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Simultaneous equations for hazards
Journal
of Econometrics
56 (1993) 1899217. North-Holland
Simultaneous Marriage
duration
equations
for hazards
and fertility
timing*
Lee A. Lillard
The concept of simultaneous equations has been a fundamental contribution of econometrics. While the concepts have been extended quite fruitfully to limited and qualitative dependent variable models. they have been rarely applied to hazard or duration (failure time) models. This paper develops an approach to simultaneity among hazard equations which is similar in spirit to simultaneous Tobit models. It introduces a class of continuous time models which incorporates two forms of simultaneity across related processes when the hazard rate of one process depends (I) on the hazard rate of another process or (2) on the actual current state of or prior outcomes of a related multi-episode process. This paper also develops an approach to modeling the notion of ‘multiple clocks‘ in which one process may depend on the duration of a related process, in addition to its own. Maximum likelihood estimation is proposed based on specific parametric assumptions. The model is developed in the context of and emptrically applied to the joint determination of marital duration and timing of marital conceptions.
1. Introduction While there is now a substantial econometric literature concerned with the specification and estimation of simultaneous equation models involving qualitative and limited dependent variables,’ there has been limited consideration of simultaneity in ‘failure time’, or hazard, processes. This paper is concerned with two other forms of simultaneity among repeated dynamic discrete choices; that is, with simultaneity generated when the hazard rate of one multi-episode process depends on (f) the (log) hazard rate or (2) the actual preceding outcomes of a related multi-episode process.
C~,r~~.\portt/u,l~~ to: Lee A. Ltllard, The RAND Corporation. 90407~2138. USA.
1700 Main Street. Santa
Monica,
CA
*Thts research was supported by NICHD. Center for Population Research. Grant No. P-50-12639. I want to thank Constantijn Panis for exceptional skill in developing the software for the model and Mary Layne for careful preparation of the complex marital and fertility histories. ‘See Maddala
J.Econ
G
(1983) for numerous
examples.
Section 2 introduces the basic elements and assumptions of the model in the context of a particular application which exhibits the essential features of the approach. The application is concerned with the joint determination of marital duration and the timing of marital births (conception dates). Section 3 develops the conditional (on heterogeneity) survivor functions and duration density functions, develops the conditional and marginal probabilities of sequences of multi-episode events. and presents the marginal likelihood function for various types of data: censored and complete durations in the joint sequences of multi-episode events. Section 3 introduces the data, presents the estimates. and discusses results for the application.
2. The model Our purpose is to develop a model of related dynamic discrete choices (multi-dimensional hazard processes) incorporating forms of simultaneity which conform naturally to intuitive notions in classical simultaneous equations and to generalizations related to qualitative and limited dependent variable models. These notions are developed in the context of a particular application but illustrate the features of the model. The application jointly considers the dynamics of marriage durations and marital fertility, i.e., the timing of marital conceptions. The model of the application may be characterized as follows.’ An individual (woman in this case) may be observed over the duration of one or more marriages: and within the marriage(s) the dates of marital conceptions are known. From the beginning of a marriage the individual (couple) is at risk of splitting up (dissolution of the marriage) and at risk of conceiving a child. The hazard of dissolution is influenced by a number of time-related factors including calendar time. the woman’s age, and the duration of the marriage. by a set of exogenous covariates, and by a set of endogenous covariates the presence of children from the current and/or previous marriages ~ determined by a potentially correlated process (fertility). Issues include the extent to which children born to a marriage reduce the hazard of dissolution and the extent to which children from previous marriages increase the hazard of dissolution. As long as the marriage continues the individual (couple) is at risk to conceive a(nothcr) child. The hazard of conception is influenced by the same set of time-related factors, plus the duration since the last birth (omitting the interval of infertility during pregnancy). a set of exogenous covariates. and by the endogenous hazard of the marriage breaking up. Issues include the extent
‘The thcorctxal arguments underlying this hpecilication derive from Becker, Landca. and Michael (I 977) and is discussed more fully in Lillard and Waite (I 992). The onl! other attempt to cmplrical model mnrltal duration and fertility jointly IS Koo and Jano~~tr (1’383).
to which individuals in marriages with a higher risk of dissolution have reduced fertility (conception hazard). Consider the following log hazard equations which characterize these processes” - first for dissolution of marriage m (m = 1, . , M) and then for conception ,j (,j = 1, , J,) in marriage IH: In /r:,(t) = x0 + xi T(t) + x;A(t) + lnhkj(t)
=
+ x;M,(r)
4Xd(f) + &K,(t)
Po + /j; T(t) + + /J;X’r)
+
(1)
6.
/l>A(t) + /3;M,(t) +
+ i,lnh~(r)
PkD,j(f)
(2)
+ v.
The terms T(t), A(t), M,,(t), and DMj(t) represent the separate ‘clocks’ of time/duration dependence of the hazards which determine the baseline hazard; Xd(t) and X’(t) are vectors of exogenous covariates; K,(t) is a vector of endogenous prior outcomes of the fertility process which affect the hazard of marital dissolution; In hi(t) is the endogenous hazard of marital disruption which contemporaneously affects the fertility hazard; and E and q are heterogeneity components such that
These log hazard equations are defined for appropriate ranges of time: for marriage durations from the time the marriage begins, denoted rd,,, until it ends either at the time of dissolution (separation or divorce), denoted t&,, or interval duration from of censoring, denoted t&,; and for the jth conception either the time of marriage (for first birth intervals with no pre-marital conception) or the time of birth of the last, (j - I)th, child, denoted r~,j, to either the conception of the jth child, denoted t~,j, or the time of censoring, denoted r:,,,;. 2.1. Titne/ciuwtion
depmdenc~e - Multiple
clocks:
T(t), A(t), M(t), D(t)
This model is essentially one of proportional hazards, with exogenous and endogenous covariates shifting the baseline hazard. The baseline log hazard is
“Subscripts convenience.
denoting
the mdividual
for whom the process
applies are suppressed
for notational
the sum of the effects of the various sources of time dependence. Each source of duration dependence is represented by a piecewise linear (on the log hazard) spline function of time.’ The terms rJI T(t) and /Y, 7’(t) represent dependence of the respective hazards on calendar time” via piecewise linear splines, i.e.. r(t) is a vector of N7. + I spline variables which sum to calendar time but which allow the slope coefficients to differ within ranges of time separated by i\ir nodes” /L,_.The spline variable for the hth interval between /lr , and j(I is given by
Therefore,
for example.
The vectors of spline variables for age A(t). for marriage duration M,,(t), and for conception episode duration D,j are similarly defined, but with their own spline nodes and slope coetticients. Note that the calendar time and age ‘clocks’ arc not dependent on which marriage or birth interval is being considered. Rl~r~itr
‘This specllication is simllnr an spirit IO other llexible forma for the bn~cl~nc hazard. See for example Han and Hauhman (IYYO). Earlier in\eatigation led to a clear reJection uf the Wclbull and other widely used parametric forms. Additionally. those forms do not allow the slmplc closed form slmultancous equations models proposed here. ‘Since January
lY5S
111the apphcatmn
“No&s need not be the same m the scparatc equations. but the same nodes have been used hcrc Ibr simphcity. Implicitly. !l,, = 0 and or,, _ , = -r This Rex~blc functional form includes the hnear tlmc trend as a ~pccial cast r’, 7;lr) = xl I if x,~ = 2,. for all L. The result IS a Gompert7 form of the hazard. ‘Including
onI> the time dcpendcncc
portion.
and omitting
?(,,(I) and K,,,(t)
193
marriage
A(fd0,).8 That
is,
In h&Jr&,) = r0 + 2; T(ti,)
+ C&A(&),
which incorporates the cumulative effects of the passage of calendar time and age up until the time of marriage. Once a marriage has begun. the time dependence is the combined effect of the further passage of time 7(t), of aging A(t). and of longevity of the marriage M(t). The resulting time dependence,
+
4(A(r) - A(fdo,))+ r;M,(t),
is also piecewise linear in time since the episode began, but with nodes at the combined locations of the nodes of the three splines and coefficients between nodes which are the sums of coefficients of the three splines. The baseline hazard is thus piecewise linear Gompertz. Conceptions. Within a conception episode of the fertility process, say the jth, the duration of the episode, the duration of the marriage, and the age of the woman all change perfectly collinearly with calendar time. Again, their separate effects can be distinguished because their initial values at the beginning of the episode differ; and they also differ between conception episodes even within a marriage. Except for the first conception episode which usually begins at the beginning of the marriage, other conception intervals begin at different points in time, at different ages of the woman, and at different marriage durations. During the first conception interval at the beginning of a marriage,” the marriage duration coincides with the duration of the conception interval. However, the two effects are separately identified since first conception intervals end at various durations (some very short) and the effect of the marital duration ‘clock’ represented by M(t) is invariant to which conception interval is being considered. Since we are concerned with marital fertility, marriage duration M(t) plays a particularly relevant role in that the marriage must survive in order for the marital fertility to occur. That is, the end of the marriage represents potentially nonrandom censoring of conception intervals.
“Initial
marriage
duration
is obviously
zero.
‘Some women are pregnant when they marry. The next conception interval is considered a second and begins at the birth of the pre-maritally conceived child; and the premarital conception episode is not included in the model.
The intercept of the ‘baseline’ log hazard equation for the ,jth conception episode. in the IpIth marriage beginning at time tb,,,,. is 21 function of the calendar time the episode begins, the age of the woman, the duration of the marriage, and the log hazard of divorce when the conception episode begins. For the conception of the ,jth child in the r,lth marriage, the intercept is given by
Once a conception cpisodc has begun. the time dependence is the combined effect of the further passage of time T(t). of aging ,4(t). of longevity of the marriage ,%f(1). and the duration of the conception interval. The resulting time dependence.
is also piecewise linear in time since the episode began, but with nodes at the combined locations of the nodes of the four splines and coefficients between nodes which are the sums of coefkients of the four splines. The baseline conception hazard is thus also piecewise linear Gompertz. The terms D,,,,(r) represent the dependence of the hazard of conception on the duration of the conception episode at time t via piecewise linear splines.‘”
-7.3. I Efftav
o/ tlw c’i/r/‘tg/lt .vttitc, or pio~ outcot1w.~: K,,,(t)
Some of the explanatory variables in a hazard equation may be jointly determined and potentially correlated with the residual or heterogeneity component of the log hazard equation. In this application the vector K,,(t) includes the potentially endogenous set of time-varying variables representing the prior outcomes of the joint process’ ’ indicator variables for the presence
of children born to this and prior marriages” may not be unrelated to the unobserved component of the hazard of marital dissolution. That is, if the behavioral processes are correlated, i.e., po,, # 0, then K,(t) will not be independent of the residual term c. This specification of the effects of the prior outcomes of related processes is similar in spirit to the specification of the effects of outcomes of prior episodes of the same process ~ past durations, number of past occurrences, the current state in multi-state processes, and so forth.13 Conditional on the residual heterogeneity components, the observed completed durations and outcomes are independent. Thus, no further identifying restrictions (e.g., exclusion restrictions on exogenous covariates) are formally required. However, such restrictions may be desirable. -7.22.
Efjk~ts of'
N
sdoted log hrrzusd ruts: In hi(t)
The risk or hazard rate associated with one process may have a direct effect on the occurrence and timing of another event, i.e., on the hazard rate of another process. The explanatory hazard rate is a latent variable and not directly observable. l4 In this application the term In /r:(t) represents the risk of (potential) marital dissolution of marriage 111which may influence the decision to bear children in that marriage, i.e., the hazard of a first and subsequent conceptions j within marriage tn. Women in higher-risk marriages may reduce the risk of conception, i < 0. The latent log hazard of marital dissolution may be substituted into the equation for the log hazard of conception to obtain the ‘reduced form’ conception hazard, In hi,(t) = /&, + +
i.sco + (pi
+ ix;)T(t)
(0; + j.r;)M,(t) +
+ i.r\Xd(f)
p;D,j(t)
+ k&Km(t)
analyses
have been proposed
(1980) and Flinn and Heckman
for labor (I 982).
+
+ q+
“In this application we have not considered children be endogenous. This is a subject for further research. “Related and Borjaa
+ (/I; + i.a;)A(t) /l\Xc(f)
is.
(3
from prior extra-marital
force dynamics.
relationships
See for example
to
Heckman
“Analogous to latent variables in qualitative and limited dependent variable models. e.g. stmultaneous probit or Tobit models. Our treatment of the latent variable is similar to the treatment in these models. See Maddala (1983). While specification of the model in this application is structurally triangular in the log hazards, in general they may be fully simultaneous. As discussed below, the usual exclusion conditions for identification suffice, even though the hazard rates are latent variables.
The hazard of conception of its own past outcomes.
then becomes ;I function in this cast the presence
of K,,,(t). the occurrences of childrenI
regressors. The The log haLards are also a function of other L’XO~~~IIOIIS vectors X”(t) and X’(r) represent the time-varying (or constant) exogenous covariates affecting the hazards of marital disruption and conception, rcspectively. Exclusion restrictions on these covari;\tes, X’(t) relative to M”(r 1. may form the basis of identification of i, the effect of In hi,(t) on In h;,;(r). These issues are discussed in the section on identification following the discussion of estimation.
The unobserved residual components of the hazard equations. i: and !I. ma! be correlated. We assume joint normality of these residual terms in the log hazard equations. That is.
Even if the residual heterogeneity terms of the haLard processes are not correlated. a structural effect of one hazard on another, 2. will introduce correlation in the proccsscs. The reduced form residuals >:and r/ + ii: are also jointly normally distributed.
2.5. ~‘ldifiOf,l/~
JYYhl~ld fiJf'J,l /lti:~lYl/S
The hazard equations for marital disruption respectively. arc given by the following:
and for marital
conceptions,
L. A. Lillurrl. Sirmdrrrr~rouseyuu~iotts h~j(t, X(t).
tl + AC) =
exp(P,
+ i.X” + (/I; + k;)r(t)
+ (pi + i.zc;)M,(t) + (/I;
where X(r) includes
/iw l~u~urd~
+ ia;)X(t)
all the covariates
197
+ (/I; + kZZ;)A(t)
+ /IkD,j(t) + &K,(t)
+ q + k),
(7)
in Xd(t) and X’(r).
3. Estimation
For any given observation, conditional on measured covariates and on the heterogeneity components E and ‘I, the outcomes of each marriage and conception process are independent, and the joint probability of any sequence or combination of outcomes is simply the product of the probabilities of the individual outcomes. Denote the baseline survivor functions for each marriage and conception episode by
~, exp(r, 10,,,
+ 4 T(r) + &A(r)
, (8)
+ a;M,(r))dr
I
S;;“,,(t)
=
exp
(i -
~ exp(j& + i.ro + (/I; + E.a;)T(r) f,,,,,,
+ (/I; f j.J;)A(T)
+ (fl; + j-CC;)M,(T) + /jkD,j(T))dT
1
(9)
respectively, for t > tt, (the date the marriage began) and for t > tfo,j (the date the jth conception episode began, i.e., the date of the last birth or the beginning of the marriage for the first conception). One of the attractive features of the piecewise linear spline formulation is that the baseline survivor functions have easily computable closed forms. Following this notation the ‘conditional’ survivor functions” for each marriage and conception episode may be defined as follows, where x(t) denotes the full history of Xd and Xc covariates from the beginning of the episode. The integrated hazard is subdivided into ranges of time within which the time-varying covariates are constant. The following survivor functions
‘“Conditional
on the value of the heterogeneity
components
E and q
then follow from the application of the proportionality of the effects of covariates and the heterogeneity components. Conception equations are written in reduced form as a function of structural parameters. For marital durations the probability that marriage 111will last at least to time I. duration I ~ r$,,, conditional on the sequence of covariates Xd(t) up to that time [represented by z(r)] and on the unobserved residual component x. is given by the conditional survivor function
where I,,, is the number of subintervals within which covariates Xd(r) arc constant, and tf,,s>z. , = I. This is also the conditional (on C) likelihood of a marital duration censored at the time of the final survey. i.e., t = t,d,,,.The conditional likelihood that marriage ITI will end at duration r:,,, is given by
./:!(f:f,,,.z(L).
i:) =
S:I,(r;,,,.%(r:f,,,), 4 /I%!,,,. .Y“(t:1,),). I:).
(11)
which is also the conditional density function for marital durations. For birth intervals. the probability that the ,jth conception will not have occurred by time t. conditional on the sequence of covariates up to that time. and the unobserved residual components i: and rf. is
where I,,lj is the number of subintervals within which covariates X’(t) are constant. and td,,,,,,,,+, = t. This is also the conditional (on rl + if:) likelihood of the Jth censored birth interval at the final survey, t,‘,,,,(t), or at the time of marital dissolution. tk(t:,,). The conditional likelihood that the ,jth birth interval of marriage 1~1 will be of completed duration tz,,j is given by
X
h~,j(t::“,j,
X(t:“,j),
Jl +
j.i:),
(13)
which is also the conditional density function of conception interval durations. For each marriage the final conception episode (which might bc the first) is always in progress (censored) at the end of the marriage or the final survey date if the marriage is still in progress. The probability of the sequence of
199
J,,, observed
conception
QL(f*, I,
episodes
within
the /nth marriage
is given by
r\ + ;A) = S>,,,(t*. z(r*), 11+ ;.a) n rmj((t&j, x(f&j), v + ;J), i= I
(14)
where t* = r:,,, in the case of a completed the case of a final survey date at t,.
marriage
at time t&, and t* = t, in
The marginal likelihood is obtained by combining (multiplying) the conditional probabilities of any particular sequence of marital and conception episode outcomes and integrating the resulting conditional likelihood over the range of the unobserved heterogeneity components t: and ‘I. There are two cases depending on whether the last Mth marriage is in progress (censored) as of the final period of observation. If the last Mth marriage has ended, the marginal likelihood (joint marginal density function for marriages and conceptions) is given by
and if the last marriage
is in progress,
the probability
of the observed
closed
t ix)
Parameters are estimated by maximum likelihood (FIML). While the marginal likelihood equations above are specified for each type of observation, the aggregate likelihood equation is the product of the individual likelihood functions over all observations. Empirically we use analytic derivatives and scoring methods for second derivatives.”
“The bicariate normal integration is performed using multi-dimensional Hermite Quadrature methuds suggested by statistician Lionel Galway. Also see Naylor and Smith (1982). The software used in this application. designed by the author and implemented by Constantijn Panis, is more general than this application. allowmg (I) the effect of a duration dependence to begin at any time. including after the episode has begun. and (2) allowing larger numbers of jointly determined hazard processes.
3.3. ltkPl7tific~rrtiotr
If the sources of correlation in the processes are not present (L = 0 and I7i,l = O), then the parameters of the two processes may be estimated separately or jointly with the same result. The marginal likelihood function separates into the product of the separate marginal likelihood functions for the two processes. The potentially cndogenous variable K,,,(t) is effectively exogenous. However, the multiple marriage episodes must be considered together. and the multiple conception episodes must be considered togcthcr, to account for heterogeneity and dependence of the log hazard on prior own process outcomes. If. however, tither is nonzcro. then the paramctcrs of the two processes must be estimated jointly.” If i = 0 but o,,, # 0. than the puramctcrs may be estimated without further restriction using the joint process marginal likelihood function. Integration over the joint distribution of heterogeneity components ‘accounts for’ the correlation in the processes. If i # 0. then some indentifying restriction is required and the processes should be considered jointly. We may jointly cstimatc the parameters of the log haLard of marital dissolution equation. the parameters of the reduced form log hazard of conception equation. and the reduced form variances and correlation of the hcterogcneity distribution. If oIII = 0, then the system is recursive; the parameter i is idcntiiied from the covariance or correlation between reduced form residuals (being the only sourcc of correlation). If I),,, # 0, then i must bc rccovcrcd from the reduced form parameters of the conception log hnrard equation. One possibility is through the effect(s) of prior conception outcomes K,,,(t) a in the reduced form equation presented above. However. in some models there may not be any variables K,,,(t) (Y;, = 0) or the prior outcomes may ha\,c direct effects on the conception hazard. In the application considered here. the number of children in the marriage K,,>(t) is the same as the birth order (parity) of the conception interval which is expected to have ;I direct effect on the probability of another conception and its timing. Assuming that there are direct effects of calendar time. age. and marital duration on the log hazard of conception. restrictions on the set of exogenous covariates will sufhce. If there is at least one covariate X”(t) affecting the hazard of marital dissolution but not the conception 10s hazard [not in the vector X’(t)]. the parameter i is estimable from its coet‘ficient in the conception equation.
3.4. Sitnulutiot~
of‘ illustratiw
relutiotwhips
To illustrate the effects of various forms of time dependence and the effects of endogenous and exogenous time-varying covariates, we will present simulations of the time path of aggregate survivor functions and of aggregate hazard functions. These calculations are performed as follows. First the ‘baseline’ survivor function is computed conditional on the initial values of calendar time and age (and marital duration in the case of conception episodes). Then the aggregate survivor function is computed conditional on the time pattern of covariates and integrated over the relevant heterogeneity components. Finally, the hazard aggregate function is computed as the relative decline of the aggregate survivor function.”
4. Data,
estimates, and results for the application
4.1. Duta
This application is based on longitudinal data on marriages and births constructed from the the Panel Study of Income Dynamics (PSID) which began in 1968 with about 5,500 households. The PSID is well-suited to analysis of the relationship between fertility and marital disruption. The sample has been resurveyed each year since that time, generating panel data on marital status and fertility. Each sample member is followed whenever a primary couple (husband, wife) separates or children and other family members exit the household.” The inclusion of the children’s marital histories guarantees a wide range of marriage year and marriage age cohorts; the long panel and retrospective period means that multiple marriages and conception episodes are observed. The PSID contains longitudinal information on both marital status and fertility, allowing us to trace the experiences of individual couples over a substantial period. The sample of married couples is large, with a sizable number of disruptions and remarriages during the period of observation. Information was obtained on marital status at each interview, minimizing recall bias, together with a complete retrospective marital and fertility history in 1985. The availability of the two types of information allows us to date marriages, separations, divorces, and births with a good deal of precision.”
‘“In addition,
some points
will be illustrated
using simple plots of the log hazard
‘“The new households that they form are added interviewed in the same way as original households.
to the sample
‘ISee Lillard and marital histories.
of the retrospective
Waite
(1988) for a comparison
as new sample and
panel
equation. units reports
and of
The sample used for this analysis is comprised of 2.376 female respondents. of which 84.3 percent were married only once during the period of observation, 13.9 percent were married twice. and 1.7 percent were married a third represented in all. See table 1. The time. There are thus 2,789 marriages sample used in this analysis is restricted to respondents who responded to the 1985 retrospective marital and fertility history, to allow us to date the events of interest precisely and to include information on marriages and fertility that began and/or ended prior to the first interview in 196X. We consider all marriages of female respondents that began in 1955 or later. The data includes the begin date of the marriage, and either the end date of the marriage (if the couple divorces or separates before the 1985 survey). or the date of censoring of the marriage by attrition. death of the spouses, or the 1985 survey. We observe the dissolution of 34 percent of first marriages. 12 percent of second marriages, and 34 percent of the third marriages in our sample. See table 3. The remaining couples. not divorced within the sample. represent censored observations on marital duration because we do not observe the dissolution of the marriage and know only that. if it occurred. it happened after the last survey period. For each marriage of each respondent we use the 1985 retrospective fertility nine months prior to the reported birth date. history to date conceptions There are up to nine conception episodes, eight births. per respondent. The distribution of conception episodes in all marriages is presented in table 3.
203
Table Conception Number of conception episodes
Number
Percent
639 604 607 317 124 41 30 13 I
6.9 25.4 25.5 13.3 5.2 1.7 1.3 0.5 0.0
2,376
100.0
I 2 3 4 5 6 7 8 9 Total
3 marriages.
The variables used in the model are defined in table 4, which also gives the specification of the dissolution and conception log hazard equations. The measures of past marital fertility outcomes K,(t) include (1) whether a child has been born to the marriage, K-lst, and the number of children beyond the first” born to the couple, K-2nd + The coefficient is then the marginal effect of another child on the log hazard of disruption. We also include with a separate effect the number of children born to the woman in previous marriages, which are included as own children in the previous marriage episode. Children conceived by the woman out-of-wedlock, K-llleg, are included but considered exogenous in this application. 23 This includes any premarital conception leading to the birth of a child in the current marriage, K-Premr. We include an indicator for second or later marriage in both the marriage and conception hazard equations. This represents ‘own process occurrence dependence’ for the marriage disruption process and ‘related process occurrence dependence’ for the conception process. Another form of ‘own process occurrence dependence’ is the inclusion of indicator variables for birth order on the current conception episode ~ again representing the marginal effect on the the log conception hazard of reaching a higher birth order. The sources of
“‘Further significant. “The research
decomposition
of the number
endogeneity of extramarital and not considered here.
beyond
conceptions,
the first versus later born was not statistically and of age at marriage,
is a subject
for further
704
Table Variables
Variable
Descr~pt~vn
affecting various
4 model components.,’ Dissolution In /l;,(r 1
x x
x
x
duration dependence or ‘multiple clocks’ discussed in the context of the model specification are of course included. The model includes a number of demographic variables that previous race, religion, and research has found to influence risk of marital disruption education. In addition. the model includes a number of exogenous variables
that should affect risk of divorce or separation directly, without affecting the risk of conception. These include measures of the legal ease, SD12, or difficulty, SD.56, of divorce in the state of residence in the current year. an index of the cost of lawyers. MSTN, the aggregate per capita divorce rate in the state in the current year, MNDR.14 4.2. E.~timttc.r mtl effkcts of’ .spc~fjcutio~t Table 5 presents estimates of the model of marital dissolution, table 6 presents estimates of the model of conception timing, and table 7 presents estimates of the variances and correlation of the heterogeneity components. In each case, column 1 presents estimates of the mode1 assuming no correlation between the hazards. f),,, = 0, and no effect of the marital dissolution hazard on the hazard of conception. i = 0. Column 2 presents estimates assuming 1. = 0 but allowing for correlation between the heterogeneity components of the two processes, /Jo,,. Finally, column 3 presents estimates allowing both sources of relationship between the processes. First, both heterogeneity components 1: and rl exhibit significant variation, and the correlation between them is highly significant. Second, the effect of the hazard of marital disruption on the conception hazard, j.. is also significant.‘” Couples in marriages with a higher hazard of dissolution exhibit significantly delayed childbearing and smaller completed family size (reduced hazard of conception). Second. consider the effect of these coefficient restrictions on the parameter estimates. In the marriage dissolution equation, allowing for the correlation between processes significantly affects the estimates of the effects of children on the hazard of marital dissolution. A significant detrimental effect of children from previous marriages on the hazard of disruption is revealed, as is a similar effect of second- and high-order children in the current marriage. In the conception hazard equation, several effects are evident of accounting for the endogenous effect of the hazard of dissolution. Most notably, the previously signihcant direct effects of the following variables are eliminated: (I) birth order beyond the third child, (2) children from previous marriages, (3) illegitimate children, (4) second marriage. and (5) being Catholic. The effects
“And a variable indtcating whether the aggregate missing in less than 2 percent of marriage years.
divorce
rate is missing. The divorce
rate was
“While the r-statistic is only marginally signilicant. the likelihood ratio statistic indicates greater significance. zf = 14.2. Similarly. while the Identifying effects of divorce laws, lawyer costs, and aggregate divorce rates have only marginal r-statistics when i is a free parameter, the joint likehhood ratio ;(: statistic is highly significant. These differences in test satistics are largely due to collinearity among the parameter estimates (and among the state-year divorce climate covariates).
free i free
I’, t, = 0
I’,,,
; = 0
Md-I
Md
-3
:
5
Md ~ I(1 +
0.7705. (0.3788)
0.763h” (0.2795)
0.043x (O.OXX2)
0.057x (0.0X89)
0.0209 (0.076 I 1
0.0143 (0.07671
0.009x (0.0329)
~ (1.0I74 (O.Oi.ihl
0.0027 (0.020 I )
- 0.0226 (0.020X)
(o.ox42l
0.0740 (0.0X5 I )
0.0753” l0.01671
O.OX?h~’ (0.0I 70)
0.0803 MA
‘0 ~ ill
MA
30 ~ 40
MA ~ 40 +
1-d ~ x0 t
~~0.04X6~’ (0.01 78)
~~o,0437h (O.OlXl)
0.06X3’
0.0800” (O.(E!43I
(0.0243)
0.0330 (0.15431
0.0730 lO.1556)
0.0479 (0.0750)
0.0326
0. I I69’, (0.050 I )
0.12X2” (0.0504)
0.00I 3 (00340)
O.OI66 (0.0348)
0.070(~ (0.0270)
0.07’) 2,’ (0.0271)
O.OX3.5~’ (0.0297)
0.0x5 I” (0.030 I )
207
rable
5 (continued) p,, free i = 0
0: ,, =O / = 0
K ~ Mar
II,, free i free
0.181 I” (0.0426)
0.2510” (0.0474)
0.26X7” (0.0484)
0. I 2I 7 (0.07X9)
0.3226” (0.0X8X)
0.3428“ (0.0896)
K ~ Premr
-- 0.1650’ (0.0864)
~ 0.0835 (0.0908)
~ 0.0840 (0.0906)
K ~ 1st
~ 0.5578” (0.0996)
- 0.3342 (0. I 102)
~ 0.3227 (0.1094)
0.269 I” (0.0716)
0.3060 (0.0699)
K ~ 2nd
- 0.0065 (0.046 I
+
Jewish
0.0272 (0.1045)
- 0.9653’ (0.5043)
)
~ 0.0766 (0.1085)
- 0.0282 (0.1062)
-
~ 0.9618’ (0.5228)
l.0291h (0.5188)
Edrll2
0.0050 (0.1035)
- 0.0253 (0.1090)
~ 0.0605 (0.1082)
Edrl6
~ 0.371X” (0.1378)
- 0.3340b (0.1447)
~ 0.3189h (0.1444)
Black
0.39’7.’ (0.08x;)
0.3624’ (0.0918)
0.3352” (0.0906)
Mar7p
0.0706 (0.2280)
- 0.2622 (0.2365)
~ 0.2618 (0.2383)
0.605 I’ (0.2299)
0.6592” (0.2383)
~ 0.1213 (0.1044)
Mndr
0.0894” (0.0320)
0.0898” (0.0335)
0.0235 (0.0168)
Mmndr
0.617”’ (0.2505)
0.6182” (0.2559)
0.3806’ (0.21 IO)
Sdl2
0.0152 (0.1086)
0.03 I6 (0.1108)
~ 0.0970 (0.0596)
Sd56
~ 0.0029 (0.0942)
0.0125 (0.0969)
- 0.0754 (0.0528)
‘Significant hSignificant ‘Significant
at the I% level. at the 5% level. at the 10% level
free
,I,,, I
0.0
=
0
0.0
lO.XY3’l
~ 0.331’) 10.23571 KBt
0
KU7
0
KHZ
2
KHZ-
5i
t
~ 0.0x54 IO. 1 t 701
0.5
0.5
KB2
MC
2
I
i
~ 0.0693 (0. t I671
~ 0. I 3xX.’ l0.01751
0. 1372.’ (0.0473)
0.60 I 5‘ (0.36301
0.%3X (0.3603)
’ 3’OX. -.. _ (0.430 t I
(0.4303,
2.3393’ (0.43 t I I
0.0927 (0.22Ohl
0.00 79 (02X6)
0.09Y2 (0.2’Ohl
0.1571‘ (0.09041
0. I 60 I L (O.OYO2)
0.1607’ IO.0903 1
0. I xxw (0.03Y’l
0. I xx I ” ((1.0.394t
0. IX 15” (O.OiY5l
0.0x03” (0 oix7t
0.07x I h (0 03x31
2.3652~’
~ 0.1672” (0.054 I I ~ O.I3Xo” (0.0473l
0.5Y63
l0.3605)
0. t 605 (0 207 t )
l.?56Y 10.X4651
0.2_154~ (0.06391
0.397 IL (0.1616)
0.0026
(0.029 I ) IO i
(0.2346)
~ O.I662~’ (0.0542t
0. t 3.37, (0.0404)
RIG
0.0754 (0.1 1671
~ 0.3023
0. I6Y3” lO.05JS I
~~ 0 0X00” (O.O?X9,
?
~ 0.77‘M (0.23 15)
0.02I17 (0.03331
209
Table
6 (continued) free i = 0
/),,i free i. free
- 0.0395 (0.0343)
~ 0.0505 (0.0334)
- 0.1732 (0.1561)
AC - 20 ~ 30
- 0.0163’ (0.0096)
- 0.0160’ (0.0095)
~ 0.1513 (0.0792)
Ac - 30 ~ 40
- 0.1386’ (0.0168)
- 0.1354” (0.0169)
~ 0.7025” (0.0493)
AC ~ 40 +
~ 0.2397” (0.0409)
- 0.236X” (0.0408)
- 0.3450’ (0.0X03)
/‘“, = 0 i = 0
II,,,
.4
14-20
Time-lY55:
Tc - 55 ~ 60
0.0467 (0.0377)
0.0510 (0.0373)
- 0.12 13 (0.2676)
Tc ~ 60 ~ 65
- 0.0826” (0.0208)
- 0.0787” (0.0206)
- 0.0236 (0.1291)
Tc - 65 ~ 70
- 0.0968” (0.0195)
- 0.0962” (0.0193)
0. I67 I (0.1707)
Tc - 70 ~ 75
~ 0.0406b (0.0199)
- 0.0482h (0.0198)
0.0148 (0.0635)
Tc ~ 75 - 08
0.0668” (0.0190)
0.0666” (0.0189)
0.1867” (0.0830)
~ 0.16X0 (0.0299)
- 0.1689” (0.0298)
- 0.0304 (0.093 I )
~ 1.092X” (0.1255)
- 1.0580 (0.1264)
- 1.2067” (0.2030)
~ 1.3160 (0.1512)
- 1.2498” (0.1514)
~ 1.7910” (0.3713)
- 1.4137” (0.0978)
- 1.3645” (0.0969)
~ 0.8942” (0.3004)
Child
- 0.3159” (0.1083)
- 0.2978“ (0.1071)
0.1761 (0.3042)
Fifth + Child
~ 0.1295’ (0.0749)
-0.1158 (0.0744)
0.3664 (0.2941)
- 04138” (0:0420) - 0.5079” (0.0870)
- 0.4038” (0.0416) - 0.5345” (0.0873)
0.0 I44 (0.2478) 0.0060 (0.3485)
Tc - 80 +
Second Third Fourth
Child Child
Kc - Mar
710
rrcc ; rrec
,I>,,
0.200’1~’ (0.0635 I
Fdrll?
0.1507
0.I523
1.3Y32
(O.NO5)
I I .23341
0.13X3”
Ih
((I.OXOO) 0.x-1
I ,’
(0.05YY~ 0.x00
0.0357
)
(O.IX22l
~ O.IiJ3’
0.657’)
(0.07YS)
10.37Y4)
0.25 1 Y,’ I0.06001 0.5323’
(0 1261)
~
0.1233’ (0 065 I
~ 0.1451’
Sib
0. Ii06 (0. I 76X I
lO.lY72l
I0.0646l EdI-
0.10’) I ‘* 10.063 I I
IO.lh5il
0.x l-w (0.22 I 7) ().I554 (0.4545)
0.0075
0.003x
0.0056
IO.02541
(O.O’JOI
(0.02501
0.000x” l0.0002l
0.6OYo” (0. I Y-IS) 0.65 I Y.’ 10.055Yl 0.0
737y3 _..,_,
0.0007~ (0.0001)
4’)
0.771X”
0.7hYa
(0. I700)
(O.I7Oh)
0.6450~’ (0.0554)
0.053’ (0.0523)
0.74X4”
0.7X72,’
(0. IYhO)
IO.19731
13371.26
22364.lb
of these variables operate marital dissolution. 4.3. Discussion
only indirectly
through
their effect on the hazard
of
qf’ results
Estimates of parameters for the preferred model, with both pE,, and i. free, are presented in column 3 of tables 5-7. Results are often discussed in terms of illustrative simulations of aggregate hazard and survivor functions, computed as noted earlier. 4.3. I. Murriuge
chratiotis
First consider the multiple sources of duration dependence. The ‘multiple clocks’ include calendar time, age of the woman, and marital duration. All three show very striking and significant duration dependence. The individual effects of these sources of duration dependence, and of their sum, on the log hazard are illustrated in fig. 1. The top panel shows the effect of calendar time. The second panel shows the effects of aging for two values of initial age (age at marriage), repeated for three values of initial year (year of marriage) so the summation of effects is apparent. The third panel shows the effect of marital duration, repeated for three values of year of marriage. The hazard of marital disruption declines monotonicly with age, especially after age twenty. The direct effect of marital duration is a very large and significant increase over the first year of marriage, followed by a relatively flat pattern thereafter. If the dependence of the hazard on age were ignored, the estimated shape of the marital duration effect would include the age effect. Even if ‘age at marriage’ were controlled, the estimated marriage duration pattern would reflect the effects of aging as the marriage endured. The hazard of marital dissolution shows a marked and significant pattern of increase over calendar time. The final fourth panel of fig. 2 shows the combined effects of calendar time, aging, and marital duration on the log hazard of marital disruption. The aggregate survivor functions and aggregate hazard functions for these six combinations are presented in fig. 2 for comparison. These patterns would represent the apparent effects of marital duration if age and calendar time were not included, and the apparent interaction of age at marriage and calendar time with marital duration. 4.3.2.
Murriuge
hrutions
Next consider the direct marital duration on the log direct effect of calendar time in the top panel along with
and indirect effects of calendar time, aging, and hazard of conception, as illustrated in fig. 3. The P(t), which is a slight positive trend, is presented the total effect, T’(t) + I.Td(t), which includes the
-1
1,
1955
1960
1965
1970
1975
1960
1965
1960
1965
Time (years)
-L
-I
1955
1960
,~
1965
IT
~.
1970
1975
Time (years)
Duration
1955
1960
1965
1970
1975
1960
Time (years)
(5)
-3 (3)
1965
213
1 i ?a I e i .>
c.Fj.6 ‘6
i
e
.6
a
.4 1
-----
T1960
1950
-
1970
i---1980
~~
T
1990
Time (years)
06
1
r\ (5)
.04 ; (3) iI I .02 &ik-
(2) 0 I~, 1950
1960
1970
1980
1990
Time (years) Fig. 2. Aggregate survivor and haLard function3 of marital dissolution for sik sccnarioh. (I I marrted in 19.5.5 at age 20: (2) married in 1955 at age 30; (3) married in 1965 at age 10: 14) marrxd in 1965 at age 30: (5) married in 1975 at age 20: (6) married 1111975 at age 30. First marriage; M hitc protestant. high school graduate. tir\t marruge al mean of other cartable.
strongly offsetting effect of the increasing log hazard of marital dissolution. with the net effect of being a negative trend. The direct effect of aging on log hazard of conception is strongly negative from age fourteen. but is offset below age thirty by the early negative effect of age on the the log hazard of marital dissolution, but then drops dramatically with age after age thirty as the effect of age on marital dissolution declines. The direct effect of marital duration on the log hazard of conception is a dramatic increase over the first five years with a slow decline over the remaining years, but which is substantially offset by the dramatic increase in the hazard of marital disruption over the initial year of marriage. The remaining pattern is the same. but dampened.
-2
-
-T-
--
1955
/1960
-7--
-r-
1965
-r
1970
--
IT
1975
---
1
1990
1985
35
40
45
35
40
25
Time (years)
-5
-6 I
-..
~__. 20
15
.--
._~_ --
25
30 A@
15
20
25
-
-
c
(ye-9
30 AwW=s)
Fig
3. Log
hazard
of conceptw~~:
Direct
and total
tune-relaled
etkctr
215
0
5
10
15
10
15
Time (years)
.6
i
0
5
Time (years) Fig. 4. Hazard of first marital conception: erects of legal environment. (I) low divorce per capita in state-year. lawyers expensive: (2) low divorce per capita in state-year. lawyers inexpensive; lawyers expensive; (4) high divorce per capita m (3) high divorce per capita in state-year, state-year. lawyers inexpensive.
To further illustrate the indirect effects of the hazard of marital dissolution on the hazard of conception, figs. 4 and 5 illustrate the effects of differences in the state-year environment for divorce on the hazard of marital disruption and on the hazard of conception. The hazard of marital disruption is increased when the cost of lawyers is reduced and when aggregate divorces per capita is high. Various combinations are illustrated in fig. 4 for a particular combination of other covariates. The illustrated effects are significant.
‘If!
Time (years)
.04
Iozipm
i
Furthermore, the effects of these variables on the hazard of conception operate only through their effect on the hazard of marital disruption, so that the indirect effect is the total cfrect. This effect on the hazard of conception for the first conception episode in the marriage is illustrated in fig. S.
5. Conclusions
This paper introduces a class of models exhibiting simultaneity among the hazards of related multiple processes, each with ‘proportional’ hazard rates. with multiple episodes of occurrence and with unobserved constant heterogeneity in the hazard rate. Simultaneity between processes includes (I) occurrence dependence on prior outcomes of the related process when the heterogeneity components are correlated across processes and (2) dependence of one hazard rate on a related hazard rate. Maximum likelihood estimation procedures are developed based on the marginal likelihood of observed outcomes, given the assumption of joint normality of log heterogeneity components.
In addition, the paper introduces an approach to specification of models with multiple sources of duration dependence. or ‘multiple clocks’, which are also consistent with the simultaneous equations specifications. These multiple sources of duration dependence are each specified as piecewise linear splines. The model is applied to data on marital durations and timing of marital conceptions. Each form of simultaneity is found to be significant in this application, as are the multiple forms of duration dependence ~ calendar time, age of the woman, marital duration, and time since last birth. This approach to simultaneous equations for hazard processes promises to have wide applicability to a variety of substantive areas of research.
References Becker, Gary S.. Elizabeth Landes, and Robert T. Michael. 1977. An economic analysis of marital instability. Journal of Political Economy X5, 114~1 187. Flinn. C.J. and J.J. Heckman. 1982. Models for the analysis of labor force dynamics, Advances in Econometrics I, 3595. Han. A. and J.A. Hausman. 1990, Flexible parametric estimation of duration and competmg risk models, Journal of Applied Econometrics 5. l-28. Heckman. J.J. and G.J. Borjas. 1980, Does unemployment cause future unemployment? Detinitions. questions. and answers from a continuous time model of heterogeneity and state dependence. Economica 47, 2477283. Koo. Helen P. and Barbara K. Janowitz. 1983, Interrelationships between fertility and marital dissolution: Results of a simultaneous logit model, Demography 20. 129-145. Lillard, Lee and Linda Waite. 1988, Panel versus retrospective data on marital histories: Lessons from the PSID, in: Individuals and families in transition: Understanding change through longitudinal data (US. Bureau of the Census. Washington, DC). Lillard, Lee and Linda Waite. 1992. A ioint model of childbearing and marital disruotion. Working paper (Rand, Santa Monica.*CA). Maddala. G.S.. 1983. Limited-dependent and qualitative variables in econometrics (Cambridge University Press. Cambridge). Naylor. J.C. and A.F.M. Smith, 1982. Applications of a method for the efficient computation of posterior distributions. Applied Statistics 31. 214-225.
of Econometrics
56 (1993) 1899217. North-Holland
Simultaneous Marriage
duration
equations
for hazards
and fertility
timing*
Lee A. Lillard
The concept of simultaneous equations has been a fundamental contribution of econometrics. While the concepts have been extended quite fruitfully to limited and qualitative dependent variable models. they have been rarely applied to hazard or duration (failure time) models. This paper develops an approach to simultaneity among hazard equations which is similar in spirit to simultaneous Tobit models. It introduces a class of continuous time models which incorporates two forms of simultaneity across related processes when the hazard rate of one process depends (I) on the hazard rate of another process or (2) on the actual current state of or prior outcomes of a related multi-episode process. This paper also develops an approach to modeling the notion of ‘multiple clocks‘ in which one process may depend on the duration of a related process, in addition to its own. Maximum likelihood estimation is proposed based on specific parametric assumptions. The model is developed in the context of and emptrically applied to the joint determination of marital duration and timing of marital conceptions.
1. Introduction While there is now a substantial econometric literature concerned with the specification and estimation of simultaneous equation models involving qualitative and limited dependent variables,’ there has been limited consideration of simultaneity in ‘failure time’, or hazard, processes. This paper is concerned with two other forms of simultaneity among repeated dynamic discrete choices; that is, with simultaneity generated when the hazard rate of one multi-episode process depends on (f) the (log) hazard rate or (2) the actual preceding outcomes of a related multi-episode process.
C~,r~~.\portt/u,l~~ to: Lee A. Ltllard, The RAND Corporation. 90407~2138. USA.
1700 Main Street. Santa
Monica,
CA
*Thts research was supported by NICHD. Center for Population Research. Grant No. P-50-12639. I want to thank Constantijn Panis for exceptional skill in developing the software for the model and Mary Layne for careful preparation of the complex marital and fertility histories. ‘See Maddala
J.Econ
G
(1983) for numerous
examples.
Section 2 introduces the basic elements and assumptions of the model in the context of a particular application which exhibits the essential features of the approach. The application is concerned with the joint determination of marital duration and the timing of marital births (conception dates). Section 3 develops the conditional (on heterogeneity) survivor functions and duration density functions, develops the conditional and marginal probabilities of sequences of multi-episode events. and presents the marginal likelihood function for various types of data: censored and complete durations in the joint sequences of multi-episode events. Section 3 introduces the data, presents the estimates. and discusses results for the application.
2. The model Our purpose is to develop a model of related dynamic discrete choices (multi-dimensional hazard processes) incorporating forms of simultaneity which conform naturally to intuitive notions in classical simultaneous equations and to generalizations related to qualitative and limited dependent variable models. These notions are developed in the context of a particular application but illustrate the features of the model. The application jointly considers the dynamics of marriage durations and marital fertility, i.e., the timing of marital conceptions. The model of the application may be characterized as follows.’ An individual (woman in this case) may be observed over the duration of one or more marriages: and within the marriage(s) the dates of marital conceptions are known. From the beginning of a marriage the individual (couple) is at risk of splitting up (dissolution of the marriage) and at risk of conceiving a child. The hazard of dissolution is influenced by a number of time-related factors including calendar time. the woman’s age, and the duration of the marriage. by a set of exogenous covariates, and by a set of endogenous covariates the presence of children from the current and/or previous marriages ~ determined by a potentially correlated process (fertility). Issues include the extent to which children born to a marriage reduce the hazard of dissolution and the extent to which children from previous marriages increase the hazard of dissolution. As long as the marriage continues the individual (couple) is at risk to conceive a(nothcr) child. The hazard of conception is influenced by the same set of time-related factors, plus the duration since the last birth (omitting the interval of infertility during pregnancy). a set of exogenous covariates. and by the endogenous hazard of the marriage breaking up. Issues include the extent
‘The thcorctxal arguments underlying this hpecilication derive from Becker, Landca. and Michael (I 977) and is discussed more fully in Lillard and Waite (I 992). The onl! other attempt to cmplrical model mnrltal duration and fertility jointly IS Koo and Jano~~tr (1’383).
to which individuals in marriages with a higher risk of dissolution have reduced fertility (conception hazard). Consider the following log hazard equations which characterize these processes” - first for dissolution of marriage m (m = 1, . , M) and then for conception ,j (,j = 1, , J,) in marriage IH: In /r:,(t) = x0 + xi T(t) + x;A(t) + lnhkj(t)
=
+ x;M,(r)
4Xd(f) + &K,(t)
Po + /j; T(t) + + /J;X’r)
+
(1)
6.
/l>A(t) + /3;M,(t) +
+ i,lnh~(r)
PkD,j(f)
(2)
+ v.
The terms T(t), A(t), M,,(t), and DMj(t) represent the separate ‘clocks’ of time/duration dependence of the hazards which determine the baseline hazard; Xd(t) and X’(t) are vectors of exogenous covariates; K,(t) is a vector of endogenous prior outcomes of the fertility process which affect the hazard of marital dissolution; In hi(t) is the endogenous hazard of marital disruption which contemporaneously affects the fertility hazard; and E and q are heterogeneity components such that
These log hazard equations are defined for appropriate ranges of time: for marriage durations from the time the marriage begins, denoted rd,,, until it ends either at the time of dissolution (separation or divorce), denoted t&,, or interval duration from of censoring, denoted t&,; and for the jth conception either the time of marriage (for first birth intervals with no pre-marital conception) or the time of birth of the last, (j - I)th, child, denoted r~,j, to either the conception of the jth child, denoted t~,j, or the time of censoring, denoted r:,,,;. 2.1. Titne/ciuwtion
depmdenc~e - Multiple
clocks:
T(t), A(t), M(t), D(t)
This model is essentially one of proportional hazards, with exogenous and endogenous covariates shifting the baseline hazard. The baseline log hazard is
“Subscripts convenience.
denoting
the mdividual
for whom the process
applies are suppressed
for notational
the sum of the effects of the various sources of time dependence. Each source of duration dependence is represented by a piecewise linear (on the log hazard) spline function of time.’ The terms rJI T(t) and /Y, 7’(t) represent dependence of the respective hazards on calendar time” via piecewise linear splines, i.e.. r(t) is a vector of N7. + I spline variables which sum to calendar time but which allow the slope coefficients to differ within ranges of time separated by i\ir nodes” /L,_.The spline variable for the hth interval between /lr , and j(I is given by
Therefore,
for example.
The vectors of spline variables for age A(t). for marriage duration M,,(t), and for conception episode duration D,j are similarly defined, but with their own spline nodes and slope coetticients. Note that the calendar time and age ‘clocks’ arc not dependent on which marriage or birth interval is being considered. Rl~r~itr
‘This specllication is simllnr an spirit IO other llexible forma for the bn~cl~nc hazard. See for example Han and Hauhman (IYYO). Earlier in\eatigation led to a clear reJection uf the Wclbull and other widely used parametric forms. Additionally. those forms do not allow the slmplc closed form slmultancous equations models proposed here. ‘Since January
lY5S
111the apphcatmn
“No&s need not be the same m the scparatc equations. but the same nodes have been used hcrc Ibr simphcity. Implicitly. !l,, = 0 and or,, _ , = -r This Rex~blc functional form includes the hnear tlmc trend as a ~pccial cast r’, 7;lr) = xl I if x,~ = 2,. for all L. The result IS a Gompert7 form of the hazard. ‘Including
onI> the time dcpendcncc
portion.
and omitting
?(,,(I) and K,,,(t)
193
marriage
A(fd0,).8 That
is,
In h&Jr&,) = r0 + 2; T(ti,)
+ C&A(&),
which incorporates the cumulative effects of the passage of calendar time and age up until the time of marriage. Once a marriage has begun. the time dependence is the combined effect of the further passage of time 7(t), of aging A(t). and of longevity of the marriage M(t). The resulting time dependence,
+
4(A(r) - A(fdo,))+ r;M,(t),
is also piecewise linear in time since the episode began, but with nodes at the combined locations of the nodes of the three splines and coefficients between nodes which are the sums of coefficients of the three splines. The baseline hazard is thus piecewise linear Gompertz. Conceptions. Within a conception episode of the fertility process, say the jth, the duration of the episode, the duration of the marriage, and the age of the woman all change perfectly collinearly with calendar time. Again, their separate effects can be distinguished because their initial values at the beginning of the episode differ; and they also differ between conception episodes even within a marriage. Except for the first conception episode which usually begins at the beginning of the marriage, other conception intervals begin at different points in time, at different ages of the woman, and at different marriage durations. During the first conception interval at the beginning of a marriage,” the marriage duration coincides with the duration of the conception interval. However, the two effects are separately identified since first conception intervals end at various durations (some very short) and the effect of the marital duration ‘clock’ represented by M(t) is invariant to which conception interval is being considered. Since we are concerned with marital fertility, marriage duration M(t) plays a particularly relevant role in that the marriage must survive in order for the marital fertility to occur. That is, the end of the marriage represents potentially nonrandom censoring of conception intervals.
“Initial
marriage
duration
is obviously
zero.
‘Some women are pregnant when they marry. The next conception interval is considered a second and begins at the birth of the pre-maritally conceived child; and the premarital conception episode is not included in the model.
The intercept of the ‘baseline’ log hazard equation for the ,jth conception episode. in the IpIth marriage beginning at time tb,,,,. is 21 function of the calendar time the episode begins, the age of the woman, the duration of the marriage, and the log hazard of divorce when the conception episode begins. For the conception of the ,jth child in the r,lth marriage, the intercept is given by
Once a conception cpisodc has begun. the time dependence is the combined effect of the further passage of time T(t). of aging ,4(t). of longevity of the marriage ,%f(1). and the duration of the conception interval. The resulting time dependence.
is also piecewise linear in time since the episode began, but with nodes at the combined locations of the nodes of the four splines and coefficients between nodes which are the sums of coefkients of the four splines. The baseline conception hazard is thus also piecewise linear Gompertz. The terms D,,,,(r) represent the dependence of the hazard of conception on the duration of the conception episode at time t via piecewise linear splines.‘”
-7.3. I Efftav
o/ tlw c’i/r/‘tg/lt .vttitc, or pio~ outcot1w.~: K,,,(t)
Some of the explanatory variables in a hazard equation may be jointly determined and potentially correlated with the residual or heterogeneity component of the log hazard equation. In this application the vector K,,(t) includes the potentially endogenous set of time-varying variables representing the prior outcomes of the joint process’ ’ indicator variables for the presence
of children born to this and prior marriages” may not be unrelated to the unobserved component of the hazard of marital dissolution. That is, if the behavioral processes are correlated, i.e., po,, # 0, then K,(t) will not be independent of the residual term c. This specification of the effects of the prior outcomes of related processes is similar in spirit to the specification of the effects of outcomes of prior episodes of the same process ~ past durations, number of past occurrences, the current state in multi-state processes, and so forth.13 Conditional on the residual heterogeneity components, the observed completed durations and outcomes are independent. Thus, no further identifying restrictions (e.g., exclusion restrictions on exogenous covariates) are formally required. However, such restrictions may be desirable. -7.22.
Efjk~ts of'
N
sdoted log hrrzusd ruts: In hi(t)
The risk or hazard rate associated with one process may have a direct effect on the occurrence and timing of another event, i.e., on the hazard rate of another process. The explanatory hazard rate is a latent variable and not directly observable. l4 In this application the term In /r:(t) represents the risk of (potential) marital dissolution of marriage 111which may influence the decision to bear children in that marriage, i.e., the hazard of a first and subsequent conceptions j within marriage tn. Women in higher-risk marriages may reduce the risk of conception, i < 0. The latent log hazard of marital dissolution may be substituted into the equation for the log hazard of conception to obtain the ‘reduced form’ conception hazard, In hi,(t) = /&, + +
i.sco + (pi
+ ix;)T(t)
(0; + j.r;)M,(t) +
+ i.r\Xd(f)
p;D,j(t)
+ k&Km(t)
analyses
have been proposed
(1980) and Flinn and Heckman
for labor (I 982).
+
+ q+
“In this application we have not considered children be endogenous. This is a subject for further research. “Related and Borjaa
+ (/I; + i.a;)A(t) /l\Xc(f)
is.
(3
from prior extra-marital
force dynamics.
relationships
See for example
to
Heckman
“Analogous to latent variables in qualitative and limited dependent variable models. e.g. stmultaneous probit or Tobit models. Our treatment of the latent variable is similar to the treatment in these models. See Maddala (1983). While specification of the model in this application is structurally triangular in the log hazards, in general they may be fully simultaneous. As discussed below, the usual exclusion conditions for identification suffice, even though the hazard rates are latent variables.
The hazard of conception of its own past outcomes.
then becomes ;I function in this cast the presence
of K,,,(t). the occurrences of childrenI
regressors. The The log haLards are also a function of other L’XO~~~IIOIIS vectors X”(t) and X’(r) represent the time-varying (or constant) exogenous covariates affecting the hazards of marital disruption and conception, rcspectively. Exclusion restrictions on these covari;\tes, X’(t) relative to M”(r 1. may form the basis of identification of i, the effect of In hi,(t) on In h;,;(r). These issues are discussed in the section on identification following the discussion of estimation.
The unobserved residual components of the hazard equations. i: and !I. ma! be correlated. We assume joint normality of these residual terms in the log hazard equations. That is.
Even if the residual heterogeneity terms of the haLard processes are not correlated. a structural effect of one hazard on another, 2. will introduce correlation in the proccsscs. The reduced form residuals >:and r/ + ii: are also jointly normally distributed.
2.5. ~‘ldifiOf,l/~
JYYhl~ld fiJf'J,l /lti:~lYl/S
The hazard equations for marital disruption respectively. arc given by the following:
and for marital
conceptions,
L. A. Lillurrl. Sirmdrrrr~rouseyuu~iotts h~j(t, X(t).
tl + AC) =
exp(P,
+ i.X” + (/I; + k;)r(t)
+ (pi + i.zc;)M,(t) + (/I;
where X(r) includes
/iw l~u~urd~
+ ia;)X(t)
all the covariates
197
+ (/I; + kZZ;)A(t)
+ /IkD,j(t) + &K,(t)
+ q + k),
(7)
in Xd(t) and X’(r).
3. Estimation
For any given observation, conditional on measured covariates and on the heterogeneity components E and ‘I, the outcomes of each marriage and conception process are independent, and the joint probability of any sequence or combination of outcomes is simply the product of the probabilities of the individual outcomes. Denote the baseline survivor functions for each marriage and conception episode by
~, exp(r, 10,,,
+ 4 T(r) + &A(r)
, (8)
+ a;M,(r))dr
I
S;;“,,(t)
=
exp
(i -
~ exp(j& + i.ro + (/I; + E.a;)T(r) f,,,,,,
+ (/I; f j.J;)A(T)
+ (fl; + j-CC;)M,(T) + /jkD,j(T))dT
1
(9)
respectively, for t > tt, (the date the marriage began) and for t > tfo,j (the date the jth conception episode began, i.e., the date of the last birth or the beginning of the marriage for the first conception). One of the attractive features of the piecewise linear spline formulation is that the baseline survivor functions have easily computable closed forms. Following this notation the ‘conditional’ survivor functions” for each marriage and conception episode may be defined as follows, where x(t) denotes the full history of Xd and Xc covariates from the beginning of the episode. The integrated hazard is subdivided into ranges of time within which the time-varying covariates are constant. The following survivor functions
‘“Conditional
on the value of the heterogeneity
components
E and q
then follow from the application of the proportionality of the effects of covariates and the heterogeneity components. Conception equations are written in reduced form as a function of structural parameters. For marital durations the probability that marriage 111will last at least to time I. duration I ~ r$,,, conditional on the sequence of covariates Xd(t) up to that time [represented by z(r)] and on the unobserved residual component x. is given by the conditional survivor function
where I,,, is the number of subintervals within which covariates Xd(r) arc constant, and tf,,s>z. , = I. This is also the conditional (on C) likelihood of a marital duration censored at the time of the final survey. i.e., t = t,d,,,.The conditional likelihood that marriage ITI will end at duration r:,,, is given by
./:!(f:f,,,.z(L).
i:) =
S:I,(r;,,,.%(r:f,,,), 4 /I%!,,,. .Y“(t:1,),). I:).
(11)
which is also the conditional density function for marital durations. For birth intervals. the probability that the ,jth conception will not have occurred by time t. conditional on the sequence of covariates up to that time. and the unobserved residual components i: and rf. is
where I,,lj is the number of subintervals within which covariates X’(t) are constant. and td,,,,,,,,+, = t. This is also the conditional (on rl + if:) likelihood of the Jth censored birth interval at the final survey, t,‘,,,,(t), or at the time of marital dissolution. tk(t:,,). The conditional likelihood that the ,jth birth interval of marriage 1~1 will be of completed duration tz,,j is given by
X
h~,j(t::“,j,
X(t:“,j),
Jl +
j.i:),
(13)
which is also the conditional density function of conception interval durations. For each marriage the final conception episode (which might bc the first) is always in progress (censored) at the end of the marriage or the final survey date if the marriage is still in progress. The probability of the sequence of
199
J,,, observed
conception
QL(f*, I,
episodes
within
the /nth marriage
is given by
r\ + ;A) = S>,,,(t*. z(r*), 11+ ;.a) n rmj((t&j, x(f&j), v + ;J), i= I
(14)
where t* = r:,,, in the case of a completed the case of a final survey date at t,.
marriage
at time t&, and t* = t, in
The marginal likelihood is obtained by combining (multiplying) the conditional probabilities of any particular sequence of marital and conception episode outcomes and integrating the resulting conditional likelihood over the range of the unobserved heterogeneity components t: and ‘I. There are two cases depending on whether the last Mth marriage is in progress (censored) as of the final period of observation. If the last Mth marriage has ended, the marginal likelihood (joint marginal density function for marriages and conceptions) is given by
and if the last marriage
is in progress,
the probability
of the observed
closed
t ix)
Parameters are estimated by maximum likelihood (FIML). While the marginal likelihood equations above are specified for each type of observation, the aggregate likelihood equation is the product of the individual likelihood functions over all observations. Empirically we use analytic derivatives and scoring methods for second derivatives.”
“The bicariate normal integration is performed using multi-dimensional Hermite Quadrature methuds suggested by statistician Lionel Galway. Also see Naylor and Smith (1982). The software used in this application. designed by the author and implemented by Constantijn Panis, is more general than this application. allowmg (I) the effect of a duration dependence to begin at any time. including after the episode has begun. and (2) allowing larger numbers of jointly determined hazard processes.
3.3. ltkPl7tific~rrtiotr
If the sources of correlation in the processes are not present (L = 0 and I7i,l = O), then the parameters of the two processes may be estimated separately or jointly with the same result. The marginal likelihood function separates into the product of the separate marginal likelihood functions for the two processes. The potentially cndogenous variable K,,,(t) is effectively exogenous. However, the multiple marriage episodes must be considered together. and the multiple conception episodes must be considered togcthcr, to account for heterogeneity and dependence of the log hazard on prior own process outcomes. If. however, tither is nonzcro. then the paramctcrs of the two processes must be estimated jointly.” If i = 0 but o,,, # 0. than the puramctcrs may be estimated without further restriction using the joint process marginal likelihood function. Integration over the joint distribution of heterogeneity components ‘accounts for’ the correlation in the processes. If i # 0. then some indentifying restriction is required and the processes should be considered jointly. We may jointly cstimatc the parameters of the log haLard of marital dissolution equation. the parameters of the reduced form log hazard of conception equation. and the reduced form variances and correlation of the hcterogcneity distribution. If oIII = 0, then the system is recursive; the parameter i is idcntiiied from the covariance or correlation between reduced form residuals (being the only sourcc of correlation). If I),,, # 0, then i must bc rccovcrcd from the reduced form parameters of the conception log hnrard equation. One possibility is through the effect(s) of prior conception outcomes K,,,(t) a in the reduced form equation presented above. However. in some models there may not be any variables K,,,(t) (Y;, = 0) or the prior outcomes may ha\,c direct effects on the conception hazard. In the application considered here. the number of children in the marriage K,,>(t) is the same as the birth order (parity) of the conception interval which is expected to have ;I direct effect on the probability of another conception and its timing. Assuming that there are direct effects of calendar time. age. and marital duration on the log hazard of conception. restrictions on the set of exogenous covariates will sufhce. If there is at least one covariate X”(t) affecting the hazard of marital dissolution but not the conception 10s hazard [not in the vector X’(t)]. the parameter i is estimable from its coet‘ficient in the conception equation.
3.4. Sitnulutiot~
of‘ illustratiw
relutiotwhips
To illustrate the effects of various forms of time dependence and the effects of endogenous and exogenous time-varying covariates, we will present simulations of the time path of aggregate survivor functions and of aggregate hazard functions. These calculations are performed as follows. First the ‘baseline’ survivor function is computed conditional on the initial values of calendar time and age (and marital duration in the case of conception episodes). Then the aggregate survivor function is computed conditional on the time pattern of covariates and integrated over the relevant heterogeneity components. Finally, the hazard aggregate function is computed as the relative decline of the aggregate survivor function.”
4. Data,
estimates, and results for the application
4.1. Duta
This application is based on longitudinal data on marriages and births constructed from the the Panel Study of Income Dynamics (PSID) which began in 1968 with about 5,500 households. The PSID is well-suited to analysis of the relationship between fertility and marital disruption. The sample has been resurveyed each year since that time, generating panel data on marital status and fertility. Each sample member is followed whenever a primary couple (husband, wife) separates or children and other family members exit the household.” The inclusion of the children’s marital histories guarantees a wide range of marriage year and marriage age cohorts; the long panel and retrospective period means that multiple marriages and conception episodes are observed. The PSID contains longitudinal information on both marital status and fertility, allowing us to trace the experiences of individual couples over a substantial period. The sample of married couples is large, with a sizable number of disruptions and remarriages during the period of observation. Information was obtained on marital status at each interview, minimizing recall bias, together with a complete retrospective marital and fertility history in 1985. The availability of the two types of information allows us to date marriages, separations, divorces, and births with a good deal of precision.”
‘“In addition,
some points
will be illustrated
using simple plots of the log hazard
‘“The new households that they form are added interviewed in the same way as original households.
to the sample
‘ISee Lillard and marital histories.
of the retrospective
Waite
(1988) for a comparison
as new sample and
panel
equation. units reports
and of
The sample used for this analysis is comprised of 2.376 female respondents. of which 84.3 percent were married only once during the period of observation, 13.9 percent were married twice. and 1.7 percent were married a third represented in all. See table 1. The time. There are thus 2,789 marriages sample used in this analysis is restricted to respondents who responded to the 1985 retrospective marital and fertility history, to allow us to date the events of interest precisely and to include information on marriages and fertility that began and/or ended prior to the first interview in 196X. We consider all marriages of female respondents that began in 1955 or later. The data includes the begin date of the marriage, and either the end date of the marriage (if the couple divorces or separates before the 1985 survey). or the date of censoring of the marriage by attrition. death of the spouses, or the 1985 survey. We observe the dissolution of 34 percent of first marriages. 12 percent of second marriages, and 34 percent of the third marriages in our sample. See table 3. The remaining couples. not divorced within the sample. represent censored observations on marital duration because we do not observe the dissolution of the marriage and know only that. if it occurred. it happened after the last survey period. For each marriage of each respondent we use the 1985 retrospective fertility nine months prior to the reported birth date. history to date conceptions There are up to nine conception episodes, eight births. per respondent. The distribution of conception episodes in all marriages is presented in table 3.
203
Table Conception Number of conception episodes
Number
Percent
639 604 607 317 124 41 30 13 I
6.9 25.4 25.5 13.3 5.2 1.7 1.3 0.5 0.0
2,376
100.0
I 2 3 4 5 6 7 8 9 Total
3 marriages.
The variables used in the model are defined in table 4, which also gives the specification of the dissolution and conception log hazard equations. The measures of past marital fertility outcomes K,(t) include (1) whether a child has been born to the marriage, K-lst, and the number of children beyond the first” born to the couple, K-2nd + The coefficient is then the marginal effect of another child on the log hazard of disruption. We also include with a separate effect the number of children born to the woman in previous marriages, which are included as own children in the previous marriage episode. Children conceived by the woman out-of-wedlock, K-llleg, are included but considered exogenous in this application. 23 This includes any premarital conception leading to the birth of a child in the current marriage, K-Premr. We include an indicator for second or later marriage in both the marriage and conception hazard equations. This represents ‘own process occurrence dependence’ for the marriage disruption process and ‘related process occurrence dependence’ for the conception process. Another form of ‘own process occurrence dependence’ is the inclusion of indicator variables for birth order on the current conception episode ~ again representing the marginal effect on the the log conception hazard of reaching a higher birth order. The sources of
“‘Further significant. “The research
decomposition
of the number
endogeneity of extramarital and not considered here.
beyond
conceptions,
the first versus later born was not statistically and of age at marriage,
is a subject
for further
704
Table Variables
Variable
Descr~pt~vn
affecting various
4 model components.,’ Dissolution In /l;,(r 1
x x
x
x
duration dependence or ‘multiple clocks’ discussed in the context of the model specification are of course included. The model includes a number of demographic variables that previous race, religion, and research has found to influence risk of marital disruption education. In addition. the model includes a number of exogenous variables
that should affect risk of divorce or separation directly, without affecting the risk of conception. These include measures of the legal ease, SD12, or difficulty, SD.56, of divorce in the state of residence in the current year. an index of the cost of lawyers. MSTN, the aggregate per capita divorce rate in the state in the current year, MNDR.14 4.2. E.~timttc.r mtl effkcts of’ .spc~fjcutio~t Table 5 presents estimates of the model of marital dissolution, table 6 presents estimates of the model of conception timing, and table 7 presents estimates of the variances and correlation of the heterogeneity components. In each case, column 1 presents estimates of the mode1 assuming no correlation between the hazards. f),,, = 0, and no effect of the marital dissolution hazard on the hazard of conception. i = 0. Column 2 presents estimates assuming 1. = 0 but allowing for correlation between the heterogeneity components of the two processes, /Jo,,. Finally, column 3 presents estimates allowing both sources of relationship between the processes. First, both heterogeneity components 1: and rl exhibit significant variation, and the correlation between them is highly significant. Second, the effect of the hazard of marital disruption on the conception hazard, j.. is also significant.‘” Couples in marriages with a higher hazard of dissolution exhibit significantly delayed childbearing and smaller completed family size (reduced hazard of conception). Second. consider the effect of these coefficient restrictions on the parameter estimates. In the marriage dissolution equation, allowing for the correlation between processes significantly affects the estimates of the effects of children on the hazard of marital dissolution. A significant detrimental effect of children from previous marriages on the hazard of disruption is revealed, as is a similar effect of second- and high-order children in the current marriage. In the conception hazard equation, several effects are evident of accounting for the endogenous effect of the hazard of dissolution. Most notably, the previously signihcant direct effects of the following variables are eliminated: (I) birth order beyond the third child, (2) children from previous marriages, (3) illegitimate children, (4) second marriage. and (5) being Catholic. The effects
“And a variable indtcating whether the aggregate missing in less than 2 percent of marriage years.
divorce
rate is missing. The divorce
rate was
“While the r-statistic is only marginally signilicant. the likelihood ratio statistic indicates greater significance. zf = 14.2. Similarly. while the Identifying effects of divorce laws, lawyer costs, and aggregate divorce rates have only marginal r-statistics when i is a free parameter, the joint likehhood ratio ;(: statistic is highly significant. These differences in test satistics are largely due to collinearity among the parameter estimates (and among the state-year divorce climate covariates).
free i free
I’, t, = 0
I’,,,
; = 0
Md-I
Md
-3
:
5
Md ~ I(1 +
0.7705. (0.3788)
0.763h” (0.2795)
0.043x (O.OXX2)
0.057x (0.0X89)
0.0209 (0.076 I 1
0.0143 (0.07671
0.009x (0.0329)
~ (1.0I74 (O.Oi.ihl
0.0027 (0.020 I )
- 0.0226 (0.020X)
(o.ox42l
0.0740 (0.0X5 I )
0.0753” l0.01671
O.OX?h~’ (0.0I 70)
0.0803 MA
‘0 ~ ill
MA
30 ~ 40
MA ~ 40 +
1-d ~ x0 t
~~0.04X6~’ (0.01 78)
~~o,0437h (O.OlXl)
0.06X3’
0.0800” (O.(E!43I
(0.0243)
0.0330 (0.15431
0.0730 lO.1556)
0.0479 (0.0750)
0.0326
0. I I69’, (0.050 I )
0.12X2” (0.0504)
0.00I 3 (00340)
O.OI66 (0.0348)
0.070(~ (0.0270)
0.07’) 2,’ (0.0271)
O.OX3.5~’ (0.0297)
0.0x5 I” (0.030 I )
207
rable
5 (continued) p,, free i = 0
0: ,, =O / = 0
K ~ Mar
II,, free i free
0.181 I” (0.0426)
0.2510” (0.0474)
0.26X7” (0.0484)
0. I 2I 7 (0.07X9)
0.3226” (0.0X8X)
0.3428“ (0.0896)
K ~ Premr
-- 0.1650’ (0.0864)
~ 0.0835 (0.0908)
~ 0.0840 (0.0906)
K ~ 1st
~ 0.5578” (0.0996)
- 0.3342 (0. I 102)
~ 0.3227 (0.1094)
0.269 I” (0.0716)
0.3060 (0.0699)
K ~ 2nd
- 0.0065 (0.046 I
+
Jewish
0.0272 (0.1045)
- 0.9653’ (0.5043)
)
~ 0.0766 (0.1085)
- 0.0282 (0.1062)
-
~ 0.9618’ (0.5228)
l.0291h (0.5188)
Edrll2
0.0050 (0.1035)
- 0.0253 (0.1090)
~ 0.0605 (0.1082)
Edrl6
~ 0.371X” (0.1378)
- 0.3340b (0.1447)
~ 0.3189h (0.1444)
Black
0.39’7.’ (0.08x;)
0.3624’ (0.0918)
0.3352” (0.0906)
Mar7p
0.0706 (0.2280)
- 0.2622 (0.2365)
~ 0.2618 (0.2383)
0.605 I’ (0.2299)
0.6592” (0.2383)
~ 0.1213 (0.1044)
Mndr
0.0894” (0.0320)
0.0898” (0.0335)
0.0235 (0.0168)
Mmndr
0.617”’ (0.2505)
0.6182” (0.2559)
0.3806’ (0.21 IO)
Sdl2
0.0152 (0.1086)
0.03 I6 (0.1108)
~ 0.0970 (0.0596)
Sd56
~ 0.0029 (0.0942)
0.0125 (0.0969)
- 0.0754 (0.0528)
‘Significant hSignificant ‘Significant
at the I% level. at the 5% level. at the 10% level
free
,I,,, I
0.0
=
0
0.0
lO.XY3’l
~ 0.331’) 10.23571 KBt
0
KU7
0
KHZ
2
KHZ-
5i
t
~ 0.0x54 IO. 1 t 701
0.5
0.5
KB2
MC
2
I
i
~ 0.0693 (0. t I671
~ 0. I 3xX.’ l0.01751
0. 1372.’ (0.0473)
0.60 I 5‘ (0.36301
0.%3X (0.3603)
’ 3’OX. -.. _ (0.430 t I
(0.4303,
2.3393’ (0.43 t I I
0.0927 (0.22Ohl
0.00 79 (02X6)
0.09Y2 (0.2’Ohl
0.1571‘ (0.09041
0. I 60 I L (O.OYO2)
0.1607’ IO.0903 1
0. I xxw (0.03Y’l
0. I xx I ” ((1.0.394t
0. IX 15” (O.OiY5l
0.0x03” (0 oix7t
0.07x I h (0 03x31
2.3652~’
~ 0.1672” (0.054 I I ~ O.I3Xo” (0.0473l
0.5Y63
l0.3605)
0. t 605 (0 207 t )
l.?56Y 10.X4651
0.2_154~ (0.06391
0.397 IL (0.1616)
0.0026
(0.029 I ) IO i
(0.2346)
~ O.I662~’ (0.0542t
0. t 3.37, (0.0404)
RIG
0.0754 (0.1 1671
~ 0.3023
0. I6Y3” lO.05JS I
~~ 0 0X00” (O.O?X9,
?
~ 0.77‘M (0.23 15)
0.02I17 (0.03331
209
Table
6 (continued) free i = 0
/),,i free i. free
- 0.0395 (0.0343)
~ 0.0505 (0.0334)
- 0.1732 (0.1561)
AC - 20 ~ 30
- 0.0163’ (0.0096)
- 0.0160’ (0.0095)
~ 0.1513 (0.0792)
Ac - 30 ~ 40
- 0.1386’ (0.0168)
- 0.1354” (0.0169)
~ 0.7025” (0.0493)
AC ~ 40 +
~ 0.2397” (0.0409)
- 0.236X” (0.0408)
- 0.3450’ (0.0X03)
/‘“, = 0 i = 0
II,,,
.4
14-20
Time-lY55:
Tc - 55 ~ 60
0.0467 (0.0377)
0.0510 (0.0373)
- 0.12 13 (0.2676)
Tc ~ 60 ~ 65
- 0.0826” (0.0208)
- 0.0787” (0.0206)
- 0.0236 (0.1291)
Tc - 65 ~ 70
- 0.0968” (0.0195)
- 0.0962” (0.0193)
0. I67 I (0.1707)
Tc - 70 ~ 75
~ 0.0406b (0.0199)
- 0.0482h (0.0198)
0.0148 (0.0635)
Tc ~ 75 - 08
0.0668” (0.0190)
0.0666” (0.0189)
0.1867” (0.0830)
~ 0.16X0 (0.0299)
- 0.1689” (0.0298)
- 0.0304 (0.093 I )
~ 1.092X” (0.1255)
- 1.0580 (0.1264)
- 1.2067” (0.2030)
~ 1.3160 (0.1512)
- 1.2498” (0.1514)
~ 1.7910” (0.3713)
- 1.4137” (0.0978)
- 1.3645” (0.0969)
~ 0.8942” (0.3004)
Child
- 0.3159” (0.1083)
- 0.2978“ (0.1071)
0.1761 (0.3042)
Fifth + Child
~ 0.1295’ (0.0749)
-0.1158 (0.0744)
0.3664 (0.2941)
- 04138” (0:0420) - 0.5079” (0.0870)
- 0.4038” (0.0416) - 0.5345” (0.0873)
0.0 I44 (0.2478) 0.0060 (0.3485)
Tc - 80 +
Second Third Fourth
Child Child
Kc - Mar
710
rrcc ; rrec
,I>,,
0.200’1~’ (0.0635 I
Fdrll?
0.1507
0.I523
1.3Y32
(O.NO5)
I I .23341
0.13X3”
Ih
((I.OXOO) 0.x-1
I ,’
(0.05YY~ 0.x00
0.0357
)
(O.IX22l
~ O.IiJ3’
0.657’)
(0.07YS)
10.37Y4)
0.25 1 Y,’ I0.06001 0.5323’
(0 1261)
~
0.1233’ (0 065 I
~ 0.1451’
Sib
0. Ii06 (0. I 76X I
lO.lY72l
I0.0646l EdI-
0.10’) I ‘* 10.063 I I
IO.lh5il
0.x l-w (0.22 I 7) ().I554 (0.4545)
0.0075
0.003x
0.0056
IO.02541
(O.O’JOI
(0.02501
0.000x” l0.0002l
0.6OYo” (0. I Y-IS) 0.65 I Y.’ 10.055Yl 0.0
737y3 _..,_,
0.0007~ (0.0001)
4’)
0.771X”
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of these variables operate marital dissolution. 4.3. Discussion
only indirectly
through
their effect on the hazard
of
qf’ results
Estimates of parameters for the preferred model, with both pE,, and i. free, are presented in column 3 of tables 5-7. Results are often discussed in terms of illustrative simulations of aggregate hazard and survivor functions, computed as noted earlier. 4.3. I. Murriuge
chratiotis
First consider the multiple sources of duration dependence. The ‘multiple clocks’ include calendar time, age of the woman, and marital duration. All three show very striking and significant duration dependence. The individual effects of these sources of duration dependence, and of their sum, on the log hazard are illustrated in fig. 1. The top panel shows the effect of calendar time. The second panel shows the effects of aging for two values of initial age (age at marriage), repeated for three values of initial year (year of marriage) so the summation of effects is apparent. The third panel shows the effect of marital duration, repeated for three values of year of marriage. The hazard of marital disruption declines monotonicly with age, especially after age twenty. The direct effect of marital duration is a very large and significant increase over the first year of marriage, followed by a relatively flat pattern thereafter. If the dependence of the hazard on age were ignored, the estimated shape of the marital duration effect would include the age effect. Even if ‘age at marriage’ were controlled, the estimated marriage duration pattern would reflect the effects of aging as the marriage endured. The hazard of marital dissolution shows a marked and significant pattern of increase over calendar time. The final fourth panel of fig. 2 shows the combined effects of calendar time, aging, and marital duration on the log hazard of marital disruption. The aggregate survivor functions and aggregate hazard functions for these six combinations are presented in fig. 2 for comparison. These patterns would represent the apparent effects of marital duration if age and calendar time were not included, and the apparent interaction of age at marriage and calendar time with marital duration. 4.3.2.
Murriuge
hrutions
Next consider the direct marital duration on the log direct effect of calendar time in the top panel along with
and indirect effects of calendar time, aging, and hazard of conception, as illustrated in fig. 3. The P(t), which is a slight positive trend, is presented the total effect, T’(t) + I.Td(t), which includes the
-1
1,
1955
1960
1965
1970
1975
1960
1965
1960
1965
Time (years)
-L
-I
1955
1960
,~
1965
IT
~.
1970
1975
Time (years)
Duration
1955
1960
1965
1970
1975
1960
Time (years)
(5)
-3 (3)
1965
213
1 i ?a I e i .>
c.Fj.6 ‘6
i
e
.6
a
.4 1
-----
T1960
1950
-
1970
i---1980
~~
T
1990
Time (years)
06
1
r\ (5)
.04 ; (3) iI I .02 &ik-
(2) 0 I~, 1950
1960
1970
1980
1990
Time (years) Fig. 2. Aggregate survivor and haLard function3 of marital dissolution for sik sccnarioh. (I I marrted in 19.5.5 at age 20: (2) married in 1955 at age 30; (3) married in 1965 at age 10: 14) marrxd in 1965 at age 30: (5) married in 1975 at age 20: (6) married 1111975 at age 30. First marriage; M hitc protestant. high school graduate. tir\t marruge al mean of other cartable.
strongly offsetting effect of the increasing log hazard of marital dissolution. with the net effect of being a negative trend. The direct effect of aging on log hazard of conception is strongly negative from age fourteen. but is offset below age thirty by the early negative effect of age on the the log hazard of marital dissolution, but then drops dramatically with age after age thirty as the effect of age on marital dissolution declines. The direct effect of marital duration on the log hazard of conception is a dramatic increase over the first five years with a slow decline over the remaining years, but which is substantially offset by the dramatic increase in the hazard of marital disruption over the initial year of marriage. The remaining pattern is the same. but dampened.
-2
-
-T-
--
1955
/1960
-7--
-r-
1965
-r
1970
--
IT
1975
---
1
1990
1985
35
40
45
35
40
25
Time (years)
-5
-6 I
-..
~__. 20
15
.--
._~_ --
25
30 A@
15
20
25
-
-
c
(ye-9
30 AwW=s)
Fig
3. Log
hazard
of conceptw~~:
Direct
and total
tune-relaled
etkctr
215
0
5
10
15
10
15
Time (years)
.6
i
0
5
Time (years) Fig. 4. Hazard of first marital conception: erects of legal environment. (I) low divorce per capita in state-year. lawyers expensive: (2) low divorce per capita in state-year. lawyers inexpensive; lawyers expensive; (4) high divorce per capita m (3) high divorce per capita in state-year, state-year. lawyers inexpensive.
To further illustrate the indirect effects of the hazard of marital dissolution on the hazard of conception, figs. 4 and 5 illustrate the effects of differences in the state-year environment for divorce on the hazard of marital disruption and on the hazard of conception. The hazard of marital disruption is increased when the cost of lawyers is reduced and when aggregate divorces per capita is high. Various combinations are illustrated in fig. 4 for a particular combination of other covariates. The illustrated effects are significant.
‘If!
Time (years)
.04
Iozipm
i
Furthermore, the effects of these variables on the hazard of conception operate only through their effect on the hazard of marital disruption, so that the indirect effect is the total cfrect. This effect on the hazard of conception for the first conception episode in the marriage is illustrated in fig. S.
5. Conclusions
This paper introduces a class of models exhibiting simultaneity among the hazards of related multiple processes, each with ‘proportional’ hazard rates. with multiple episodes of occurrence and with unobserved constant heterogeneity in the hazard rate. Simultaneity between processes includes (I) occurrence dependence on prior outcomes of the related process when the heterogeneity components are correlated across processes and (2) dependence of one hazard rate on a related hazard rate. Maximum likelihood estimation procedures are developed based on the marginal likelihood of observed outcomes, given the assumption of joint normality of log heterogeneity components.
In addition, the paper introduces an approach to specification of models with multiple sources of duration dependence. or ‘multiple clocks’, which are also consistent with the simultaneous equations specifications. These multiple sources of duration dependence are each specified as piecewise linear splines. The model is applied to data on marital durations and timing of marital conceptions. Each form of simultaneity is found to be significant in this application, as are the multiple forms of duration dependence ~ calendar time, age of the woman, marital duration, and time since last birth. This approach to simultaneous equations for hazard processes promises to have wide applicability to a variety of substantive areas of research.
References Becker, Gary S.. Elizabeth Landes, and Robert T. Michael. 1977. An economic analysis of marital instability. Journal of Political Economy X5, 114~1 187. Flinn. C.J. and J.J. Heckman. 1982. Models for the analysis of labor force dynamics, Advances in Econometrics I, 3595. Han. A. and J.A. Hausman. 1990, Flexible parametric estimation of duration and competmg risk models, Journal of Applied Econometrics 5. l-28. Heckman. J.J. and G.J. Borjas. 1980, Does unemployment cause future unemployment? Detinitions. questions. and answers from a continuous time model of heterogeneity and state dependence. Economica 47, 2477283. Koo. Helen P. and Barbara K. Janowitz. 1983, Interrelationships between fertility and marital dissolution: Results of a simultaneous logit model, Demography 20. 129-145. Lillard, Lee and Linda Waite. 1988, Panel versus retrospective data on marital histories: Lessons from the PSID, in: Individuals and families in transition: Understanding change through longitudinal data (US. Bureau of the Census. Washington, DC). Lillard, Lee and Linda Waite. 1992. A ioint model of childbearing and marital disruotion. Working paper (Rand, Santa Monica.*CA). Maddala. G.S.. 1983. Limited-dependent and qualitative variables in econometrics (Cambridge University Press. Cambridge). Naylor. J.C. and A.F.M. Smith, 1982. Applications of a method for the efficient computation of posterior distributions. Applied Statistics 31. 214-225.