorcedand SeparatedWomen
In thissection,we examinetheremarriage
and welfare
choicesofdivorcedorseparated
women.We presentestimates
ofthreemodels-a standard
MNLGT modelwithindividual
characteristics
as explanatory
variables,a pure CLGT model withcharacteristics
of the
alternatives
as explanatory
variables,and a mixedmodelthatincludescharacteristics
ofboth
theindividualand thealternatives.
Our analysisis based onldata fromthe Panel Studyof IncomeDynamics(PSID) on
whitewomenunderthe age of 45 who becamedivorcedor separatedbetween1969 and
1982. Each womanis observedfromthedateofherdivorceor separation
untilremarriage,
theend ofthepanelobservation
period,orthesixthpost-divorce/separation
year,whichever
comesfirst.
Our dataare in person-year
event-history
definedovera spellofbeing
format,
"unmarried."Formally,we are estimating
a discrete-time
hazard model of time until
remarriage,
usinlg
an MNLGT or CLGT modelas theestimation
procedure.Time-varying
independent
variablesare measuredas oftheperson-year
used in theanalysis.See Allison
(1982, 1984) fora generaldiscussionofdiscrete-time
hazardmodels.1
In each year,a womanis observedin one ofthreealternative
states:she can remarry,
she can remainsingleand receivewelfare,or she can remainsinglewithoutreceiving
welfare.We use functionial
ratherthan legal definitionsof marriage,divorce,and
remarriage.
Unmarried
couplesaretreated
as married
bythePSID iftheyresidetogether
for
twoconsecutive
interviews;
giventhisdefinitioni,
we can analyzethe"remarriage"
choiceof
separatedwomen.Welfarereceiptis definedas receiving
a dollaror moreof incomefrom
AFDC or the "otherwelfare"category
used in thePSID. 12
We are interestedin analyzingthe determinanits
of the trichotomous
choice of
remarriage,
welfarereceipt,and remaining
singlewithoutwelfarereceipt.The motivation
includesunderstanding
the potentialroleofAFDC incomein discouraging
remarriage
as
wellas themoregeneraldeterminants
of remarriage
decisions.
One explanation,
castin termsofindividualcharacteristics,
mightfocuson suchthings
as a woman'sage and education,thenumberofchildrenshe has, and whether
she resides
in an urbanarea. We estimatethismodelwiththeMNLGT model.
A different
explanationmightconsider,instead,the exogenousincome(the income
availableto a womanat zerohoursofwork)and hernet(after-tax)
wageratein each ofthe
threealternatives.
This corresponds
to a modellikeequation(6) in whichthevalue of an
alternative
is a functionof itscharacteristics,
hereexogenousincomeand prices.Technically,we are usingtheconceptof indirectutility
in whichthemaximumutility
functions
(satisfaction)
availableto an individualin an alternative
dependson itsexogenousincome
and thesetofprices(in our model,thewagerate)it provides.
Consider,first,
theexogenousincomeavailableto a womanin each alternative.
While
someincome,suchas childsupport,
and incomefromdividends
and interest
areunaffected
by her choice of alternative,
othercomponenits
varysystematically
by alternatives.
For
instance,ifshe wereto acceptwelfare,
shewouldreceivethelegallymandatedbenefits
paid
in herstateofresidence,givenherfamilysize and otherincome.Ifshe wereto marry,
she
13 butshe wouldhaveaccessto someportion
wouldbe ineligibleforwelfareand benefits,
of
her new husband'sincome. If she remainedsinglewithoutacceptingwelfare,she would
receivealimonyand/orchildsupportincome,ifany,plusanyincomefromdividendsand
interest.
Her after-tax
wage ratewouldalso differ
acrossalternatives,
even thoughthe market
422
Demography,
Vol. 25, No. 3, August1988
(pretax)wageratefora particular
womanis likelytobe constant
acrossalternatives.
After-tax
wagesare particularly
low in thewelfarealternative
byvirtueofthehighbenefitreduction
rateappliedto earnedincomein welfare
and theexistence
ofan earnings
ceilingto maintain
welfareeligibility.
This model,in whichchoiceamongalternatives
is a function
oftheexogenousincome
and wagecharacteristics
of each alternative,
is estimated
withthe CLGT model. We also
estimatea mixedmodelthatincludestheindividualvariablesfromtheMNLGT modeland
thealternative-specific
variablesfromtheCLGT model. In boththepureCLGT and the
mixedmodels,we allow the numberof alternatives
to varyacrossindividuals.Women
withoutdependentchildrenand womenwithsubstantial
nonlaborincomeare ineligiblefor
welfare,and thusthatalternative
is notavailableto them.
The characteristics
ofthesample,includingsamplesize and mean value ofall ofthe
in Table 2. The MNLGT resultsappearin Table 3,
independent
variables,are presented
columns1 and 2. The coefficients
relativeto the omittedcategory,
expresseffects
single/
welfare.We findthatmoreeducatedwomenare morelikelyto be eithermarriedor single/
no welfare
thantobe receiving
welfare,
whereasresidencein an urbanareaand havingmore
childrenbothdecreasethoseprobabilities.
Olderwomenaremorelikelyto be singleand not
buttheyareno morelikelytobe married.Despitethestatistically
receivewelfare,
significant
it is noteasyto explainwhythesevariableshavetheimpactsthattheydo: Do
coefficients,
Table2. Characteristics
ofPSID Sampleof
WhiteWomenUndergoing
Divorce
orSeparation,
1969-1982
Characteristic
No.
Samplesize
Persons
Person-years
alternatives
Person-year
Married
welfare
Single/no
Single/welfare
Individual
characteristics
460
1,269
3,304
1,269
1,269
766
Age
Years ofeducation
No. of children
Urban residence
Economiccharacteristics
ofthealternatives
($)
AFDC income
Mean
30.9
12.1
1.4
0.28
(thousands)
3.68
(thousands)
16.26
Spouse incomea
Wage ratea
Married
Single/nowelfare
Single/welfare
4.80
5.59
1.30
Note: Allfigures
inthetableare weighted
to adjustfor
and nonresponse
differential
rates.
samplingproportions
Alldollarfigures
are expressedin 1982 dollars.AFDC =
Aidto FemalesWithDependent
Children.
a Computed
on an after-tax
basis.
Multinomial
andConditional
LogitModels
423
Table3. EstimatesofRemarriage
and Welfare
ChoicesofDivorcedand SeparatedWhiteWomen,
PSID, 1969-1982
Variable
Constant
No. of children
Age
Education
Urban residence
Husband's
MNLGTmodel
CLGTmodel
Mixedmodel
Married Single
Married Welfare Single
Married Welfare Single
-0.680*
(0.093)
-0.818*
(0.076)
(0.018)
(0.016)
(0.020)
(0.065)
(0.057)
0.017
0.363*
-0.731 *
(0.238)
0.094*
0.024
-0.216*
-0.249*
-0.613*
- 0.578*
- 0.548*
(0.203)
(0.093)
Wage ratea
-
1,269
(0.093)
(0.270)
-0.018
(0.017)
1,269
-950.4
0.081*
(0.018)
0.435*
incomea
Nonlabor
- 3.415*
(0.922)
0.31 7*
(0.080)
AFDC income
income
Number
of
cases
Log-likelihood
- 2.587* - 3.004*
(0.499) (1.001)
-2.918*
-4.682* -2.408*
(0.873) (0.783) (0.542)
(0.247)
0.047*
0.192*
(0.022)
0.202*
(0.051)
1.102* 1.102*
(0.107) (0.107)
(0.055)
1.102* 1.492* 1.492*
(0.107) (0.162) (0.162)
1.492*
(0.162)
(0.090)
(0.076)
(0.076)
-0.011
1,269
766
-828.4
0.215*
1,269
-0.102
(0.092)
1,269
766
-770.0
0.182*
1,269
Predicted
value,after
tax.
*Significant
atthe5 percent
level.
a
the resultsreflectdifferences
in opportunity,
or does behaviordiffer
even givenlsimilar
opportunities?
The negativeeffect
ofchildrenon remarriage
is illustrative,
sinceone might
well hypothesize
thatmarriage
wouldbe especiallyattractive
to womeniwithchildren.The
estimated
coefficients
are usefulfordetermininig
whomakeswhichchoice,buttheyare less
usefulforexplainingwhyshe does so.
The pureCLGT modelis shownin columns3-5. We see therethatthe incomeofa
woman's(potential)new husbanddoes not have a significant
effect
on the probability
of
theeffect
remarriage;
is, in fact,negative
butverysmall.14 In contrast,
theamountofAFDC
benefitshas a positiveand significant
effect
on the probability
thata womanwill be on
AFDC. We findthatnonlaborincome(mostlycomposedofalimonyand/orchildsupport)
has no effect
on theprobability
ofbeingmarriedrelativeto welfare,
butthatitincreasesthe
probability
thata womanwillbe singleand notreceiving
15
welfare.
Interpretation
oftheestimated
effect
ofa woman'swagerateis illustrative
oftheCLGT
approach.As shown,thewagecoefficient
is large,positive,
and significant.
Itscocfficient
has
been constrained
to be equal acrossalternatives,
the assumption
reflecting
thata dollarof
after-tax
incomeis equallyvaluablein each alternative.
The positivecoefficient,
therefore,
indicatesthathigherwagesincreasethevalue ofan alternative.
Althougha woman'swage ratehas thesame effect
on utilityin each alternative,
this
does notmeanthatithas no effect
on herchoiceamongthealternatives.
The effect
ofany
variableon choiceprobabilities
derivesfromthedifference
in itsvalueacrossalternatives
[see
424
Vol. 25,No. 3, August1988
Demography,
eq. (5)]. Thus a woman'swage rateaffects
herchoice,dependingon how herwagevaries
across alternatives.
That variation,in turn,dependson the estimatedincome of her
prospective
husband,thescheduleofwelfarebenefits
in herstate,and herown wagerate.
For example,becauseafter-tax
marriedand singlewagesaresimilarformostwomen,16 the
wageratedoes notgreatly
affect
thechoicebetweenmarrying
and remaining
single.It does,
however,substantially
affect
thechoicebetweenthosetwoalternatives
and welfarebecause
after-tax
welfarewagesare sharplylower.Moreover,the difference
betweenwelfareand
nonwelfare
wagesis greatest
fortwogroupsof women:womenin statesthatprovidelow
welfarebenefits,a practicethateffectively
imposesa verylow maximumwage rate,and
high-wage
women,sincetheabsolutedifference
betweenwelfareand nonwelfare
wagerates
is greatest
forthem.17
Finally,considerwhatwouldhappeniftherewerea $1 increasein a woman'spretax
wagerate.Utilitywouldrisein each alternative
by 1.102 (thecoefficient
on thewage rate
fromTable 3) timesthe resulting
increasein after-tax
wages.Thus, forexample,utility
wouldincreaseleastin thewelfarealternative,
againbecauseofitshightaxrate.Although
utilityin each alternative
is now higher,it is, ofcourse,impossiblefortheprobability
that
each alternative
is chosento increasesimilarly.Rather,the resulting
choice probabilities
wouldbe calculatedbyusingequation(2) andsubstituting
thenewsetofafter-tax
wagerates.
In thiscase, theprobability
ofchoosingwelfare
wouldfall,sinceitsutility
levelis nowlower
relativeto theothertwoalternatives.
Estimatesof the mixedmodel are presentedin columns6-8. Coefficients
on the
individualcharacteristics
nlow showthe impactof thesecharacteristics,
net of a woman's
economicopportunities;
as in theMNLGT model,theyaremeasuredrelativeto thesingle/
welfarealternative.
are now estimatedto have substantially
Many of thesecharacteristics
different
and more readilyinterpretable
effects
on remarriage
and welfarechoices. For
example,the numberof children'8a womanhas is now seen to increasethe utilityof
marriage
and hencetheprobability
thatshe willremarry,
aftercontrolling
forhermarriage
This finding
opportunities.
nicelyseparatesthe negativeimpactof childrenon remarriage
19
fromtheir positiveimpact onl remarriage,given those opportunities.
opportunities
Additional
yearsofeducationnowreduce,rather
thanincrease,theprobability
ofbothbeing
marriedand beingsingle/no
welfare,relativeto beingon welfare.Urban residencestill
lowerstherelativeprobability
ofbeingeithermarriedorsingle/no
and age similarly
welfare,
increasesthe probabilities.
As forthe characteristics
of the alternatives,
the income of a
woman's potentialspouse is now estimatedto be positiveand statistically
significant,
smallin magnitude.
The effect
ofAFDC is virtually
althoughitis stillrelatively
unchanged
bytheadditionoftheindividualcharacteristics.
Finally,we notethatboththe pureCLGT model and the mixedmodel are ideally
suitedto simulationof policychangeswhenever,
as in thiscase, thecharacteristics
of the
alternatives
aredetermined
bygovernment
policy.One can easilyassignnewvaluesto reflect
thepolicychangeofinterest
and thenrecalculatetheappropriate
probabilities
byusingthe
20 One can alsodo thisforthepureMNLGT model,butthe
estimated
structural
parameters.
results
of,forinstance,simulating
theeffect
ofa changein thenumberofchildrena woman
has or in her education are less informative
and less directlyamenable to policy
manipulation.
Summary
This articlehas providedan introduction
to and illustration
ofthe use of conditional
logitto estimatemultiple-category
discrete-choice
problems.CLGT is closelyrelatedto the
better-known
MNLGT model,but it derivesfromdifferent
and is
behavioralassumptions
estimatedin different
form.The CLOT modelis appropriate
wheneverit is reasonableto
425
Multinomialand ConditionalLogitModels
are a functionof the relevant
assumethatindividualchoicesamongavailablealternatives
oftheindividual.In thelatter
thantheattributes
rather
ofthosealternatives,
characteristics
We argue,however,thatsucha modelis usually
is appropriate.
case, MNLGT estimation
We
modeland thusis of somewhatmorelimitedinterest.
nonbehavioral
a reduced-form
fallnaturally
and othersocialscientists
to demographers
believethatmanyissuesofinterest
intoa CLGT model.
thekeydifference
betweenthetwomodelsinvolvestheunitofanalysis:in
Statistically,
an MNLGT model,theindividualis theunitofanalysis,whereasin a CLGT model,the
variablesof a CLGT model are
is the unitof analysis.The explanatory
setof alternatives
variables,such as personal
butindividual-level
ofthealternatives,
characteristics
primarily
attributes,
can be readilyaccommodatedin a CLGT model. Anotherusefulfeatureof a
among
in the availablealternatives
CLGT model is its abilityto allow fordifferences
individuals.
thepostdivorce
betweentheseapproachesbyconsidering
We illustrated
thedifference
maritalstatusand welfarereceipt.Estimatesof threemodels
choicesof womenregarding
as explanatory
(1) an MNLGT modelthatused individualcharacteristics
werepresented:
wage rateand exogenousincome
variables;(2) a CLGT model in which the after-tax
variables;and (3) a
weretheexplanatory
availableto a womanin each ofthreealternatives
twomodels.
mixedlogitmodelthatincludedthevariablesfromthefirst
(as measuredbytheincome
opportunities
In themixedmodel,we foundthatmarriage
ofremarriage
spouse)havea modestpositiveimpacton theprobability
ofa woman'spotential
Interestingly,
havea slightly
impacton remarriage.
stronger
negative
andthatAFDC benefits
once we controlfor
we also findthatwomenwithmorechildrenaremorelikelyto remarry,
theirpoorermarriageopportunities.
Notes
(publishedbetweenFebruary1984 and May 1986)
A reviewof 10 issues of Demography
researchusinga logitmodel. Seven of the 10 involved
produced10 examplesof discrete-choice
variables-Masseyand Mullen(1984)analyzedthepresenceofyoungchildren
dependent
two-category
in a household,Landale and Guest (1985) mobilityplans and actions,Tienda and Glass (1985)
Entwisleand her colleagues(Entwisleet al., 1984; Entwisle,
women'slabor forceparticipation,
behavior,DaVanzo and Habicht(1986) infantmortality,
Mason, and Hermalin,1986)contraception
award.Examplesof three-category
and Bellerand Graham(1986) the presenceof a child-support
choice models include Lehrerand Kawasaki's(1985) analysisof a child care modal choice and
decisionmaking.Robinsand
Leibowitz,Eisen, and Chow's (1986) analysisof teenagepregnancy
modelofwelfareand childsupport.
a four-category
Dickinson(1985) estimated
logit.
2 The threemultiple-category
in note 1 are all examplesofmultinomial
modelsidentified
Leibowitz,Eisen, and Chow (1986) used conditionallogitto describewhat is more commonly
logit.
consideredmultinomial
3 Although
ofsomeofthe
methodsdescribedhereis new,and discussions
noneofthestatistical
and Lerman,1985;judgeetal.,
texts(see Ben-Akiva
and econometrics
issuescan be foundin statistical
on theissuesdiscussed
1980;Maddala, 1983),we knowofno applieddiscussionthatfocusesexplicitly
here.
4 For modelingand estimation
drawnbetweenMNLGT and CLGT is
purposes,thedistinction
see thethird
The modelsdo, however,sharea commonlikelihoodfunction;
usefuland instructive.
sectionfora discussionofthis.
as theunitofanalysis.Whatwe call
5As in CLGT, themixedlogitmodelusesthealternative
logit,"withthepureMNLGT and CLGT models
referred
toas "multinomial
mixedlogitis sometimes
are used (see
or ofindividuals
ofalternatives
treatedas specialcases in whichonlythecharacteristics
sinceitsuggests
confusing,
Amemiya,1985;Ben Akivaand Lerman,1985).We findthisterminology
different
pure MNLGT
model in questionis the morefamiliarand significanitly
thatthe statistical
modelof equation(1).
426
Demography,
Vol. 25, No. 3, August1988
6 BoththeCLGT and MNLGT modelsare basedonltheassumption
thattheerrortermsfollow
and are independent
acrossalternatives.
See thethirdsectionfordetails
an extremevaluedistribution
ofthisassumption.
on the implications
7 For instance,
iftheoriginalprobabilities
forsomeindividualarePI = 0.4, P2 = 0.4, and P3 =
cause P2 and P3 to fall to 0.32 and 0.16,
0.2, then an increasein PI to 0.52 would necessarily
and notforthe
foran inidividual
holdsonlyforthe ratioof probabilities
respectively.
This property
choice.
makinga particular
aggregate
proportion
of individuals
8 The discussion
ofLIMDEP's DiscreteChoice program;
thatfollowsis basedon therequirements
Mlogitproceedssomewhatdifferently.
9 Assigning
forthoseindividuals.
As can
0 valueswillnot,in fact,producethecorrect
probability
thenexp(Za) = exp(0)= 1. Sincethisalternative
be seenin equation(2), ifZ = 0 forsomealternative,
ofitsselectionshouldbe zero;insteadit
does notexistfortheindividualin question,theprobability
is takenoverall otheralternatives.
As
wouldturnoutto be 1/(1+ , exp(Zikcx),
wherethesummation
a result,theotherprobabilities
wouldbe too low.
10This can be done readilywithLIMDEP's DiscreteChoice program.
The same thingcan be
it.
butwe knowofno software
thatfacilitates
done forMNLGT programs,
II A similarprocedureusinga moreelaboratemixedmodelis detailedin Hoffman
and Duncan
(1987).
12 The "otherwelfare"category
AFDC
includesGeneral Assistanceand some misreported
issomewhat
arbitrary,
Ellwood(1986)showedthatas a practical
income.Eventhoughthe$1 threshold
betweenvariousthresholds.
matter,thereis littledifference
13 There is a minorexception
to thisin statesthatpermitotlherwise
eligiblemarriedcouplesto
rare(about 150,000cases nationally
undertheAFDC-UP program.It is sufficiently
receivebenefits
peryearduringtheperiodwe analyze)thatour datasetprovidestwofewcasesto permitanalysis.
14Although
tothedecisionofall women,
theincomeofa newhusbandispresumed
tobe relevant
Thus we use an estimated
valueofnewhusband'sincome
itis observedonlyforwomenwho remarry.
The
model fiton the womenwho remarried.
forall womenin the sample,based onla regression
(age,
as a functioni
ofherown personalcharacteristics
incomeofa woman'snewspouseis estimated
numberof children,residence,etc.) and thoseof her formerhusband,includinghis incomeand
womenmaynotbe a randomsample,evenofthosewhoare observationeducation.Since remarried
ally identical,we also correctforpossibleselectionbias, usinga techniqueoutlinedby Lee (1983).
(Estimatesofthespouseincomeequationare availablefromtheauthors.)
15 The effect
thevariable
ofnonlaborincomeis measuredrelative
tobeingon welfare.
We treated
in nonlaborincomeacross
in thisway(likeMNLGT estimation)
variation
becausethereis insufficient
alimony-weassumedthattheywould
thealternatives.
Nonlaborincomediffers
onlyforwomenwitlh
fewwomenreceivedanyalimony.
lose theiralimonyiftheyremarried-andrelatively
16 They differ
in our analysisbecause we treather incomeas marginalto her husband'sand
wagerateonlthatbasis.Averagemarriedwagesare about90 percentofaverage
computeherafter-tax
singlewages,althoughthereis some variation,dependingon the incomeof a woman'sprospective
spouseand theirnonlaborincome.
17 These twoeffects
betweenwelfare
elaboration.
First,theabsolutedifference
mayrequirefurther
and nonwelfare
after-tax
wages. Second, when
wagesis largerforwomenwithhighernonwelfare
welfarebenefitsare relatively
low, the maximumincome thatcan be earnedwhile maintaining
is also low. Since a womancannotearnmorethanthisamount,she faceswhatamountsto
eligibility
in low-benefit
a zerowagerateon welfare
once she reachesthatlevel.This effect
is stronger
statesand
forhigh-wage
womenthanlow-wagewomen,sincein bothcases themaximumearningsamountis
morereadilyattained.
18 Note thatwe have assumedthatthe numberof childrena womanhas affects,
otherthings
We note,in passing,thattheabilityto so constrain
alternative.
equal, onlythevalue ofthemarriage
a coefficient
is an advantageoftheCLGT model.
19 We foundthateach additional
childreducedtheincomeofa potentialspouseby 7 percent.
20 See Hoffman
and Duncan (1987) fora simulationoftheeffects
ofchangesin welfarebenefits
rates.
on cumulativeremarriage
Multinomialand ConditionalLogitModels
427
Acknowledgments
This articlcis basedon researclh
bv NationalInstitute
supported
forChild I-Health
and Human Developmeicnit
GrantIROl HD 19339-01.It has beniefited
bv Johln
fromhelpfulcommcnits
Bounid,DorothvDuncani,Robert
We also thanik
Hutchens,WillardRodgers,GarvSolon, and ArlandThornitoni.
twoanonivmous
referees
and the
deputveditorforthcircommncits.h'lec
orderoftheauthors'namtes
was determinied
randomlv.
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