[This is a very preliminary draft. Please do not quote without an author’s permission]
Estimating lifecycle labor supply model for men using sample selection corrected panel
data models: Canadian evidence
Eyob Fissuh1
This version: April 2008
Department of Economics
University of Manitoba
Winnipeg, Canada
R3T 5V5
Phone: 204-474-6291
E-mail: umghebr2@cc.umanitoba.ca
1
Eyob would like to thank Wayne Simpson for his continued guidance and advice during the various stages of this
paper. I received many useful comments and suggestions from L.Brown and L.Wang. Eyob would like also to
thank University of Manitoba Research Data Centre staff –Ian Clara and Kelly Cranswick- for their help in
accessing internal files of Survey of Labor and Income Dynamics (SLID). The financial assistance from the
University of Manitoba Graduate Fellowship (UMGF) and RDC graduate research award are acknowledged. All
remaining errors are mine.
1
Estimating lifecycle labor supply model for men using sample selection corrected panel
data models: Canadian evidence
Abstract
This paper estimates lifecycle labor supply model for men in Canada using the second panel of Canadian
Survey of Labor and Income Dynamics (SLID) spanning from 1996 to 2001. This paper explores the
application of several panel data models in estimating a lifecycle labor supply model. The estimated
intertemporal labor supply substitution is compared across the different panel data estimators. The results
indicate that lifecycle labor supply estimates are sensitive to the econometric specification of unobserved
individual heterogeneity and non-random sampling. This paper confirms that failure to control for the
individual heterogeneity and sample selection bias produces upward biased parameter estimates. Panel data
models which do not control for sample selection bias gave an intertemporal labor substitution between
0.23 and 0.48. Using a model which controls for sample selection problem and individual heterogeneity,
intertemporal substitution elasticity was found to be 0.16. Moreover we find the wealth effects to be
statistically insignificant implying that the intertemporal elasticity is a very good approximation of both the
own compensated and uncompensated labor supply elasticity. Our empirical results reveal that it is
important to distinguish between the effect of evolutionary and parametric wage changes on labor supply
decisions of individuals over their lifecycle.
I. Introduction
Lifecycle labor supply decision-making implies that economic agents aim at maximizing
lifetime utility subject to their resource constraints. Situating the labor supply decision of an
individual in a lifecycle environment allows estimation of econometrically meaningful
parameters (MaCurdy, 1981). The proper estimation of a lifecycle labor supply model permits
the estimation of the intertemporal elasticity of substitution. The estimates of the
intertemporal substitution elasticities are important for at least the following three reasons.
Firstly, they can be used for welfare analysis in studies such as income tax, social benefit,
subsidies, pension, childcare and the like2. Secondly they are very important links to the
macroeconomics of fluctuations in the Real Business Cycle (RBC) theory.3 Intertemporal
elasticity of substitution is also important in government policies which aim at influencing
2
See Kumar (2005) for an elegant application of the effect of taxation on female labor supply in the USA. The
third proposal aims at examining maternal employment and child care mode choice. In doing so, we will use
Frisch labor supply estimates and their associated wealth effects.
3
As it is well known in the literature one of the biggest challenges of the RBC theory is the empirically small
intertemporal substitution elasticity of labor (Romer, 2006; Rebello, 2003). There are some studies that produce a
sizable intertemporal elasticity of substitution by employing biannual data as opposed to annual data which are
common in national surveys. Kimmel and Ksiener(1998) argue that the intertemporal elasticity of substitution is
buried in the data and they suggest for the use of disaggregated data to produce the size and sign of the
intertemporal elasticity of substitution to resurrect the RBC theory. Overall, however, the intertemporal elasticity
has been very small ( See Table 7 in the text for summary of some studies conducted in North America, Europe
and Asia).
2
savings (Altonji and Ham, 1990). This paper contributes to the literature by estimating,
among others, theory consistent intertemporal substitution elasticities using a rich Canadian
panel data by employing appropriate techniques.
The lifecycle labor supply model shows the presence of person-specific effects in labor
supply decisions. The econometric implication is that cross-sectional regressions are
contaminated with omitted variables bias and inconsistency problems. The fact that the fixed
effects are composed of observables and non-observables calls for panel data models to
control individual heterogeneity. For this reason there has been an increased employment of
panel data in the labor supply literature (Jakubson, 1988).
Despite the surge of interest on labor supply behavior of men in the USA in the last three
decades4, the research on lifecycle labor supply behavior of men in Canada is scanty. Reilly
(1994) is the first attempt in estimating lifecycle labor supply model of men in Canada. Reilly
employs a dynamic model on the basis of annual hours and weeks per year worked which is
an extended version of Hanoch(1980). While the estimated intertemporal elasticity with
respect to annual number of hours was not statistically significant, Reilly finds the
intertemporal substitution elasticity with respect to number of weeks worked to be 0.6. Reilly
uses cross-sectional data from a limited sample of nongovernmentally employed men between
the ages of 25-55 in the Maritime Provinces of Canada. Our study however employs a
representative panel data from the Survey of Labor and Income Dynamics(SLID) from five
provinces of Canada for all men between age 25 and 60 years old both employed and
unemployed. Our panel data allows us to control for individual heterogeneity by correcting
for sample selection bias using a fixed effects sample selection model. Furthermore, in
contrast to what Reilly reports, our results suggest the intertemporal substitution elasticity
4
Pencavel (1986) presents an excellent review of studies on labor supply behaviour of men in Anglo-American
countries. Apart from the general discussion on the trends of labor force participation, the survey does not contain
any Canadian study. Frequently cited studies on lifecycle labor supply of men in the USA include MaCurdy
(1981, 1983), Altonji(1986), Ham(1986), and Abowd and Nard(1989).
3
with respect to annual hours worked to be in the range of 0.16 to 0.48 with the lower bound
being the estimate from the model which controls for individual heterogeneity and sample
selection problem. We also find the wealth effects, as well as the elasticity of wages with
respect to the marginal utility of wealth to be negative but statistically insignificant. Thus our
results suggest that the intertemporal substitution elasticity is a reasonable approximation to
the uncompensated and compensated labor supply elasticities.
Any stochastic specification of labor supply will have to account for nonrandom sampling of
any empirical data. Positive number of hours worked is only reported for those who have
worked during the reference year and we do not observe the wages if the number of hours
worked is zero. This selection problem has been long identified in the literature (Wales and
Woodland 1980, Gronau 1974, Lewis 1974, Heckman 1980). Failure to account for this
problem of sample selection will produce biased and inconsistent estimates (Heckman 1980).
This study experiments with different variants of Tobit models in panel data which control
endogeneity of wages and individual heterogeneity and correct for sample selection bias.
This study will estimate a wage offer equation to predict the missing wage and use the
predicted wages as instruments for actual wage. Most previous studies use OLS to predict
wages for the missing wage. However the inconsistency of the OLS regression on the
censored data will contaminate the labor supply estimates withinconsistency and inefficiency
(Vella 1998). In this paper the wage offer equation will be modeled as Type II Tobit model
(Amemiya 1980) and the labor supply equation-reduced number of hours equation will be
estimated using Type II Tobit model in a panel data framework. The current study will
employ a very rich panel data from SLID to estimate lifecycle labor supply model parameters
for Canada by formally testing for the presence of sample selection in the data and will make
the necessary corrections. This study will also estimate variants of the Tobit model in a panel
data context such that it controls for the heterogeneity that is implied by the theoretical
lifecycle labor supply model. This will help to facilitate the comparison of our models to
those already existing in the literature.
4
The objectives of the paper are three-fold. Firstly, the paper estimates labor supply elasticities
using theory consistent econometric methods. Secondly, the paper aims at investigating the
sensitivity of lifecycle labor supply parameter estimates to the changes in econometric
estimation strategies. To achieve this goal we will estimate different limited dependent
variable models in a panel data context. Mainly we employ Tobit and sample selection
models that control for unobservable individual effects in the labor supply equation. To
increase the reliability of our estimates we try to improve on the instrumentation of wages, as
it is well known wages will be observed if and only if the number of hours worked is above
zero. To this end we model the wage offer equation as a Type II Tobit model where the
selection equation is reduced form number of hours equation. This will be a good contribution
to the literature as it will be the first of its kind to circumvent the missing wages for nonworkers and control endogeneity of wages using the SLID data set. Unlike most of the
literature we include tenure and tenure squared as instruments in our wage prediction.
Thirdly, the paper aims to understand the labor supply behavior of men in Canada using the
rich data set from SLID. To the best knowledge of the author no one has estimated an explicit
lifecycle labor supply model for men using SLID data by employing sample selection
models.5 Thus this paper will help to fill the gap on the hardly existent study of lifecycle labor
supply behavior of men in Canada.
The rest of the paper is organized as follows. Section two presents the theoretical
underpinning of the lifecycle labor supply model. Section three develops the econometric
strategy. In this section we discuss variants of Tobit and sample selection6 models. Section
four presents estimation results. Section five concludes.
II. Theoretical framework
5
Hum, Simpson and Fissuh (2006) use SLID data to study the effect of health on labor supply behavior of men in
Canada. They estimate logit models which are theory consistent.
6
Sample selection model can be considered to be a generalized Tobit model (Heckman 1978).
5
The theoretical setting of our paper is based on basic theory of consumer behaviour where a
consumer is faced with a problem of maximizing lifetime utility subject to wealth constraint.
A similar theoretical framework has been employed by Heckman (1976), Heckman and
MaCurdy (1980), MaCurdy (1981) Jackubson (1988) Card (1998) French (2004), Cahuc
Zygelberger(2004) and other authors. Hence the paper does not aim at detailed derivation of
the model. We will only note the main results.
Assume that a consumer faces a choice between leisure and work over his lifetime. Assume
also that the utility function is quasiconcave at age t which is given by U [Ct , Lt ); X t )]
where Ct , Lt and H t are within-period consumption, leisure time and a vector of observed
time variant individual characteristics7 of the agent respectively. Thus lifetime utility of a
typical consumer to time t is given by
U [(ct , lt ; xt , ct +1 , lt +1 ; xt +1 ,..., cT , lT ; xT ]
8
1.0
The consumer is limited by the time path of his wealth constraint9:
at = (1 + rt )at −1 + d t + wt ht − pt ct
[1.1]
where ht = (l * −lt ) such that l * is total number of hours available for work; at is real asset
at time t ; at −1 is real asset endowment of the consumer at time t − 1 ; wt is exogenously given
within period wage level; pt is within period price of consumption ct ; d t is within period
non-wage income10. In this dynamic setting we also assume that the consumer can freely
borrow and save at each point in time at an interest rate rt and the time discounting factor
is ρ .
Hence the utility maximization problem of our consumer is one of
7
X t Contains observable individual characteristics.
8
Limitation of our utility function among others includes the persistence of consumption habit, human capital
formation, trade off between leisure, training and consumption (See Hotz et al 1988).
9
This formulation or set up ignores the role of taxes (progressive) which might impact the separability of choices.
See Blomquist(1985) for the effect of progressive taxes on the separability of choices and Kumar(2005) for the
effect of relaxing the linearity in the budget constraint on the labor supply response parameters.
10
Strictly speaking non-wage income is
d t + rt a t −1 .
6
T
Max
= ∑ U (ct , lt ; xt )(1 + ρ ) −t
U
{ct ,lt ,at }
[1.02]
t =0
T
Subject to
∑ κ [a
t
t
− (1 + rt )at −1 − d t − wt ht − pt ct ](1 + rt ) −t
t =0
This problem could be solved by Lagrange technique. The first order conditions for
maximization, normalizing the price of consumption to 1, are given by
U c [ct , lt ; xt ] −
1+ ρ
κt = 0
1+ r
U l [ct , lt ; xt ] − κ t
[1.03]
1+ ρ
wt ≥ 0
1+ r
[1.04]
κ t = (1 + rt +1 )κ t +1
[1.05]
where U c (.) and U l (.) are the marginal utilities with respect to within period consumption
and leisure. In the optimal path, conditions (1.3)-(1.5) unambiguously suggest that the
marginal rate of substitution between consumption and leisure remains the same at each time
period. Note that this optimal path equilibrium is an outcome of the additive separability
assumption invoked onto the utility function and is not a general result. Given the first order
conditions, explicit solutions for leisure11 and consumption demand can be written as:
ct = c( wt , (1 + ρ ) t (1 + r ) − t κ t , pt ; xt )
[1.06]
ht = h( wt , (1 + ρ ) t (1 + r ) − t κ t , pt ; xt )
[1.07]
ht > 0 if − ∂∂Uht |κ t =κ 0 = ( 11++ρr ) t wt

where ht = 
1+ ρ t
∂U
ht = 0 if - ∂ht κ t =κ 0 > ( 1+ r ) wt
The above solutions for consumption demand and labor supply functions show that regardless
of the participation decision of our consumer over the lifetime, labor supply and consumption
demand are entirely determined by the functional form of the utility functions, κ 0 , taste
factors, the interest rate, ,the discount rate ρ and contemporaneous wage rate. Most
11
Recall that ht
= (l * −lt ) .
7
importantly the consumption demand and labor supply functions given by (1.6) and (1.7) are
defined for a given marginal utility of wealth ( κ t ). In the literature these demand functions
are knows as κ constant demand functions (Browning and Meghir 1991, Heckman and
MaCurdy 1980).
It is worth noting that κ t summarizes the effect of lifetime tastes and prices and other
variables included in the preference function. It is clear from equation (1.3) and (1.4) the
reservation wage in the lifecycle model depends on lifetime tastes and prices summarized by
the marginal utility of wealth, κ t . The marginal utility of wealth in turn depends on the other
variables included in the consumer problem. To see this, substitute the demand functions into
the constraint to get an implicit expression for the optimal κ t :
T
∑ κ [a
t
t
− (1 + rt )at −1 − d t + wt ht ( wt (1 + ρ ) t (1 + r ) −t κ t , pt ; xt )
t =0
t
−t
[1.08]
−t
+ pt c( wt , (1 + ρ ) (1 + r ) κ t , pt ; xt )](1 + rt ) = 0
The κ constant demand functions help us to explore the responsiveness of labor supply to
changes in wages. However, unlike the Marshallian demand or Hicksian demand elasticity
functions which keep income and utility constant respectively, the elasticity computed from
this function keeps marginal utility of wealth, κ t ,constant. The Frisch elasticity measure from
(1.7) is given by
∂ht wt
∂wt ht
[1.09]
κ t =κ
It will prove empirically useful to flesh out expression [1.05] which is the time path of the
Euler equation for the optimal path of marginal utility of wealth, κ t , for our estimation
strategy. Repeated iteration of the Euler equation yields
8
ln κ t = ln κ 0 −
∑ ln(1 + r )
[1.10]
t
It is evident from [1.10] that the path of the marginal utility of lifetime wealth has a fixed
effects component which is specific to an individual, ln κ 0 , and a component which is path
dependent representing a common effect. Note that equation (1.10) suggests that the time path
of κ t is independent of variations in wages which implies that the appropriate elasticity
measure to gauge the effect of wage variation at a certain point in the lifetime is the Frisch
elasticity. This way of decomposing the marginal utility of wealth proves to be empirically
very useful. Expression (1.10) has an important econometric implication. Lifecycle labor
supply models which ignore individual heterogeneity are intrinsically flawed. 12As we have
discussed above, κ t is a function of individual attributes and other taste shifters. Thus,
assuming no correlation of the error term and regressors will produce biased and inconsistent
estimates. In fact, our preliminary results show that the random effects estimates are
downward biased and Hausman test rejects the null hypothesis of no systematic difference
between the random effects and fixed effects estimates. This study uses a sample selection
model which controls for individual heterogeneity to circumvent the aforementioned problem.
There are some important propositions that follow from expressions (1.6)-(1.7). Assume that
leisure and consumption are normal goods and the utility function is concave. Thus, interior
solution conditions (1.6)-(1.7) imply:
∂ct
∂c
∂h
∂ht
∂ht
< 0 , 1+ ρt t < 0 , t > 0,
> 0,
>0
1+ ρ t
∂κ
∂ ( 1+ r )
∂κ
∂ ( 1+ r )
∂w t
13
(1.11)
The above propositions have important testable implications. For example in the empirical
analysis we would a priori expect the coefficient of the wage variable in the Frish labour
12
This result implies that nonexperimental cross-sectional studies of labor supply are theory inconsistent and their
estimates are biased and inconsistent. This nullifies all the labor supply estimates prior to 1970 and beyond which
constitute a lion’s share of the labor supply studies in the literature. Experimental studies might get around this
problem through proper randomization.
13
The proofs for these propositions are available in Heckman (1974).The reading material for this section could
also be found in MasCulel and Green (1995).
9
supply function to be positive. On the other hand we would expect the sign of the wage
variable in our estimation of the “fixed effects” to be negative because the second derivative
of the utility function with respect to wealth should be negative.
Our Empirical Model
Let the utility function of our individual consumer take the following form:14
ui (cit , lit ) = θ1it [cit ]Φ1 − θ 2it [hit ]Φ 2
where θ1i (t ) > 0 and
[1.12]
θ 2i (t ) > 0 are time-specific age shifters. It imperative to note that
θ1i (t ) > 0 and θ 2i (t ) > 0 are functions of individual background variables such as
human capital endowments (say health and education) , race, gender, age, number of children
under the age of five, region of residence and the like. Given the above preference function,
our consumer faces a maximization problem of her lifetime utility subject to her wealth
constraint. The Lagrangian function for the problem looks like:
L=∑
θ1it [cit ]Φ − θ 2it [hit ]Φ
(1 + ρ ) t
1
2
(1.13)
T
− ∑ κ t [at − (1 + rt )at −1 − d t − wt ht − pt ct ](1 + rt )
−t
t =0
Assuming no corner solutions and normalizing price of consumption to 1,15 the first order
conditions of the above optimization problem are:
Φ θ c Φ1 −1 (t )
∂L
=> 1 1 it t − κ it (1 + rt ) = 0
∂cit
(1 + ρ )
14
MaCurdy (1981), Chamberlain (1984), Jakubson (1988) and Blundell and MaCurdy (1999) use similar empirical
models. However our estimation strategy differs from those employed in these studies. We work in situation of
complete certainty. But incorporation of uncertainty into the model is straightforward and it does not change the
essential results of the model.
15
We will relax this assumption in our effort to test for separability of the preference function. We will test the
additive separability assumption in our model using the approach by Ham and Reilly (2002) and Browning (1991).
Moreover the implicit assumption of no liquidity constraint might cause some bias in our model. Domeij(2001)
argues that failure to control for the possibility of borrowing constrains causes a downward biased labor supply
estimates. More specifically he argues that the estimates will be about 50 downward biased. Unfortunately SLID
does not have information on asset and wealth of individuals which could be used to relax this assumption. If the
claim by Domeij92001) true this downward biases will be off set by the sample selection bias with model which
don not control for sample selection bias. Comparison of our estimated β with that his reveals that the results are
similar. Domeij92001) reports the estimated
β for the USA using PSID data between 0.24 and 0.56.
10
Φ −1
Φ 2θ 2it hit 2
∂L
=>
− κ it wit (1 + rt ) −t = 0
t
∂hit
(1 + ρ )
(1.14)
∂L
=> κ it = (1 + rt +1 )κ i (t +1)
∂ait
The above first order conditions imply the following Frisch labor supply equation (assuming
marginal utility of wealth and the interest rate are constant in the lifecycle) is
ln hit =
1
[ln κ i − ln Φ 2 − ln θ 2it + t ( ρ − r ) + ln wit ]
Φ2 − 1
(1.15)
If we assume that taste factors towards leisure are independently distributed as
θ 2it = exp(−αxit − f i + eit ) we can write the labor supply equation of an individual as
follows
ln hit = β [ln κ i − ln Φ 2 − αxit + t ( ρ − r )] + β ln wit + uit
(1.16)
where β = 1 /(Φ 2 − 1) and uit = −eit β , which is a random error that varies over time with
mean zero. We can rewrite the individual κ constant supply equation as follows
ln hit (t ) = ci + δt + µxit + β ln wit + uit
(1.17)
where ci = β [ln κ i − ln Φ 2 + f i ] , δ = β ( ρ − r ) and µ = βα
Expression (1.17) is the key equation of interest. However equation (1.17) assumes
exogenously given wages which is not plausible. As it was mentioned in the introduction
wages are not observed for individuals who do not work. In other words, wages will only be
observed if annual number of hours worked is positive. This is a typical incidental truncation
problem where wages are a function of number of hours worked. To deal with this
nonrandom sampling and endogeneity of wages, we propose that the wage offer equation
adopts the following structure:
11
ln w *it = D 1 xit + ci + uit ,
i = 1,..., n
t = ti ,..., Ti 16
(1.17)
s *it = D 2 zit + ς i + vit vit | zi ~ N (0,1)
(1.19)
sit = 1
(1.20)
if
s *it > 0
Where ln w *it is a latent endogenous wage with an observable counterpart ln wit .
s *it is
latent participation decision with an observable counterpart sit . Equation (1.17) is the wage
offer equation which is the equation of interest and equation (1.19) is a reduced form for the
propensity to participate in the labour market. xi
and zi contain vectors of exogenous
individual characteristics such as tenure, years of experience, years of schooling, health, and
others. It is conceivable that most of the variables that enter the wage equation will also
determine participation in the labour market. In our empirical models we will impose some
exclusion restriction. D 1 and D 2 are vectors of unknown parameters and uit and vit are
random error terms with E (uit / υ it ) ≠ 0 . We assume that (u , υ ) is independent of zi (where
zi might contain elements of xi 17), ci and ς i are individual fixed effects which are time
invariant. As suspected, the empirical analysis provides an evidence of sample selection bias
(see Table 5 and Table 6). Thus, our employment of a sample selection model will help to
predict wage offers for non-workers and possibly circumvent the problem of endogenous
regressors in our labor supply equation.
It is clear that ci is individual specific component which does not vary with time. Also note
that f i
represents unobservable variables which constitute the unmeasured component of
individual preference. It is very useful to remember that ci contains κ i which is the marginal
utility of wealth and hence we cannot assume that ci is exogenous to the other factors in the
16
Note also that
t = ti ,..., Ti implies that the panel structure could be unbalanced. We conduct a test for attrition
bias and the null hypothesis could not be rejected after taking the sample selection problem.
17
It is conceivable that most of the variables which influence participation in the labour market will also affect
wages hence we can expect that z contains most of elements of x . Ideally, we would like to have some
exclusion rule here for efficiency reason.
12
labor supply equation. As we have seen in our theoretical discussion ci is a function of the
interest rate, time preference, taste shifters and all other variables included in the wealth
constraint. This study will treat ci as a fixed effect which is person-specific and the
econometrics estimation will explicitly account for this. This shows that cross-sectional
studies which do not control for the unobservable individual specific factors are contaminated
with the problem of bias and inconsistency as E[ci | x) ≠ 0 . Appropriate estimation of the
empirical labor supply model (1.17) gives the intertemporal substitution elasticity ( β ). Also
note that expression (1.17) suggests that age will directly enter as an argument in the labor
supply model if and only if ρ ≠ r . We include age and age squared in our empirical model
and they were statistically significant with the expected size and sign. If we want to compare
the effect of wage variation across consumers we need to specify a certain mathematical
function of the fixed effects model.
As it was shown in the theoretical model, the marginal utility of lifetime wealth is a function
of future wages and initial wealth and some other characteristics. Unfortunately theory does
not specify the exact functional form of the relationship. In this study we assume that the
fixed effects component of the labor supply model follows the following empirically tractable
form18:
T
ci = xiφ + ∑ γ (t ) ln wit + ait ζ + bi
(1.21)
t =1
The concavity assumption for the utility function implies that γ (t ) < 0 and ζ < 0 .19 These
imply that marginal utility of wealth diminishes with increases in wealth and wages. It is
crucial to recall that the marginal utility of wealth is like a permanent income in the
permanent income hypothesis of consumption since it remains constant throughout the
18
MaCurdy (1981), Blundell and MaCurdy (1999) and Cahuc and Zylberberg (2004) use similar specification.
The second derivative of the utility function must be negative by assumption. Note that in expression (2.20) the
dependent variable includes a marginal utility of wealth.
19
13
lifecycle of an individual (Friedmand 1957; Heckman, 1974).20 In this equation xi 21
represents the time invariant observed characteristics of the individual. In this study xi
contains education, father’s education, mother’s education and ethnicity.
φ, γ, and ζ are
unknown parameters of the model. bi is assumed to be identically and independently
distributed across consumers. The fixed effects can be imputed from the estimation of the
equation (1.17). However, we need to make some assumptions to proceed with the estimation
of (2.21). Empirical estimation of the time path of wages is usually not available and we need
to use some projections. As in MaCurdy (1981) and Cahuc and Zylberberg (2004) we propose
that lifetime path of wages is a quadratic function of age and some other age invariant
characteristics.
wit = ω0i + tω1i + t 2ω 2i + vi (t )
(1.22)
Where ω ji = zi d j
(1.23)
zi is a vector of time invariant exogenous determinants of lifetime wages. The time invariant
variables included in our empirical model are education, education squared and background
variables (education level of mother, education level of father, race) and time dummies for
each year.
d j is vector of unknown parameters.
t is the age of an individual t = 0,1,2,..., T + 1 . This specification will be tested for
sensitivity.
We need also to specify the equation for the initial wealth. We will assume that wealth can be
approximated by the time path of lifetime investment income. That is, we assume that
20
As claimed in the theoretical section the marginal utility of wealth is like a sufficient statistic that will
summarize all historical effects and, in a world with no uncertainty about the path of wages, it does not need to be
revised once it is set at the initial period. See Heckman (1974), Heckman and MaCurdy (1980) and MaCurdy
(1981) for more discussions.
21
xi contains the time invariant component of xit
in the utility function (1.01).
14
rt ait = iit where iit is within-period investment income of an individual. In this study we
employ investment income as property income is missing from SLID.
As in Cahuc and Zylberberg (2004)22 we assume that wealth is a quadratic function of age and
time invariant background variables of an individual:
iit = α 0i + tα1i + t 2α 2i + vit
(1.24)
where t stands for age
α ji = g i p j
j = 0,1,2,3,4,..., n
(1.26)
Where p j is a vector of unknown parameters. g i = vector of background variables which
include education level of individuals, mother’s education, father’s education, race, and time
dummies for each year in the panel. I i (t ) is investment income vi (t ) is the identically and
independently distributed error term. Note that α 0i = Ai (0)r , which is the initial wealth.
Given equation (1.22) and equation (1.24) we can rewrite (1.21) as follows
T
ci = xiφ + ∑ γ (t ) ln wit + ait ζ + bi
t =1
ci = xiφ + ω0iγ 0 + ω1i γ 1 + σ 2iγ 2 + α 0iζ + η i
where γ j =
∑t
j
(1.26)
γ (t )
In reduced form equation (1.26) can be represented as
1.2723
ci = Gi Ω + ui
Joint estimation of (1.22) and (1.25) and (1.26) provide all the parameters needed to gauge the
labor supply response to parametric changes in wages.
We can get three types of elasticity estimates from the models that we have developed so far.
The first one is intertemporal labor substitution elasticity. As it was mentioned above the
22
MaCurdy (1981) and Blundell and MaCurdy (1999) also follows similar specification.
23
We are implicitly assuming that the error terms across the equations are independent. If they are not
independent it would be difficult to stack them in (1.27).
15
intertemporal substitution elasticity is given by β and the value of β would be positive if
leisure is normal good.24 This provides a direct elasticity of substitution for hours work in any
two periods. It calculates the elasticities for evolutionary change in wages by keeping the
marginal utility of wealth constant. Using the Slutsky equation, we can easily derive the own
price and cross price elasticities. The compensated own price elasticity is
∂hit
∂wit
∂hit
∂hit
=
ui =u
−
ai 0 = a
∂ ln hit
∂ ln hit
=
∂ ln wit ∂ ln wit
∂hit
hit
∂a0 i
−
a0 = a
∂ ln hit wit
hit
∂ai 0
= β + γ (t) - ζhit wit
And the cross compensated effect is
∂ ln hit
∂ ln w jt
=
ui =u
∂ ln hit
∂ ln w jt
−
A ( 0 )i = A
∂ ln hit
hit w jt
∂ai 0
= γ (t) - ζhit wit
It is very useful to differentiate the intertemporal elasticity ( β ) the uncompensated own price
effect ( β + γ(t ) ) and the uncompensated cross price effect ( i.e. γ(t ) ). The compensated
elasticities are useful to compare the performance of different categories of the labor force.
For instance we can compare the performance of immigrants and natives with different wage
profiles but the same lifetime utility. The parameter γ (t ) measures the labor supply response
to parametric change in the wage profile. It is imperative we remind ourselves that
compensated elasticity and intertemporal elasticities may not be the same. Despite this clear
theoretical distinction it is not uncommon to see confusion in the literature in interpreting the
coefficients of wages in the lifecycle labor supply model. Assuming leisure is normal in all
time periods β > β + γ (t) - ζN(t)W(t) > β + γ (t) .If the wealth effect is zero three of these
elasticities reduce to β .
24
That is to say if wage increases in a certain period this causes an increase in the price of leisure; if leisure
is a normal commodity the substitution effect will lead to more work. Note that we do not have income
effects here because if there is any effect it should operate via the fixed effect by impacting the marginal
utility of wealth which is assumed to be constant.
16
III. Methodology: Estimation Strategy
We will employ different variants of nonlinear panel data regression models. In the linear
panel fixed effects model estimation is made easier by time demeaning variables. In this case
the constraint or person-specific effects are wiped out. Unfortunately in the nonlinear panel
data case there is the notorious incidental parameter problem that will force us to estimate a
huge number of constants.25 We will discuss this in a sequel.
It is not the aim of the current study to provide detailed proofs of the econometric models to
be employed. Since the detailed proofs of the models that are employed can be found in
standard econometric text books such as Wooldridge (2002), Greene (2003) and Baltagi
(2005), we will develop the models without providing detailed proofs.
A. Tobit model
Basic Model
Consider the following estimable version of equation (1.17). Rewrite expression (1.17) for
each individual man as follows
y *it = µxit + β ln wit + ci + uit
Where
(1.28)
yit* is the latent number of hours worked, µ = βα , ui (t ) = −ei (t ) β
and
ci = β [ln κ i − ln Φ 2 + f i ] , which is the person-specific effect from equation (1.16) and
x it and µ are conformable vectors. The list of wage predictors included in equation (1.18)
includes years of schooling, tenure (in years), tenure squared , race, regional dummies which
25
However, the problem is inherently statistical but not practical.
17
are supposed to capture the variations in the labor markets across provinces and price
differences, and dummies for each year in the panel26.
ci contains any individual
heterogeneity in preferences or skills and the like which can be taken to be constant over time
t . As was clear from the theoretical discussion above, the wealth effect and non-wage income
effects will only enter into the picture through this individual effect. uit is the idiosyncratic
error structure from equations (2.17) to (2.26) and are assumed to be identically and
independently distributed with N (0, σ 2 ) . If we allow for the possibility of a corner solution
we may rewrite (3.1) as
µx + β ln wit + ci + uit if yit* > 0,
yit =  it
otherwise .
0
(1.29)
We assume that there is random sample of N men over T periods of time which may be
balanced or unbalanced. In one case where the sample is nonrandom ( such as sample
selection problem) our estimation strategy will have to be adjusted (Wooldridge 1995 and
2002). We will discuss more about this in the sample selection model. However it is essential
to assume that E[uit | ci , xit ] = 0 . In other words, we assume that the error term is
independent of the regressors and individual effects. Equation (3.2) defines a multivariate
correlated Tobit system if we stack the equations over time. The usage of Tobit model is
tantamount to assuming that the data are truncated because of corner solutions. We will see in
the sample selection model that the truncation could be an outcome of participation decision
in the labor market. Note that
yit*
is the latent variable with its observed counterpart of yit .
There are three basic ways of estimating the Tobit model (3.2) depending on the assumptions
made about the individual effects and the regressors in the system. The first basic estimation
technique is the pooled cross-section model. In this model we treat each cross-section as a
26
For notational convenience we have dropped the age variable. However, expression (2.17) suggests that
age will enter as an argument in the labor supply model if ρ ≠ r . We include age and age squared in our
empirical model.
18
separate observation and we estimate the model using OLS over the NT observations. In this
model the individual effects will be absorbed by the common constant term. It goes without
saying that this approach is not theory consistent. However, this approach will help us to
assess the extent of bias introduced by ignoring the person-specific effect. The estimates from
this approach will also allow us to compare our estimates with some cross-sectional studies in
the literature.
The second Tobit specification is the random effects(RE) model. In the random
effects model we assume that the unobserved individual effects are uncorrelated with the
regressors in the model. The advantage of random effects over the pooled cross-section panel
data is that it is relatively efficient under the some assumptions. However, given the theory
that we have discussed in the previous section, it is less likely that the unobserved
characteristics are to be uncorrelated with the independent variables in the model. Also, as
opposed to the traditional treatment of random effects model, we model the group specific
constants as randomly distributed cross-sectional units. However, theory suggests the
presence of person specific effect which contradicts with the basic assumption of the random
effects model. We have seen from equation (1.8) that the individual effect is a function of all
variables in the model.27 Thus it would be implausible to assume uncorrelated random effects
model. Rather we consider the “correlated random effects” model of Chamberlain (1984).
The advantage of the “correlated random effects” model is that the conditional assumption
about the distribution of ci allows us to reduce the number of parameters to a constant
number which does not grow with sample size. Note that this approach allows us to get
27
To be more precise it will be a function of all the variables in the model and some other non-observables
as the utility function only includes some key variables. We can modify this by including a variable which
represents all other relevant variables as arguments in the utility function. Thus the fixed effect will be a
function of all variables which are observable and non-observable. Given that our utility function has only
three arguments and the budget line, the marginal utility of wealth will be a function of the variables in the
maximization problem (budget set and utility function variables.
19
around the incidental parameter problem but at a cost of additional assumptions about the
distribution of ci .28
The third specification is the fixed effects (FE) model. In this variant of the estimation
strategy we can directly tackle, at least theoretically, the incidental parameters problem. The
advantage of this approach over the random effects estimation is that we are not required to
make orthogonality assumption about c directly. The disadvantage is that the number of
parameters to be estimated increases with sample size. This approach required T to be very
large to get consistent estimates of c . As was discussed above with fixed T, as is the case in
our sample, c is not only inconsistent but also it contaminates ( µ , β ) withinconsistency and
bias.29 One can only hope that the bias is not inherently large. Greene (2001) shows using
Monte Carlo study that the bias is not very large with T=8. A comparison of the random
effects model and the fixed effects model should provide a good sense of the problem. If the
results from these two approaches do not differ significantly, it may be the case that the
incidental parameter problem is not very serious. However, it is also possible that the RE and
FE are similar because the incidental parameter problem is very serious but is offset by the
bias arising from E[ci | xit ] ≠ 0 .
B. Sample Selection Model
By definition, equation (2.16) represents lifecycle labour supply decision of every man of
working age whether or not a person was working at the time of the survey. However,
because we can only observe the number of hours worked for the employed, we select our
sample on the basis of participation in the labor market. This is a typical case of incidental
truncation problem. The data on number of hours worked is missing because of the outcome
of participation in the labor market. In the literature this is sometimes called sample selection
problem. This selection problem has been long identified in the literature (Wales and
28
29
It is also possible to employ minimum distance estimation ( Wooldridge, 2004; Jakubson, 1988)
See Chamberlain(1984), Wooldridge(2002) or Greene(2003) for detailed discussions and proofs.
20
Woodland 1980, Gronau 1974, Lewis 1974, Heckman 1978, Vella 1998). Our estimation
technique should account for the non-random nature of the sample as failure to do so may
yield inconsistent estimates (Heckman 1978). To deal with this infamous problem of
nonrandom sampling we propose that the labor supply equation follows the following panel
data30 structure:
ln y *it = µxit + ci + uit ,
s *it = zitψ + ςi + vit
sit = 1
t = ti ,..., Ti 31
i = 1,..., n
vit | zi ~ N (0,1)
(1.36)
s *it > 0
if
(1.37)
yit = ( y *it )( sit )
Where y *it
(1.35)
(1.38)
is a latent endogenous number of hours with an observable counterpart yit .
s *it is latent participation decision with an observable counterpart sit . Equation (1.35) is the
labor supply function which is the equation of interest and equation (1.36) is a reduced form
for the propensity to participate in the labor market. xi and zi contain vectors of exogenous
individual characteristics such as , age, years of schooling, health, imputed wages and others.
It is conceivable that most of the variables that enter the wage equation will also determine
participation in the labor market. In our empirical models we will impose some exclusion
principle. µ and ψ are vectors of unknown parameters and uit and vit are random error
terms with E (uit / υ it ) ≠ 0 . Note that, for the sake of convenience, unlike the previous section
µ now includes β , the coefficient of ln Wi (t) . We assume that (u ,υ ) is independent of
See Baltagi (2005), Hsiao(2003), Greene(2003), Wooldridge(2002) for discussion on panel data modeling.
See Vella (1998) for a readable survey of sample selection models.
31
Note also that t = ti ,..., Ti implies that the panel structure could be unbalanced.
30
21
zi (where zi might contain elements of xi 32), ci and ς i are individual fixed effects which
are time invariant.
Define the following deviation forms of the variables as
&x&it = xit −
∑ xir sir
∑ sir
if
∑s
&y&it = y it −
∑y s
∑s
if
∑s
ir ir
ir
ir
>0
(1.39)
>0
(1.40)
ir
Hence the fixed effects estimator for unbalanced ( µ FE (u ) ) and balanced panel ( µ FE ( B ) ) are
as follows
−1
 N τ
  N τ

µ FE (u ) =  ∑∑ &x&it ' &x&it sit   ∑∑ &x&it ' &y&it sit 
 i =1 i =1
  i =1 i=1

 N
µ FE ( B) = 
 i=1
τ
∑∑
i =1

&x&it ' &x&it d i 


−1
 N


 i =1
(1.41)

τ
∑∑ &x& ' &y& d 
it
it i
i =1
(1.42)

{
where d i = (Π tτ=1 s it ) = 1}
For the consistency of our fixed effects estimator we require that
E[u&&it | &x&it , sit ] = 0 . In
other words, the sample selection should operate via individual fixed effects which will be
removed by time demeaning the variables. Note that this assumption will break down if the
selection process is operating through some time invariant unobservable.
To get the random effects model we follow Vebeek and Nijiman(1992) . Define
yit = ( yi1 ,..., yiτ )' , xit = ( xi1 ,..., xiτ )' , and u it = (u i1 ,..., u iτ )' . Assume that all the variables
in the labor force participation equation are available and define the number of units
sit = 1 as Ti and define Ti xTi matrix Ri transforming yit into the Ti dimensional vector of
It is conceivable that most of the variables which influence participation in the labor market will also
affect wages; hence we can expect that Z contains most of elements of X. Ideally, we would like to have
some exclusion rule here for efficiency.
32
22
observed y io . Note that matrix Ri is obtained by deleting the rows of the T dimensional
identity matrix corresponding to sit = 0 . Defining the unit vector I , the variance covariance
of the error term in equation (1.35) can be written as
Ω = σ c2 ii '+σ u2 I . Given this random error structure, the random effects estimator for the
unbalanced and balanced panel case are

µ RE (u ) = 


∑

µ RE ( B) = 


∑
xio ' Ω i −1 xio




−1
xio ' Ω i −1 xio d i









∑ xio ' Ωi −1 yio 
−1
(1.43)






∑ xio ' Ωi −1 yio di 
(1.44)

where Ω i = Ri ΩR 'i and xio = Ri xi R ' i
For the consistency of (1.40) and (2.17), we need E[u it + ci | xit , sit ] = 0 . Thus, the random
effects estimator will be inconsistent if the selection is operating either through the individual
or/and the idiosyncratic error. Note that the beauty of the fixed effects model is that it does
not require a selection equation and it does not impose any distributional assumption about
the error terms.
These are the traditional panel data estimators. As it was discussed above, it is most likely
that the assumptions required for consistent estimation by the aforementioned models will be
violated. In light of these, Wooldridge (1995) developed an estimator for testing and
correcting a selection problem.33 Under some general mild conditions, this model will
produce consistent estimates of µ .
Testing and correcting for sample selection
33
The basic testing procedure is similar to that of Ridder (1990) Nijiman and Verbeek(1992) and Vella
and Verbeek(1994), who base on simple variable addition test but provide a more general way of
obtaining consistent estimates.
23
There have been a number of suggestions on the detection of sample selection bias in panel
data models (Wooldridge 1995, Verbeek and Nijman 1992 1996, Vella 1998, Vella and
Verbeek 1994) 34. The basic premise of this approach is that it parameterises the conditional
expectation required for the consistency of the pooled estimator:
E[ci + uit | xi , sit = 1] = E[ci | xit , sit = 1] + E[uit | xit , sit = 1] = 0 ∀t
(1.45)
That is to say, the approach parameterises assumption (1.45) and adds the parameters as
additional regressors in the main equation. After several manipulations the equation of
interest can be written as:
E[ yit | xi , vit ] = β xit + πx + ξ t λ ( xiψ t )
(1.49)
It is possible to obtain a consistent estimate of µ by first estimating a labor force participation
equation using probit model sit on xi for each panel j and saving the Inverse Mills ratios
λ̂it for all i and t . The next crucial step is to run pooled OLS regression using the selected
sample: yit , on xit , xi , λˆit , d t λˆit ,..., d T λˆit for all sit = 1 where d t − d T are the time dummies.
Wooldridge (1995) shows that we can get consistent estimates of (1.49) using either OLS or
minium distance estimator.35 Note this approach allows for the correlation between the
unobservables in the selection equation, vit and the unobservables in the wage offer
equation (ci , u it ) the selection process might operate via both the error term from the main
equation uit and the unobservable individual effect ci . However, we need to adjust the
standard errors for general hetroscedasticity and autocorrelation and for the first stage
estimation.
Wooldridge(1995) adopts a variable addition test to detect the presence of sample selection
problem. Under the null hypothesis of E[uit | xi , sit , c] = 0, t = 0,1,..., T the inverse Mills
34 Wooldridge (1995) claims that the estimation of the selection model is of second order importance as the
objective is to derive a test and we are not interested in the selection equation parameters per se. We will
test this claim concerning whether the different specifications of the selection make any difference or not.
35
See Wooldridge (1995) for the derivation of the variance covariance matrix.
24
ratios from equation (1.36) (for each cross-section) should not be significant in an equation
estimated by the fixed effects method. To elaborate, in the first step estimate the Inverse Mills
ratio from (1.36) for each cross-section. The next step is to estimate equation (1.35) using
fixed effects model on the selected sample by including the Inverse Mills Ratios as additional
regressors and then testing the sample selection using t test for the Inverse Mills Ratios in
this fixed effects model. Wooldridge (1995a) shows that the limiting distribution of t under
the assumption E[uit | xi , sit , c] = 0 is not affected by the estimation adopted for the
participation equation (1.36). As long as the standard errors are robust and adjusted for
hetroskedasticity, we can trust the student t test.
IV. Data
The data employed for the estimation of lifecycle labor supply functions are drawn from the
second panel (1996 to 2001) of Survey of Labor and Income Dynamics (SLID). Our sample
contains individuals between the ages of 21 to 65 of men. We focus on 6 years: 1996, 1997,
1998, 1999, 2000 and 2001. Unlike most studies (For example Reilly 1994, MaCurdy 1981,
Osberg and Phipps 1993) in the literature, we do not restrict our sample to those who reported
non-zero number of hours. We argue that that such a restriction causes sample selection bias,
yielding inconsistent parameter estimates. The SLID is a continuing panel of Canadian
households which began in 1993. It combines the former Labor Force Activity Survey, an
intermittent series of panel surveys conducted during the 1980s, with the Survey of Consumer
Finance, a regular cross-sectional survey conducted annually. The SLID design is a series of
overlapping 6-year panels, with a new panel enrolled every three years.
It is common in the literature to calculate the annual number of hours worked by multiplying
the number of weeks worked and the usual number of hours worked per week. Furthermore
hourly wage is calculated by dividing reported annual earnings by estimated annual number
of hours worked. These calculations usually introduce measurement bias and yield
25
inconsistent parameter estimates. Any measurement error in the annual number of hours
worked will carry over to the hourly wage rate (Ziliak and Kniesner 1999, Mroz, 1987).Smith
Conway and Kniesner (1999) argue that the choice of wage measure matters. In SLID the
annual number of hours and the composite hourly wage are calculated from very extensive
interview with detailed questions on each job and payment that individuals get in the survey
period. Questions on number of jobs held and the hours worked pay by pay, number of weeks
worked, number of weeks absent from work and others where respondents are walked by
detailed questionnaire to retrieve information and where access to the income files of
respondents is obtained are supposed to produce reliable information on number of hours
worked and hourly wage.
The wage predictors include years of schooling, years of experience, years of experience
squared, tenure, tenure squared, a dummy variable for visible minority, regional dummies and
time dummies. We introduce tenure variable to improve the instrumentation of wages and the
precision of the structural parameter. Tenure is defined as the number of years worked with
the current employer. We impute wages using the sample selection model indicated in
equation (2.18-2.20) but this time the selection equation is a probit model. Other income is
calculated as the difference between individual total annual wages and total household
income.
The health variable is the self reported health status. However, because of the established
research, our health variable is instrumented. We follow the lifecycle consistent health
model.36 The health predictors include imputed wage, age, and age squared, time dummies
and level of self-assessed stress level. The age variables are expected to proxy depreciation in
health stock and the time dummies to capture technological changes and other production
functions relationships. The health equation was estimated using fixed effects logit model (see
36
Dustmann and Windmeijer (2000) derive lifecycle consistent demand for wage following the same
approach that we adopted to derive our lifecycle labor supply model.
26
Appendix for the results of the model).37 However, we will also use the instruments without
including imputed wage to explore the sensitivity of our results. The descriptive statistics of
the key variables is in Table 1.
-------------------Table 1 here-------------------------
V. Estimation results
In this section we present the estimation results using the different models discussed above.
We first present the results from a sample of individuals who reports non-zero number of
hours using standard panel data models. This serves a double purpose. Firstly, we will be able
to compare our results to those in the literature because most of the old literature uses a
sample of employed men only in their analysis. Secondly, these results can be used as a
benchmark to explore the extent of bias introduced by ignoring the sample selection problem.
Table 2 presents estimation results from traditional panel data models using the selected
sample only. The first column of Table 2 reports the pooled OLS results. The second and
third columns report the results for the random effects (RE) and fixed effects (FE) models. As
Table 2 demonstrates, three of these panel data models produce remarkably different
estimates. An LM test has rejected the null hypothesis of variance constancy in favour of the
RE model implying that the classical regression model with a single constant is inappropriate
for our data. But this does not necessarily imply that the RE model is the best model which
describes the data. There is a competing FE specification that could induce the sample data. In
fact, a Hausman test with a null hypothesis of no systematic difference between the RE and
the FE estimates ( H 0 : β FE = β RE ) was rejected in favour of the FE model. Therefore the
preferred model according to both econometric and economic theory is the FE model. A
comparison of the implied intertemporal labour elasticity reveals that the FE model produces
37 This might introduce the incidental parameter problem but we hope that the incidental parameter
problem has not affected our results significantly. Greene (2001) argues that the extent of the bias is not
terribly bad when T is larger than 2.
27
the largest estimate followed by the RE model. More specifically, the implied intertemporal
labor substitution for the pooled, RE and FE model are 0.23, 0.27 and 0.31 respectively.
These results are very similar to those reported in the literature by employing a sample of
employed men only. Altonji(1986) presents the survey of labor supply model for men in the
USA and reports the intertemporal labor substitution to be between 0 and 0.35.
-----------------------Table 2 here-------------------------
We now turn to the standard panel data models without excluding those who reported zero
hours. Table 3 summarise the results from the standard OLS, RE and FE estimates without
taking into account the truncation in the data. In Column 1 and 2 we present the standard OLS
and Random Effects models. The results are similar but not the same. For example, the
implied intertemporal elasticity of substitution are 0.48 and 0.39 for the RE and OLS
respectively. However, both the RE and pooled OLS models are not consistent with our
theory because they ignore individual heterogeneity. Column 3 of Table 3 presents the FE
estimates. A Hausman test of the correlation between individual heterogeneity and the
regressors rejects the null hypothesis of H 0 : β FE = β RE . Table 3 demonstrates that the FE
estimates are lower relative to the pooled OLS and the RE point estimates. For example, the
coefficients of the imputed wage are 351.72, 707.98 and 591.36 for the FE, RE and pooled
OLS respectively. One explanation for the lower values of the FE estimates might be that the
partial elimination of the upward bias that could be represented by the fixed effect component
of the selection mechanism. Of course, the traditional OLS and RE are biased because they
fail to account for the individual heterogeneity if E[ci | xi ] ≠ 0 .
---------------------------------Table 3 here---------------------------
So far, there has not been an explicit account of those with zero hours reported in our sample.
In this section we treat zero hours
as an outcome of a corner solution and hence, we
28
estimate different versions of panel data Tobit models. Table 4 presents the results from the
three traditional Tobit models: pooled, RE and FE models. The second column of Table 4
contains the results from the pooled Tobit model. This model ignores the panel structure and
treats the data as extended cross-section. However, this form of estimation does not exploit
the panel nature of the data and is inconsistent with our economic theory. Comparison of
these results with the cross-sectional Tobit models employed in the literature reveal that the
implied intertemporal substitution elasticity is in the range of those reported in the literature.
The implied intertemporal elasticity is 0.41.
Column 1 of Table 4 reports the results of the uncorrelated RE model. This model assumes
that the idiosyncratic error terms are independent of the regressors, which is not consistent
with our economic theory. Table 4 demonstrates that the RE model estimates are significantly
different from that of the pooled OLS model. Also note that most of the marginal effects
from the RE model are smaller compared to that of the pooled OLS. For example, the
marginal effect of “imputed wage” has decreased by about 40 percent, which implies a Frisch
labor supply elasticity of 0.25 but statistically insignificant at less than 10 % level. The
correlated RE (Chamberlain type) was also estimated but it gave results similar to that of the
FE model that is reported in Column 3. For this reason, we only report the results of the FE
model. Note that for reasons which are not yet clear all the standard errors of the marginal
effects in the uncorrelated RE model are relatively so big that all the marginal effects turned
out to be statistically insignificant.38
Column 3 of Table 4 presents the point estimates from FE Tobit model. The FE approach
models the fixed effects directly. In this model the fixed effects are assumed to be
uncorrelated with the regressors in the model. This formulation is theory consistent.
38
Table 3
I checked this issue with William Greene who is the author of LIMDEP ( the econometric package
which I am using) he suggested that it probably is related to the model specification among which is the
exogeneity assumption.
29
demonstrates that the FE estimates are significantly different from the RE model estimates.
Unfortunately, there is no statistical test available to compare the RE and FE estimates as they
are based on different statistical grounds. The best test available is economic theory which is
in favour of the FE model. But examination of some of the key variables would facilitate the
comparison. For example, the coefficient of “Education” is positive and significant in the
pooled and RE Tobit models but it changes to a statistically significant negative coefficient as
we move to the FE model. The same is true with some of the marital status indicators. The
implied Frisch elasticity from the FE model is calculated to be 0.34 which is about 0.07,
lower than that of the pooled Tobit model estimate. This could be explained by the partial
elimination of upward bias which could have been introduced by any form of incidental
truncation.
Now we turn to the sample selection models. The sample selection model of labor supply
effectively separates the participation decision from the choice of number of hours of labor
supply, conditioning on participation. We estimate the pooled OLS and FE version of the
sample selection model. The last two columns in Table 4 report the results of pooled and FE
sample selection models respectively. Comparison of the standard pooled OLS with that of
the FE estimates reveals that there is sharp difference on the estimated parameters. Note that
the standard sample selection model is theory inconsistent and the fixed effects model is more
attractive from the theoretical point of view as it controls for the individual specific effects
explicitly.
-----------------Table 4 here------------------------------------------------------------------------------
According to Table 4 the Frisch elasticity for the FE and pooled OLS sample selection models
are calculated to be 0.32 and 0.25 respectively. This could imply that the bias introduced by
E[ci | xit ] ≠ 0 is an important source of bias and inconsistency. We observe also a sharp
30
difference in the parameter estimates for the marital status variables in both models. While the
pooled sample selection model implies that widowed, separated and divorced men work more
hours than single unmarried men, both the FE Tobit and FE sample selection models imply
the opposite. As it has been mentioned time and again, from the point view of our economic
theory, the FE model is more attractive than the pooled sample selection model.
However the traditional FE sample selection model is consistent only if the selection process
is not correlated with the idiosyncratic error. The hope is that the selection problem will be
wiped out by time demeaning the variables during the estimation process as indicated in
expression (1.41). If the selection problem is also correlated with time variant unobservables,
then we need to be concerned about consistency of our results. In the next section we formally
test for sample selection bias with a view to correct any that may appear.
The test results for the presence of sample selection problem in our data are reported in Table
5 and Table 6. Table 5 presents the λt estimated from cross-sectional selection equations. As
expected, all the λt are negative and statistically significant at less than 1% level. We also test
for the presence of sample selection bias in our panel data in Table 6. According to Wooldrige
(1995) the test should be conducted as follows. We first estimate the λt from the selection
equation estimated for each cross-section. In the next step estimate the fixed effects model
using the selected sample only but including the inverse mills ratios as additional regressors.
The first column of Table 6 reports the results of a FE model on the selected sample and most
of the λt turn out to be statistically significant at less than 5 percent level of significance. A
joint test of significance rejects the null hypothesis of jointly λt are zero, H 0
: λi = 0 . These
tests provide sample evidence that our panel data is contaminated with some sort of sample
selection problem.39
39
There might also be attrition. In this paper we hope that by controlling sample selection problem we are
capturing attrition bias if there is any.
31
-------------------------------------------Table 5 here-----------------------------
Having established the presence of sample selection problem, the next step is to address the
problem using some sort of correction mechanism. The second column of Table 6 reports the
results of the Wooldridge’s(1995) estimator (WE). According to the results in Table 6 we see
stark differences between the FE estimator and that of the WE. To facilitate our comparison
we compare the implied elasticities from both models. The implied intertemporal elasticity
from the FE model is 0.36 and from that of WE is only 0.16. Note that the WE produces a
value of β which is lower than that of the traditional FE sample selection model by 0.09. But
the WE produces a value of β
-----------------------------------------------Table 6 here---------------------------------------
which is less than 50 percent of β implied by the FE on the selected sample reported in
Table 6. The upward bias in the FE model on the selected sample with the λt as additional
regressors could be a support for Wooldridge’s (1995) claim that this FE model cannot be
consistent under any circumstance. It is interesting to note the similarity between the
intertemporal substitution elasticity implied by FE sample selection model( β =0.32) and that
of the FE model on the selected sample( β =0.36) with the λt as additional regressors. This is
not unexpected as both models control the selection problem partially.
When we turn to the issue of sensitivity of parameter estimates to the change in estimation
models the results in this paper demonstrate that parameter estimates are very sensitive to the
econometric modelling adopted. Examination of the implied effects of selected variables
would facilitate our discussion. Let us take the effect of number of kids in a family. Contrary
to what has been documented by most researchers we find a negative effect of fatherhood on
labor supply in most of models which do not account for sample selection problem. In most of
32
the cases the parameter estimates of number of kids are negative and significant at less than
1% level of significance. This might be an indication of a change in the role of men in home
production. This is not without any empirical support. Vere(2005) finds that married men’s
labour supply falls with a second child in 1980s and 1990s in the USA. Vere argues that the
home intensity effect dominates over the specialization effect and most of the adjustment with
fatherhood lies on earnings and not labor supply. However once we control for individual
hetrogenity and correct for sample selection bias the WE model implies the opposite. This
seems to agree with the majority view. For example, Lundberg and Rose (1999, 2002) report
positive effect of fatherhood on labor supply and earnings. Angrist and Evans (1998) report
that male labor supply is not affected by childbearing and if there is any it is done by
sacrificing male leisure time.
Another variable worth examining is “health”. The coefficient of health was negative in all of
the models which do not correct for sample selection bias. However in the WE model the
coefficient of health is positive and statistically significant at less than 1% level of
significance. This is indeed very interesting because the effect of health was negative but
insignificant in the fixed effects Tobit and other traditional panel data sample selection
models at 10% level of significance. This further reinforces the claim that the effect of health
on labor supply is sensitive to the instrumentation and estimation technique employed (Currie
and Madrian 1999). The positive relationship makes more sense as Figure 4B in Appendix B
shows that the number of hours worked is higher for healthy people than that of the
respondents who reported poor health. Overall these findings demonstrate that the point
estimates of a labor supply model are sensitive to the econometric modelling, and accounting
for sample selection bias matters.
Finally we want to compare our results with those in the literature. Table 1A summarizes
some previous related studies on the responsiveness of men’s labor supply. We mainly
33
summarize the Frisch elasticity, uncompensated wage elasticity and wealth elasticity. Of
course, there are obvious differences in the estimation strategies and data employed in these
studies and any strict comparison should take this into account. The aim here is to see how
our results fare with those documented in the literature. Overall our results are not very
different from previous studies and are in the range of the survey of American studies in
Altonji (1986). The implied Frisch elasticity from the preferred model (WE) is marginally
similar to that of MaCurdy (1981) which employs similar modelling framework but different
estimation strategy. The advantage of this study is that it controls for the individual fixed
effects and the sample selection problem which is ignored in many studies. However the
findings in this paper differ from the sole Canadian study by Reilly(1994). Reilly reports the
intertemporal elasticity with respect to number of weeks worked in a year to be 0.6 and
intertemporal substitution elasticity with respect to annual hours worked to be as large as 0.9
but statistically insignificant. Our study reveals that the value of β ( intertemporal elasticity
with respect to annual hours worked) is statistically different from zero in a range of 0.16 to
0.48. The differences could mainly be attributed to the sample selection problem inherent in
his modeling. Note that the upper bound for our estimates comes from the models which do
not account for individual heterogeneity and sample selection bias and the lower bound for
our estimates comes from the WE model which controls for both individual heterogeneity and
sample selection bias. Thus this study finds that failing to control for individual heterogeneity
and sample selection problem produces upward biased estimates of intertemporal substitution
elasticity.
Wage shift parameters
The previous section estimated intertemporal elasticity of labor. In this section we will
estimate the wage shift parameter. These estimates will help in providing additional
information about the responsiveness of number of hours worked to wage changes. The
estimation of the parameters which measure the parametric change in the wage profile is not
straightforward. Hence we need to obtain the observable counterparts by manipulating or
34
exploiting the observable components of (2.17). From equation (2.17) it is possible to predict
ci as follows.
E[ln hit ] = E[ci + δt + µxit + β ln wit + uit ]
E[ln hit ] = E[ci ] + E[δt ] + E[ µxit ] + E[ β ln wit ] + E[uit ]
cˆi =
1 J
∑ (ln hit ) −δˆt − µxit − βˆ ln wit )
J j =1
(1.50)
j = 1,..., J
Where the value of j stands for a cross-section of a panel and in our case it assumes values of
1-6 corresponding to each year of the panel. Note that, from the law of large numbers,
asymptotically E[cˆi ] = ci .For the other parts of equation (2.1)
we follow MaCurdy (1980)
and Dustmann and Windmeijer (2000). To get an observable counterpart of the wage growth
equation, we use:
d k ln Wi ( j ) / k = ω1i + ω2i [2t ( j ) − k ] + d k vi ( j ) / k
(1.51)

 d k ln wij
d k vi ( j )
d v (2)
1
− 1 i
− d k ln wi 2  = ω 2i +
(1.52)

−
−
−
−
2j −k −3 k
j
k
k
j
k
(
2
3
)
2
3

where d is a difference operator. Note that the average value of (1.52 ) is ω2i . Hence
ω2i =

 1 d j +1 ln wi ( j +2)
1
{
− d1 ln wi 2 }
∑

τ −1  j
j +1

(1.53)
Given (1.50), from (1.52) we find
ω1i =
ω0i =

 d j ln Wi ( j + 1)
1
− ω 2i [2t ( j + 1) − j ]
∑

j
τ −1 

1
∑ {ln w
τ
ij
2
− ω1i + t ( j ) − ω 2i [t ( j )] }
(1.54)
(1.55)
It can easily be shown that, using the law of large numbers, E[ωih ] = ωhi . We can employ the
same strategy to get observable counterparts of the initial wealth by replacing ln wij by ii ( j )
in (5.5) as α 0i =
1
∑ {i ( j ) − α
τ
i
1i
( j ) − α 2i [t ( j )]2 } . Then we can estimate the following
system of simultaneous equations to find the complete parameters to gauge the wage shifters:
35
cˆi = X i φ + ω0i γ0 + ω1i γ1 + ω2i γ2 + α0i ζ + η1
(1.56)
ωhi = g i qh + η3
(1.57)
where h = 0,1,2
α0i = g i p0 + η4
(1.58)
Where g i contains exogenous variables and q0 , q1 , q2 and p0 are defined as in equations
(2.10) and (2.13). The structural parameters of interest are γ0 , γ1 , γ2 and ξ . The g i vector
determines lifetime income and wage path. The variables included in the execution of the
actual empirical analysis include education level of individuals, mother’s education, father’s
education, race, and time dummies for each year in the panel. This system of equations could
be estimated by two stage least square (2SLS). Of course the standard errors should be
corrected for the first stage estimation. Note that the endogenous variables are ω hi and ĉi .
Table 7 presents the estimation results. According to Table 7 almost all of the coefficients
have the expected sign but not all of the coefficients were statistically significant at less than
10 percent level of significance. We report the results for two sets of intertemporal elasticity
values, β =0.16 and β =0.42. Table 7 reveals that the value of γ (t) decreases with an
increase with a value of β used to impute the fixed effects. This is expected as a constancy in
the uncompensated elasticity requires inverse relationship
-------------------------Table 7 here ----------------------------------------------------
between β and γ (t) . As it was discussed in the theoretical section of the paper, to find the
uncompensated elasticties we need to find β + γ (t) . Assuming that the average working time
for a typical man is 40 years, dividing γ0 by 40 gives an average cross uncompensated
elasticity of 0.00075. If we add this average cross-uncompensated elasticity to β =0.16 we
get a value of 0.15925 which implies that the average own-uncompensated elasticity is
approximately 0.16. Note that, assuming leisure is normal in all time periods, the theoretical
36
prediction is that β > β + γ (t) - ζht w t > β + γ (t) . Thus the own-compensated elasticity of
labor supply is in fact 0.16. This can be interpreted as follows. A 100 percent increase in
wages at time t will induce about a 16% increase in labor supply ( number of hours worked)
in time t but leaves the number of hours worked in all other periods almost unaltered.
Table 7 suggest that ζ is negative but statistically not significant at less than 10 percent level
which suggests that three of the elasticities discussed in this paper are almost the same. The
measure of initial wealth employed in this study is derived from investment income. This has,
of course, a number of problems as it does not truly measure actual wealth without error. This
could be a partial explanation for the insignificant coefficient. We also experimented with
property income and other family income but the results were very similar to what is reported
in Table 7. Unfortunately SLID does not contain detailed information on asset and wealth.40
Overall our results imply that equating intertemporal labor substitution and the
uncompensated elasticity would not be too bad as an approximation as is done in crosssectional studies. As it can be verified from Table 8 our results are very similar to those in the
literature. Smith Conway and Kniesner (1998), MaCurdy (1981) and Tries (1990) report
similar results for the USA.
Adding the value of γ0 to β gives the effect of a parallel shift in the wage profile of an
individual over the lifecycle. According to the empirical results γ0 + β (0.16-0.03) is found
to be 0.13. This can be interpreted as follows. A 1 percent increase in all wages over the
lifecycle leads to an increase in lifecycle labor supply in all ages by 0.13 percent. Similarly,
we can gauge the effect of a change in the slope of the wage profile by adding γ 1 and γ 2
with β . Unfortunately these slope parameters are not statistically significant at less than the
40
In fact the internal file for asset and wealth in SLID is blank.
37
10 percent level in cases where β =0.16. However, the negative values of γ 1 and γ 2 imply a
backward bending labor supply.
VI. Conclusion
This paper attempts to estimate lifecycle labor supply model for men in Canada using
sample selection corrected panel data models that control for individual heterogeneity.
We employ data from the second panel of Canadian Survey of Labor and Income
Dynamics (SLID) spanning from 1996 to 2001. The results confirm that lifecycle labor
supply estimates are very sensitive to the specification and changes in the econometric
methods employed. Failure to control for the individual heterogeneity as in crosssectional models, and sample selection in most cases produces upward biased
intertemporal labor substitution elasticity. The models which do not account and correct
for sample selection bias produce intertemporal labor substitution elasticity ranging
from 0.23 to 0.48. However, using WE estimator after correcting for sample selection
problem and individual heterogeneity intertemporal elasticity we find intertemporal
labor substitution elasticity of 0.16. Moreover we find the wealth effects to be
statistically insignificant implying that the intertemporal elasticity is a very good
approximation of both the compensated and uncompensated labor supply elasticity.
The key message of this study is that situating the labor supply decision of an individual in a
lifecycle setting produces meaningful parameter estimates and controlling for individual
heterogeneity and sample selection bias matters. As it was demonstrated in this paper it is
very crucial to understand the parameter estimates from a lifecycle labor supply model. For
example, the empirical results show that the effect of a 100 percent increase in wages at a
typical age would induce an increase in labor supply at that particular age by 16 percent but
would leave all labour supply decisions in all other ages unchanged. On the other hand, a 100
38
percent increase in wages over all ages (parallel shift of the wage profile) would induce a
13% increase in labor supply in all ages.
However, our results should be interpreted with caution. This study implicitly assumes that
individuals make their labor supply decisions freely. This may be a problem in light of the
institutional constraints in our modern capitalists systems. This study also does not take taxes
and fixed costs of work into account. We also assume in this study that individuals are not
liquidity constrained and there is no human capital formation which affects labor supply
decisions. Furthermore, the assumptions that we adopted to project lifetime wages and initial
wealth should also be remembered when we interpreted the results. Lastly it would be useful
to experiment with different utility functions to check the robustness of our results to changes
in the utility function.
39
Appendix A
Table 1. Descriptive statists of the key variables
Variable
Observations
Mean
Std. Dev
Annual Hours worked
Imputed log wage
Health
Other income
Age
Age squared (Age 2)
Education
Marital status variables
Common-law=1 if in common law relationship
zero otherwise.
Married=1 if married and zero otherwise.
Separated=1 if separated zero otherwise
Divorced=1 if divorced and zero otherwise
Widow=1 if widow zero otherwise.
Single
Visible minority
Children age 0 -5 years
Children age 5-15 years
45455
45455
45455
45455
45455
45455
45455
1487.64
2.92
0.22
38759.06
43.70
2021.38
13.23
1055.33
0.20
0.16
46771.90
10.57
949.59
3.89
45455
45455
45455
45455
45455
45455
45455
45455
45455
0.08
0.70
0.03
0.04
0.01
0.14
0.04
0.24
0.61
0.27
0.46
0.17
0.20
0.10
0.35
0.21
0.57
0.94
40
Table 2. Panel data estimates of men labor supply using selected sample
Pooled
RE
Education
Age
Age2
Married
Common-law
Separated
Divorced
Widow
Imputed wage
Health
Other income
Children age 0 -5 years
Children age 5-15 years
Constant
Hausman test (FE Vs RE)
-9.34
18.59
-0.33
235.08
108.48
65.95
55.90
118.78
458.13
-47.62
-0.002
6.21
5.44
527.89
β
0.23
(6.45)
(5.01)
(7.92)
(18.74)
(6.79)
(2.89)
(2.63)
(2.5)
(14.62)
(1.93)
(29.27)
(0.91)
(1.25)
(6.47)
-10.61
33.42
-0.51
230.22
112.64
39.50
1.88
60.63
534.70
104.43
-0.004
1.73
-6.72
-18.24
(4.71)
(6.39)
(8.79)
(12.98)
(5.33)
(1.41)
(0.07)
(0.98)
(13.39)
(7.87)
(44.12)
(0.21)
(1.14)
(0.18)
0.27
FE
-47.61
54.95
-0.82
83.95
63.44
-100.99
-113.79
-43.89
616.66
-99.45
-0.005
-19.57
-33.12
221.32
(3.49)
(3.06)
(5.88)
(2.75)
(1.99)
(2.49)
(2.62)
(0.52)
(3.34)
(2.64)
(46.41)
(1.91)
(3.90)
0.99)
0.31
LM(Pooled Vs non pooled)
Data Source: (SLID 2001). Absolute value of t-statistics in parentheses.
Table 3. Standard panel data labor supply model estimates ( full sample)
Variable
Pooled
Education
Age
Age2
Married
Common-law
Separated
Divorced
Widow
Imputed wage
Health
Other income
Children age 0 -5 years
Children age 5-15 years
Constant
2.62
59.60
-1.05
379.58
219.98
144.72
74.63
218.33
591.36
472.34
-0.004
-79.77
-48.32
-1263.78
Hausman test (FE Vs RE)
414.94
(1.46)
(13.86)
(-22.63)
(24.22)
(10.73)
(5.04)
(2.92)
(4.44)
(15.45)
(31.90)
(50.43)
(9.07)
(8.87)
(13.76)
β
RE
1.31
56.63
-1.01
248.74
144.11
47.09
-34.80
131.50
707.98
176.13
-0.005
-38.34
-41.95
(0.48)
(9.92)
(16.60)
(11.68)
(5.72)
(1.49)
(1.07)
(2.17)
(15.79)
(14.09)
(57.58)
(4.19)
(6.26)
0.39
0.48
LM(Pooled Vs non pooled)
36204.93
Data Source: SLID (2001) Absolute values of t-statistics in parentheses.
41
FE
-36.93
96.07
-1.24
58.80
44.71
-111.42
-160.30
0.05
351.72
94.42
-0.005
-23.00
-45.56
0.25
(2.55)
(5.29)
(9.15)
(1.78)
(1.29)
(2.62)
(3.58)
(0.00)
(1.85)
(7.05)
(53.96)
(2.12)
(5.24)
Table 4. Samples selection model estimates of lifecycle labor supply model( reported estimates are marginal effects)a
Constant
Education
Age
Age2
Married
Common-law
Separated
Divorced
Widow
Imputed wage
Health
Other income
Children age 0 -5 years
Children age 5-15 years
Random Effects
-1010.86 (-15.30)
7.13 (5.73)
52.40 (18.11)
-0.89 (-27.80)
263.91 (26.75)
154.51 (11.5)
90.40 (4.66)
26.50 (1.71)
118.83 (3.64)
363.79 (0.14)
-146.00 (13.12)
-0.02 (-7.73)
-70.67 (-116.86)
-40.94 (-10.09)
Tobit Model
Pooled OLS
(12.89)
12.12 (5.86)
88.82 (17.59)
-1.51 (27.29)
447.04 (24.82)
261.59 (11.19)
153.02 (4.65)
44.76 (1.51)
201.23 (3.36)
615.41 (13.75)
-247.38 (7.20)
-0.005 (50.184)
-119.67 (11.99)
-69.35 (11.13)
β
Fixed Effects
-47.9
160.87
-2.17
52.47
42.36
-197.95
-237.55
-32.27
500.57
-40.4
-0.005
-49.71
-74.1
(3.04)
97.69
(13.39)
(1.46)
(1.13)
(4.17)
(4.70)
(0.33)
(2.33)
(0.93)
(66.19)
(4.12)
(7.46)
Sample selection modela
Fixed Effects
Pooled OLS
-47.77 (11.75)
56.44 (6.88)
-0.84 (143.26)
83.95 (6.91)
62.98 (1.52)
-103.12 (2.62)
-115.92 (2.19)
-43.40 (0.48)
619.10 (3.04)
-98.92 (1.84)
-0.005 (2.34)
-20.28 (1.72)
-33.82 (2.77)
-2.79 (1.85)
40.69 (11.07)
-0.68 (16.49)
305.07 (23.04)
153.28 (9.10)
91.18 (3.88)
54.99 (2.56)
144.08 (3.46)
479.39 (15.61)
-50.06 (2.27)
-0.003 (59.369)
-23.60 (3.12)
-12.53 (2.81)
0.25
0.41
0.34
0.32
0.25
N
947
TN
45455
Data Source (SLID 2001) Absolute values of t-statistics in parentheses.
Note: Health is endogenous. We estimate health equation with predictors including age, age squared, imputed wage, self reported stress level, and
time dummies.
The reported wage is imputed wage from sample selection corrected wage offer equation.
a
The sample selection RE model could not converge and was not estimated.
42
Table 5. Estimated λt from the estimated cross-section selection equations
λt
λ1996
λ1997
λ1998
λ1999
λ2000
λ2001
-0.42 (6.00)
-0.48 (5.24)
-0.44 (4.84)
-0.38 (4.22)
-0.38 (4.00)
-0.31 (3.54)
Data source: (SLID 2001). Absolute value of t-statists in parentheses.
43
Table 6. Fixed effects labor supply model using the selected sample only with the Inverse
Mills ratios included (to test for sample selection).
Variables
Constant
Education
Age
Age2
Children age 0 -5 years
Children age 5-15 years
Imputed wage
Other income
Health
Married
Common-law
Separated
Divorced
Widow
λ1996
λ1997
λ1998
λ1999
λ2000
λ2001
β
N
TN
F Test for H 0 : λi = 0
Fixed Effects b
-111.46
11.44
-0.78
363.77
-45.62
732.56
-0.008
-88.98
-14.52
73.30
-131.06
-120.65
9.33
(6.70)
(0.52)
(3.97)
(5.13)
(3.80)
(4.54)
(55.83)
(2.38)
(0.43)
(2.28)
(3.21)
(2.75)
(0.11)
18236.94 (8.77)
Wooldridge’s(1995)a
-35.29
-4.60
-0.13
8.03
212.78
315.65
-0.0027
15.71
173.81
95.56
61.05
29.20
79.58
(10.74)
(0.96)
(2.31)
(0.74)
(8.54)
(7.32)
(29.45)
(0.25)
(13.03)
(5.91)
(2.63)
(1.35)
(1.61)
6632.59 (18.89)
-12791.08 (5.64)
-6942.56 (13.1)
-1548.07 (0.84)
1603.74 (3.66)
-1254.27 (1.38)
-600.76 (2.92)
10414.88 (6.00)
5003.56 (10.35)
-14583.49 (7.03)
0.36
9947
33555
-6961.90 (16.92)
0.16
F(6,35476) 11.8
224.14
Data Source (SLID 2001)
Note: Health is endogenous. We estimate health equation with predictors including age, age
squared, imputed wage, self reported stress level, and time dummies.
The reported wage is imputed wage using sample selection corrected wage offer
equation.
Time dummies indicating each year of survey were included in the above labor
supply equation.
a
The full set of results are available in the Appendix. The interpretation of the actual
size of the
coefficients for most of the variables requires knowledge of the full set of results.
b
we have also conducted a test following Nijman and Verbeeck (1992) by including
the lagged Inverse Mills’s ratios only and we get similar results.
44
Table 7. Implied uncompensated elasticities
γ0
γ1
β =0.16
γ2
-0.03
-0.22
-0.94
(2.69)
(0.98)
(0.98)
β =0.30
-0.015
-0.12
-0.78
(2.09
(2.98)
(5.74)
Absolute value of t-statistics in parentheses
Note: we employ two stages least square instrumental variable estimation
ξ
0.00
(0.00)
0.00
(0.00)
Table 8. Survey of selected studies labor supply of men
Author
Country of study
Substitution
elasticity
[0.0,0.35]
Uncompensated
wage elasticity
Wealth
elasticity
Altonji ( 1986)
USA
Hausman(1981)
US
[0, 0.03)
[-0.95, -1.03]
Blomquist (1983)
Sweden
0.08
[-0.03, -0.04]
Blundell and Walker (1996)
UK
0.024
-0.287
Triest (1990)
US
0.05
0
Van Soest et al(1990)
Netherlands
0.12
-0.01
MaCurdy (1981)
USA
[0.1,0.3]
[0.1, 0.3]
0
Reilly(1994)
Canada
0.6a
[0.05,0.26]
Hum, Simpson & Fissuh (2007)
Canada
Kumar(2005)
USA
[0.5,1.26]
[0.9,1.0]
Ham & Reilly(2006)
USA
Kniesner & Ziliak (2006)
[0.2,0.5]
Ziliak & Kniesner (1999)
USA
[0.12, 0.15]
Ghez & Becker (1975)
USA
[-0.068,0.44]
Smith(1977)
USA
0.32
Smith Conway& Kniesner(1999)
USA
[-0.024, 0]
Kimmel and Kniesner(1998)
USA
0.39
Kuroda and Yamamoto (2007)
Japan
[0.1,0.2]
This study
Canada
[0.16,0.48]
[0.16,0.48]
0.00
Notes:Most of the studies adopt different estimation strategies and any comparison of the
estimates must be cautious.
C
It is some times called income elasticity. However since there is no as such income effect
and if there is any it should operate via the marginal utility of wealth the designation of
income effect is misleading.
a
Elasticity is with respect to number of weeks worked.
45
Appendix B
1500
1000
500
0
Annual Hours worked
2000
Age hours worked profile over the life cycle
20
30
40
Age
50
60
70
Figure 1B
1400
1200
1000
800
2
2 .5
3
I nw a g e
3 .5
4
Figure 2B
2000
Education- hours worked profile
0
500
Hours worked
1000
1500
Annual Hours worked
1600
L o g w a g e h o u r s w o r k e d p r o f ile o v e r t h e lif e c y c le
0
10
20
Years of Schooling
Figure 3B
46
30
40
Annual
Hours
Worked
Figure 4 B - Age -Hours Worked Profile by Self Reported Health Status
2000
1500
1000
500
0
20
30
40
Age
Good health
50
60
Poor healths
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