~c Natioml hngimdfial Sweys (NLS) prog,m s“ppons many st”dics designed

m ticr=e md=stmdtig of tie labor m~kcl md metiods of data mlleaio”. ~e DiscwsionPaperseriesrnm~ratessomeof tie rcsmch comkg out of tie progrm.

Mmy of the P.PH in tis se,ies have bmn delivered m find rqorIs co-issio”d m

pmt of m exum”rd ~mt ~ogrm, under which grm~ me awmdd tio”gh a

com~titive process. ~ s~ies is in[mded 10circulate he rmd,ngs to titer~td readers

wititi md o“mide tie Bwea” of tir Swtis!ics.

Pasms titmeswd in obtatimg a mpy of my Discussbn Paper,olher NLS reprti, or

gmerd tiomtiion abouLtie ~ Fogrm md s“weys should W“IXI tie B“rea” of

hhr Statktics. Office of Em”omic Rcse~ch, Wmhhgto”, DC 20212.0001 @02-606-

7405).

Material i“ his p“btica[ion k ti he public domtin md, wihappropriate&it, may

be =~oduced witiout ~missio”.

%mions ad Comlusiom ex~essed in his docmmt xc tiIOSeof tie anti.,(,)

d do not necess=ily rqr~mt m official position of theBureau of Lahr S1~istim

or tie U.S. DepULme”tof Labor.
.

NLS

Discussion

Papers

●
Dynamic Models of the Joint Determination of

Labor Supply and Family Structure

Saul Schwatiz

Carleton University and TURSUniversity

Roben Hutchens

Cornell University

George Jakubson

Cornell University

June 1991

.

~s project wasfundedbytheU.S.DepartmentofLabor,BureauofLabr Statistics undergrant

numberE-9-J-8-WO.me authorswishtotiafitheparticipants inseminaratBLS,Tufts

University, CarletonUrdveraity, andComeUUtiveraity. h addition, someofthisresearch was

mnductedatme ComeUNation~Supercomputer Facifity, Centerfor~eoryandSimulation in

ScienmandEn@rteering, whichisfundedinpartbytheNationaJScienceFoundation. New State,

andIBM Corporation. Opiniomstatedintispaprdonotnecessarily represent theofficial

position orpoficyoftheU.S.DepartmentofLabor.
Dynamic Models of the Joint Determination

Labor Supply and Family Structure

Saul Schwartz

of

Carleton University and Tufts University

Robert Hutchens

Cornell University

George Jakubson

Cornell University

This project was funded by the U.S. Department of

Labor, Bureau of Labor Statistics under Grant Number E9-J-8-0090. Opinions stated in this document do not

necessarily represent the official position or policy

of the U.S. Department of Labor.

We would like to thank a nutier of individuals for

their assistance during this project. They include Dr.

Hichael Perganit, our project officer at the Bureau of

Mor Statistics (BLS), as well as participants in

seminars at BM, Tufts University, Carleton University,

and Cornell University. In addition, some of this

research was conducted at the Cornell National

Supercomputer Facility, Center for Theory and

Simulation in Science and Engineering, which is funded

in part by the National Science Foundation, New York

State, and IBM Corporation.
T~LES OF

Executive

Dynamic Models of the Joint Determination

Labor Supply and Family Structure

Summary ...................................

Introduction .........................................

of

. . . . . . . . . . . . 1

. . . . . . . . . . . . 4

Chapter 1

I.

II.

III.

Chapter 2

I.

II.

III .

IV.

Conclusion

Cross-sectional Models of Labor Force Participation

and Marltal “ status ...................................g

Single-E~ation Models ..............................10’

Single-E~ation Models with

Lagged Dependent Variables. .........................15

Simultaneous Probit Models ..........................lg

A~ oice Mode of

bbor Force Participation and Marital Status. .......34

Theoretical Model ...................................37

Analytic Forms ......................................3g

Estimation of the Dynamic Model .....................41

Results from the Dynamic Model ......................44

..................................................i..48

Appendix A Va~ able PreDarat.on Issues ....53

Appendix B Additional Models of Labor Force Partici Datibn

and Marital Status ..................................6s

Appendix C FORT~ Proqrams Used for the Dvnamic Model .........77

Bibliography ...................................................78
DYNANIC AND SIMULTANEOUS MODELS ‘OF THE JOINT DETERMINATIoN

OF LABOR SUPPLY AND F~ILY STRUCTURB

Executive SUary

-. The past twenty-five years have seen the emergence of

number of important longitudinal data sets. Foremost among

is tie Set of surveys known, collectively, as the National

a

these

Longitudinal Surveys (N=). The availability of nationallyrepresentative, longitudinal data has spawned a variety of

econometric methods designed to study the economic behavior of

individuals over time. These include hazard rate analysis, event

history studies and techni~es for pooling time-series and crosssectional data.

This report deals with another econometric model developed

to exploit longitudinal data - dynamic stochastic discrete ckoice

models (Eckstein and Wolpin, 1989) . We use data from the

National Longitudinal Survey - Youth Cohort (NLS-Y) to e~lore a

dynamic discrete choice model of the labor force participation

and marital status of young mothers. The theory underlying such a

model is guite appealing. Expectations

allowed to influence tirrent decisions

maximization framework. In that

one.

The econometric estimation

solution to a recursive dynamic

maximization of a multi-period,

sense,

of our

about the future are

in an eqlicit utilityour model is a structural

model rewires the

programming problem and the

multi-state likelihood function.
2

The programs remired to estimate the parameters of our model are

available from the authors upon reguest (see Appendix C) -

Because the dynamic discrete choice model 2s “relatively new

and somewhat complicated, our work (and this report) moves from

simple models to more complex models. In the first part of

Chapter 1, we estimate relatively simple models of labor force

participation and marriage using standard discrete-choice

techniques. Then,,we exploit the longitudinal nature of the N=-

Y by adding lagged values of the two dependent variables to the

simple models. The most complicated model in Chapter 1 is a twoe~ation simultaneous probit model of labor force participation

and marriage.

Aside from the development and implementation of our dynamic

discrete choice model using the N--Y, several other interesting

results arise from of our work on this project:

(1)

(2)

we found no evidence of any interdependence between marital

status and labor force participation. In particular, in the

simultaneous probit model estimated in Chapter 1, current

labor force participation did not affect current marital

status nor did current marital status affect current labor

force pa*icipation. ~rthermore, we could not reject -e

hypothesis that the covariance between the error terms of

the e~ations representing labor force participation and

marital status was zero;

adding lagged dependent variables as explanatory variables

to the models estimated in Chapter 1 indicated that there is 

a higher-than-e~ected correlation between past status and

current status. This leads to the conclusion that, except

for unobserved factors, the determinants of the “initial

conditionvt - the marital status/labor force participation

prevailing at the time the woman first had a child - seems

to persist over time. For example, the most important

determinant of whether a woman worked in 1985 was whether

she worked in 1984. BY contrast, demographic variables pale

in significance beside lagged dependent variables. For

example, once we account for past pa*icipation in the Aid
3

(3)

to Families with Dependent Children (AFDC) program, the race

and ethnicity of the young mother becomes irrelevant to her

labor force participation decision. mile there is

substantial change in labor force participation and marital

status - 50% of the young mothers in the sample change one

or the other before 1985 - these changes seem to be the

result of factors that we cannot obse~e. These conclusions

are buttressed by similar results from the dynamic model of

Chapter 2;

the dynamic stochastic discrete choice model did not lead to

any results that were substantially different tha~he

results obtained from the simpler Chapter 1 models. Past

values of the Ststateatvariable were ~ite important while

current demographic characteristics were relatively

unimportant.

Our model posits rational decision-making by these young

mothers. Since this assumption is not directly tested in our

model, our results may be affected if the assumption is

incorrect. Others researchers, however, have found this model

more useful than standard models in other coritexts (see Eckstein

and Wolpin, 1989, for a review of this literature).

The constraints imposed by tha computational burden of the

estimation forced us to keep our model ~ite simple. The

similarity of results across dynamic and static models may

indicate only that simplicity. Wile the model may be too simple

to capture behavior ade~ately, it is a step in the right

.

direction. If there is to be progress in modeling labor force

participation, we believe that a structural approach is

absolutely essential.
DYN~IC AND SINULT_EOUS MODELS “OF THE JOINT DETE~INATION

OF LABOR FORCE SUPPLY AND FANILY STRUCTU=

Introduction ,“

The past twenty-five years have seen the emergence of a

ntier of important longitudinal data sets. Foremost among these

is the set of surveys known, collectively, as the National

Longitudinal Surveys (NLS). The availability of nationallyrepresentative, longitudinal data has spawned a variety of

econometric methods designed to study the economic behavior of

individuals over tine. These include hazard rate analysis, event

history studies and techni~es for pooling time-series and crosssectional data.

This report deals with another econometric model developed

to exploit longitudinal data

models (Eckstein and Wolpin,

National Longitudinal Survey

- dynamic stochastic discrete choice

1989) .

- Youth

dynamic discrete choice model of the

and marital status of young mothers.

discrete because in any time period,

.

We use data from the

Cohort (NE-Y) to explore a

labor force participation

The womangs choices are

she is either part of the

labor force or she is not; in any time period, she is either

married or she is not. ~rthermore, these models are stochastic

in the sense that observably identical individuals may not behave

in identical ways because of factors that are unobservable to the

researcher.

The advantages of such a dynamic model are best understood

when contrasted with a static model. A static model e~lains the
5

labor force participation and marital status decisions in terms

of current variables (such as current wages, current n-ers of ,

children and current age) . In contrast, a dynamic model such as

ours explains labor force participation in terms of both current

variables and the emected values of future variables. For

example, a dynamic model would incorporates the idea that todayts

decision to participate in.the labor force will affect future

ievels of income, children, and schooling. Moreover, one’s

e~ectations about these potential conse~ences feed back into

todayls decision to pa*icipate in the labor force. Thus, a

dynamic model has the advantage of yielding a more realistic

picture of actual behavior.

Because of the e-licit utility-maximization that is its

theoretical base, we thinK. of ‘our‘model as a ~tnctural one. An

alternative to structural estimation, an alternative explored in

Chapter 1 of this report, is the estimation of a model that

“approximates” the reduced form of the structural model. Under

this alternative strategy, one implicitly solves the dynamic

structural model for its reduced form, in which the endogenous

variables are a function of current and past realizations of the

exogenous variables. Altiough the explicit reduced form solution

to a structural dynamic model is usually nonlinear and ~reme~Y

complex, it is always possible to take a Taylor expansion to

obtain a linear approximation of this reduced form. In the

reduced form, each endogenous variable is a function of a linear
6

combination of coefficients and exogenous

error term. The coefficients are then the

variables as well as an

object of estimation.

The principal advantage to the ‘lapproximationlrapproach is

that it is less restrictive. As such, it may provide estimates of

how a large variety of exogenous variables affect endogenous

variables. The structural approach, implemented in Chapter 2,

involves using an iterative maximization routine to solve

system of nonlinear e~ations in each time period. This

complexity limits the range of explanatory variables that

a

can be

incorporated into the analysis. The ‘qapproximationttapproach is

computationally simpler, reguiring less programming and computer

time.

The structural approach has, however, other advantages.

First, and perhaps most important, estimation is focused on

utility functions and constraints. In contrast to the

lrapproximation“ approach, the assumptions underlying the

estimation are explicit. Second, the structural approach can

provide more precise parameter estimates and stronger (more

restrictive) tests of the theory. Wolpin (1984) argues ~ite

forcefully that the structural model, if correctly specified,

implies restrictions that permit more precise inference and a

more parsimonious representation of complex relationships. But,

he goes on to say, if the model is incorrectly specified, all

statistical inferences may be contaminated, regardless of the

offending assumptions (p. 854).
7

Our work involves specifying four possible choices for each

woman - two labor force participation l’states”’(in the labor

force and not in the labor force) as well as twm marital statuses 

(married or not married). The estimation of the parameters of

such a model involves complicated and computer-intensive maximum-

‘likelihood techni~es. The relevant programs, which we have used

to estimate models of up to six Sastatesstwere developed by Ge0r9e

Jakubson and are available upon re~est.

Working with longitudinal data also rewires considerable

effort to ensure that survey responses. are consistent over time.

This is especially true of the N~-Y codes for the interrelationships among the individuals who move in and out of

families over time. We spent a great deal of time “cleaning” the

data as pa* of this project and as part of another, related

project (Hutchens, Jakubson and Schwartz, 1990b). The result Of

those effo*s is a relatively Itclean” set of data on the fam~ly

structure of N=-Y female respondents (see Appendix A) . We were

ably assisted in that effo* by Angela Mikalauskas.

Since our focus is primarily methodological, we do not spend

a great deal of space reviewing the vast literature on the

determinants of labor force participation or on the smaller

literature concerning the determinants of marital status. For

reviews of that literature, see Johnson and Skinner (1986) ,

McElroy (1985), Gonul (1989),

Because

and somewhat

dynamic discrete

complicated, our

and Killingsworth (1980).

choice models are relatively new

work (and this repo*) begins with
simple ‘rapproximationfa

Itst~ctural” model. In

8

models and then moves to the more complex

the first pa* of Chapter 1, we estimate ,

relatively simple models of labor force participation and

marriage using standard discrete-choice techniques. Then, we

eXPloit the longitudinal nature of the N~-Y by adding lagged

values of the two dependent variables to

most complicated model in Chapter 1 is a

simultaneous probit model of labor force

marriage.

the simple models. The

two-eguation

participation and

Chapter 2 contains the theoretical and empirical versions of

our

the

structural dynamic model. The chapter begins by laying out

utility maximization assumptions that underlie the later

empirical worK. After describing the statistical issues involved

in estimating the dynamic programing model implied by the

theory, we ’present our empirical parameter estimates. A short

summary concludes the report.

We would like to thank a number

assistance during this project. They

Pergamit, our project officer at the

of individuals for their

include Dr. Michael

Bureau of ~bor Statistics

(BE), as well as participants in seminars at BM, Tufts

University, Carleton University, and Cornell University. In

addition, some of this research was conducted at the Cornell

National Supercomputer Facility, Center for Theory and Simulation

in Science and Engineering, which is funded in part by the

National Science Foundation, New York State, and IBM Corporation.
C~PTER 1 ‘

Cross-sectional Models of Labor Force

Participation and Marital Status

.“

In this chapter, we begin our examination of the

relationship between marital status and labor force

participation. We start by selecting a sample of woman

children. We then estimate a series of single-equation

with

models of

the two dependent variables. In the next part of the chapter, we

estimate the structural parameters of a cross-sectional twoeWatiOn system in which one equation represents the marriage

decision and the other equation represents the labor force

participation decision. This bivariate simultaneous equations

mOdel enables us to estimate, for these young mothers, the impact

of labor force participation gn marriage and the effect of

marriage on labor force participation.

We exploit the time-series nature of the N~-Y data by

introducing past values of the dependent variables into our

econometric models. Adding this dynamic element to the model

allows us to account for the impact of past decisions on current

status. For example, we can answer questions such as “Does last

yearss marital status affect this year’s labor force

participation decision?”

The AFDC program (which is an important source of financial

support for these women) enters the model in that past AFDC

pa*icipation, treated as an exogenous variable, is allowed to

affect both current labor force participation and current marital

status.
10

We estimate three sets of models in this chapter. The first

set are essentially single-e~ation models of marriage and labor .

force participation. In that context, we look at how past marital

status and labor force pa*icipation affect current marital

status and labor force participation.

The second set of models are bivariate simultaneous”

e~ations models without any dynamism - only current values of

the variables appear. The major difference between this model and

a standard simultaneous e~ations model is that both dependent

variables - marital status and labor force participation - are

dichotomous. Thus our model is a *simultaneous probit” model. The

third set-of models combines the first two by introducing lagged

dependent variables into the simultaneous probit models.

I. sinqle-E quation Models

In this section, we estimate two single-e~ation, reducedform models of labor force participation and marital status,

respectively. As noted in the introduction, these are linear

“approximations” to the reduced form of a structural model,

described in Chapter 2.

The basic facts about marital status and labor force

participation in our sample are straightforward. The women in the 

sample were all 14-21 years of age in 1979, when the NU-Y began.

In 1985, the year to which these estimates apply, the women were

aged 20-27. The sample consists of 2,221 women with children:

of these, about 47% were working in 1985 and 65% were married (or
.,

living with a

working while

L.

‘lpartnertl). Of the narr,ied women, almost 49% were

38% of the unmarried women were working. The

. patterns are ~ite different across racial and ehnic groups.

-Ong the 715 black women in the sample, almost 62% are

unmarried. By contrast, of the 1,506 women who are not black,

only 23% were unmarried.

The models we estimate are cross-sectional, dichotomous

probit models. These single-eqation reduced-form models are too

simple, in theoretical terms, to capture some potentially

important links between the two decisions. For example, the two

decisions are assumed to be made independently. Despite their

simplicity, these models form a baseline from which we can assess

the gain to be realized from more complicated models.

We assume that labor force participation depends, in

general, on a woman’s comparison of her reservation wage to the

market ,wage available to her. Market wages depend, in.turn,’ on

previous labor force experience; past labor force participation

should thus increase the probability of current labor force

pa*icipatian,

also depend on

*lhighest grade

on the womanas

holding other variables constant. Market wages

educational attainment, represented here by

completedmr by the woman. Reservation wages

tastes for leisure as well as her other

the

depend

obligations. Foremost among those obligations for our sample is

childcare. The more children a woman has, and the younger the

children are, the less likely a woman is to work.
12

A controversial and still unsettled

to which past AFDC participation affects

participation. If AFDC pa*icipation, in

~estion is the e=ent

current labor force

and of.itself, reduces a 

woman’s inclination or ability to work, then past AFDC

participation should decrease the current probability of labor

force participation.

To an unknown tient, past marital status may influence

current labor force participation. As in the case of past AFDC

participation, the causal links between past marital status and

current labor force pa*icipation are unclear.

Theoretical e~lanations for labor force participation are

considerably more developed than theoretical e~lanations for

marriage. In patiicular, the literature on labor force

participation leads.to.clear implications for the specification

of an econometric model. The theoretical rationale for marriage

depends on the productivity gains potentially available to both

parties. Marriages break do- whenever those gains are

insufficient (or whenever the gains are divided in such a way

that one party is.worse off than they would be outside the

marriage) . But since the productivity gains are unobservable, as

is the distribution of the gains between pa*ners, empirical

analysts must be satisfied with an econometric model that asse*s

the fiportance of a number of observed variables (such as

education and labor force ~erience) in terms of their potential

contribution to marital l’productivity.al
13

Race is clearly an important correlate of the probability of

marriage, simply because of the observation, noted above, that a ,

much smaller proportion of black mothers are narrie’d.

Table 1.1 contains parameter estimates from two simple

.

reduced-fore cross-sectional models of marital status and labor

force participation. Variable definitions are shown in Appendix

Table Al. As we would expect, the number and ages of children

are correlated both with being married and with working. Having

children under 3 years of age substantially increases the

probability that a woman is married and substantially lowers the

probability that she works. To a lesser extent, this is also true

Of children between the ages of 3 and 6. Once a womants children

are of school age, however, they affect neither the probability

of working nor the probability of being married. Black women are

much less likely to be married, even in the ,multivariate context.

Hispanic women are also somewhat less likely to be married than

white women.

Turning to the determinants of labor force participation for

women with children, we find, as expected that women with more

years of education are significantly

those with fewer years of education.

less likely to be working than white

more likely to work than

Black women are somewhat

women, while Hispanic women

are somewhat more likely to working than white women. Those who

live in the South and those who live in SMSAS are more likely to

be working than those who do not. The AFDC system does not have

a significant impact on either marriage or working.
are

14

As a method of describing the characteristics of women who

married or who are working, these simple cross-sectional

models are mite informative. They

the number and ages of children as

and working and they point to race

important correlates.

But as a way of understanding

illustrate the importance of

correlates of both marriage

and ethnicity as two other

the underlying relationship

between marriage and working, these models are too simple because

that relationship, if any, is not made explicit. For example,

one might think that unmarried women were more likely to work

than married women since they lack the economic support that

night be provided by a husband or pa*ner. But looking at the

bivariate relationship between marriage and working, one might

come to the opposite conclusion. After all, as noted above, more

married women were working than unmarried women. But this

obseNation may be explained by the fact that unmarried women

with children are eligible for economic support from the AFDC

program, suppo* that bight enable then to stay out of the labor

market and in their homes with their children.

Our immediate goal is to see if there is a relationship

between marriage and work (and AFDC participation). We expect

that there are unobsemed factors affecting both the decision to

work and the decision to marry, so we do not include “workinga’ as

an e~lanatory variable in the marriage e~ation, nor do we

include “married” as an e~lanatory variable in the e~ation for

labor force participation. These variables are endogenous and
15

their inclusion on the right-hand side of the simple probit

models would lead to inconsistent parameter estimates, estimates .

that would compromise our assessment of the relationship between

marriage and work. Moreover, because almost all women receiving

AFDC are unmarried and because the decision to participate in the

the marriage and work decisions, we cannot include AFDC status

an ~lanatory variable in either eguation.

Using linear models of this type, we have two ways of trying

disentangle the relationship between marriage and working. The

AFDC program involves the same unobserved factors as play a role

in

as

to

first is to -ploit the time-series nature of the NLS-Y by

including lauaed values of the dependent variables as explanatory

variables in the e~ations for current marital status and current

labor force participation. Furthermore, we can include lagged

=DC participation as a way of accounting for the availability of

financial suppo* for unmarried women. The second method is to

use simultaneous e~ations techni~es to test the hypothesis that

unobserved factors affect both the labor force participation and

the marital status decisions. The following two sections attempt

those two extensions of the simple single-e~ation reduced-form

models.

II. ~Sin e-E at.o ed DeDendent Variab es

The results of our inclusion of lagged dependent variables

in our model are shown in Table 1.2. Lagged AFDC participation

has a dramatic impact on both marriage and work. Those who

received AFDC in 1984 were very much less likely to be married in
1985. This

unmarried.

16

is not surprising since most AFDc recipients are

And, since relatively few women become married over a .

one year period, being unmarried in 1984 is a guod “pro~tf for

not being married in 1985.

In the marital Status eguation, women who worked in 1984

were not significantly more or less likely to be married in 1985.

The inclusion of lagged AFDC and lagged labor force participation

does not change the numerical magnitude of the other significant

coefficients, reported in Table 1.1. Those with young children

remain more

less likely

In the

variable is

likely to be married; blacks and Hispanics remain

to be married.

labor force participation e~ation, the lagged AFDC

large and its standard error is small. Those on AFDC

in 1984 were ~ite unlikely to be working in 1985. That is, very

few of these mothers leave AFDC and enter the labor force in any

one year. The coefficient on lagged marital status is

significantly different from zero but very small. The lack of

importance of lagged marital status in the labor force

pa*icipation e~ation and of lagged labor force participation in

the marital status e~ation is an early indication of the seeming

independence of those two decisions. That independence is a theme

that runs through the entirety of this report.

Interestingly, when we condition on lagged AFDC, race is no

longer important. ~t differently, when past AFDC pa*icipation

is nOt accounted for, as in Table 1.1, it would seem that blacks

are leSS likely to work than whites. AFDC participants are less
17

likely to work than non-AFDC participants and the seeming

correlation between race and working is caused by the correlation .

between race and AFDC participation. The inclusion” of the lagged

dependent variables also reduces the numerical magnitude of the

regional variables, SOUTH and SMSA. In and of themselves, these

variables seam not to affect the probability of working. Instead,

they appear to affect the probability of receiving AFDC and

determine the probability of working only through that indirect

channel.

The importance of including lagged variables is not only to

estimate coefficients more accurately in cross-sectional models.

As can be seen above, marriage and work do not adjust

instantaneously. In that sense, ll~hereYOU are depends on where

youSve been*l. Marital status in 1985 and labor force

pa*icipation in 1985 are greatly influenced by past marital

status and past labor force participation.

That notion ie an integral part of the

Chapter 2, where we make explicit the links

decisions and current states. To the -tent

structural model in

between past

that current

variables are correlated with past dependent variables, their

impact will be overstated in cross-sectional models. The example

of how including past AFDC participation eliminates the

correlation between race and labor force participation

illustrates this idea.

Table 1.3 chows the effect of including a complete set of

lagged dependent variables, variables going back over the 1979-
18

1985 NU-Y survey period. ,Sixyears of lagged

(1979-1984) as well as corresponding years of

AFDC participation

lagged labor force .

participation and lagged marital status are inc~uded along with

the same set of current exogenous variables as appear in Table

1.1 and 1.2.

The inclusion of a more complete history for the young woman

further reduces the impact of current variables. For example,

with that history in place, only the presence of very young

children (less than three years old) affects

working. Women who have a history of working

unless they have very young children.

Similarly, once ‘Ihistoryamhas been held

the probability of

continue to work

constant, the

presence of school-age children actually increases the

probability of working. This is entirely plausible; having older

children (compared to

is Ilfarther alongll in

More importantly

marital status in the

otherwise similar women) means than a woman

the life-cycle and is returning to work.

for our purposes, the inclusion of lagged

marriage e~ation and the inclusion of

lagged labor force participation in the current labor force

participation e~ation points up how strongly current status

depends on past status. The single

whether a woman worked in 1985 was

far the most important determinant

most impo*ant determinant of

whether she worked in 1984. By

of whether a woman was married

in 1985 was whether she was

The policy implication

marriage and labor force of

married in 1984.

here is that the immediate impact on

almost any macroeconomic policy is
19

likely to be ~ite small. But any effects that policy does have

will continue reverberate into the future.

In cross-sectional models, correcting for simultaneous

e~ations bias when trYin9 tO ascertain the impact of, say, labor

force participation on the probability of being married is very

important. Part of the task is accomplished by including lagged

dependent variables in the model and

dependent variables. This is because

current labor force participation on

excluding the current

part of the impact of

current marital status is

really the impact of past labor force participation (which is

being picked up the current value of labor force participation).

III . Simultaneous Probit Models

In this section, we estimate simultaneous e~ations models

of marital status and labor force participation. In light of our

effort (in Chapter 2) to construct a structural model of discrete

decisions, these simultaneous e~ations models need some

explanation.

As discussed in the introduction, the model in Chapter 2 is

a structural model in the sense that estimation is predicated on

the maximization of utility functions subject to constraints. We

think of all of the models estimated in this Gapter as linear

approximations to a !Istnctural modelS8 of the SOrt

tiapter 2. The models in the last section are also

models in the sense that only exogenous and lagged

variables appear on the right-hand side.

laid out in

‘qreduced-form”

endogenous
20

But what about the models in this section, where we estimate

simultaneous eguations models in which current endogenous

variables can appear on the right-hand side. In’standard textbook 

discussions, such models are portrayed as ‘structural” but we

reserve that tem for models such as”that presented in Chapter 2.

Econometrically, a simultaneous e~ations model simply

imposes constraints on the reduced form parameters and then tests’

those restrictions. For example, the coefficient, on an eX09en0us

variable might be constrained to be zero (to have no impact)

one e~ation while it is allowed to be nonzero in another.

Rather than thinking of the simultaneous e~ations models as

trUe ‘astructuresmof the joint decisions, we’view them simply

in

the

as a

way of testing a set of plausible constraints on the reduced-form

parameters. That is, our simultaneous e~ations model is another

Ilreduced formmfmodel.

While that view of such models seems simple, the actual’

estimation process is not. Both labor force participation and

marital status are dichotomous variables, so standard

simultaneous e~ations techni~es must be modified in order to

estimate the parameters of the models. We now discuss some of the

econometric issues that arise in making those modifications.

The unifying principle of most models with limited dependent

variables is the notion that while we, as researchers, might be

able to observe only a limited number of values for a dependent

variable (such as working or not working, or being married or
21

unmarried), these observed values have been generated by

continuous, unobserved latent underlying variables.

In our case, there are two such unobserved”latent variables.

Let A* be an unobse=ed, continuous index of a young motherSs

desire to be married. Let L’ be another unobserved, continuous

index of a womanss propensity to participate in the labor force.

As noted earlier, we have a fairly clear theory about how L’

‘~works’t- if a woman is offered a wage higher than her

reservation wage (which is a function of her marginal

productivity at home) then she works; if her wage offer is less

than her reservation wage she stays home. L* is a function OE the

difference between the woman~s market wage and her resonation

wage. This difference is unobserved since reservation wages are

unobsemed.

Suppose that both A’ and L* are both functions of a set of

variables X. At this point, suppose X might include current and ‘

lagged exogenous variables as well as current and lagged

endogenous variables. The coefficients on X for A’ and L* will

be denoted 8A and fl~,respectively. So,

(1) A* = X’@A - 6A

(2) L* = X’@L - 6L

where C& and EL are unobserved serially uncorrelated errors whose

distribution is bivariate nomal. The variances of e~ and •~ are

unidentified and set to unity while their covariance is a

parameter to be estimated.
Wile the woman is

researchers can observe

22

assumed to tiow both A“

only whether or not the

and L*, the

woman is married .

and whether or not the woman patiicipates in the iabor force. Let

the variables observed by the researcher be A and L where:

(3) A=lifA*>O; A = O otherwise;

(4) L = 1 if L* > O; L = O otherwise.

The models in the last section (Tables 1.1-1.3) treated the

A* and L“ variables in isolation using the NS-Y variables

defining A and L. The probability that A = 1 (that the N=-Y

respondent was married) was assumed to be a

vector that included only current exogenous

variables.

We now want to e~and the scope of our

exPlain joint decisions concerning not only

also labor force participation. Thus we are

effect of current labor force participation

function of an X

and lagged endogenous

model in order to

marital status but

concerned with the

on current marital

status and with the effect of current marital status on current

labor force participation. In addition, we want to allow the

unobserved factors that influence labor force participation and

marital status (that is, the error terms in e~ations (1) and

(2)) to be correlated. This correlation, if any, would imply that

there are common unobserved factors influencing both decisions.

These additional considerations add up to a hypothesis that

decisions about marital status and labor force participation are

made simultaneously. In terms of specifying an econometric

model, that hypothesis implies that when we include “labor force
23

participation” in X, its parameter estimate in eguation (1) will

be significantly different from zero. When we include “marital

statustt in X in the estimation of eguation (2), its coefficient

should also be significant. The way in which 811abor force

participation” and ‘Bmarital statusfs enter e~ations (1) and (2),

however, makes a difference in estimating the models.

One specification is that it is the ~alitative variable -

for example, whether or not one is in tie labor force - that

influences marital status, rather than the continuous underlying

variable.

Another plausible specification is that the latent variable

is the important determining factor; that, for example, the value

of L* is important in determining the value of A*. A variant on

this second model, is perhaps m“ost plausible. In that variant, the

observed ~alitative variables depends on the current latent

variable and the lagged ~alitative variables. mat is, in making

current decisions about labor force participation, a woman

considers the current value

current value of the latent

of past marital status. In

of her propensity to be married (the

variable) but only the observed value

other words, last year’s marital

status has a

(last year’s

irrelevant.

O-1 impact but last

value of the latent

year’s propensity to be married

variable) is now forgotten or

We label these two models *8A*’and ‘tB.lrNot

model specified differently, but each model must

differently.

only is each

be estimated
24

In Model A, using the definitions of E~ations (l)-(4):

(5) A* = XS$A + ~L - ●A

,’

(6) L* = X’~L + AA - EL

where X and the two corresponding vectors of parameters have been

redefined to exclude A and L, the observed ~alitative variables

for marital status and labor force participation, respectively.

Unfortunately, as specified in E~ations (5) and (6), this

model is underidentified, regardless of the exclusions that might

be imposed on the X vector. The reason for the

underidentification is that the model is logically inconsistent

(Schmidt,- 1982) . The problem can be seen as follows.

Following Maddala (1983), suppose the vectors p~ and p’ are

all zero and that ●A and EL are independent, normal variates: the

argument holds even when these assumptions are not made but the

point will be clearer if we make them. With these assumptions,

(7) Pr(A=l, El) = Pr(A*>O, L*>O) = pr(~ - EA > 0, X - ●L > O)

=’pr(eA c ~, CL < i) = FA(~) * FL(~)l

(8) Pr(A=l, &O) = Pr(A*>O, L-cO) = Pr( - 6A > 0, A - EL c O)

= pr(6A c O, e’ > X) = FA(0) * [1 - FL(A)]

(9) Pr(A=O, &l) = Pr(A*cO, L*>O) = Pr( T - EA c O, - EL > O)

= Pr(cA > T, ~’ < O) = [1 - FA(T)I * FL(0)

(10) Pr(A=O, =0) = Pr(A*<O, L.<O) = Pr( - CA c O, - ●L c O)

= pr(eA > 0, 6L > O) = [1 - FA(0)]*[l - F’(o)]
The four probabilities in

unity since they represent the

25

eguations

only four

(7)-(10) must

possibilities

add up to

for any

given woman. The sum of the four probabilities is:’

(11) 1 + FA(0)*FL(O) - FA(T)*FL(0) - FA(0)*FL(A) + FA(7)*FL(A)

E~ation (11) will e~al unity if 7 or A or both are e~al to

zero but not otherwise.

In general, a latent variable cannot be a function of its

observed indicator in a single e~ation model and, in a twoe~ation model, only one of the O-1 obsened dependent variables

can appear on the right hand side.

Thus , looking back to e~ations (5) and (6), either observed

marital status (A) cannot be in the labor force participation

e~ation o? observed labor force participation (L) cannot,be in

the marital status e~ation. The constraint is imposed by the

econometrics of the models and not by any economic reasoning. To

make the model both econometrically estimable and economically

plausible, we have to make

“comes firstgg.

For example, we could

decision is made first, as

an assumption about which ‘decision

assume that the marital status

a function of only age, education,

race, region of residence and the AFDC parameters and ~ as a

function of labor force participation. Then, the labor force

participation decision could be made as a function of the same

demographic variables plus observed marital status.

We begin by estimating two versions of eguation (5) and (6),

denoting them as Model Al and Model A2. In both cases, the vector
X contains

endogenous

26

the same list or cument exogenous and lagged

variables defined in Ap”pendix Table Al and used in the

simple reduced-form

models Al and A2 is

zero (Model Al) and

zero (Model A2).

e~ation models. The difference between

that one constrains the parameter Y to be

the other constrains the parameter A to be

~ Model B

Model B is a cross-sectional simultaneous probit model of

the 1985 living arrangements and labor force participation of

young women with children. The dependent variables are the O-1

labor force participation status and the O-1 living arrangement

status of a sample of women

Longitudinal Survey - Youth

The “strengthlr of each

with children drawn from the National

cohort .

womanas decision - represented by the

amount by which A* exceeds zero - is irrelevant in Model A. A

woman whose labor force participation decision is Steasysf(because

her market wage is much higher than her reservation wage) is no

more or less likely to be in the labor force than a woman for

whom the decision to work was marginal (in the sense that her

resemation wage is close to her market wage) .

In Model B, we assume that it is not the 0-1 labor

participation decision that is relevant but that it is rather the

~lStr~gthll of fiat decision that is inpOtiant. “strength” is

captured by the values of the unobserved latent variables, A* and

L*. Algebraically,

(12) A. = Xr@A + T*L* - CA
27

(i3) L* = x,~~ + A*A’ - EL

mile this model seems only slightly different from Model A, ,

as represented by e~ations (5) and (6), it does n’othave the

(econometric) problem of logical inconsistency.

Models A and B were both estimated using the LIS~L based

estimation package known as LISCOMP (Muthen, 1988) . Using

methods-of-moments type estimators, LISCOMP provides a flexible

environment for estimating the parameters of latent variable

models.

Tables 1.4 and 1.5 show the

those simultaneous (and dynamic)

force participation.

results of our estimation of

models of marriage and labor

The major result is easily stated. There does not seem to be

any simultaneous e~ations bias to be corrected. Current labor

force participation and current marital status seem to be

independent. Furthermore, there seems to be no correlation

between the current error terms of the two eguations. This result

echoes the lack of importance of lagged labor force participation

and lagged marital status in the single eguation models (see

p.16) . Tables 1.4 and 1.5 make this guite clear by putting three

e~at~ons s~de-by-side for marriage and labor force

participation, respectively.

In column (1) of each table is a reduced-form probit model

of the dependent variable, with only current ~ogenous variables

and lagged values of the dependent variable. COIUm (2) shows the

eWation from either Model Al or Model A2 in which the current
28

vaiue of one dependent variable appears as an independent

variable in the eqation for the other dependent variable. For .

example, oolumn (2) of Table 1.4 allows observed .ctirrentlabor

force participation to affect current marital status. Finally,

column (3) of each table is one of the two e~ations from Model

B, the fully simultaneous model. In that model, the latent labor

force participation variable is allowed to affect current marital

status and the latent marital status variable is allowed to

affect current labor force participation.

The thrust of both Tables 1.4 and 1.5 is that there is

little or no evidence of any simultaneity between labor force

participation and marriage, once the ‘thistoryqtof labor force

participation and marriage is included in the models.

The simple reduced-form probit coefficients are essentially

unchanged when we allow for a nonzero covariance between the

error terms of the two eguations and when we include current

labor force participation in the marital status e~ation and

vice-versa. This is true regardless of which method we use to

introduce the simultaneity - using the current observed value

LFP or marital status (Models Al or AZ) or using the current

latent LFP or marital status (Model B).

of

In no case is the estimated error covariance significantly

different from zero: even the point estimates are ~ite smallFurthermore, the coefficients on LFP in the marital status

ewation and on marital status in the LFP e~ation are also very

small in magnitude and not significantly different from zero.
29

Table 1.1

Single-E~ation Cross-sectional Models,of

Marital Status and Labor Force Participation

Women Aged 20-27, with Cbildr’en, in 1985

Dependent Variables: LFP85 = 1 if respondent is working in 1985:

0 othemise

~Y285 = lif respondent is married or

lives with a “partners’;

O otherwise

Coefficient Estimates (Standard Errors)

Independent

Variables

AGE

AGESQ/100

EDUC

EDUCSQ/100

BLACK

HISPmIC

SOUTH

S~A

KIDS2185

KIDS2285

KIDS2385

AFDCG85

AFDW85

Constant

Sample Size

Nean of Dep. Variable

-2 log likelihood

Marital Status

(1)

0.17 (.32)

-0.18 (.66)

-:.;: (.09)

-1:17 [::;]

-0.25 (.08)

:.:; (.09)

0:40 [:::;

0.19 (.05)

-0.02 (.05)

-0.11 (.37)

* *

Labor Force

Participation

(2)

0.48 (.30)

-0.92 (.62)

0’.26 (.10)

-0.42 (.45)

-0.25 (.07)

0.15 (.08)

0.29 (.08)

0.16 (.06)

-0.46 (.05)

-0.21 (.05)

-0.07 (.05)

0.10 (.37)

* *

-2.16 (3.9) -8.49 (3.7)

2,221 2,221

0.47 0.65

629.7 253.1

A 18*s1indicates that the coefficient eStimate was less than 0.005.
30

Table 1.2

Models of Labor Force

Participation and Marriage

with Lagged Dependent Variables

Women with Children, Aged 20-27, in 1985

Dependent Variables: LFP85 = 1 if respondent is working in

O otherwise

~Y285 = 1 if respondent is married

Independent

Variables

LFP84

~Y284

AFDC84

AGE

AGESQ/100

EDUC

EDUCSQ/100

B~CK

HISP~IC

SOUTH

SMSA

KIDS2185

KIDS2285

KIDS2385

AFDCG85

AFDCW85/100

Constant

Sample Size

Mean of Dep. Variable

-2 log likelihood

lives ;ith a “partnertt:

O otherwise

Coefficient Estimates

Marital Status

(1)

-0.09 (.07)

-1.28 (.08)

-0.02 (.34)

_~.;~ (.69)

0:58 ~:;~~

-0.94 (.07)

-0.30 (.09)

-0.02 (.09)

0.00 (.07)

0.46 (.06)

‘“”0.28 (.06)

0.07 (.06)

0.29 (.31)

0.04 (.30)

0.55 (4.1)

2,221

0.47

728.4

1985;

or

(Standard Errors)

Labor Force

Participation

(2)

-0.13 (.07)

-0.96 (.08)

0.36 (.31)

-0.71 (.64)

0.26 (.10)

-0.53 (.44)

-0.06 (.07)

0.14 (.08)

0.17 (.09)

0.16 (.06)

-0.45 (.06)

-0.16 (.05)

-0.05 (.05)

0.35 (.29)

-0.60 (.27)

-6.51 (3.8)

2,221

0.65 ‘“”

433.8
31

Independent

Variables

LFP84

LFP83

LFP82

LFP81

LFP8 o

LFP79

~Y28 4

=Y283

~Y282

~Y28 1

=Y28 O

mY279

AFDC84

AFDC83

AFDC82

AFDC81

AFDC8 o

AFDC79

AGE

AGESQ/100

EDUC

EDUCSQ/100

BLACK

HISPAWIC

- SOUTH

- SMSA

KIDS2185

KIDS2285

KIDS2385

Constant

Table 1.3

Models of Labor Force Participation and

Marital Status with Lagged Dependent Variables

Women with Children, Aged 20-27, in 1985

Coefficient Estimates (Standard Errors)

Labor Force

Marital Status Participation

(1) (2)

0.12 (.09) 0.90 (.07)

-0.05 (.10) 0.31 (.08)

-0.11 (.10) 0.23 (.08)

0.09 (.09) 0.19 (.08)

-:.;: (.09) 0.16 (.08)

(.09) 0.15 (.07)

1.80 (.10) -0.13 (.10)

0.42 (.11) 0.05 (.10)

-0.03 (.12) 0.00 (.10)

-0.00 (.12) 0.16 (.10)

-:.:: (.12) 0.03 (.10)

(.12) 0.08 (.10)

-:.:: (.12) -0.61 (.11)

0.00 (.12)

0:00 [:::; 0.29 (.12)

-0.22 (.14) 0.13 (.12)

0.24 (.14) -0.08 (.13)

0.07 (.15) -0.12 (.14)

-:.~~ (.42) -0.28 (.34)

0.46 (.70)

-0:16 ~:;~] 0.16 (.10)

0.86 (.59) -0.35 (.46)

-0.41 (.10) 0.14 (.08)

-0.18 (.11) 0.26 (.09)

0.03 (.11) 0.10 (.09)

0.10 (.09) 0.14 (.07)

0.24 (.07) -0.33 (.06)

0.09 (.07) -0.04 (.06)

0.08 (.07) 0.10 (.06)

6.23 (5.1) 1.90 (4.2)

Sample Size 2,221 2,221

Mean of Dep. Variable 0.47 0.65

-2 log likelihood 1486.4 869.2
32

~IT~ STATUS

Independent

Variables

LFP85

~Y284

~Y28 3

~Y282

~Y281

~Y280

~Y27 9

AFDC84

AFDC83

AFDC82

AFDC81

AFDC8 o

AFDC79

AGE

AGESQ/100

EDUC

EDUCSQ/100

BUCK

HIsPmIc

SOUTE

SNSA

KIDS2185

KIDS2285

KIDS2385

Error Covariance

Constant

Table .1.4

Simultaneous and Dnamic Models

of Marital ‘Status .’

Women with &ildren, Aged 20-27, in 1985

Coefficient Estimates (Standard Errors)

Reduced-Form

Probit

(1)

1.79 (.10)

0.41 (.11)

-0.01 (.12)

0.00 (.12)

-0.07 (.12)

0.19 (.12)

-0.70 (.12)

0.01 (.13)

0.03 (.14)

-0.21 (.14)

0.24 (.14)

0.07 (.15)

-0.55 (.41)

1-08 (.84)

-0.16 (.13)

0.86 (.58)

-0.40 (.10)

-0.18 (.11)

0.04 .11)

0.10 (.09)

0.23 (.07)

0-10 (.07)

0.08 (.07)

6.61 (5.0)

Model A2

(2)

0.12 (.19)

1.80 (.10.)

0.42 (.12)

-0.02 (.13)

0.00 (.12)

-0.06 (.12)

0.18 (.12)

-0.67 (.13)

0.02 (.14)

0.02 (.15)

-0.21 (.15)

0.24 (.14)

0.08 (.17)

-0.58 (.42)

1.2 (.9 )

-0.17 (.16)

0.9 (.8 )

-0.40 (.10)

-0.19 (.11)

0.03 (.12)

0.09 (.09)

0.25 (.08)

0.11 (.07)

0.08 (.07)

0.01 (.09)

-7.04 (5.2)

Sample Size 2,221 2,221

Mean of Dep. Vari&le 0.47 0.47

Model B

(3)

-0.01 (.06)

1.76 (.10)

0.38 (.12)

0.06 (.13)

0.01 (.12)

-0.08 (.12)

0.19 (.12)

-0.78 (.13)

0.10 (.14)

0.06 (.15)

-0.25 (.15)

0.31 (.14)

0.04 (.17)

-0.46 (.42)

0.9 (.9 )

-0.18 (.16)

0.9 (.7 )

-0.38 (.10)

-0.24 (.11)

0.08 (.12)

0.14 (.09)

0.22 (.08)

0.08 (.08)

0.09 (.07)

-0.06 (.08)

5.64 (5.2)

2,221

0.47
~OR FORCE

PARTICIPATION

Independent

Variables

~Y285

LFP84

LFP83

LFP82

LFP81

LFP8 o

LFP79

AFDC84

AFDC83

AFDC82

AFDC81

AFDC80

AFDC79

AGE

AGESQ/100

EDUC

EDUCS~100

BLACK

HISPANIC

SOUTH

SMSA

~.DS2185

KIDS2285

KIDS2385

Error Covariance

Constant

Sample Size

33

Table 1.5

Simultaneous and Dynamic Models

of Labor Force Participation

Women with tiildren, Aged 20-27, in 1985

Coefficient Estimates (Standard Errors)

Reduced-Form

Probit

(1)

0.91

0.31

0.22

0.18

0.15

0.15

-0.61

0.01

0.28

0.11

-0.10

-0.15

-0.20

0.32

0.16

-0.41

0.01

0.24

0.10

0.14

-0.33

0.00

0.11

(.07)

(,08)

(.08)

(.08)

(.07)

(.07)

(.10)

(.12)

(.12)

(.12)

(.12)

(.13)

(.34)

(.70)

(.10)

(.46)

(.08)

(.09)

(.09)

(.07)

(.06)

(.06)

(.06)

0.68 (4.2)

Model Al

(2)

-0.04 (.13)

0.91 (.07)

0.31 (.08)

0.22 (.08)

0.18 (.08)

0.15 (.08)

0.15 (.07)

-0.63 (.10)

0.00 (.11)

0.28 (.12)

0.11 (.12)

::.;; (.13)

(.14)

-0.20 (.35)

0.3 (.7 )

0.16 (.11)

-0.4 (.5 )

0.09 (.09)

0.24 (.09)

0.10 (.10)

0.14 (.07)

-0.33 (.07)

0.01 (.06)

0.11 (.06)

0.01 (.09)

-0.74 (4.2)

2,221 2,221

Mea~ of Dep. Variable 0.65 0.65

Model B

[3)

-0.01

0.91

0.32

0.25

0.18

0.14

0.15

-0.65

0.07

0.34

0.10

-0,11

-0.23

-0.10

0.1

0.25

-0.8

0.11

0.24

0.13

0.14

-0.38

-0.03

0.14

-0.06

-0.97

2,221

0.65

(.04)

(.07)

(.08)

(.08)

(.08)

(.o8)

(.07)

(.11)

(.12)

(.12)

(.13)

(.13)

(.14)

(.35)

(.7 )

(.11)

(.5 )

(.09)

(.09)

(.10)

(.07)

(.07)

(.06)

(.07)

(.08)

(5.2)
Chapter 2 ‘

A Dynamic Stochastic Discrete Choice Model

of Labor Force Participation and Marital Status

.

.

In ~apter 1, we estimated the parameters

static models of labor force participation and

Here, we estimate the dynamic four-alternative

same decisions.

As before, the relevant theoretical model

,“

of one-period

marital status.

version of the

refers to a young

mother who chooses among four alternative states, defined by

whether the women is married and whether she participates in the

labor force. The four alternatives states are:

(1) married and in the labor force;

(2) married and not in the labor force;

(3) not married and in the labor force; and

(4) not married and not in the labor force.

Our model is explicitly dynamic. In choosing alternative i, the

woman not only considers her utility in that alternative today,

but also the utility she can expect to obtain in the future.

The model begins with a rational young woman with a timeinvariant utility function and accurate forecasts of her expected

utility (that is, forecast errors have zero mean) . She exercises

choice among the four alternative states, recognizing that

today’s decisions may have long-term effects. For example,

. choosing not to work in any period may reduce her future income

(inside and outside of marriage). If she has a high discount

rate, however, such future consequences may carry little weight

in her decision-making.
35

The work reported in this chapter should be viewed as an

~loration of multiple-alternative discrete “choice models.

While our model focuses on four alternatives, it can be expanded

to any number of alternatives.1

In theoretical models of this type, current actions affect

future decisions in two different ways. First, current actions

might affect the returns to future actions or the constraints

faced by the decision-maker in the future. Second, the

decision-maker recognizes that fact and takes account of the

probable future effects of current actions when deciding on the

current action.

period no longer

current period.

incorporates the

Thus the object of maximization in the current

involves only the utility function in the

Instead it is a ‘fvalue function” which

current utility function’and also the discounted

e-ected value Of next periodls utility function. The decisionmaker calculated the #*discounted expected valueut conditional on

what she knows in the current period (the current periodas

information set) .

There are two basic approaches to build~ng an estimating

model that respects the above discussion. If the dependent

variable is continuous, the “~ler e~ationsuv approach is

appropriate.2 But the Euler e~ation approach is not

applicable when, as in our model,

discrete rather than continuous.

the possible actions and evaluate

This entails evaluating, for each

the action to be explained is

Instead, one must enumerate all

the value function for each.

possible action in the current
36

period, the expected future return of each possible future

action. A rational decision-maker then takes that action which .

maximizes the value function. This approach is typically termed

,,dynamic progrmin9.”

There are then two approaches to implementing a dynamic

programming model. In some cases the environment is stationary

and one can show that the value function takes the same form over

time (for example, some of the simple job search models take this

form) . In such cases one can work with the same function in each

period.

In our problem the environment is not stationary so we

cannot use this approach. For example, the number of children

changes over time. Hence we must use a solution techni~e known

as ‘backwards recUrsion.#l We fi-rst‘pick a teminal date, say T.

Given her position at date T-1, the woman then faces a static

optimization problem. We can thus

decision at T as a function of the

in period T-1 along with any other

Since T is the terminal period, no

need to be calculated.

Now , at T-1, we calculate the

characterize the optimal

values of the state variables

exogenous variables at date T.

e~ectations of future events

expected value of making

alternative decisions, conditioning ,onthe values of the state

variables at T-2. These are expected values since they include

the (discounted) expected value of the period T value function,

Eor the different possible period T-1 decisions. We must

calculate the expected value associated with all possible
37

decisions at date T-1. We then move backwards through time.

In general, at

associated with the

conditioning on the

and calculating the

the choices made at

time t we calculate the e~ected values

possible decisions which can be made at t,

levels of the state variables at time t-1,

e~ected values at time t+l, conditioning on

t.3

Consider a young woman with at least one child and a timeinvariant utility function. In period t, that woman chooses among

our four possible marital status/labor force participation

alternatives. In making her decision, she considers both the

utility available in each alternative (UO(t), U,(t), u2(t) and

U3(t)), and the utility she can expect in the’ future given her

chosen alternative in period t. It is this last element that

distinguishes this model frotithe static model presented in the

last chapter.

Introducing

chosen at time t

Alternatives are

notation, let

and di(t) = O

d~(t) = 1 if alternative i is

otherwise, where i = 1,...,4.

mutually exclusive; that is, Z di(t) = 1.

We assume that Ui(t) is a linear function of a vector of

exogenous variables that are the same for all alternatives

(X(t)), and a vector of dummy vari

alternative in period t-1. Thus,

(1) u;(t) = ~ix(t) + QfD(t-l) + U~

where:

kbles indicating the womnts

t) + E(t), i=O,...,3; t = l,T.

Ui(t) - the womanas utility in alternative i in period t;
x(t) -

D(t-1) -

●(t) -

Ui(t) -

38

a vector of exogenous characteristics (mostly

demographic) of the woman in period t:

(dO(t-l), dl(t-l), dz(t-l); d (t-l)).is a 4

variable vector indicating ti e alternative chosen

in period t-1:

a normally distributed random error that is

uncorrelated with X(t), D(t-1) , and c(t!) ;

an error term that is drawn from an’extreme value

distribution of the ‘form, F(ui)=e~(-exp(-uj/~} }.

It is pure white noise - E(ui(t,j),u.i’(t’,g ’)}=O,

i+ i’, j +jc,t+tf. Moreover, itls

uncprrelated with D(t-1), X(t), and e(t).

Note that the error term, e(t) is not subscripted - it does not

depend upon the alternative chosen in period t. pi and ai are

vectors of parameters to be estimated. Also note that since D(t1) enters into the U(t) function, past choices influence today’s

utility, and today~s choices influence future utilities.

The woman’s objective at any time t = 0,1,... ,T, is tO

maximizer

{

(2) E ~ pi-’~ Ui(j)di(j)ln(t)

j=t i=o }

where,

p is the woman’s discount factor, and

n(t) is her information set at time t.

The woman maximizes (2) by choosing the optimal se~ence of

control variables for all future periods. Thus, she chooses the

. .

opt~mal df(~), 1 = 0,...,3; g = t, t+l,.. ., T.

This problem can be solved through backward se~ential

solution of Bellman’s ewation (Bellman, 1957). In particular,

let the value of choosing alternative i at time t be written,
39

(3) LiV(~(t)) = Ui(t) + pE(V(n(t+l)

where V(n(t+l)) = Maxi(LiV n(t+l) ).

ld~(t) = 1), t = I,...,T-l,

Thus, E(V(~(t+l)) Idi(t) = 1} is the maximum expected value

of utility in period t+l given that the individual has chosen’

alternative i in period t. In period T, the value of choosing

alternative i is simply,

(4) LiV(n(T)) = Ui(T)

AS demonstrated below, the solution for Liv(n(t)) is obtained by

substituting recursively from T.

11. Analvtic Forms.

To estimate this model one needs analytic forms for (3) and

(4) as well as an expression for the probability of choosing

alternative i. To that.end it is simplest to rewrite the value of

choosing alternative i at time t as,

(5) L~V(Q(t)) = LiV(t)* -+ e(t) + Ui(t),

where

ui(t) is the i.i.d. extreme value error,

e(t) is the normally distributed random error,

LiV(t)* = B{X(t) + a~D(t-l) + pE(V(n(t+l) )ldi(t)=l),

fort=l ,...,1,1, and

L~V(T)* = piX(T) + aiD(T-l) .

The term, LiV(t)*, is obtained by substituting (1) into e~ations

(3) and (4).

Since ui(t) is distributed i.i.d. extreme value, the

probability that the woman chooses alternative i can be written

as a logit. To see this, let P(i,tlD(t-1)) be the probability
.-

40

that the woman chooses alternative i in period

the alternative chosen in the previous period.

P(i,tlD(t-1)) =

=

=

Note that since e(t)

out of the last line

Prob(LiV(~(t)) > LjV(n(t))

t conditional on

Then,

for j # i}

Prob(LiV(t)* + c(t) + Ui(t) >

LjV(t)* + e(t) + uj(t) for j + i)

Prob( uj(t) - ui(t) <LiV(t)* - LjV(t)*}

is identical

of the above

distributed i.i.d. extreme value,

(6) P(i,t]D(t-1)) =

To compute this

for all alternatives, it drops

expression. Since ui(t) is

3

eW{Lfv(t)*)/ .x exp(LjV(t)*)

3=0

probability, one must first

And from e~ation (5), that rewires information

compute LiV(t)*.

on

E(V(n(t+l)) Idi(t) = 1). The extreme value distribution

implies an expression for E(v(n(t+l)) Idi(t)“= 1). From

and Stern (1988), p. 8,

(7) E{V(~(t+l))ldi(t) = 1) = T(7 + K + in (~exp

j=0

where,

7 is Euler’s constant (= .5772): and

of Ui(t)

Berkovec

DjV(t+l)*/T)) },

K is a constant e~al to the expected value of ln(4e~{e(t)}).

A solution is obtained by computing LiV(t)*, i = 0,...,3 for

the last period (period T), and then using e~ation (7) to

compute LiV(t)* for the neti to the last period, T-1. Continuing

this backward recursion, one obtains values of LiV(t)* for all
41

time periods. &d given the values of LiV(t)*, one can compute

the probabilities in e~ation (6) for all time periods.

III. Eetima tion of the Dvnamic Model

The goal of estimation is to use data on the exogenous

variables (X(t) and D(t-1)) and the endogenous variables, D(t) to

estmate the parameters r, p, pi, ai,l = 0,...,3. To that end,

note that the likelihood that the woman chooses alternative i in

period t is,

di(t)

(8) ~ P(i, tlD(t-1))

i=O

Generalizing slightly, the likelihood that she chooses the

se~ence of contiol variables di(j), i = 0,...,3; j = 1,..,T is,

T di(t)

(9) n ; P(i,tlD(t-1))

t=l i=o

To estimate the model, we first form a sample likelihood

function by taking the product of these individual likelihood

functions, and then use a maximization routine with numerical

derivatives to find the parameters, r, p, Pi, and a~, i = O, ....3

which maximize the likelihood function.

Several of our data handling procedures must be discussed,

however, before we describe the actual estimation process.

First, we have assumed, from the onset, that a woman without
42

children is very different than a woman with children, so we have

excluded women without children from the analysis. For women

with children, we beqin the problem when the woman’ first has a

child.

Specifically, we start by determining the first year in

which the woman has a child of her own in the household. The

Womangs status in the year prior to that year is then her

,Iinitial condition.B14 Since the N~-Y respondents had children

at different times, the number of observed statuses (the di(t))

will vary across individuals; different women will contribute

different numbers of decisions to the statistical problem.

Having assiqned each woman an initial state, we then

determine her status in each year, along with the vector of

ewlanatov variables corresponding to each year.

We delete any observation for which any of the data we

rewire is missing.s We are left with a sample of 1,983 young

women. The upper panel of Table 2.1 shows how many of the

respondents were in each of the four statuses in each year. Also

shown, for each year

not yet have a child

likelihood function.

The lower panel

prior to 1985, is the number of women who do

and who are not yet included in the

of Table 2.1 displays the distribution of

these women by the number of decision periods. The sample is

weighted (but not heavily weiqhted) in favor of those with a

greater number of decision periods. This is partly because two

qroups of women contribute six decision periods, those with a
43

child already present in 1979 and those whose first child appears

on the record in 1980.

Table 2.2 displays means and standard deviations for the

demographic variables used here. Since different women contribute

different numbers of decisions, Table 2.2 displays the means of

these variables for the whole sample (upper panel) and for the

samPle whi~ is ,,activetlduring each year (lOwer Panel) .

The key pa*, of our maximization of the likelihood function

(shown in e~ation (9)) is the backwards recursion that takes

place for a given woman.6 There are two types of parameters

there: (1) the status-specific coefficients on explanatory

demographic variables (the pi in the theoretical discussion

above) and; (2) the status-specific coefficients on the woman’s

status in the previous period (the ai in the theoretical

discussion above).

The scale parameter of the extreme value distribution, ?, is ‘

not identifiable and is normalized to unity. This

underidentification is common in models of discrete choice. For

example, the probit model normalizes its variance to unity also.

The discount factor, p, is identifiable in principle. We found

it impossible, however, to identify this parameter-7

For each sample observation, we begin witi the terminal date 

and calculates the value function, LjV(T), associated wi~ each

possible choice, as a function of the previous period’s state.

As shown in the discussion after eguation (3), this value is a
44

function only of terminal period demographic variables, X(T), and

the woman’s status in the last period, D(T-1).

Having calculated LiV(T), we then move to the’previous time

period. For periods other than T, the maximization routine must

first calculate, for each status, the e~ected value of utility

in the next period for each possible choice. That is, the

routine must calculate the values of E(V(n(t+l)) ) for the periods

other than T.

Using the values of E(V(n(t+l))}, we can then calculate the

value for the current period, LiV(t). We continue this process

until we have etiausted all the decision points for this

observation. Using the calculated values of LiV(t), for all four

statuses, we then calculate the choice

(7). These probabilities represent the

observation to the log likelihood.

IV. Results from the Dvnamic Model

Table 2.3 shows our estimates for

probabilities in eguation

contribution of each

the parameters ai and pi

in eguation (1); we show the absolute value of the asymptotic

normal statistic, for the null hypothesis that the coefficient is

zero, below each parameter estimate.

The estimates of ~i are presented as a matrix in the upper

panel of Table 2.3. The rows of this matrix indicate the

characteristic under consideration while the columns indicate the

status whose utility function is being estimated. Each

coefficient is an estimate of the effect of individual
45

d&ographic characteristics on the utility

the four labor force participation/marital

of being in any one of

status categories. The

estimates of ai are presented as a matrix in the lower panel of

Table 2.3. The rows of the matrix indicate the previous year”s

status while the columns again indicate the status whose utility

function is being estimated. Each coefficient

effect of last periodas status on the utility

of the four labor force participation/marital

indicates the

of being in any one

status categories

this period.

As an example of the

the negative coefficients

interpretation of the

on the variable BUCK

estimates of pi,

in the first two

CO1-S ‘of Table 2.3 indicate that black mothers gain less

utility from being married than comparable white mothers,

regardless of labor force pa*”icip”ation. The positive

coefficients on the variable BUCK in third and

Table 2.3 indicate that black mothers gain more

being married than white mothers, regardless of

fourth COl_S Of

utility from ~

labor force

participation. By contrast, the across-the-board positive

coefficients on the variable HISP~IC indicate that Hispanic

mothers receive have higher uttility than white mothers in all

four statuses, ceteris paribus. None of the coefficient

estimatee, however, allow us to rej,ect the hypothesis that the

coefficients are zero in the population.

Our estimates of ~,, the coefficients on the demographic

variables, are uniformly insignificant. tioking at the

algebraic sign of the coefficient estimates, we see that age
46

(AGE/10) has a small positive effect on the

the labor force. Number of children (KIDS1)

on the utility of all statuses except being

utility of being in

has a positive impact .

unmarried and in the

labor force. The coefficients on race/ethnicity were discussed

above.

As an example of the interpretation of the estimates of ffi,

note the large and statistically significant coefficient (2.27)

in the first column and first row of Table 2.3. This coefficient

indicates the high utility associated with being married and in

the labor force for those women who were also married and in the

labor force in the previous period.

In the conteti of a dynamic programming model, the estimates

for ffireflect the value of remaining in the same status as in

the previous period. If one made the optimal choice in the

previous period, then

current period is the

of the disturbance or

variables. Therefore

positive, or at least

is strongly supported

the only reason to change status in the

arrival of new information, either

in terms of one of the explanatory

we expect the diagonal elements to

not negative and significant. This

in terns

be

pattern

by the coefficient estimates in Table 2.3:

all of the coefficients on the diagonal of the ai matrix are

large, positive and statistically significant.

The off-diagonal elements, which represent the change in

mean utility from changing status, should be negative. The

argument is the same. If the previous decision were optimal,

then the mean change in utility from the change in status should
47

be negativ=. .~anges do occur, but” only in

un~ected information, represented here by

response to new

the disturbance term. .

As indicated in Table 2.3, though, some of the off-diagonal 

elements of the table are both positive and statistically

significant. Entering the labor force increases utility. The

change in utility when a women moves from being married and out

of the labor force to being married and in the labor force is

1.44 with a normal statistic of 3.09. For unmarried women moving

intO tbe labor force, the relevant coefficient is 1.25 (2.68).
Conclusion

In this

of the labor

report, we have estimated a dynamic stochastic model

force participation and marital status decisions of

young mothers. In implementing that model empirically, we used

data from the on-going National Longitudinal Survey - Youth

Cohort .

The major advantage of such a model is theoretical. It

incorporates the appealing notion that young mothers think about

the future in making decisions today. The model uses an explicit

utility-maximization framework, in contrast to less ‘Istmctural!t

models as have been more comonly used.

Empirically, the model we use estimates the parameters of a

four-state model. The same programs, however, can be used to

estimate the parameters of larger models; we report its use in a

six-state model in Appendix B to this report.

In order to assess the usefulness of the dynamic model, we

have estimated a series of models, of increasing complexity. In

this particular context, there does not seem to be much gain in

using more complicated cross-sectional mcdels. In particular, the

earlier models, discussed in Chapter 1, indicate that the labor

force participation and marital status decisions are independent

of each other. These indications first appear in cross-sectional

models using data for 1985.

The cross-sectional models also suggested that once past

values of labor force participation and marital status are

included in the analysis, demographic variables (such as race,
49

etihnicity, age and education) are relatively unimportant in

determining current labor force participation and marital status.

This cross-sectional conclusion appears again in.the dynamic

model, whid uses data from all years.

There are two ways to view that result. One is that there is

little to be gained from using the dynamic model because the same

conclusion can be drawn from the simpler model. The other view is

that the dynamic model is working properly becau se it leads to

the same conclusion as the simple model.

OUr view is that the dynamic model is the theoretically

appropriate model in this context; The lack of appreciable ‘Igainvt

(in the fo= of more precise and plausible parameter estimates)

should not impede its adoption.

The constraints imposed by the computational burden of the

estimation forced us to keep our dynamic model mite simple. The

similarity of results across dynamic and static models may

indicate only that simplicity. mile the model may be too simple

to capture behavior ade~ately, it is a step in the right

direction. If there is to be progress in modeling labor force

patiicipation, we believe that a structural approach is

absolutely essential.
50

Table 2.1

Descriptive Statistics for the Sample of

Young Mothers Used in the Dynamic Model

A. Classification of Women with ~ildren by Year, by Marital

Status and -bor Force Participation

1985

Status

Married

In Labor Force

Not in Labor Force

Not Married

In Labor Force

Not in bbor Force

Without Children

Total

1985

Status

Married

In Labor Force

Not in Labor Force

Not Married

In Labor Force

Not in Labor Force

Without tiildren

Total

Number of Women in Each Category

1980 1981 1982 1983 1984 1985

128 210 256 355 460 561

248 331 434 503 535 571

116 152 192 247 323 377

230 322 412 447 456 474

1261 968 689 431 209 0

1983 1983 1983 1983 1983 1983

Percentage of Women in Each Category

1980” 1981 1982 1983 1984 1985

6.5 10.6 12.9 17.9 23.2 28.3

12.5 16.7 21.9 25.4 27.0 28.8

5.8 7.7 9.7 12.5 16.3 19.0

11.6 16.2 20.8 22.5 23.0 23.9

63.6 48.8 34.7 21.7 10.5 0

100.0 100.0 100.0 100.0 100.0 100+0

B. The Distribution of Women with Children BY Nutier of

Available Decision Periods in Dynamic Model

Number of Periods

Total 1 2 3 4 5 6

Number of Women 209 222 258 279 293 722 1,983

Percentage 10.5 11.2 13.0 14.1 14.8 36.4 100.0
51

Table 2.2

Means and Standard Deviations for Independent

Variables in tie Dynamic Programming ’Model

Women with ~ildren in 1985

Independent Standard

Variable ~ ~iation

AGE 24.45 2.21

B~CK 0.32 0.47

HISP~IC 0.18 0.38

KIDS1 1.55 0.78

Sample Size = 1,983

Means for Independent Variables

in the Dynamic Programing Model ##ActivellDecision Makers, bY year

Independent

Variable m m

AGE 20.57 21.17

B~m 0.38 0.36

HISP~IC 0.14 0.16

KIDS1 1.08 1.18

Sample

Size 722 1015

Year

= w = m

21.93 22.68 21.55 24.45

0.34 0.33 0.33 0.32

0.17 0.18 0.18 0.18

1.28 1.38 1.46 1.55

1294 1552 1774 1983
52

Table 2.3

Coefficient Estimates for a Four State

Dynamic Programming Model of Marriage

and Labor Force Participation

Coefficients on Demographic Variables, pi

value of asymptotic normal statistic)

Harried Married Not Married

In LF Not in LF In LF

-.398 .869 -.847

(.610) (1.342) (1.293)

-.025 -.332 .316

(.050) (.662) (.630)

.147 .078 .026

(.294) (.156) (.052)

.441 -.363 .783

(.869) (.716) (1.541)

.397

(:::2)

-.315

(.793) (.629)

Status-Specific

(absolute

Not Married

Not in”LF

CONSTANT

B~CK

HISPANIC

AGE/ 10

.772

(1.189)

.437

(.871)

-.453

(.894)

KIDsI .234

(.467)

Status-SDecific Coefficients on Past Status Variables, Ui

(ab~oltitevalue of asymptotic normal statistic)

Current Status

Not Married

In LF

Previous

Status t

Married

In LF

Married

Not in LF

Not Married

Not in LF

Married

In LF

2.269

(4.872)

1.420

(3.041)

.124

(.262)

-.47i

(.980)

Married

Not in LP

1.444

(3.093)

2.634

(5.672)

-.447

(.941)

.734

(1.563)

Not Married

In LF

.392

(.833)

.114

(.242)

2.127

(4.567)

1.580

(3.377)

Not Married

Not.in ~

-.599

(1.263)

.606

(1.296)

1.254

(2.685)

2.833

(6.099)

Value of Log Likelihood Function: -7209.10038
Appendix A ‘

Variable Definitions and Data Preparation Issues

This Appendix begin with the definition of the variables

appearing in the body of our report. The definitions appear in

Table Al. The remainder of the Appendix discusses, in

substantially greater detail, some of the problems in using the

NLS-Y for time-series analysis of decisions concerning family

structure.

The Problems in Defininq Family Structure over Time in the NLS-Y

In order to make our results comparable to those of earlier

work done on Current Population Survey (CPS) cross-sections, we

decided to construct CPS-type marital status and living

arrangement definitions, such as ltprimary family-r, lrsubfamilyfl

and ‘*unrelated individuals.fg A description of the available

variables in the NLS-Y documentation suggested that these living

arrangement definitions were feasible and would reguire fairly

“straightforward manipulations of the data. Unfortunately, we

encountered numerous problems in the construction of our marital

status and living arrangement measures because of inaccuracies in

the documentation or niscodings in the data themselves. The

latter problem diminishes in the later years of the survey, but

is particularly prevalent during the early years of the survey

(1979-1981).
AFDCXX

AGE (SQ)

EDUC(SQ)

B~CK

HISPAWIC

SOUTH

SMSA

HDS2185

~DS2285

KIDS2385

=DS 1

54

Table Al

Variable Definitions for

tibor Force Participation and

Models of

Marital StatUS

-or force participation variable defined as 1 if

respondent is in the labor force in year xx and O

otherwise.

Marital Status variable defined as 1 if respondent

married in year m and O otherwise.

Marital Status variable defined as 1 if respondent

married or livinq with a ‘epartnertfin year & and O

the

is

is

otherwise.

Welfare participation variable defined as 1 if the

respondent received income from AFDC in year xx and O

otherwise.

The respondent’s age in years, measured continuously

from birth. AGESQ is AGE s~ared.

The highest grade completed by the respondent as of the

date of interview in 1985. EDUCSQ is EDUC s~ared.

Takes the value 1 if the respondent reports her race as

black; O otherwise.

Takes the value 1 if the respondent reports her

ethnicity as Hispanic: O otherwise.

Takes the value 1 if the respondent’s

the south in 1985; O otherwise.

Takes the value 1 if the respondent’s

an SMSA in 1985: 0 Othe~iSe.

residence is in

residence is in

The number of children (own, adopted or partners) of

age O, 1 or 2 years in 1985.

The number of children (own, adopted or partners) of

age 3, 4 or 5 years in 1985.

The number of children (own, adopted or partners) 6

years old or more in 1985.

The total number of children, between O and 3 years of

age, present in the respondents household.
55

.

Table Al

Variable Definitions for Models Of.

tibor Force Participation and Marital Status

(Continued)

AFDCG The relevant 1985 AFDC maximum payment, for the

respondents geographic state and family size.

~D~ The estimated difference, in 1985, between AFDC

payments for a household head and a subfamily head.
.

56

The N=-Y survey gathers information on all individuals (to

a maximum of 15) who live in the same household as the respondent ,

and classifies household members into families. The information

collected includes each household memberss sex, age, relationship

to tie respondent. In addition, the N=-Y documentation indicates

that the first individual in the household record is the

household head. Taken together, this information should have

been sufficient to construct definitions of living arrangement

measures that are consistent with the CPS.

After some data manipulation, however, it became clear that

there were serious inconsistencies in the data. First, the

individual who appears in the first position of the household

record cannot be reliably declared as the household head. This

was later confirmed. by the N~-Y data archivists at Ohio State.

Household head information was consistently collected in 1979, by

means of a separate survey ~estion. In subse~ent survey years,

however, the interviewer became responsible for correctly placing

the household head in the first position of the household record

data. Unfortunately, this approach has proved to be unreliable.

Some attempt to use mortgage information to identify the

household head was made, but this also proved to be unsuccessful.

This inability to identify the household head has limited the

atent to which the living arrangement measures created from the

N=-Y parallel the CPS definitions.

Second, each household member is assigned a family unit

number which identifies the family to which the household member
57

belongs. Theoretically, the family ‘unit number could then be

used to determine the number of families within a dwelling unit .

as well as identifying members within a family. Individuals are

considered to be members of the respondents family if they are

related by blood or marriage. Unrelated individuals, including

cohabitation partners, should not be coded as members of the

respondents family. This information, however, was found to

fairlY inconsistent. For example, unrelated individuals were

often given the same family unit nutier as the respondent

suggesting a single family unit in the household. Yet, a

be

respondent living with siblings or other relatives did not share

the same family unit number suggesting multiple families within

the dwelling unit. These inconsistencies were sufficiently

common that any systematic use”of the family unit number was

abandoned.

Given the problems associated with identifying the

household head and using the family unit number to unravel

multiple family households and their members, it became necessary

to base the living arrangement measures solely on the

Ilrelationship to youthsm codes. This task was further complicated

by the fact that individuals may appear in any order within the

fifteen household records for a single year. For example, there

may be information in positions one, three and six of the

household record, with no information in any other positions.

Further, the positions with data are not consistent from year to

year. In one year the respondent may be in position three and
58

the spouse in position one, yet the following year the respondent

is in position one and the spouse in position two, with no change .

in overall family composition. In addition, the creation of

marital status and living arrangement measures was further

complicated by the need to allow

A respondent can declare an

the marital union is,not legally

can be legally married or simply

they are cohabiting as a spouse.

is an individual of the opposite

for partners as well as spouses.

individual as a spouse even if

binding. That is, a respondent

regard the individual with whom

A partner, on the other hand,

sex who lives with the

respondent as a cohabitant and is identified as such by the

respondent.

Essentially, for each year, it was necessary to loop

through all fifteen household records and classify any individual

who resided in the household into the relevant categories of

living arrangement. This included whether or not the respondent

was living with parent(s) or parent(s)-in-law; living with

relatives over 18: living with nonrelative; living with a spouse

or partner; living with own, step or adopted children; living

with partnerfs children. Given the complexity of the task at

hand, and cognizant of the apparent limitations of the data

themselves, other variables were used to cross-check the living

arrangement measures which had been created using the

Ilrelationship to youth~~ codes.

One of the relationship to youth codes specifically

categorizes an individual within the household as being the
59

respondents 8Bpartner.tt.There were numerous cases, however,

where an interviewer check ~estion indicated that the respondent

was currently living with an individual of the opposite sex as a 

partner but the ‘#relationship to youthsg code revealed no partner

living in the household. To resolve this inconsistency, it was

necessary to look more closely at individuals in &e household

who were coded as nonrelative of the respondent. To check if an

individual coded as a nonrelative was really a partner, one of

two routes was taken. The first systematically looked at

nonrelative when there was no partner or spouse in the

household. Specifically, if there was only one individual

household who was coded as a nonrelative, and was an adult

then that ,individual was reclassified as e ~tpartner.ll The

in the

male,

second

route consisted of dumping data records and hand-coding the

relevant variables for that observation when inconsistencies were

found between interviewer checks and the relationship to youth

codes. Hand-coding of observations will be described more fully

in a subse~ent section of this appendix.

Having identified a Iapartner,gtwe made an att-pt tO

determine if any of the children in the household who were coded

as nonrelative could be reclassified as the partnerss children.

If an expanded definition of being married includes partners

along with spouses, then the partner”s children

classified as part of the respondents family.

some data manipulation because the relationship

should be

This reqired

codes offer no

clue as to the parenthood or guardian relationship of nonrelative
60

c“hildren to the respondent or other members of the household. An

individual was defined as a partnerls child if all of the

following conditions held: .’

(a)

(b)

(c)

(d)

(e)

The partner’s family unit number was different from the

respondent ‘s. If the partner and the respondent shared the

same family unit number, it was assumed that all children

relevant to their family would have been coded as the

respondents own, step or adopted children:

When the respondent and partner had different family unit

numbers, the individually family unit number had to be the

same as the partner$s. That is, any potential child of the

partner should be coded as belonging to the partnerts

family:

In the “relationship to youthva code, the individual was

coded as being a nonrelative. Any potential child of the

partner should have no family relationship to the

respondent;

The individual was under 18 years of age;

The partner was at least 16 yea= older than his Dotential

child.

There are serious limitations with the approach used to

identify the partnerts children which must be pointed out.

First, this is

household, who

conceivably be

. are simply not

at best an educated guess of which children in the

are not related to the respondent, could

the partner$s children. The relationship codes

sufficiently detailed to be able to determine the

identity of the partnerss children without error. Second, it was

necessary to use the family unit numbers in this endeavor and the

limitations of those numbers have already been described. We

hope that the criteria used were sufficiently stringent that the

probability of error in classifying partner’s children was

minimized.
61

~Ca .n and D ta Errors

After repeated iterations of doing consistency checks and

printing out inconsistent records, we were able.to program nany

of the corrections. However, for a subset of observations this

proved to be impossible. We therefore recoded these observations

manually after examining the records closely.

The inconsistencies and errors appear to occur most

fre~ently in households with large groupings of individuals

where the possibility of shared living arrangements with family

members and/or nonrelative was the highest. Also, many of the

inconsistencies were related to difficulties in correctly

identifying the respondents partner. To simply delete these

records from the sample would have resulted in disproportionately

dropping those cases in which the respondent was in a shared

living arrangement or cases where a partner, was present in the

household. Yet, these were exactly the cases of primary interest

to the analysis.

A number of data checks were used to validate some of the

living arrangement measures created. One of the data checks used

initially was the recorded household record type. Three versions 

of household records are used by the N=-Y. Version A is used if

the respondent is living with parent(s) or parent(s)-in-law. In

this case, the household interview, which collects information

about the occupants of the household, is conducted with one of

the parents. Version B is used if the respondent is living in a

temporary dwelling unit such as a sorority, fraternity, dormitory
62

.

o“rmilitary ~a*ers. These respondents are not considered to be

within the sample relevant to this analysis and were dropped at ,

the beginning of the analysie. Version C is used’if the

respondent is living in their own dwelling unit or is the head of

a family unit. Attempts to compare household record type against

the created measure of whether the respondent is living with

parent(e) or parent(s)-in-law based on the relationship to youth

codes proved to be futile. The NU-Y allowed interviewers to use

household record version C even when the respondent was under 18

and living with parent(s) or parent(s)-in-law if the interviewer

ascertained that contacting the parent would be awkward or there

was reason to suspect the parent would not consent to the

interview. Other exceptions are based on the respondents age

(either younger or older than’18) ‘and whether they have lived

continuously with parent(s) or parent(s) in-law. These

exceptions made it impossible to use this variable as a check

against whether the respondent was sharing the household with

parent(s) (in-law) based on the relationship to youth codes.

Three specific checks of the constructed living arrangement

measures were made. They were concerned with the correct

identification of spouses and partners, and accurately

distinguishing partners from nonrelative. The first two checks

were constructed from NU-Y interviewer check ~estions.

Specifically, they ask “is the respondent married and the spouse

listed on the household record” and l’doesthe respondent live

with an adult nonrelative of the opposite sex.st After 1981, the
63

latter guestion becomes more specific and asks ‘*is

currently living as a partner with an opposite sex

the respondent

adult.tT The ,

answers to both these ~estions were compared toconstmcted

variables concerned with whether the respondent had a spouse or

partner based on the Vlrelationship to youthsm codes. The last

check was concerned with flagging any respondent which reported

multiple spouses or partners or both a spouse and a partner in

the household.

men an inconsistency was found, data from multiple years

was printed. Specifically, data from the year in which the

inconsistency was found (year t) , as well as data from the

previous -(year t-l) and subsegment (t+l) years was printed. If

the inconsistency occurred in the first year of the data survey

(1979), however, the two subse~ent years (t+l, t+2) were

printed. While some inconsistencies could have been resolved from

a single year~s data, others could only be resolved by obseNing

the age and sex composition of the household in past or future

years. A total of 210 records were examined and corrected.

of

t,

in

Some attempt has been made to construct general categories

errors found when looking at the printed records for the years

t-1 and t+l (or t, t+l and t+2 when the inconsistency occurred

1979) . It should be noted that corrections were only made to

constructed variables in order to maintain the integrity of the

original data set. mat follows is a discussion of each type of

error in descending order of fre~ency.
64

that

as a

First and most fre~ently, the interviewer check indicated

the respondent was living with an adult of the opposite sex ,

partner. Yet, according to the ‘Relationship to youth”

codes, no individual was coded as a partner. Multiple

.

nonrelative lived in the household, however. This particular

scenario led to four sub-categories of problems and solutions.

(a)

(b)

(c)

(d)

.

Amongst the male nonrelative(s) in the household, no single

individual could be discerned as being the respondents

partner even after comparing the sex and age composition of

the household in year t with years t-1 and t+l. In these

cases (fre~ency=52) , the respondent was recoded as being

single;

The respondents partner could be discerned from the male

nonrelative(s) in the household after comparing the sex and

age composition of the household in year t with years t-l

and t+l. In these cases (fre~encp21), the respondent was

recoded as living with a partner. In a similar case, the

interviewer check indicated there was no opposite sex adult

livin’g with the respondent. Yet, a single individual was

coded as being the respondents partner from the

~lrelatiOnship to youthvt codes. In this case (fre~ency=l),

the respondent was recoded as living with a partner:

Al,l the nonrelative in the household shared the game family

unit number, suggesting they formed a single family that was

unrelated to the respondent. It was very difficult to

discern, however, if one of the male family members was the

respondents partner. As aresult, the respondent was

recoded as being single in these cases (freguency=lO) ;

A1l of the nonrelative in the household were females. In.

these cases (freguency=7), the respondent was recoded as

being single.

Second, the marital status of the respondent (single,

married with spouse present or living with partner) as detemined

from the ‘relationship to youth” codes was inconsistent with one

or both of the interviewer checks. Sub-categories of this problem

are discussed below. In general, discrepancies were resolved by

ignoring the interviewer checks and classifying marital status
65

tiased on who

relationship

(a)

(b)

(c)

(d)

(e)

resided in the household, as determined from the

codes.

The interviewer checks indicated that boththe respondent’s 

spouse and partner were present in the household. ‘Accordinq

to the relationship codes, however, only a spouse resided in

the household. In these cases (fre~enc~20), the respondent 

was recoded as married, spouse present;

The inteniewer checks indicated that both the respondent’s

spouse and partner were present in the household.

~rthermore, the relationship codes found either a spouse

and partner or a spouse and male nonrelative residing in the

household with the respondent. In these cases

(fre~enc~12), the respondent was coded as married, spouse

present. It was assumed that the male nonrelative in these

cases were not living as partners with the respondent. In

addition, any individual coded as a partner was viewed as a

miscode and subsequently counted as a male nonrelative.

This was done because it was difficult to imagine a

household where the respondent was living with a spouse and

a live-in companion of the opposite sex simultaneously. The

spousal relationship took precedence over the partner

relationship because the spousal relationship has generally

been less difficult to discern in this data set;

The interviewer check indicated the respondent was living

with an opposite sex adult as a partner. No partner or male

nonrelative was identified, however, from the relationship

codes. In these cases (fre~ency=17), the respondent was

recoded as being single:

The interviewer check indicated the respondent was married,

with spouse present, but according to the relationship

codes, no spouse was present-in the household. In these

cases (fre~ency=ll), the respondent was recoded as beinq

sinqle:

The interviewer codes indicated the respondent had a spouse

but no Dartner. Yet. the relationship codes revealed a

partner-but no spouse. In these case= (fre~enc~–6), the

respondent was recoded as livinq witi a partner.

Third, and ’most disturbing, were errors in the relationship

codes found through various discrepancies in one or more of the

three data checks outlined earlier. A total of 42 relationship

coding errors were found. All could be corrected using
66

.

.

information on the sex and age OE household members as well as

the composition of the household in years t, t-l and t+l. Some .

examples of the types of miscodes which occurred include:

(a) The relationship codes revealed multiple spouses, where

extra spouses were determined to be a child, sister,

brother, cousin or other relative of the respondent;

(b) The respondent’s spouse or partner was erroneously miscoded

as some other relative (for example, as a sister, brother,

father, mother, daughter-in-law or foster child) . The most

striking example of this type of error were individuals

coded as the respondents sister (relationship code 7) who

were also males. Looking at information from years t-l and

t+l, these individuals were subse~ently recoded as the

respondent’s spouse (relationship code 1) . It is obvious

that relationship codes 7 and 1 were transposed while being

transcribed from the original interview sheets:

(c) The last grouping contains miscellaneous coding errors such

as a nonrelative miscoded as a foster child; a partnervs

child miscoded as a partner; a spouse miscoded as another

respondent; daughters miscoded as sisters; brothers and

sisters miscoded as partners and other-in-laws; partners

miscoded as boarders.

Fourth, the relationship codes reveal ,the respondent was

LiVing with more than one partner, where some or all of these

pa*ners were really nonrelative. In these cases (fre~ency=7),

information from years t-1 and t+l as well as sex and age

information from year t was used to try and discern a true

partner. Those determined not to be the respondents partner

were recoded as nonrelative.

Lastly, in a few cases (fre~ency=2) the relationship codes

were either completely missing or so badly miscoded that the

entire observation was set to missing for that year. In other

cases (fre~ency=2), only some of the relationship codes were

missing and it was possible to reconstruct the composition of the
67

household based on sex, age and househo~d composition information

in years t, t-l and t+l.

mile many inconsistencies and coding errors”were

corrected, it should be noted that the final data set may still

contain errors. Of particular concern is any undetected

*relationship to youthta coding errOrs. These would affect

cross-sectional analysis as well as transition rates for marital

status and living arrangements. One concern is mitigated,

however, by the tiowledge that the majority of the coding errors

which were found in the 210 cases examined manually occurred in

the first three years of the data survey.
Appendix B

Additional Models of Labor Force

Participation and Marital Status

This appendix present two additional models or labor force

participation and marital status. The first is a model of the

Ilinitial cOnditionsl~ for the women in Our .Sample. since the

models presented in the text suggest that a young motherts labor

force participation and marital status tend to remain constant,

except for random “factors, it is of some interest to examine the

demographic correlates of those initial conditions. The second

model illustrates the geferal applicability of the multi-state,

multi-period dynamic model developed by George Jakubson for this

project. In that second model, we estimate a ~ state labor

force participation and marital status model: Labor force

participation remains as a O-1 variable but ‘Imarital Statusgn can

now take on three values - married, unmarried and heading onems

own household and unmarried and living with relatives.

AKoeo ll~n.t.a ~

The results presented in the text highlight the importance

of looking at where a woman has been in order to describe where

.

she is now. Legged marital status is the most important

determinant of current marital status; lagged labor force

participation is,the most important determinant of current labor

fOrCe participation. -St year’s AFDC participation has important

negative impacts on current marital status and current labor

force participation.
69 >

In one sense, this emphasis on”laqqed dependent variables

simply pushes the problem back a few steps. If race affects labor .

force participation only because black women are ~ore likely to

participate in the AFDC program, then why are black women more

likely to participate in the AFDC program?

Though these ~estions are not amenable to statistical

analysis, this section addresses the ~estion ~~whatare the

correlates of the initial conditions?.,t The dynamic programing

model presented in Chapter 2 deals with four marital status/labox

force participation states,

four states here as well.

The dependent variable

(1)

(2)

(3)

(4)

so we restrict our attention to those

can take four values:

married and in the labor force;

married and not in the labor force;

not married and in the labor force: and

not married and not in the labor force.

The variable is defined at the time when an w-Y respondent,

first reports having a child of her own in her household. For

example, if the woman first reports having a child in 1982, tien

our dependent variable and all other variables in the model are

given their 1982 values. If the woman first reports having a

child in 1984, then all variables take on their 1984 values.

By defining the variables in this way, we are tryinq to look 

at the correlates of a marital status/labor force participation

variable, at .me a wo a irst has ~.

Consider the following tabulation of the four labor force

pa*icipation/narital status states in the woman~s initial

condition as compared with the same four states in 1985.
.

.

70

Ebor Participation/Marital Status in 1985

1 2 3 4 Total s

Initial Condition

1.

2.

3.

4.

Married, in LF 251 125 36 20 432

Married, Not in U 142 259 62 125 494

Not Married, in LF 92 62 155 8a 397

Not Married, Not in LF 76 125 140 319 660

Total 561 571 377 474 19a3

Of the l,9a3 women who report having a child in their

household between 1979 and 19a5, inclusive, almost 50% remain in

the status that they *#started” in.

Our initial conditions model, reported in Table Bl,

considers the marital status/labor force participation variable

as a function of only age, race, ethnicity and the number of

young children present;

limited to correspond to

model in Chapter 2.

The coefficients in

the probability of being

probability of not being

The number of independent variables is

the’variables included in the dynamic.

the first column enable us to compare

married and in the labor force to the

married and not in the labor force. For

example, the negative coefficient on BUCK is large (in absolute

value), and statistically significant, indicating that being

black significantly reduces the probability of being married (at

the time when the

unmarried and not

since only 14% of

almost 60% of the

first child “appears”) as compared to being

in the labor force. This is not surprising

the married women were black as compared to

unmarried women. The same story holds for
71

Hispanic women as well, though the coe~ficient is considerably

smaller.

The coefficients in the second column compare the

probability of being married and not in the labor force to the

probability of being not married and not in the labor force (at

the time of when the respondents first child was born). Here

again, black and Hispanic women are more likely, compared to

white women, to be unmarried and not in the labor force than to

be pa~ and not in the labor force.

Age plays a powerful role here, especially considering the

limited age range of the NLS-Y respondents. As indicated.by the

positive and statistically significant coefficients on AGE in

Table Bl, older women are considerably less likely to be

unmarried and out of the labor” force. Greater numbers of children

under three lowers the probability of being in the labor force.

This overall picture drawn by these ~ainitial conditionslt

models is not particularly surprising. Roughly put, if a woman is

in an economically healthy position before she has a child - in

the labor force or married or both, then she is in an

economically healthy position after she had a child. The thrust 

of our other modelling efforts is to show that once an initial

condition is established it tends to be perpetuated. Thus our

story seems to b“ethat the eventual economic health of women with

children is established early on and then tends to persist over

time.

A~ S.X-St e ode iciDat.on and ital Sta us
six

The results of extending

states are shon in Table

?2

our four-state dynamic model to a

B2. These results are ~ite similar

to the results of the four state model; we

indicate the practicality of extending the

multiple states. The six states are:8

show them largely

dynamic model to

to

1 = Married, in labor force (MLF)

2 = Subfamily Head, in labor force (SHLF)

3 = Household Head, in labor force (HHLF)

4 = Married, not in labor force (~LF)

5 = Subfamily Head, not in labor force (SHNLF)

6 = Household Head, not in labor force (HHNLF)

As was true in the four state model, the demographic

variables have virtually no significant impact on the utility

of being in any one of the six states. The parameter estimates

for pj (Panel 1 of Table B2) are uniformly insignificant. This

parallels a similar result from the four-state model.

men we turn to the estimates of the ai vector, we see again

that the “previous statusat has the most value. The coefficients

on the diagonal (in Panel 2 of Table B2) are the ones that have

large normal statistics (indicating a significant difference from

zero). The only exception to this is that being a subfamily head

and not in the labor force seems to be of value in making the

transition to being a subfamily head and in the labor force (the

coefficient estimate is

vice-versa (coefficient

3.8).

Our interpretation

that, aside from random

2.4 with a normal statistic of 3.9) and

estimate of 2.3 and normal statistic of

of the results of the six state model

factors that are unobserved by the

is
researcher,

determining

only past status

current status.

73

plays an

,

important role in
74

Table B1

Four-valued Logit Model of Marital,

Status and AFDC Participation’

Initial Conditions Model

NL8-Y Respondents men the First

Report Having a Child in Their Household

Fre~ency Count of Dependent Variable

(1) Married and in the labor force 432

(2) Married and not in the labor force 494

(3) Not married and in the labor force 397

(4) Not married and not in the labor force 660

Total Sample Size 1983

Married and Married and

In the Not in the

Labor Force Labor Force

(1) (2)

BLACK

HISPANIC

AGE

KIDS1

Constant

-2.27 (12:3) -2.22 (13.1)

-0.78 ( 3.9) -0.19 ( 1.1)

0.60 (15.6) 0.33 ( 9.4)

-0.36 ( 2.3) 0.21 ( 1.6)

-11.58 (14.8) -6.09 ( 8.8)

Not Married

and in the

Labor Force

(3) .

-0.64 (4.4) ,

-0.17 (0.9)

0.28 (8.0)

-0.50 (3.4)

-5.47 (8.0

.

.
75

Table B2

Coefficient Estimates for

Dynamic Programming Model

a Four State

of Marriage

and Labor Force Participation -

~Ur~DhiC Pa el . Variables. Bi

Dependent Variable:

1=

2=

s=

4=

5=

6=

Harried, in labor force (MLF)

SUbfamily Head,

Household Head,

Married, not in

Subfamily Head,

Household Head,

in labor force (SHLF)

in labor force (HHLF)

labor force (~LF)

not in labor force (S~LF)

not in labor force (~LF)

Status-Specific Coefficients on Demographic Variables, pi

(absolute value of asymptotic normal statistic)

Status

l=~F

2=SHLF

3=HHLF

4=mLF

5=smLF

6=~LF

Constant

-1.4

( .7)

-0.2

(0.2)

(:::)

(:::)

(:::)

-1.5

(0.6)

Black

-0.5

(1.2)

(::;)

(:::)

-0.5

(1.8)

(:::)

(:::)

Hispanic

(:::)

-0.1

(0.1)

(:::)

(:::)

-0.1

(0.2)

(::;)

Age/10

(:::)

(:::)

-0.1

(0.2)

-0.2

(0.3)

-0.7

(1.3)

(!::)

KIDS 1

(:::)

-0.2

(0.4)

(:::)

(;::)

-0.1

(0.2)

-0.2

(0.4)
76

Table B2

(continued)

Coefficient Estimates for a Four State

Dynamic Programming Model of Harriage .

and Labor Force Participation

. ~iPanel .

Dependent Variable:

1 = Married, in labor force (MLF)

2 = Subfamily Head, in labor force (SHLF)

3 = Household Head, in labor force (HHLF)

4 = Married, not in labor force (NNLF)

5 = Subfamily Head, not in labor force (SHNLF)

6 = Household Head, not in labor force (HHNLF)

Status-Specific Coefficients on Past Status Variables, a,

(ab~olute value of asymptotic.normal statistic) ‘

Previous

Status l=MLF

2 = SHLF

(:::)

. 3=HHLF

(::$)

.4=~LF

(:::)

5 = SHNLF

(:::)

6=HHNLF -7.0

(0.4)

Current Status

2=SHLF 3=HHLF 4=MNLF

-0.3

(0.3) (:::) (:::)

(:::) (;::) (:::)

(:::) (:::) (::?)

-0.8 -0.5

(0.9) (0.6) (::;)

(:::) (:::) (:::)

-0.2

(0.2) (:::) (!::)

5=SHNLF 6=HHNLF

-0.2 -8.7

(0.2) (0.4)

(:::) (:::)

(:::) (;::)

2.0 ,

(!::) (0.6)

(:::) (:::)

(:::) (:::)

Value of Log Likelihood Function: -760.8
.

The

estimate

pages to

here.

77

Appendix C

FORTRAN Programs for Dyna~ic Mod&l’

inclusion of the actual FORW programs used to

the parameters of our dynamic would add approximately 40

this report. For that reason, we do noti’%nclude them

However, those programs can be obtained by sending a blank,

formatted IBM-compatible diskette to:

Professor George Jakubson

Ives Hall

1~-Cornell

Cornell University

Ithaca, New York 14853-3901

The programs can also be obtained electronically by sending email to AK5J at CORNELLA.BITNET.

.
Bellman, R.,

78

Bibliography

~ minq, Princeton, NJ: Princeton

University Press, 1“957.

.’

Berkovec, James and Steven Stern, ‘rJobEx”itBehavi6r of Older

Men,l’ Jefferson Center for Political Economy Discussion

Paper #169, University of Virginia, February, 1988.

Eckstein, Zvi and Kenneth I. Wolpin, #rTheSpecification and

Estimation of Dynamic Stochastic Discrete choice :<odels,at

Journ al of Euman Resources, Fall, 1989, p.562-598.

Gonul, Fusun, ‘lDynamic Labor Force Participation Decisions by

Males in the Presence of Layoff and Uncertain Job Offers,”

Journal of Human Resources, Spring, 1989, p.195-220.

Hotz , V. Joseph and Robert A. Miller, llAnEmpirical Analysis Of

Life Cycle Fertility and Female Labor Supply,” Econometrics,

January, 199S, p.91-118.

Hutchens~ Jakubson and Schwartz, Preliminary Final Report on

Grant #E-9-J-8-0090, Au~st, 1990a.

Hutchens, Jakubson and Schwartz, Final Report to the Department

of Health and Human Services, 1990b

Johnson, William R and Jonathan Skinner, “Labor Supply and

Marital Separation,” American EcOnom ic Review, June, 1986

Killingsworth, Mark, Labor SUDDIY, Cambridge: Cambridge

University Press, 1983.

Maddala, G.S., ~m. ~ e - e ende t

~, Cambridge: Cambridge University Press, 1983.

McElroy, Marjorie, l!~e Joint Determination of Household

Membership and Market Work: The Case of Young Men,” Journal

of Labor Economics, 1985

Muthen, Bengt O., LIS ~ OMP: Ana ine r Structural

E~,

Mooresville, IN: Scientific Software, 1988.

Schmidt, Peter, ~aconstraints on the Parameters in simultaneous

Tobit and Probit Models, e,in Manski, Charles and Daniel

McFadden, Strut ~i ur Ana isc ete Data: W th

~ometric ADDlications. Cambridge, MA: MIT Press, 1982.

Wolpin, Kenneti I., ,v~ Estimable Dynamic stochastic Model Of

Fertility and Child Mortality,” ~Journa

~conomv, 1984, P.852-874.
79

Endnotes

1. In work reported in Appendix B, we show the results of our

estimation of the parameters of a six state moael.

2. There are two general methods for solving a dynamic

programming problem. When the control variable is continuous

and there are no censoring or truncation issues, it is

straightforward to make use of Bellmanes E~ation to solve the

problem. There are two sets of first order conditions to

maximize the value function. The first set essentially mean

that there are no within period possibilities for utility

increasing reallocations. These are the same first order

conditions @at arise in a static problem. The second, the

‘-Euler E~ations,S~ mean that there are no between period

possibilities for utility increasing reallocations, that is,

no arbitrage possibilities across time periods.

The.model is typically closed with a rational e~ectations

assumption, so that the forecast errors (for the next period

state variables) have zero mean. This allows one to combine

the two sets of first order conditions’ to define a set of

e~at~ons for the forecast errors. Since these have zero mean

(by assumption), this provides a tractable method for

specifying an estimating model. TBere are many examples of

this approach in the literature.

Alternatively, under some conditions the value function

defines a contraction mapping. In these cases one can

literally compute the value function by iterating the

contraction mapping to convergence. The contraction can then

be built into the computation of the likelihood of a given

sample.

Unfortunately neither of these two approaches are available to

us, because the choices with which we are dealing (e.g., the

marriage choice) are intrinsically discrete. Because of the

discreteness, tie value function is not differentiable with

respect to the choice variable, ruling out the Euler E~ation

approach. And our environment is not stationary because, for

example, the number of children varies over time, so we cannot

make use of the contraction mapping approach.

3. An important issue here is the choice of terminal date T. In

a stationary environment this is not as difficult, so long as

there is discounting. As the terminal date T is moved farther

into tie future, the contribution to current period value of

the expected future events grows smaller and smaller. By
pushing T far enough away, any errors made by ignoring time

periods later than T become negligible. One can then

implement the backwards recursion, since the stationarity

assumption implies that the same decision making environment

exists at all time periods.

If the environment is not stationary the problem is harder.

For example, if there are exogenous variables which affect

ztility which change over time in a manner which is not

completely predictable, then one cannot simulate the decision

making environment in periods for which one does not have

data. Therefore, the terminal date T cannot be pushed

arbitrarily far into the future, but rather must be the latest

date for which data are available.

4. men we observe a woman who already has a child in 1979! the

first year of the data, we cannot determine her ‘finztial

conditions because we do not have the data. We therefore use

1979 as the initial condition for these women and use her

decisions from 1980 to 1985 in the estimation.

In the context of our model, there is no harm in doing this.

The disturbances in the model are independent and identically

distributed across women, choicest and tine periods. The

previous period state variables, D(t-1), and the current

period exogenous variables, X(t).r characterize the decision

making environment, so that while we do not see all tbe

choices made by a woman who had a child with her in 1979,

those which we do see are made in the same way as those for

the women for whom we observe the first appearance of a child.

5. Thus any woman who was not interviewed in each year is

deleted. In principle, it is possible to deal with ‘eholes~’in

the record by integrating over alz the possibilities in the

missing year(s) . The probability weighted values would then

be used in the calculations for future years.

is,

This approach

unfortunately, computationally intractable.

6. To maximi~:e the likelihood function in e~ation (9], we

utilize hill-climbing routine GQOPT (written and

maintained by Professor Richard Quandt, Department of

Economics, Princeton University) because of its flexibility.

Within the pactige are a number of different algorithms:

Davidson-Fletcher Powell (DFP), ~adratic hill-climbing

(G~DX), a simplex search, a conjugate gradient method, and

others. This flexibility is important because the log

likelihood function is difficult to maximize. We found it

necessary to start from many different places to ensure that

we found the values of the parameters for which the function

attains its maximum. Different algorithms performed well or

poorly in different regions of the parameter space. The
81

.

<

7.

8.

FORT- code we use to maximize the likelihood function

appears in an appendix.

When we free up that parameter, we have serious convergence

problems. A grid search over the concentrated log likelihood

function (concentrating out all the’ other parameters) shows

that the likelihood value is very insensitive to the value of

the discount factor. In the results below, then, the discount

factor has been fixed at 0.9.

Strictly speaking, our classifications here are not identical

to the Census definition of ‘tsubfamilyrsas it appears in the

Current Population Survey. Using the N=-Y, we have no

uniformly reliable way to know who owns the dwelling unit or

whose name is on the lease. Hence, unlike those working with

the CPS, we cannot distinguish between the following two

situations:

1. Respondent and her children live

2. Respondents parents live in the

the respondent and her children.

In the first case, the respondent is a

in her parent’s hone.

respondents ‘home with

subfamily head. In the

second case, the respondents parents form the subfamily.

Since our sample is young, we suspect that the vast majority

of shared Living arrangements are of the first type and we

therefore use the ted ‘#subfamilys*head.

*’