Simulation Estimation of Two-Tiered Dynamic Panel Tobit
Models with an Application to the Labor Supply of Married
Women
Sheng-Kai Chang ∗
Abstract
In this paper a computationally practical simulation estimator is proposed for the two-
tiered dynamic panel Tobit model originally developed by Cragg (1971). The log-likelihood
function simulated through procedures based on a recursive algorithm formulated by the
Geweke-Hajivassiliou-Keane simulator is maximized. The simulation estimators are then
applied to study the labor supply of married women. The rich dynamic structure of the
labor force participation decision as well as hours worked decisions that are conditional on the
participation of married women are identified by using the proposed simulation estimators.
The average partial effects of the participation and hours worked decisions for married women
in response to fertility decisions and increases in the husband’s income are also investigated.
It is found that the hypothesis that the fertility decision is exogenous and the hypothesis
that the husband’s income is exogenous to the married women’s labor supply function are
both rejected in the dynamic and static two-tiered models. Moreover, children aged between
6 and 13 years old may have a negative impact on the hours worked decision for married
women that is conditional on their participation. However, these children may provide some
positive incentives for married women to participate in the labor force.
JEL classification: C15; C23; C24; J13; J22.
Keywords: Two-tiered dynamic panel Tobit models; GHK simulator; Correlated random
effects; Initial conditions problem.
∗Corresponding author. Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei
100, Taiwan. Tel.: +886-2-2351-9641 ext 376; fax: +886-2-2351-1826. Email address: [email protected].
1 Introduction
In the literature on panel data models, one critical issue is the estimation of limited dependent
variable (LDV) models characterized by the presence of lagged dependent variables and serially
correlated errors. A dynamic panel Tobit model is a leading example. The conventional tech-
niques used in the estimation of linear panel data models are not applicable to the estimation
of dynamic panel Tobit models due to the nature of the Tobit structure. Furthermore, the
introduction of lagged dependent variables makes conventional estimation techniques even more
difficult to apply.
One possible method for estimating the dynamic panel Tobit model is the fixed effects
approach. The fixed effects model is valid under weak restrictions on the unobserved individual
heterogeneity. For example, Honore (1993) estimates the panel Tobit model with lagged observed
dependent variables through the fixed effects approach by creating orthogonality conditions for
method of moments estimators. A set of identification conditions for Honore’s model is provided
by Honore and Hu (2004). See also Hu (2002) for the estimations of the censored panel data
model with lagged latent dependent variables by applying the fixed effects method.
The other method proposed to handle the dynamic panel Tobit model is the random effects
approach. By specifying the distribution of the error conditional on the regressors, the random
effects estimators can be obtained through maximizing the corresponding likelihood function.
However, the likelihood function of the dynamic panel Tobit model is usually intractable since
the dimension of an integral involved in its calculation is as large as the number of censoring
periods in the model. Under such circumstances, simulation-based inference methods can be
extremely useful.
The impact of simulation methods on the analysis of LDV models is profound, especially
under recent advances in computing technologies. Various simulation estimation methods and
procedures for drawing random variables have been proposed in the econometrics literature.
For instance, Lerman and Manski (1981) and Gourieroux and Monfort (1993) suggest adopting
the maximum simulated likelihood method (MSL), McFadden (1989) and Keane (1994) propose
the method of simulated moments (MSM), and Hajivassiliou and McFadden (1998) present the
2
method of simulated scores (MSS). See also Hajivassiliou (1993, 1994) for their applications.
Different simulators have also been proposed for simulating multinomial probabilities in LDV
models. Among multivariate normal probability simulators, Hajivassiliou et al. (1996) suggest
that the Geweke-Hajivassiliou-Keane (GHK) simulator is, in terms of root mean squared errors,
the most reliable simulator among the thirteen simulators they examined for approximating the
multivariate normal distribution and its derivative.
This paper studies a practical, operational and versatile maximum simulated likelihood pro-
cedure through the correlated random effects approach for two-tiered dynamic panel Tobit mod-
els using GHK simulation estimators. It is known that one potential restriction of the traditional
Tobit models is that the choice between the dependent variable y = 0 versus y > 0 and the de-
cision regarding the amount of y given that y > 0 is determined by a single mechanism. This
is not always reasonable. Cragg (1971) proposed a two-tiered model to allow the parameters
which characterize the decision regarding y = 0 versus y > 0 to be separate from the parameters
which determine the decision regarding how much y is given that y > 0. The traditional Tobit
models can be viewed as a special case of Cragg’s two-tiered model.
Moreover, the correlated random effects approach is attractive in several respects. First,
time-invariant, time-varying, and time-dummy variables can be incorporated into the model and
they can be consistently estimated using the proposed simulation estimators. Most importantly,
the approach allows some degree of dependence between the individual unobserved heterogeneity
and exogenous explanatory variables. Although stronger distributional assumptions are made for
the random effects approach compared with the fixed effects approach, the proposed simulation
estimation method with correlated random effects approach allows for complicated dynamics.
The introduction of lagged latent dependent variables and lagged observed dependent variables,
possibly with more than one lag, is straightforward. It is also easy to accommodate serial
correlations in errors. Modifying the estimators to accommodate such specifications is done
3
fairly easily, and in an intuitive manner.1 In addition, the two-tiered Tobit model proposed
by Cragg (1971) can be easily extended and adopted by the simulation estimators through the
correlated random effects approach. The predictions of the model can also be generated by the
estimation results through the correlated random effects approach for two-tiered dynamic panel
Tobit models.
The proposed simulation estimators are applied to study married women’s labor supply.
The dynamic panel Tobit model as well as Cragg’s two-tiered model are used to study married
women’s working behavior using the data from the Panel Study of Income Dynamics (PSID).
The average partial effects of the participation and hours worked decision for married women
in response to the fertility decision and increases in the husband’s income are also investigated.
It is found that the hypothesis that the fertility decision is exogenous and the hypothesis that
the husband’s income is exogenous to the married women’s labor supply function are both
rejected in dynamic and static two-tiered models. Moreover, children aged between 6 and 13
years old may have a negative impact on the hours worked decision for married women that is
conditional on the participation. However, these children may provide some positive incentives
for married women to participate in the labor force. The Monte Carlo experiments at the end
of the empirical section also provide supporting evidence for Cragg’s two-tiered model over the
traditional Tobit model.
The remainder of this paper proceeds as follows. Section 2 proposes a practical simulation
estimator for two-tiered dynamic panel Tobit models. The likelihood function simulation through
the GHK simulator is presented and then the consistency and asymptotic normality of the
simulation estimator is demonstrated. Section 3 applies the simulation estimators to study the
married women’s labor supply function. Section 4 concludes the paper.
1It has become quite common to allow rich dynamics in economic models since Heckman (1981) pointed out the
importance of distinguishing true state dependence from spurious state dependence. For instance, Hyslop (1999)
incorporates state dependence, serial correlation, and individual heterogeneity into the labor force participation
model of married women.
4
2 Practical Simulation Estimators for Two-Tiered Dynamic Panel
Tobit Models
2.1 Correlated Random Effects
In a dynamic panel data framework, the Tobit model is described as:
y⋆it = xitβ + yi,t−1λ+ ci + uit (1)
yit = maxy⋆it, 0, t = 1, . . . , T i = 1, . . . , N
Note that model (1) is characterized by lagged observed dependent variables.2 The component
ci is an unobserved individual specific random disturbance which is constant over time, and uit
is an idiosyncratic error which varies across time and individuals. Throughout this section, I
assume that ci and uit are Gaussian conditional on xi1, · · · , xiT .
Following Chamberlain (1984), the unobserved individual heterogeneity ci can be assumed
to be correlated with the exogenous variable xit for all t in a linear way. Thus, ci in (1) can be
presented as
ci = ω0 + ω1xi1 + · · ·+ ωTxiT + di (2)
where di in (2) and uit in (1) satisfy the following properties:
E[di|xi] = 0, E[uit|di, xi] = 0, E[d2i |xi] = σ2d, E[u2it|di, xi] = σ2
u,
E[didj |xi, xj ] = 0, E[uitujs|di, dj , xi, xj ] = 0, E[uituis|di, xi] = 0 (3)
where xi = (xi1, · · · , xiT ) and xj = (xj1, · · · , xjT ), and for all i = j and t = s. Thus, the
composite error ϵit can be represented as
ϵit = di + uit
2The dynamic panel Tobit model with lagged latent dependent variables can be obtained from (1) by replacing
the first line of (1) with y⋆it = xitβ+y⋆
i,t−1λ+ ci+uit. The proposed simulation estimator is also applicable to the
model with lagged latent dependent variables. Although I present the likelihood simulation based on model (1),
the likelihood simulation based on the model with the lagged latent dependent variables can be easily modified
from the simulated likelihood function of (1).
5
Under such a specification, ω0 cannot be identified if a constant is included in xit. One
way to include a constant term in xit is to assume the independence of time constant xit and
the unobserved individual heterogeneity ci. Under such a circumstance, the coefficient of the
constant term xit can be consistently estimated using the simulation estimator. Moreover, time
dummies are also excluded from xit in ci because they do not vary across i.
For a correlated random effects approach of the Chamberlain type, however, the dimension
of the parameters to be estimated will increase at the rate T . When the time period T or the
dimension of x is small, computation is not an issue for estimating ω0, ω1, . . ., ωT , β and λ.
Otherwise, following Mundlak (1978), Hajivassiliou (1985, 1987) and Wooldridge (2002), ci can
be assumed to be as follows:
ci = ω0 + ωxi + di, xi =1
T
T∑t=1
xit (4)
and ω0 and ω can also be consistently estimated using the proposed simulation estimators.3
One advantage of using the correlated random effects approach of (4) is that, no matter
how large the time period T is, the number of parameters to be estimated will only be affected
by the dimension of xi. For a panel data set with a large T , this specification method for
unobserved individual heterogeneity will not only allow for correlation between ci and xi, but
will also make computation easier for parameter estimation. Equation (4) will be employed to
model unobserved individual heterogeneity in the empirical section.
2.2 Simulation Estimation
For the random effects plus AR(1) specification, the error term ϵit can be specified as
ϵit = di + vit
vit = ζvi,t−1 + uit. (5)
where di is defined either by (2) or (4), and uit and di continue to satisfy all of the conditions
in (3). The covariance structure of the random effects plus AR(1) specification is denoted as
ΣRE+AR(1). Moreover, the stationarity assumption |ζ| < 1 is also assumed to be satisfied for
the random effects plus AR(1) errors model.
3The time dummies and time constant variables are also not included in xi.
6
Let Iit be a censored indicator function, such that
Iit =
1 for y⋆it > 0
0 for y⋆it ≤ 0
Therefore, for individual i, if Iit = 1 then y⋆it is observed and yit = y⋆it. On the other hand, y⋆it
is censored and its value is not observed (i.e., yit = 0) if Iit = 0.
Thus, the simulated likelihood function with R simulation draws based on the GHK simulator
for individual i can be described as:
Li =1
R
R∑r=1
T∏t=1
[f (r)(yit|yi,t−1)]Iit [P (r)(Iit = 0|yi,t−1)]
1−Iit (6)
By letting li = ln(Li), the simulated log-likelihood function can be represented as:
lR = ln(N∏i=1
Li) =N∑i=1
li (7)
The error terms ϵi are assumed to have a normal distribution, that is, ϵi ∼ N(0,ΣRE+AR(1)),
where ϵi = (ϵi1, . . . , ϵiT )′ is a T × 1 column vector, and so E[ϵiϵ
′i|xi1, · · · , xiT ] = ΣRE+AR(1).
Under this assumption, the simulation estimator based on the GHK simulator is expected to be
useful for the dynamic panel Tobit models since it is well recognized that the GHK simulator
is very accurate for the simulation of a multivariate normal distribution (Hajivassiliou et al.,
1996).
Let A be the Cholesky decomposition of ΣRE+AR(1), that is, ΣRE+AR(1) = AA′, where A
is a lower-triangular matrix. Given this structure, we can write ϵi = Aηi, where ϵi = di + uit,
ηi ∼ N(0, I) and ηi = (ηi1, . . . , ηiT )′.
Let θ be the vector of parameters of interest in the random effects plus AR(1) model, i.e.,
θ = (β, λ, σ2u, σ
2d, ζ). Given the sample, suppose that for individual i the total number of instances
of censoring is mi and that censoring occurs at time t1, . . . , tmi . The random variables ξ(r)it are
drawn from the uniform random number generator on [0,1], where t = t1, . . . , tmi , i = 1, . . . , N
and r = 1, . . . R. In order to guarantee the validity of the stochastic equicontinuity condition4
for the simulation estimator, the random numbers are drawn once and kept fixed when θ varies.5
4See Hajivassiliou and McFadden (1998).5See McFadden (1989) for further details.
7
To simplify the exposition, y⋆i0 = 0 is assumed for all i. Let Φ be the cumulative stan-
dard normal function. Then, the truncated normal random variables η(r)it can be simulated or
calculated by
η(r)it =
Φ−1(ξ(r)it Φ(
−xitβ−yi,t−1λ−xiω−∑t−1
k=1Atkη
(r)ik
Att)) for t ∈ t1, . . . , tmi
yit−xitβ−yi,t−1λ−xiω−∑t−1
k=1Atkη
(r)ik
Attfor t ∈ t1, . . . , tmi
(8)
Then, the simulated likelihood function (6) can be obtained through
f (r)(yit|yi,t−1) =1
Attϕ(
yit − xitβ − yi,t−1λ− xiω −∑t−1
k=1Atkη(r)ik
Att) (9)
for t ∈ t1, . . . , tmi, where ϕ is the standard normal density function, and
P (r)(Iit = 0|yi,t−1) = Φ(−xitβ − yi,t−1λ− xiω −
∑t−1k=1Atkη
(r)ik
Att) (10)
for t ∈ t1, . . . , tmi. Thus, by combining (9) and (10) into (6), a practical simulation estimator
of θ denoted as θ can be obtained for maximizing the simulated log-likelihood function (7).
Based on the GHK simulation estimators presented above, the asymptotic variance associated
with them can be consistently estimated as follows. Let
Ω =1
NΣNi=1[si(θ)si(θ)
′]
and
J =1
NΣNi=1siθ(θ)
where si is the simulated score function associated with (7), and siθ is the first derivative of si
with respect to θ, that is, J is simply the Hessian matrix of the simulated log-likelihood function
(7). Therefore, the consistent estimator of the asymptotic variance of the simulation estimator
θ can be written as
Avar(θ) = J−1ΩJ−1. (11)
See Hajivassiliou and McFadden (1998) for more details.
The average partial effects (APEs) can also be estimated by using the GHK simulation esti-
mators θ under the correlated random effects approach. For the APEs of the decision probability
8
related to y = 0 versus y > 0, supposing that there are K regressors in x and xK is a discrete
random variable, the APEs with respect to xK when the xK is changed from xK(0) to xK(1) can
be written as
[N−1ΣNi=1Φ(x
1i θ/σϵ)]− [N−1ΣN
i=1Φ(x0i θ/σϵ)] (12)
where
x1i θ = x1β1 + ...+ xK−1βK−1 + xK(1)βK + y−1λ+ xiω
and
x0i θ = x1β1 + ...+ xK−1βK−1 + xK(0)βK + y−1λ+ xiω
xi = (xi,1, ..., xi,T ), and σϵ =√σ2d + σ2
u/(1− ζ2). Moreover, the (x1, ..., xK−1, y−1) are any
given values of the first K − 1 regressors and lagged dependent variables. Normally, the sample
averages of these random variables are used to evaluate the APEs.
The focus of attention in the Tobit models is also on the expected values E(y|x, y > 0) and
E(y|x). For a given xt, the E(yt|xt, yt > 0) and E(yt|xt) can be calculated as follows
E(yt|xt, yt > 0) = xtθ + σϵ[ϕ(xtθ/σϵ)
Φ(xtθ/σϵ)]
and
E(yt|xt) = Φ(xtθ/σϵ)xtθ + σϵϕ(xtθ/σϵ)
Thus, the APEs of E(y|x, y > 0) with respect to xK when xK is a discrete random variable
and is changed from xK(0) to xK(1) can be represented as
[N−1ΣNi=1(x
1i θ + σϵ[
ϕ(x1i θ/σϵ)
Φ(x1i θ/σϵ)])]− [N−1ΣN
i=1(x0i θ + σϵ[
ϕ(x0i θ/σϵ)
Φ(x0i θ/σϵ)])] (13)
Moreover, the APEs of E(y|x) with respect to xK when xK is a discrete random variable
and is changed from xK(0) to xK(1) can be represented as
[N−1ΣNi=1(Φ(x
1i θ/σϵ)x
1i θ + σϵϕ(x
1i θ/σϵ))]− [N−1ΣN
i=1(Φ(x0i θ/σϵ)x
0i θ + σϵϕ(x
0i θ/σϵ))] (14)
9
2.3 Two-tiered Dynamic Panel Tobit Models
One potential restriction of Tobit models is that of the choice between y = 0 versus y > 0 and
the decision regarding the amount of y given that y > 0 is determined by a single mechanism. In
other words, the decision related to y = 0 versus y > 0 is inseparable from the decision regarding
how much y is given that y > 0 in the traditional Tobit model. However, the mechanisms which
determine these two may not be the same in some economic models.
Specifically, any variable which increases the probability of a nonzero value must also increase
the mean of the positive values in the traditional Tobit models. This is not always the case. For
example, as for the application of the married women’s labor supply which is further discussed
in the empirical section that appears later, the traditional Tobit model indicates that married
women who have one more child aged between 6 and 13 years old should always decrease (or
increase) both the probability of labor force participation and the mean hours worked. However,
by using the more generalized types of Tobit model, it is found that married women with one
more child aged between 6 and 13 years old will increase their labor force participation rate but
will decrease the mean hours worked that are conditional on the participation.
In order to relax the restrictions imposed by the traditional Tobit models, Cragg (1971)
proposed a two-tiered model to allow the parameters which characterize the decision regarding
y = 0 versus y > 0 to be separate from the parameters which determine the decision regarding
how much y is given that y > 0. The traditional Tobit models can be viewed as a special case
of Cragg’s two-tiered model.6 In other words, there are basically two assumptions in Cragg’s
two-tiered model. First, the probability of a zero observation is given by a probit model with the
first tier parameters, and then the density of the dependent variable that is conditional on being
a positive observation is truncated at zero and characterized by the second tier parameters.
Cragg’s model is easily extended from the cross-sectional framework to the dynamic panel data
models using the simulation estimators proposed earlier.
There are alternatives to Cragg’s two-tiered models. One of the examples is the Heckman
6See for example Lin and Schmidt (1984) for a specification test of the Tobit model against Cragg’s model.
10
(1979) type of sample selection model which can be simply characterized as follows:
y1 = xβ1 + u1
y2 = xβ2 + u2 (15)
where only the sign of y2 is observed and y1 is observed if and only if y2 > 0. Under such a
specification, Tobit model is a special case of the Heckman’s model when β1 = β2 and u1 = u2.
The Heckman model (15) is different from Cragg’s two-tiered model in two respects. First, it is
indicated in the model (15) that there is a positive probability of observing y1 < 0. Secondly and
more importantly, the unobserved y1 is literally unobserved, rather than observed as being equal
to zero. The second difference is in any case fundamental while the first difference can be solved
by measuring y1 in logarithms, but under such a case, the model no longer includes the Tobit
models as a special case. Therefore, although Cragg’s two-tiered model is more restrictive than
the Heckman full selection model, for the economic data set in which the non-zero observations
are all positive and zero is a meaningful and common value for the dependent variable, Cragg’s
two-tiered model might be a better choice than Heckman’s sample selection model. For example,
for the married women’s labor supply data set used in the empirical section, Cragg’s two-tiered
model on the one hand provides more flexible specifications than the standard Tobit models,
and on the other hand it is also more appropriate for characterizing the data set than Heckman’s
selection models.
Specifically for Cragg’s two-tiered Tobit models, let Γ1 = (β1, λ1, ω1) be the first-tiered
parameters which determine the initial decision between y = 0 and y > 0. Let Γ2 = (β2, λ2, ω2)
be the second-tiered parameters which determine the amount of y given that y > 0.
y⋆i0 = 0 is assumed again for all i. Then, for the GHK estimators for the two-tiered Tobit
models, the truncated normal random variables η(r)it can be simulated from
η(r)it =
Φ−1(ξ(r)it Φ(
−xitβ1−yi,t−1λ1−xiω1−∑t−1
k=1Atkη
(r)ik
Att)) for t ∈ t1, . . . , tmi
yit−xitβ2−yi,t−1λ2−xiω2−∑t−1
k=1Atkη
(r)ik
Attfor t ∈ t1, . . . , tmi
Notice that the first-tiered parameters Γ1 = (β1, λ1, ω1) are used in the first line, and the
second-tiered parameters Γ2 = (β2, λ2, ω2) are used in the second line to obtain η(r)it .
11
Then the simulated likelihood function of the two-tiered version of (6) for individual i can
be described as:
Li =1
R
R∑r=1
T∏t=1
[P (Iit = 1|Γ1, yi,t−1)
P (Iit = 1|Γ2, yi,t−1)f(yit|Γ2, yi,t−1)]
Iit [P (Iit = 0|Γ1, yi,t−1)]1−Iit (16)
where the traditional Tobit models can be viewed as a special case of (16) when Γ1 = Γ2.
Specifically, f(yit|Γ2, yi,t−1), P (Iit = 1|Γ1, yi,t−1), P (Iit = 0|Γ1, yi,t−1) and P (Iit = 1|Γ2, yi,t−1)
in (16) can be calculated by
f(yit|Γ2, yi,t−1) =1
Attϕ(
yit − xitβ2 − yi,t−1λ2 − xiω2 −∑t−1
k=1Atkη(r)ik
Att) (17)
and
P (Iit = 1|Γ1, yi,t−1) = 1− Φ(−xitβ1 − yi,t−1λ1 − xiω1 −
∑t−1k=1Atkη
(r)ik
Att) (18)
and
P (Iit = 1|Γ2, yi,t−1) = 1− Φ(−xitβ2 − yi,t−1λ2 − xiω2 −
∑t−1k=1Atkη
(r)ik
Att) (19)
and P (Iit = 0|Γ1, yi,t−1) = 1− P (Iit = 1|Γ1, yi,t−1). Thus, a practical simulation estimator of θ
can be obtained for maximizing the simulated log-likelihood function by using (16). Moreover,
the consistent estimator of the asymptotic variance of the two-tiered simulation estimators can
be obtained through (11). The two-tiered dynamic panel Tobit model will be applied to study
the married women’s labor supply in the empirical section.
The APEs of two-tiered Tobit models can also be estimated by using the GHK simulation es-
timators. Let θ1 = (β(1), λ(1), ω(1)) be the estimators from the first tier, and θ2 = (β(2), λ(2), ω(2))
be the estimators from the second tier. Then the APEs of the decision probability are related
to y = 0 versus y > 0, supposing again that there are K regressors in x and xK is a discrete
random variable. The APEs with respect to xK when xK is changed from xK(0) to xK(1) can
be written as
[N−1ΣNi=1Φ(x
1θ(1)/σϵ)]− [N−1ΣNi=1Φ(x
0θ(1)/σϵ)] (20)
where it is assumed that
x1θ(1) = x1β1(1) + ...+ xK−1
βK−1(1) + xK(1)β
K(1) + y−1λ(1) + xiω(1)
12
x1θ(2) = x1β1(2) + ...+ xK−1
βK−1(2) + xK(1)β
K(2) + y−1λ(2) + xiω(2)
x0θ(1) = x1β1(1) + ...+ xK−1
βK−1(1) + xK(0)β
K(1) + y−1λ(1) + xiω(1)
and
x0θ(2) = x1β1(2) + ...+ xK−1
βK−1(2) + xK(0)β
K(2) + y−1λ(2) + xiω(2)
xi = (xi,1, ..., xi,T ), and σϵ =√σ2d + σ2
u/(1− ζ2). Moreover, (x1, ..., xK−1, y−1) are any given val-
ues of the first K−1 regressors and lagged dependent variables. Notice that only the estimators
from the first tier are used.
The APEs of E(y|x, y > 0) with respect to xK when xK is a discrete random variable and
is changed from xK(0) to xK(1) can be represented as
[N−1ΣNi=1(x
1θ(2) + σϵ[ϕ(x1θ(2)/σϵ)
Φ(x1θ(2)/σϵ)])]− [N−1ΣN
i=1(x0θ(2) + σϵ[
ϕ(x0θ(2)/σϵ)
Φ(x0θ(2)/σϵ)])] (21)
Notice that only the estimators from the second tier are used.
Moreover, the APEs of E(y|x) with respect to xK when xK is a discrete random variable
and is changed from xK(0) to xK(1) can be represented as the difference between
[N−1ΣNi=1(Φ(x
1θ(1)/σϵ)x1θ(2) +
Φ(x1θ(1)/σϵ)
Φ(x1θ(2)/σϵ)σϵϕ(x
1θ(2)/σϵ))] (22)
[N−1ΣNi=1(Φ(x
0θ(1)/σϵ)x0θ(2) +
Φ(x0θ(1)/σϵ)
Φ(x0θ(2)/σϵ)σϵϕ(x
0θ(2)/σϵ))] (23)
3 An Empirical Application: Married Women’s Labor Supply
3.1 Motivation
One of the most interesting topics in labor economics is the labor supply of married women.
Due to fertility decisions and non-labor income, the labor supply of married women is much
more complicated than the labor supply of men. Most of the empirical research conducted in
this area is focused within the context of cross-sectional data. Since a large fraction of married
women have zero working hours, the Tobit model is usually employed to perform the estimation.
13
Jakubson (1988) incorporated the life-cycle model into the married women’s labor supply
equation and estimated the model using four-year panel data. Hyslop (1999) also applied the
panel Probit model with a rich dynamic structure to study the labor force participation of
married women. It is crucial to incorporate the dynamic structure into the married women’s
labor supply function as indicated in Hyslop’s results. Moreover, in analyzing the interactions
between fertility and the labor supply decision for married women, Browning (1992) pointed out
that it is important to control for the dynamic structure of their labor supply decisions.
In a dynamic structure, it is often argued that individuals who have experienced an event
in the past are more likely to experience the same event in the future than individuals who did
not experience the event. This is what Heckman (1981) refers to as “state dependence.” There
are two types of state dependence in a dynamic structure. One is “true state dependence,”
and the other is “spurious state dependence.” True state dependence, as explained by Heckman
(1981), is when “as a consequence of experiencing an event, preference, prices or constraints
relevant to future choice are altered.” On the other hand, Heckman (1981) explains spurious state
dependence as “the phenomenon that individuals may differ in their propensity to experience
the event. If individual differences are correlated over time, and if these differences are not
properly controlled, previous experience may appear to be a determinant of future experience.”
Therefore, it is important for economists to distinguish true state dependence from spurious
state dependence in the labor supply function of married women.
The empirical specification for modeling the labor supply of married women in this section
involves the reduced form method. There is also a huge literature on structural models of female
labor supply, for example, Heckman and MaCurdy (1980, 1982) and Eckstein and Wolpin (1989).
Although the reduced form methods are useful for initial exploratory data analysis and they are
relatively easy to implement compared with the structural form methods, there are several
advantages in structural estimation. The first advantage of having a structural model behind
the estimations is that the parameters estimated can be interpreted in a way that is consistent
with the theoretical framework. In other words, the detailed implications of the theory, and
not only of the data, can be learned during the process of estimating the structural models.
Moreover, the predictions of the impacts of policy changes can be generated more accurately
14
by the structural models than by the reduced form models. See Rust (1994) for more details
regarding the structural estimation.
In this section, the two-tiered dynamic panel Tobit model is employed to study the labor
supply of married women. Unlike the models estimated by Cogan (1981) and Mroz (1987),
panel data are used in the estimation. In addition, rather than using the static panel Tobit
model estimated by Jakubson (1988), rich dynamic structures are incorporated in the model and
estimated by the proposed simulation estimators. Moreover, decisions over both participation
and hours are considered through the Tobit structure, while Hyslop (1999) only focuses on the
labor force participation decision for married women. Since Cogan (1981) and Mroz (1987) have
pointed out a possible mis-specification of the simple Tobit specification for the labor supply
of married women due to the significant fixed cost involved in their labor force participation
decisions, the Cragg type two-tiered Tobit model is also implemented to study the labor supply
of married women.
3.2 Data
The data used in the estimation consist of a nine-year panel data set from the PSID. The sample
contains observations for 1,627 women continuously married between 1984 and 1992, and aged
between 19 and 60 in 1985. Table 1 presents the summary statistics for the data used. The
husband’s income in Table 1 is expressed in constant (1985) thousands of dollars, computed as
nominal earnings deflated by the consumer price index. The numbers in the parentheses are the
standard errors corresponding to the variables.
The distribution of years worked during the period for the full sample in Table 1 also suggests
the importance of incorporating the dynamic structure for the married women’s labor supply.
For example, supposing that the individual’s participation decision is distributed according to
an independent binomial distribution, then about 9.08% of the sample would be expected to
work each year if the participation probability were fixed as 0.766 (the average participation
rate during the period), about 13.42% of the sample would be expected to work each year if the
participation probability were fixed as 0.8, and about 4.04% of the sample would be expected
to work each year if the participation probability were fixed as 0.7. In these three cases, less
15
Table 1. Sample Characteristics
Variable Full Employed Employed Employed
Name Sample 9 Years 0 Years ≥ 1 Year
Age (1985) 34.68(8.769)
34.77(8.144)
41.99(9.520)
34.22(8.723)
Education 12.89(2.071)
13.14(2.031)
12.07(1.987)
12.94(2.077)
Race (Black=1) 0.212(0.409)
0.233(0.423)
0.240(0.427)
0.210(0.408)
Husband’s Income ($1000) 31.04(29.16)
28.99(24.59)
40.51(42.81)
30.45(28.10)
Children between 1 and 2 0.218(0.466)
0.173(0.419)
0.176(0.456)
0.221(0.467)
Children between 3 and 5 0.277(0.516)
0.222(0.462)
0.242(0.523)
0.279(0.516)
Children between 6 and 13 0.712(0.891)
0.659(0.849)
0.616(0.844)
0.718(0.894)
Hours of Work 1179.88(887.01)
1695.84(593.45)
0 1253.86(914.41)
No. of years worked
zero 96 − 96 −
one 50 − − 50
two 56 − − 56
three 65 − − 65
four 66 − − 66
five 77 − − 77
six 83 − − 83
seven 115 − − 115
eight 149 − − 149
nine 870 870 − 870
Sample Size 1627 870 96 1531
16
than 0.002% of the sample would be expected not to work at all. However, the sample’s relative
frequencies are 53.47% of the sample who worked each year for 9 years and 5.9% of the sample
who did not work at all, accordingly. Therefore, Table 1 indicates that there is significant
persistence in the observed annual participation decisions of married women.
The differences in the characteristics of the married women across the sub-samples shown
in Table 1 can be interpreted as follows. Women who work each year are better educated, are
more likely to be black, have lower than average husband’s income, and have fewer dependent
children. On the other hand, women who have never been employed during the sample period
are older, less educated, are more likely to be black, have higher than average husband’s income,
and have slightly fewer dependent children.
3.3 Estimation Results
The dynamic panel Tobit model (1) with the covariance matrix ΣRE+AR(1) under the assump-
tions in (3) is employed to study the labor supply of married women. The dependent variable
is the annual hours of work. The explanatory variables include constants (Cons), years of
schooling (Edu), wife’s race (Race, black=1), wife’s age (Age), square of age (Age2), husband’s
income (Hinc), number of children aged between 1 and 2 years old (C12), number of children
aged between 3 and 5 years old (C35), and number of children aged between 6 and 13 years
old (C613). The husband’s income is used as a proxy variable for non-labor income for married
women. Furthermore, the lagged dependent variable is not included as a regressor in the static
models. The numbers in the parentheses are the estimated standard errors corresponding to
the estimators. The optimization subroutine used is the ConstrOptim procedure based on the
R software, and the BFGS algorithm is used for the maximization.7
The estimation results are presented in Table 2 for the static models. The pure random
effects plus AR(1) model which assumes that ci = di is presented as model (1) of Table 2. It is
shown that better educated or black or older women tend to work more than less educated or
white or younger women. In addition, the women who have children, especially children between
the ages of 1 and 2, tend to work less than the women who have no children. These results are
7The same procedure and algorithm are used later in the Monte Carlo experiments subsection.
17
Table 2. Static Panel Tobit Models of Married Women’s Labor Supply
Model (1) (2) (3) (4)
Correlated Correlated Correlated
Variable RE+AR(1) RE+AR(1) Cragg-1 Cragg-2 Cragg-1 Cragg-2
Cons 190.6(1.229)
−5.870(2.797)
−42.89(267.7)
36.75(264.7)
−114.9(271.3)
89.31(272.0)
Edu 44.91(9.64)
55.31(11.24)
55.61(9.482)
13.93(10.66)
82.09(10.19)
39.52(12.11)
Race 211.6(51.92)
33.75(54.78)
−68.77(44.11)
142.8(43.84)
31.44(44.48)
165.4(44.74)
Age 37.90(7.132)
53.08(7.522)
23.38(13.42)
69.72(12.56)
22.74(13.63)
66.47(12.89)
Age2 −0.630(0.100)
−0.805(0.105)
−0.362(0.169)
−0.897(0.163)
−0.384(0.169)
−0.886(0.164)
Hinc −2.651(0.487)
−1.558(0.411)
−2.267(0.465)
−2.633(0.717)
−1.461(0.361)
−1.265(0.509)
C12 −213.7(21.90)
−211.0(22.69)
−204.0(27.76)
−173.6(21.49)
−170.3(31.44)
−169.5(21.90)
C35 −191.6(18.31)
−165.9(19.42)
−115.7(22.67)
−139.1(18.84)
−127.5(27.06)
−150.1(19.73)
C613 −78.26(13.29)
−49.76(14.84)
13.75(16.31)
−54.20(13.09)
12.76(22.36)
−46.68(14.33)
Hinc − −6.117(1.717)
− − −4.825(1.464)
−6.588(2.227)
C12 − −55.33(40.92)
− − −124.8(119.4)
−45.64(126.2)
C35 − −72.57(40.74)
− − −104.6(114.1)
−127.5(127.2)
C613 − −76.71(33.71)
− − −90.02(39.24)
−113.8(37.39)
ρ 0.568 0.721 0.568 0.574
ζ 0.619 0.563 0.541 0.531
log-Like −92207 −92140 −91431 −91394
18
standard in the literature. The fraction of the variance due to individual heterogeneity, ρ, is
0.568, and the AR(1) coefficient ζ is 0.619.
The unobserved individual heterogeneity ci in the correlated random effects plus AR(1)
model (2) of Table 2 is assumed to be as follows:
ci = ω0 + ω1Hinci + ω2C12i + ω3C35i + ω4C613i + di (24)
where Hinci, C12i, C35i, C613i are time averages of the corresponding variables for individual
i. Based on model (2) of Table 2, it is shown later in Table 4 that both the husband’s income
and fertility decision variables are endogenous in the static model (1) of Table 2 at the 1%
significance level. Thus, the impact of education and age is underestimated and the impact of
race and younger children is overestimated in model (1) of Table 2 if the endogeneity problem
is ignored. Moreover, ρ increases and the AR(1) coefficient ζ decreases from model (2) to model
(1) in Table 2.
The two-tiered Cragg type panel Tobit model is estimated in model (3) of Table 2 where the
decisions on labor force participation and hours worked that are conditional on the participation
are estimated separately. Based on model (3) of Table 2, the women receiving higher education
are more likely to participate in the labor force. However, being conditional on the participation,
education has no statistically significant impact on the decision regarding hours worked. On the
other hand, in a way that is conditional on participation in the labor force, black or older women
tend to work more than white or younger women, but there is no statistically significant impact
of race or age on the labor force participation decision for married women. The husband’s income
has almost the same negative impact on both the participation and working hours decisions.
Thus, for married women, the higher the husband’s income, the less likely it is that she will
participate in the labor force, and the less hours she will work if she decides to participate in the
labor force. The children aged between 1 and 5 years old have a negative impact on the married
women in both the participation as well as hours worked decisions, and not surprisingly, the
younger the children, the stronger the negative impact will be on both margins. The children
aged between 6 and 13 years old also have a negative impact on the working hours decision that
is conditional on the participation. However, they may have a positive impact on the labor force
19
participation decision although this positive influence is not statistically significant. The ρ and
AR(1) coefficient ζ in model (3) are similar to the ones in model (1) of Table 2.
The static correlated Cragg’s two-tiered model is estimated in model (4) of Table 2 by
using (24). The husband’s income is still endogenous in both tiers of the model. However, the
coefficients of C12i and C35i are each insignificant at the 10% level. Moreover, unlike model
(3) of Table 2, education has a statistically significant positive impact on both the participation
and working hours decisions.
The dynamic panel Tobit model is employed to study the labor supply for married women
in Table 3. The initial conditions problem is dealt with in Heckman’s approach. The true state
dependence parameters λ are close to zero and are not statistically significant in the RE+AR(1)
model but are marginally statistically significant in the correlated RE+AR(1) model. Moreover,
the AR(1) coefficient ζ increases slightly from Table 2 to Table 3. The correlated random effects
plus AR(1) model in (2) of Table 3 is based on (24). The coefficients of Hinci, C12i, C35i,
C613i are marginally statistically significantly different from zero at the 5% significance level
which indicates that the husband’s income as well as fertility decisions are endogenous in the
dynamic model.
The two-tiered Cragg’s type dynamic panel Tobit model is estimated in model (3) of Table 3.
As in Table 2, it can be shown that education only has a statistically significant impact on the
labor force participation decision, but not on the hours worked decision that is conditional on the
participation. On the other hand, in a way that is conditional on the participation in the labor
force, black women tend to work more than white women. However, there is no statistically
significant impact of race on the labor force participation decision. The husband’s income also
has an equally negative impact on both participation and the hours worked decision, as shown
in Table 2. Moreover, the children between 1 and 5 years of age have a statistically significant
negative impact on both participation as well as the hours worked decision. The children aged
between 6 and 13 only have a negative impact on the working hours decision that is conditional
on the participation, but there is no statistically significant impact for the children aged between
6 and 13 on the labor force participation decision for married women.
The dynamic version of the correlated Cragg’s two-tiered model is estimated in model (4)
20
Table 3. Dynamic Panel Tobit Models of Married Women’s Labor Supply
Model (1) (2) (3) (4)
Correlated Correlated Correlated
Variable RE+AR(1) RE+AR(1) Cragg-1 Cragg-2 Cragg-1 Cragg-2
Cons 150.0(102.5)
−83.49(321.7)
−307.6(302.2)
198.5(303.8)
−572.6(302.4)
378.1(310.0)
Edu 45.45(13.33)
79.48(12.98)
54.59(10.54)
6.257(11.89)
77.28(11.27)
30.94(13.22)
Race 207.0(94.65)
131.6(57.23)
53.61(49.67)
232.4(47.51)
73.25(49.94)
239.7(48.30)
Age 42.67(24.63)
50.34(15.19)
44.98(14.88)
71.20(14.31)
59.50(14.85)
66.37(14.74)
Age2 −0.695(0.238)
−0.803(0.188)
−0.660(0.182)
−0.947(0.182)
−0.852(0.181)
−0.928(0.185)
Hinc −2.716(0.554)
−1.509(0.435)
−2.276(0.502)
−2.543(0.692)
−1.682(0.437)
−1.100(0.509)
C12 −201.1(33.01)
−140.3(24.12)
−206.9(28.91)
−168.3(22.81)
−92.75(32.50)
−121.0(24.06)
C35 −178.2(23.52)
−114.6(21.07)
−157.4(23.81)
−156.9(20.01)
−66.68(28.89)
−102.1(21.52)
C613 −69.52(34.31)
−26.15(16.03)
−20.52(17.26)
−74.55(14.08)
12.83(23.34)
−31.51(15.60)
λ −0.047(0.479)
−0.054(0.019)
−0.082(0.049)
−0.039(0.029)
−0.102(0.042)
−0.052(0.025)
Hinc − −7.010(2.002)
− − −6.164(1.532)
−7.958(2.205)
C12 − −264.1(146.3)
− − −309.7(134.5)
−121.9(139.1)
C35 − −290.8(143.6)
− − −207.4(131.8)
−356.9(143.5)
C613 − −81.74(43.05)
− − −74.40(42.16)
−86.74(40.37)
ρ 0.568 0.721 0.568 0.574
ζ 0.671 0.613 0.597 0.608
log-Like −92203 −92091 −91383 −91331
21
of Table 3. The hypothesis of the exogeneity of the husband’s income can be rejected at the
5% significance level in both the first and second tier of the estimation. The impacts of race
and age on the married women’s labor supply in model (4) of Table 3 are similar to those in
model (3) of Table 3. However, education now has a statistically significant positive impact on
both the labor force participation decision and the mean hours worked decision conditional on
the participation. Most interestingly, in the dynamic correlated Cragg’s two-tiered model it is
found that the children aged between 6 and 13 years old will give the married women a positive
incentive to participate in the labor force.
The estimators of the true state dependence parameter λ are all negative and range from
−0.102 to −0.039 in Table 3. They are statistically significant only in the correlated random
effects models. On the other hand, the estimators of the spurious state dependence parameter
ζ are slightly larger in Table 3 than in Table 2. The estimators of the ρ in Table 3 are similar
to the ones in Table 2. Thus, based on Cragg’s two-tiered models that adopt the correlated
random effects approach, there is a significant true state dependence in both the participation
and hours worked decision for the married women’s labor supply. The ignorance of the true
state dependence parameter, λ, will lead to an overestimation of the spurious state dependence
parameter, ζ, as shown in Tables 2 and 3.
In order to investigate the exogeneity of the fertility decisions and the husband’s income to
the married women’s labor supply function in various models, the Wald statistics and t statistics
of the following hypotheses are performed for the correlated random effects models in Tables 2
and 3.
The hypothesis that the fertility decisions are exogenous to the married women’s labor supply
function can be written as
H0 : ω2 = 0, ω3 = 0, ω4 = 0 (25)
where ω2, ω3 and ω4 are defined in (24). The fertility decisions are endogenous to the married
women’s labor supply function if the null hypothesis (25) is rejected. This null hypothesis can
be tested based on the Wald statistics.
Moreover, the hypothesis that the husband’s income is exogenous to the married women’s
22
labor supply function can be represented as
H0 : ω1 = 0
where ω1 is defined in (24). This null hypothesis can be tested using t statistics for the tradi-
tional Tobit model and Wald statistics for Cragg’s two-tiered model. The husband’s income is
endogenous to the married women’s labor supply function if H0 : ω1 = 0 is rejected.
The test results are reported in Table 4. At the 1% significance level, the hypothesis that
the fertility decisions are exogenous to the married women’s labor supply function is rejected
in both the dynamic and static models. Moreover, the hypothesis that the husband’s income
is exogenous to the married women’s labor supply function is also rejected in all models at the
1% significance level. Therefore, there is strong evidence that both fertility decisions and the
husband’s income may not be exogenous variables in both the static and dynamic models of the
married women’s labor supply function.
3.4 Average Partial Effects
In order to investigate the impact of the fertility decisions and the change in the husband’s
income on the married women’s participation decisions and expected hours worked, the APEs
of the fertility decisions and husband’s income changes for married women are presented using
equations (12)-(14) and equations (20)-(23). By employing both standard Tobit models and
Cragg’s two-tiered models, the following four events are considered: 1. having a child aged less
than 2 years old, 2. having a child aged between 3 and 5 years old, 3. having a child aged
between 6 and 13 years old, and 4. 10% increases in the husband’s income. Two types of APEs
of expected hours worked for married women are considered, that is, one unconditional and
another conditional on the labor force participation. It is assumed that a married woman has
no children in the base model. The age and husband’s income of the married woman are set
equal to the sample average, that is Age = 38 and Hinc = 31, 000. Moreover, two education
levels are considered for estimating the average partial effects, one being Edu = 12 (possibly
high school graduates), and the other Edu = 16 (possibly college graduates).
Three types of APEs are reported in Tables 5-7. P.P. stands for the labor force participation
23
Table 4. Tests of Exogeneity of Fertility Decisions and Husband’s Income
H0 : ω2 = 0, ω3 = 0, ω4 = 0 (exogeneity of fertility decisions)
Model Test Statistics p− value
Static RE+AR(1) 19.225 0.00025
Dynamic RE+AR(1) 43.680 0
Static Cragg (1st tier) 13.260 0.00410
Static Cragg (2nd tier) 20.260 0.00015
Static Cragg (both tiers) 25.550 0.00027
Dynamic Cragg (1st tier) 29.900 0
Dynamic Cragg (2nd tier) 36.346 0
Dynamic Cragg (both tiers) 46.492 0
H0 : ω1 = 0 (exogeneity of husband’s income)
Model Test Statistics p− value
Static RE+AR(1) 3.563 0.00037
Dynamic RE+AR(1) 3.501 0.00046
Static Cragg (1st tier) 3.296 0.00098
Static Cragg (2nd tier) 2.958 0.00310
Static Cragg (both tiers) 10.986 0.00410
Dynamic Cragg (1st tier) 4.023 0.00006
Dynamic Cragg (2nd tier) 3.609 0.00031
Dynamic Cragg (both tiers) 16.279 0.00029
24
probability for married women, E(y|x) represents the expected hours worked given the personal
characteristics x, and E(y|x, y > 0) refers to the expected hours worked given the personal
characteristics x and their being conditional on the participation. These three types of APEs
are reported for the differences from the base model in response to the fertility decision and the
change in the husband’s income, namely, events 1-4. Both the random effects and correlated
effects models are employed in Tables 5 and 6. In Table 7, only correlated effects models are
used.
The APEs of the fertility decision and the increase in the husband’s income for married
women with 12 years of education are reported in Table 5 for both the two-tiered and Tobit
models. For the labor force participation decision, it is shown in both the Tobit and two-tiered
models with either random effects or correlated random effects that blacks have higher labor
market participation rates than whites for married women with the same personal characteristics.
Moreover, blacks also have higher expected hours worked than whites with the same personal
characteristics, that are conditional or unconditional on the participation. Overall, the random
effects models which assume the exogeneity of fertility decisions and the husband’s income tend
to overestimate participation decisions and expected hours worked as well as the impacts of
fertility decisions and the increase in the husband’s income than the correlated random effects
models.
In comparing Cragg’s two-tiered models with the standard Tobit models, it is found that
the Tobit models tend to overestimate the participation probability and to underestimate the
expected working hours. One interesting observation in the correlated random effects two-tiered
models is that the married women, blacks or whites, will increase their participation probability
in the labor market but will decrease their expected hours worked if they have a child aged
between 6 and 13 years old. For all other models, the participation probability and expected
hours worked will both decrease if the married woman has a child aged between 6 and 13
years old. It is impossible for the Tobit models to detect evidence like that since those models
are restricted in that the direction of the changes in response to the fertility decision on the
participation probability and expected hours worked should always be the same. The impacts
of the increase in the husband’s income on the participation probability and expected hours
25
Table 5. Average Partial Effects Estimations with 12 Years of Education
Two-Tiered; Base Model(Edu=12, Age=38, Hinc=31,000, C12=0, C35=0, C613=0)
Whites Blacks
P.P. E(y|x) E(y|x, y > 0) P.P. E(y|x) E(y|x, y > 0)
Base Model(RE) 0.869 1361.0 1567.0 0.882 1559.0 1768.4
Base Model(CRE) 0.839 1275.8 1505.7 0.858 1479.1 1707.7
Differences from the Base Model
C12 = 1(RE) -0.059 -203.8 -137.0 -0.056 -219.8 -147.1
C12 = 1(CRE) -0.027 -118.8 -96.13 -0.025 -128.8 -103.8
C35 = 1(RE) -0.044 -174.2 -128.0 -0.041 -187.9 -137.4
C35 = 1(CRE) -0.019 -95.68 -81.44 -0.018 -103.9 -87.85
C613 = 1(RE) -0.005 -61.47 -61.70 -0.005 -66.50 -66.00
C613 = 1(CRE) 0.004 -16.39 -25.44 0.003 -18.32 -27.36
10% ↑ Hinc(RE) -0.001 -8.5 -6.6 -0.002 -9.2 -7.1
10% ↑ Hinc(CRE) -0.002 -4.8 -2.8 -0.001 -4.8 -3.2
Tobit; Base Model(Edu=12, Age=38, Hinc=31,000, C12=0, C35=0, C613=0)
Base Model(RE) 0.909 1217.6 1339.7 0.942 1409.3 1496.8
Base Model(CRE) 0.858 1183.7 1368.7 0.884 1298.4 1458.0
Differences from the Base Model
C12 = 1(RE) -0.043 -178.7 -139.6 -0.032 -186.4 -152.9
C12 = 1(CRE) -0.032 -118.1 -89.72 -0.028 -122.1 -95.01
C35 = 1(RE) -0.038 -158.8 -124.4 -0.027 -165.5 -136.1
C35 = 1(CRE) -0.026 -96.87 -73.74 -0.023 -100.1 -78.06
C613 = 1(RE) -0.014 -62.72 -49.76 -0.010 -65.12 -54.23
C613 = 1(CRE) -0.006 -22.36 -17.15 -0.005 -23.06 -18.13
10% ↑ Hinc(RE) -0.002 -7.7 -6.2 -0.002 -7.9 -6.6
10% ↑ Hinc(CRE) -0.001 -4.0 -3.1 -0.001 -4.1 -3.2
26
worked are relatively negligible compared with the impacts of the fertility decisions.
The APEs of the fertility decisions and the increase in the husband’s income for married
women with 16 years of education are presented in Table 6 for both the two-tiered and Tobit
models. It is reported in Table 6 that, with all other things being equal, the higher the education
level, the higher the labor force participation probability and expected hours worked that the
married women will have. Moreover, in response to the fertility decisions and the increase in
the husband’s income, the APEs have a smaller impact on labor force participation probabil-
ity for a married woman with 16 years of education than for one with 12 years of education.
Overall, the better educated married women will have higher labor force participation rates
and higher numbers of expected hours worked, whether they be conditional or unconditional on
the participation. Moreover, their participation decisions are less sensitive to the fertility deci-
sions than those of the less educated married women. In Table 6, blacks also have higher labor
market participation rates and expected hours worked than the whites in the case of married
women with the same personal characteristics. Furthermore, when compared with the correlated
random effects models, the random effects models also tend to overestimate the participation
probability, the expected hours worked and the impacts of fertility decisions and the increase in
the husband’s income. As in Table 5, the impacts of the increase in the husband’s income on
the participation probability and expected hours worked are also relatively negligible compared
with the impacts of the fertility decisions in Table 6.
In order to investigate the impacts of the fertility decisions on different age groups, the
APEs for married women between 20 and 40 years old are presented in Table 7 for two different
education levels. The participation probability and expected hours worked reported in Table 7
are for the married women with the same personal characteristics as in the base model, except
that C12 = 1, that is, with one child aged less than 2 years old. The numbers reported in
the parentheses are the APEs that are different from those in the base model. That is, these
numbers represent the impact of having one child aged less than 2 years old on the participation
probability and expected hours worked. For example, for a 20-year-old white married woman
with only one child aged less than 2 years old, a husband with an income of $31, 000, and 12 years
of education, the labor force participation rate is expected to be 0.75, the expected numbers of
27
Table 6. Average Partial Effects Estimations with 16 Years of Education
Two-Tiered; Base Model(Edu=16, Age=38, Hinc=31,000, C12=0, C35=0, C613=0)
Whites Blacks
P.P. E(y|x) E(y|x, y > 0) P.P. E(y|x) E(y|x, y > 0)
Base Model(RE) 0.916 1454.4 1588.0 0.925 1656.8 1790.8
Base Model(CRE) 0.910 1473.4 1608.2 0.922 1686.7 1817.3
Differences from the Base Model
C12 = 1(RE) -0.044 -190.9 -138.2 -0.041 -204.3 -148.1
C12 = 1(CRE) -0.018 -118.3 -100.3 -0.016 -126.1 -107.1
C35 = 1(RE) -0.033 -165.7 -129.1 -0.030 -177.4 -138.3
C35 = 1(CRE) -0.013 -96.59 -84.91 -0.012 -103.1 -90.64
C613 = 1(RE) -0.004 -62.78 -62.21 -0.003 -67.45 -66.41
C613 = 1(CRE) 0.002 -20.68 -26.48 0.002 -22.55 -28.19
10% ↑ Hinc(RE) -0.001 -8.2 -6.6 -0.001 -8.6 -7.1
10% ↑ Hinc(CRE) -0.001 -4.1 -2.9 0.000 -4.3 -3.1
Tobit; Base Model(Edu=16, Age=38, Hinc=31,000, C12=0, C35=0, C613=0)
Base Model(RE) 0.938 1385.6 1477.0 0.962 1582.4 1645.1
Base Model(CRE) 0.915 1466.2 1592.7 0.933 1587.9 1693.5
Differences from the Base Model
C12 = 1(RE) -0.033 -185.6 -151.3 -0.023 -191.3 -163.5
C12 = 1(CRE) -0.023 -102.3 -225.4 -0.019 -129.6 -107.2
C35 = 1(RE) -0.029 -164.8 -134.7 -0.020 -169.8 -145.5
C35 = 1(CRE) -0.018 -103.9 -84.00 -0.015 -106.1 -88.01
C613 = 1(RE) -0.010 -64.87 -53.71 -0.007 -66.63 -57.76
C613 = 1(CRE) -0.004 -23.88 -19.47 -0.003 -24.36 -20.37
10% ↑ Hinc(RE) -0.001 -7.9 -6.6 -0.001 -8.0 -7.1
10% ↑ Hinc(CRE) 0.000 -4.3 -3.5 0.000 -4.4 -3.6
28
Table 7. Average Partial Effects Estimations with Different Age Groups
Whites Blacks
Age P.P. E(y|x) E(y|x, y > 0) P.P. E(y|x) E(y|x, y > 0)
Two-tiered CRE; Base Model(Edu=12, Hinc=31,000, C12=0, C35=0, C613=0)
20 0.750(−0.032)
945.4(−109.1)
1242.1(−87.73)
0.776(−0.030)
1116.9(−121.0)
1420.4(−96.62)
25 0.787(−0.029)
1061.3(−114.6)
1331.4(−92.43)
0.811(−0.027)
1245.3(−125.5)
1518.7(−100.7)
30 0.807(−0.028)
1135.2(−117.6)
1389.4(−95.21)
0.829(−0.025)
1326.3(−127.9)
1582.0(−103.3)
35 0.814(−0.027)
1163.1(−118.8)
1412.7(−96.28)
0.835(−0.025)
1356.8(−128.8)
1607.4(−103.9)
40 0.807(−0.028)
1143.5(−118.4)
1400.2(−95.71)
0.829(−0.025)
1335.6(−128.6)
1593.8(−103.4)
Two-tiered CRE; Base Model(Edu=16, Hinc=31,000, C12=0, C35=0, C613=0)
25 0.875(−0.020)
1258.3(−115.4)
1425.9(−96.82)
0.891(−0.018)
1456.3(−124.1)
1621.7(−104.4)
30 0.889(−0.019)
1332.3(−117.4)
1486.8(−99.43)
0.904(−0.017)
1536.1(−125.5)
1687.4(−106.5)
35 0.893(−0.018)
1360.6(−118.3)
1511.3(−100.4)
0.908(−0.016)
1566.4(−126.1)
1713.7(−107.2)
40 0.889(−0.019)
1342.0(−118.1)
1498.1(−99.89)
0.903(−0.017)
1546.7(−126.0)
1699.6(−106.8)
29
working hours 945.4 hours per year, and the expected numbers of working hours 1242.1 hours
per year conditional on the participation. Moreover, because of this younger child (less than
2 years old), it is expected, compared with the base model, that her labor force participation
rate will decrease by 3.2%, the expected working hours will decrease by 109.1 hours, and the
expected numbers of hours worked conditional on the participation will decrease by 87.73 hours.
From Table 7, it is also shown that the labor force participation rate and expected hours
worked for married women peak around the age of 35, and then decrease after that. Moreover,
the impacts of a young child on the labor force participation rate seem to be larger for the
younger and less educated married women than the older and better educated ones for both
blacks and whites. Overall, blacks have a higher labor force participation rate and expected
hours worked than the same age whites with the same personal characteristics. However, the
impacts of a younger child on the labor force participation rate and the expected hours worked,
in terms of the percentage reduction from the base model for married women, are larger for
whites than for blacks for all age groups.
3.5 Fixed Costs and Labor Supply for Married Women
Cogan (1981) and Morz (1987) indicate that the restrictions imposed by the simple Tobit spec-
ification are violated because of the significant fixed costs associated with entry into the labor
market for the married women’s labor supply. Following Cogan (1981), let the working hours
equation hi and the reservation hours equation h⋆i for married women be:
hi = β′xi + λhi,−1 + ϵi (26)
h⋆i = γ′xi + δhi,−1 + νi (27)
where the xi have the same explanatory variables as in the previous subsection. Assume that
the differences in the parametric specifications of the labor force participation decision and
hours worked decision that are conditional on the participation are all due to the fixed costs
associated with entry into the labor market for married women. In other words, the simple Tobit
specification would be an appropriate model to use if there were no entry cost. Furthermore, it
is assumed that the error term in (27) ν = 0, or that ν is negligible compared to the error term
30
Table 8. Sign Estimates for the Reservation Hours Equation of Married Women
Variable (3) of Table 2 (3) of Table 3 (4) of Table 2 (4) of Table 3 Cogan (1981)
Edu − − − − −
Race + + + + n.a.
Age + + + + +
Age2 − − − − −
Hinc − − + + +
C12 + + + − +
C35 − + − − +
C613 − − − − n.a.
in (26).
Thus, a married woman decides to participate in the labor force if
hi > h⋆i
Based on Cragg’s two-tiered model in Section 2, let Γ1 = (β1, λ1, ω1) be the first-tiered pa-
rameters which determine the initial decision between participation hi > h⋆i or no participation
hi ≤ h⋆i . On the other hand, once a married woman decides to participate in the labor force,
let Γ2 = (β2, λ2, ω2) be the second-tiered parameters which determine the number of working
hours hi given hi > 0. Thus, based on the assumptions described above, β1 = β−γ, λ1 = λ− δ,
β2 = β and λ2 = λ.8 Therefore, the parameters in the reservation hours equation, γ and δ, can
be recovered by
γ = β2 − β1
δ = λ2 − λ1
The estimates of the signs of γ are reported in Table 8 from both the static and dynamic
versions of Cragg’s two-tiered models in Tables 2 and 3. It is shown in Table 8 that all models
have the same signs for Edu, Race, Age, and Age2 in the case of the reservation hours equation.
8If there is no fixed cost, that is, h⋆i = 0, then β1 = β2 and λ1 = λ2
31
The husband’s income has a negative impact on the reservation hours equation in the random
effects Cragg two-tiered model (model (3) of Tables 2 and 3), but it has a positive impact on the
reservation hours equation in the correlated random effects models. In comparing the results of
the correlated random effects models in Table 8 with the estimation results of the reservation
hours function in Cogan (1981), the signs are the same for the wife’s education, the husband’s
earnings and the wife’s age. In addition, it is found in Table 8 that black women might have
higher reservation hours than white women. Also shown in Cogan’s estimation results is that
the number of children aged between 0 and 6 years will have a positive impact on the reservation
hours. For the correlated random effects two-tiered models in Table 8, although the number
of children aged between 1 and 2 years old will increase the reservation hours, the number of
children aged between 3 and 13 years old will decrease the reservation hours for married women.
3.6 Monte Carlo Experiments
In this subsection, a small number of Monte Carlo experiments are performed in order to provide
supporting evidence for the estimation results of Cragg’s two-tiered models. For each experi-
ment, the Monte Carlo data set is generated from two-tiered models and estimated by both
standard Tobit and two-tiered models. The initial conditions problems are also investigated
using Heckman’s approach.
In order to replicate the variation in the empirical data, the exogenous variable in the Monte
Carlo experiments, Zit, is a standardized composite variable generated from the data used in the
empirical analysis. Specifically, Zit = (Xitθ−Z)/σZ , where Xit is a vector of variables in Table
3, and θ is a vector of parameters estimated in the first tier of Model (4) in Table 3. Moreover,
Z =
∑N
i=1
∑T
t=1Xitθ
NT , and σZ =√σ2d + σ2
u/(1− ζ2), where σ2d, σ
2u and ζ are estimated in Model
(4) of Table 3.9
The Monte Carlo data sets are generated from the two-tiered models. In the first tier, the
probability of a zero is given by a probit model with a parameter vector β10 for an intercept, β11
for an exogenous variable Zit, and λ1 for a lagged dependent variable. On the second tier, the
density of the dependent variable conditional on being a positive observation is assumed to be
9A similar approach is used in the Monte Carlo experiments in Hyslop (1999).
32
truncated at zero with a parameter vector β20 for an intercept, β21 for an exogenous variable Zit,
and λ2 for a lagged dependent variable. The error terms uit and di are assumed to satisfy the
conditions in (3) and are assumed to be normally distributed. The covariance matrix is assumed
to follow the random effects plus AR(1) specification with σ2u = 0.5, σ2
d = 0.5 and ζ = 0.5.
The simulation estimator is obtained by maximizing the simulated log-likelihood function of the
Monte Carlo data set. In all Monte Carlo experiments, the number of individuals, N , is set
at 1,000, and each individual is observed in five time periods. Each data generating process is
simulated 20 times, and estimated by using the GHK simulation estimators with 10 simulation
draws. The Monte Carlo data sets are estimated by both Cragg’s two-tiered model and the
standard Tobit model.
The first two Monte Carlo experiments are conducted without an initial conditions problem
in Table 9. That is, all the data are observed from the beginning of the data generating process,
and it is assumed that there is no lagged dependent variable in the first period. The censoring
frequency is 0.4328 for experiment I and is 0.5091 for experiment II in Table 9. The mean,
median, 5th percentile, 95th percentile, and standard error (S.E.) of the simulation estimators
are reported for each Monte Carlo experiment. In experiment I of Tables 9 and 10, the slope
parameters in the first and second tiers have the same sign but have different magnitudes. On
the other hand, the slope parameters in the first and second tiers have the same magnitude but
have different signs in experiment II of Tables 9 and 10.
It is shown in Table 9 that the estimators of β and λ for the Tobit model tend to be the
average of the true parameters from the first and second tiers. For the two-tiered estimators,
the signs of the parameters are all correctly estimated, although they tend to underestimate
most of the parameters. For the estimation of σ2u, σ
2d and ζ, the two-tiered estimators tend to
underestimate them. On the other hand, Tobit estimators tend to overestimate σ2u, σ
2d and ζ
due to the restrictions on the parameters imposed by the Tobit models.
For the investigation of the initial conditions problem, it is assumed that the Monte Carlo
data are observed from the middle of the data generating process. The Monte Carlo data set is
generated using the true parameter values in Table 10 for 9 periods, but only the last 5 periods
of the data are observed and used in the estimation. The Heckman approach is used to deal
33
Table 9. Random Effects plus AR(1) Model without Initial Conditions Problem
I: Parameter β10 β11 λ1 β20 β21 λ2 σ2u σ2
d ζ
TrueV alue 0.000 1.000 0.500 0.500 1.500 0.300 0.500 0.500 0.500
Two-Tiered:
Mean -0.005 0.718 0.789 1.094 1.352 0.264 0.411 0.269 0.367
Median -0.001 0.726 0.780 1.099 1.353 0.265 0.408 0.272 0.371
5thperc. -0.049 0.590 0.733 1.054 1.212 0.230 0.384 0.212 0.309
95thperc. 0.027 0.798 0.881 1.124 1.512 0.296 0.439 0.320 0.418
S.E. 0.028 0.075 0.051 0.022 0.100 0.019 0.018 0.038 0.041
Tobit:
Mean 0.231 1.531 0.307 - - - 0.709 0.752 0.567
Median 0.227 1.531 0.308 - - - 0.710 0.749 0.575
5thperc. 0.154 1.345 0.262 - - - 0.670 0.600 0.501
95thperc. 0.308 1.702 0.353 - - - 0.768 0.851 0.617
S.E. 0.048 0.129 0.027 - - - 0.030 0.079 0.042
II: Parameter β10 β11 λ1 β20 β21 λ2 σ2u σ2
d ζ
TrueV alue 0.000 1.000 0.500 0.000 -1.000 0.300 0.500 0.500 0.500
Two-Tiered:
Mean -0.067 0.587 0.746 0.310 -1.896 0.241 0.455 0.574 0.467
Median -0.060 0.580 0.760 0.292 -1.878 0.240 0.451 0.587 0.471
5thperc. -0.115 0.397 0.660 0.257 -2.146 0.209 0.432 0.486 0.401
95thperc. -0.014 0.739 0.804 0.377 -1.690 0.270 0.478 0.654 0.517
S.E. 0.034 0.108 0.057 0.044 0.136 0.020 0.017 0.061 0.040
Tobit:
Mean -0.092 0.120 0.299 - - - 0.628 0.858 0.563
Median -0.094 0.141 0.298 - - - 0.625 0.873 0.570
5thperc. -0.156 -0.120 0.270 - - - 0.601 0.703 0.497
95thperc. -0.022 0.280 0.325 - - - 0.659 1.006 0.613
S.E. 0.044 0.139 0.024 - - - 0.021 0.102 0.037
34
with the initial conditions problem.
The censoring frequency is 0.4093 for experiment I and is 0.4895 for experiment II in Table
10. Like Table 9, the estimations of β and λ for the Tobit model in Table 10 tend to be the
average of the true parameters from the first and second tiers. Moreover, the Tobit estimators
tend to overestimate σ2u, and σ2
d and ζ. Overall, the two-tiered estimators perform equally well
with or without the initial conditions problem.
It is indicated in the Monte Carlo experiments that when the true data generating process is
determined by two different sets of parameters, Cragg’s two-tiered estimators are preferable to
the traditional Tobit estimators since the Tobit estimators tend to perform like the average of the
first- and second-tiered parameters. Thus, when the true data generating process is unknown,
Cragg’s two-tiered estimators may be better choices than the Tobit estimators.
4 Conclusion
This paper has proposed a computationally practical simulation estimator for dynamic panel
Tobit models as well as Cragg’s two-tiered dynamic panel Tobit models based on maximizing
simulated log-likelihood functions. The proposed Cragg two-tiered simulation estimators are
then applied to study married women’s labor supply. The rich dynamic structures of the labor
supply of married women have been identified through the proposed simulation estimators.
By applying the two-tiered dynamic panel Tobit model based on the correlated random ef-
fects approach, it is found that education has a positive impact not only on the labor force
participation decision, but also on the hours worked decision which is conditional on the par-
ticipation. However, race may only have a statistically significant positive impact on the hours
worked decision that is conditional on the participation, but not on the labor force participation
decision. In addition, children aged between 6 and 13 years old may have a negative impact on
the hours worked decision for married women that is conditional on the participation. However,
these children may provide some positive incentives for married women to participate in the
labor force. Furthermore, the hypothesis that the fertility decision is exogenous and the hypoth-
esis that the husband’s income is exogenous to the married women’s labor supply function are
35
Table 10. Random Effects plus AR(1) Model with Initial Conditions Problem
I: Parameter β10 β11 λ1 β20 β21 λ2 σ2u σ2
d ζ
TrueV alue 0.000 1.000 0.500 0.500 1.500 0.300 0.500 0.500 0.500
Two-Tiered:
Mean 0.232 1.080 0.582 1.239 1.718 0.067 0.472 0.410 0.629
Median 0.233 1.076 0.582 1.237 1.780 0.065 0.478 0.391 0.638
5thperc. 0.157 0.906 0.507 1.179 1.502 0.022 0.438 0.326 0.580
95thperc. 0.301 1.262 0.648 1.304 1.859 0.111 0.494 0.513 0.669
S.E. 0.057 0.125 0.055 0.047 0.130 0.030 0.021 0.071 0.034
Tobit:
Mean 0.474 1.901 0.077 - - - 0.703 0.728 0.758
Median 0.476 1.898 0.076 - - - 0.699 0.744 0.764
5thperc. 0.406 1.658 0.048 - - - 0.663 0.343 0.696
95thperc. 0.555 2.098 0.106 - - - 0.748 1.029 0.809
S.E. 0.050 0.148 0.023 - - - 0.025 0.229 0.035
II: Parameter β10 β11 λ1 β20 β21 λ2 σ2u σ2
d ζ
TrueV alue 0.000 1.000 0.500 0.000 -1.000 0.300 0.500 0.500 0.500
Two-Tiered:
Mean 0.049 0.455 0.498 0.367 -1.883 0.069 0.493 0.725 0.665
Median 0.054 0.435 0.484 0.361 -1.888 0.067 0.496 0.733 0.664
5thperc. -0.009 0.344 0.431 0.285 -2.049 0.041 0.453 0.543 0.618
95thperc. 0.131 0.634 0.567 0.497 -1.727 0.098 0.524 0.867 0.720
S.E. 0.047 0.102 0.053 0.064 0.118 0.019 0.026 0.111 0.038
Tobit:
Mean 0.045 0.084 0.074 - - - 0.636 0.866 0.756
Median 0.031 0.077 0.081 - - - 0.638 0.915 0.758
5thperc. -0.029 -0.089 0.028 - - - 0.601 0.487 0.695
95thperc. 0.142 0.253 0.109 - - - 0.666 1.126 0.811
S.E. 0.053 0.129 0.024 - - - 0.024 0.226 0.039
36
both rejected in the dynamic and static two-tiered models.
Acknowledgements
I would like to thank Arthur Goldberger, Bruce Hansen, and Gautam Tripathi for their helpful
discussions and suggestions. I am especially indebted to Yuichi Kitamura for his encouragement
and invaluable guidance. I would also like to thank Professor John Rust and two anonymous
referees for their useful comments and suggestions. All errors which remain are mine.
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