Simultaneous Equation Models with Endogenous
Limited Dependent Variables: Efficiency of
Alternative Estimators (1)
Giorgio Calzolari
Università di Firenze, Department of Statistics “G. Parenti”, [email protected]
Antonino Di Pino
Università di Messina, Department “V. Pareto”, [email protected]
Abstract: A simultaneous equation model with two endogenous limited dependent
variables characterized by an identical selection mechanism is considered in this study.
The FIML procedure proposed by Poirier-Ruud (1981) for a single equation switching
model is extended to provide the estimation of two distinct simultaneous equations (for
instance, wage and reservation wage equations) joining with the selection function.
Efficiency of FIML approach with respect to the Two-Stage Heckman approach is
verified by an accurate Monte Carlo experiment taking into account endogeneity and
assuming different distributions of the error terms. Simulation results show that if
distributional assumptions of normality and homoskedasticity in the error terms are
relaxed, finite and large sample bias is produced by the two parametric procedures
above mentioned. But a semiparametric procedure as the Two-Stage CLAD (Khan and
Powell, 2001) produces consistent estimates for the equation referring to the regime
with more uncensored cases. The results of an empirical application on a sample of
Italian Graduates are quite similar for the equation with more uncensored data (the wage
equation), while FIML estimates are markedly different with respect to the T-S
Heckman in the equation in which most cases are censored (reservation wage equation).
Keywords: Selection Bias, Endogenous Switching, semiparametric regression
(1)
A. Di Pino has benefited from financial support of MIUR - National Research Project (PRIN 20082010)
1
1. Introduction: estimation of a regression model with a two-regime
specification
We consider a simultaneous equation model with two equations (Eq. (1) and Eq. (2))
whose dependent variables (for instance, individual wage and reservation wage taken
from a cross-sectional survey sample) are both partially observed, or "limited", as a
consequence of a selection mechanism that doesn’t allow to observe them together. In
particular, the observation of one of the two dependent variables doesn’t permit us to
observe the other. The selection mechanism may be specified by a third equation (Eq.
(3)) whose dependent variable is a binary dummy “indicator” that produces
two
different regimes given, in this study, by the working status of the subject1.
Furthermore, we can consider that the choice of a subject to work or not is influenced
by both wage and reservation wage, with opposite effects. The two-regime
characteristic suggests to specify the model as an endogenous two-regime “switching”
regression model (Maddala and Nelson, 1975; Lee, 1978; Poirier and Ruud, 1981;
Heckman, 1990; inter alia). In this context, we may adopt different alternative
approaches to estimate the model: i) a parametric Two-Stage procedure, defined here TS Heckman, generally used to estimate regression models with limited dependent
variable (Heckman, 1976; Lee, 1978; Lee, Maddala and Trost, 1980); ii) a parametric
Maximum Likelihood (FIML) approach utilized to estimate regression models with
endogenous switching (Poirier and Ruud, 1981), iii) a Two-Stage Censored Least
Absolute Deviations (CLAD) estimator as a semiparametric approach to estimate
censoring model without imposing distributional assumption to the error terms (Khan
and Powell, 2001).
Purpose of our analysis is to compare the performances of the three estimation methods
by means of a detailed set of Monte Carlo experiments. Using simulated data, we
analyze the finite sample bias of the estimation results and the relative efficiency of the
estimation methods when the error terms of the selection criterion equation are
1
Another relevant application of a two-regime approach may be given by a model that estimates
simultaneously labour income equations of both subjects who attend the college after high school and
subjects who decide to access directly the labour market,. The choice to attend the college or not is
considderd as a binary selection mechanism . A large discussion of the problem can be found is in the
study of Carneiro, Hansen and Heckman (2003).
2
correlated with the error terms of the individual wage and reservation wage equations
(endogeneity). Simulated data are produced with different distributions of the error
terms. Particularly interesting is the case where the usual assumption of
homoskedasticity is relaxed. To conclude, an empirical application is proposed on
individual wage and reservation wage equations, using a sample of Italian Graduates.
In the next section a brief survey of estimation methods for endogenous switching
models is reported. In the third paragraph, we provide a stochastic specification of ,
respectively, maximum likelihood (FIML), T-S Heckman and semiparametric T-S CLAD
estimators taking into account the case in which two different limited dependent
variables, explained by potentially different regressors, are observed in each regime. In
the fourth paragraph, the results of simulations based on the use of the above mentioned
parametric and semiparametric procedures are discussed. The results of an empirical
application estimating both wage and reservation wage function on a sample of Italian
graduates are reported in the fifth section. Then, the sixth paragraph reports final
observations and remarks.
2. Conceptual framework
We may consider the binary dependent variable of the selection criterion equation
analogous to a selection "rule": if the dependent variable is one, the individual is
employed, and if the variable is zero the individual is unemployed.
However, the
assignment of an individual to a working or a non-working status follows a “nonrandom” criterion as a consequence of the dependence of the selection criterion with
respect to the outcome (the wage, for instance). In this context, the effect of the
explanatory variables on the two limited dependent variables in the two different
regimes can be defined as an "endogenous switching".
Several ways have been suggested to model this causal relationship in an endogenous
switching framework. In an article of 1974, Maddala and Nelson discuss the
methodological issues of a model where an endogenous selection criterion produces a
two-regime regression model. The problem consists in the estimation of demand and
supply schedules in disequilibrium markets. In this context, the demanded quantity and
3
the supplied quantity of a commodity, during period t, are not observed, and only the
exchanged quantity and the market price during the period are observed. The price
difference observed during the period represents the selection criterion for the two
regimes. The price rises if demand is higher than supply, and we can assume that the
observed exchange quantity is equal to the supplied quantity. Analogously, the price
diminishes if demanded quantity is lower than supplied quantity, and in this case we
can assume that the demanded quantity is observed. If we consider demanded quantity
and supplied quantity as different dependent variables explained by two distinct
regression equations, the Maddala-Nelson specification of the demand-supply model
does not permit to observe at the same time both demanded quantity and supplied
quantity, but only one of these. The authors suggest to adopt a maximum likelihood
estimator by utilizing conditional densities of observed exchanged quantity (normally
distributed) to be equal to demanded quantity or to supplied quantity.
Schmidt (1978) and Lee (1978) propose two different estimation methods to explain
wage difference between unionized and non-unionized workers. Schmidt adopts a
maximum likelihood estimator given by the product of the marginal densities of two
variables that are assumed to be independent from one another: 1) the error term of the
wage equation, and 2) the difference between the expected wage of a unionized worker
and the expected wage of a non-unionized worker.
Lee (1978) suggests a Two-Stage approach, where at a first stage an indicator
equation, given by a union membership function, is estimated by a Probit reduced form
regression. At the second stage, the two equations, respectively, of wage of unionized
workers and of wage of non-unionized workers are estimated by OLS. Both the OLS
regressions are corrected for selectivity by utilizing as regressor the estimated inverse of
Mill's ratio, provided by the Probit regression at a first stage for both unionized and
non-unionized workers equations. This Two-Stage method (known as Probit Two-Stage
method) is similar to the procedure introduced by Heckman to obtain a consistent and
computationally feasible estimate of limited dependent variable equations (Heckman,
1976).
However, in the Two-Stage estimation approach to the two-regime regression models,
the selection function may be estimated by a reduced-form Tobit regression. This is the
case, for instance, of the estimation of the education return model. In this context, the
4
desired years of college education involves the two-regime selection criterion. This
variable is not dichotomous, but “limited”, and when it is equal to zero, it determines
the regime in which education return does not depend on college attendance (Kenny,
Lee, Maddala and Trost, 1979). In this case, we can estimate at the first stage the
selection criterion equation by a reduced-form Tobit regression. The predicted values of
the first stage estimation serves to obtain the inverse of the Mill's ratio estimates to
utilize at a second stage as correction term, like in the T-S Heckman model, for the
OLS estimation of education return for subjects with college attendance .
We focus our attention on the theoretical approach suggested in Poirier and Ruud
(1981). The authors show that we have an endogenous switching model only if the
specification of the regression equation in two different regimes is related to the
expected value of the dependent variable in each regime. If applied to our example, this
implies that the individual reservation wage level is given by the subject’s evaluation of
her/his own qualities and endowment, and it is not conditional on her/his non-working
status. Analogously, working income or wage depends on the individual endowment,
education and on exogenous characteristics of the labour market, but does not depend
on the specific employment condition of the subject. In this context, a FIML estimation
procedure can be based on a likelihood function given by the product of marginal
probabilities of the subjects to perceive a wage or, alternatively, to desire a reservation
wage. The Poirier-Ruud assumption involves consequences in terms of identifiability of
the model. The main consequence is that a part of the variables conditioning the
selection criterion (at least one) should be independent with respect to the outcome
variables (wage and reservation wage). A similar identification criterion is requested if a
T-S Heckman procedure is applied.
Powers (1993) applies the endogenous switching regression
estimation methods
(including the Poirier-Ruud FIML procedure) to estimate the effect of family structure
on the phenomenon of female’s early marriage and fertility. The author hypothesizes
that the binary variable indicating whether a young woman reported giving birth to a
child before her 20th birthday or not can be specified by an equation model
characterized by two regimes: the first, whether the woman lived her childhood and
adolescence apart from one or both biological parents or in a context of economic
5
deprivation; the second whether the woman grew up in a family where both biological
parents were present.
However, the validity of a FIML estimator, and of a parametric method in general,
requires a correct specification of the error distribution. Misspecification in the
distribution of the error terms in the model equations, such as heteroskedasticity or nonnormality, may cause inconsistency to FIML estimation results.
To circumvent this problem, in the last decades, several semiparametric alternatives to
estimate consistently censored regressions have been proposed. A well-known
semiparametric approach is the Censored Least Absolute Deviations (CLAD) estimation
method, proposed by Powell (1984). CLAD is a root-n-consistent estimator that allows
to obtain the regression coefficients by minimizing the sum of absolute residuals. The
adoption of this loss criterion implies to assume that the median of the error distribution
is equal to zero. Under this assumption, the median of the dependent variable, y, of the
regression equation y = x'φ + η is the regression function x'φˆ as in the quantile median
regression. The distributional assumption of "zero median" in the error terms is not
very restrictive, and permits non-normal, heteroskedastic and asymmetric errors.
However, Censoring LAD (or MAD2 ) regression runs if the uncensored distribution of
the error terms include the median (equal to zero). This implies that a semiparametric
CLAD estimator can be adopted only if the censored cases are less than the uncensored
cases. As a consequence, in our two-regime model, only the equation referring the
subsample with more observations can be estimated using semiparametric approach.
Therefore
a
comparison
of
estimation
properties
between
parametric
and
semiparametric procedures can be conducted only considering the regime with the
higher number of observations.
Regarding inferential properties of CLAD estimator, Paarsh (1984) and Moon (1989)
show that a bias characterizes the finite sample distribution of the CLAD markedly
when we have a high share of censored observations. To solve this problem, Khan and
Powell (2001) suggest a new two-step version of the CLAD estimator with the purpose
of correcting the bias effect in finite samples. The authors adopt, at the first step, a
semiparametric or nonparametric procedure to select the observations with a positive
2
MAD: Median Absolute Deviation
6
value for the regression function; at the second step, a median quantile (MAD or LAD)
regression is performed on the selected observations.
Regarding the first stage estimation of the Khan-Powell’s procedure, a problem occurs
for the computational difficulty to estimate non-parametrically functions with high
dimensional arguments. In order to circumvent this difficulty, Khan (2005) and Blevins
and Khan (2009) suggest to use, for the selection function estimation, a Nonlinear Least
Square “Sieve” estimator that allows to estimate semiparametrically the selection
criterion function. The characteristics of this estimator will be briefly discussed in the
next section.
Parametric and semiparametric approaches to a two-regimes simultaneous equations
estimation will be here compared with reference to their own respective finite-sample
inferential properties by means of a Monte Carlo experiment. FIML and T-S Heckman
can be compared in order to evaluate estimation performance in both regimes. The
model here adopted, however, has a different stochastic specification with respect to
the previous endogenous-switching model versions, because in the two regimes two
limited dependent variables equations with potentially different explanatory variables
are specified. For this reason, in the next section theoretical issues and the stochastic
specification of, respectively, the Poirier-Ruud FIML estimator, the Heckman TwoStage (or Probit Two-Stage) approach, and the Khan-Powell semiparametric procedure
are briefly illustrated.
3. Stochastic specification of the model and estimation procedures.
Firstly, we provide a generic specification of the model as follows.
Let’s start considering a three equations model in which wage, reservation wage and the
individual participation propensity in the labour market will be estimated
simultaneously.
W = x'1 α + u
if L = 1; 0 otherwise
(1)
RW = x'2 β + v
if L = 0; 0 otherwise
(2)
7
 L = 1 if L* > 0

 L = 0 otherwise
L* = z ' γ + ε
(3)
Moreover, if L = 1, ε > − z ' γ ; if L = 0, ε ≤ − z ' γ
The variable W is a vector of n1 elements, the wages and salaries perceived by the n1
employed people. The variable RW is a vector of n2 elements, the reservation wages
desired by the n2 unemployed subjects and looking for a job. The binary variable L is a
vector of n= n1 + n2 elements composed by n1 elements equal to one and n2 elements
equal to zero. Moreover, x'1; x’2 and z’ are row-vectors, respectively, of three exogenous
variables matrices X1, X2 and Z. Some exogenous explanatory variables on the right
hand side of each equation can be common to the three equations, but the following two
identification conditions must be observed: i) at least one of the regressors of L
(included in Z) must be independent with respect to W and RW, and ii) at least one of
the regressors of, respectively, W and RW, must not appear in the equation of L. The
error terms u, v, and ε are assumed normally distributed with covariance matrix given
by:
 σ u2

Σ= 0
σ uε

0
2
v
σ
σ vε
σ uε 

σ vε  .
(4)
1 
If the covariances σ uε and σ vε are assumed to be different from zero, the indicator
variable L can be considered as non-exogenous with respect to W and RW.
Consequently the "change" of regime from L = 0 to L = 1 should be considered as an
"endogenous" switching.
Given the characteristics of the model, the rationale of the three estimation procedure
previously discussed will be here briefly reported.
3.1 The FIML estimator
We start to consider the previous aleatory variables u, v, and ε. For the well-known
properties
of
density
have: ϕ (u, ε ) = ϕ (u )ϕ (ε u )
functions
and
and
of
conditional
ϕ (v, ε ) = ϕ (v )ϕ (ε v ) .
distributions are:
8
The
density
functions
probability
we
marginal
P (u ) =
z 'γ
+∞
∫ ϕ (u, ε )dε = ∫ ϕ (u , ε )dε
− z 'γ
and
P (v ) =
−∞
− z 'γ
+∞
−∞
z 'γ
∫ ϕ (v, ε )dε = ∫ ϕ (v, ε )dε
(5)
Furthermore, by substituting L = 1 or L = 0 into the Equation (3), we have, respectively:
P(u u > 0) = P(ε > − z ' γ ) = 1 − Φ(− z ' γ ) = Φ( z ' γ )
if L = 1
(6a)
P(v v ≤ 0) = P(ε ≤ − z ' γ ) = Φ(− z ' γ ) = 1 − Φ ( z ' γ )
if L = 0
(6b)
If we consider the error terms u and v normally distributed, the p.d.f. of u if u u > 0 and
of v if v v ≤ 0 are, respectively:
ϕ (u ) =
1
σu
 W − x'1 α 

 σu

Φ(z' γ )
ϕ 
if L = 1,
and ϕ (v ) =
1
σv
 RW − x'2 β 

σv


1 − Φ(z ' γ )
ϕ 
if L = 0 (7)
Furthermore, utilizing the conditional density of a bivariate normal distribution and
assuming
L (u, v, ε ) = ∏ ϕ (u, ε )∏ ϕ (v, ε ) = ∏ ϕ (u )∫ ϕ (ε u ≥ 0)∏ ϕ (v )∫ ϕ (ε v ≤ 0 )
L =1
L =0
L =1
the
L =0
Likelihood function to be maximized is:


 1  W − x' α
∏  σ ϕ  σ 1
L =1  u
u




 
σ uε
  z ' γ + 2 (W − x '1 α )  
σu
 
   ⋅  1 ϕ  RW − x '2 β
 Φ

1

  ∏
σv
2
2 2
L =0  σ v
 

σ
σ
−
u
uε



 
(
)


 
σ vε

   
(
)
z
'
+
RW
−
x
'
β
γ


2
σ v2

 


⋅ 1− Φ
1



2
2 2
σ
−
σ

v
vε

  


  
(

 ⋅

(8)
)
This function differs from the Poirier-Ruud’s because different dependent variables are
observed in the two regimes (W and RW, respectively), and the explanatory variables
may change in both respective equations.
9
3.2 The Two-Stage Heckman (or Probit Two-Stage) estimator
Considering the popularity of the T-S Heckman approach to the estimation of model
with selectivity and its large utilization in many empirical studies, here we only describe
briefly the specification of T-S Heckman model applied to a two-regime simultaneous
equation estimation ( Lee, 1978; Lee, Maddala and Trost, 1980; Heckman, Tobias and
Vytlacil, 2003).
Starting from the previous basic model, this estimation method consists in a two-stage
procedure. A first stage is given by a Probit estimation of Eq. (3), z ' γˆ , normalizing
σ ε2 = 1 . Then, conditional on working status, L = 1, the expected value of wage is
derived from Eq. (1):
E (W L = 1) = x'1 α + E (u L = 1)
(9)
E (W L = 1) = x'1 α + E (u ε > − z ' γ )
(10)
If u and ε are assumed to be normally distributed, we have: E (u ε ) =
σ uε
ε.
σ ε2
Assuming σ ε2 = 1 and considering the properties of the truncated normal distribution,
we have:
E (W L = 1) = x'1 α + σ uε
φ (z ' γ )
Φ(z ' γ )
(11)
Analogously we obtain the expected value of reservation wage from Eq. (2):
E (RW L = 0 ) = x'2 β + σ vε
φ (z ' γ )
1 − Φ(z ' γ )
(12)
The Probit estimates at a first stage can be utilized replacing
Eq. (11), and
φ (z ' γ )
φ (z ' γˆ )
with
in
Φ(z ' γ )
Φ( z ' γˆ )
φ (z ' γ )
φ ( z ' γˆ )
with
in Eq. (12). Then, the parameters α and β and
1 − Φ(z ' γ )
1 − Φ( z ' γˆ )
the error terms covariances σ uε and σ vε can be estimated consistently at a second stage
by OLS utilizing the equations:
W = x'1 α + σ uε
φ ( z ' γˆ )
+ ηu
Φ ( z ' γˆ )
on employed people only
and
10
(13)
RW = x'2 β + σ vε
φ ( z ' γˆ )
+ ηv
1 − Φ( z ' γˆ )
on unemployed people only
(14)
where ηu and ηv are iid random errors with zero mean.
This estimation method is not generally efficient, but it is consistent. Lee, Maddala and
Trost (1980) showed that the covariance matrix of the coefficients α and β is
underestimated if heteroskedasticity affects the Probit estimation at a first stage. More
recently, Heckman, Tobias and Vytlacil (2003) discussed the properties of the treatment
parameters estimators in a framework similar to a two-regime switching model when
the assumption of correct specification of the errors distribution is relaxed. Applying a
Monte Carlo experiment, they observed that the estimated treatment parameters, using
T-S Heckman procedure, generally have small bias even if error terms are generated by
heavy-tailed Student-t. But when data are generated from highly asymmetric
distributions, as Chi Square(d.o.f. 3), estimates show large bias.
3.3 The Two-Stage CLAD semiparametric estimator
The semiparametric estimator proposed by Khan and Powell consists in a two-step
procedure. At a first step, the observations with a positive value for the regression
function are selected utilizing a non-parametric or a semiparametric estimator of the
selection equation (Eq. (3)) like, for instance, the Manski's Maximum Score estimator
(Mansky, 1985). This procedure, in particular, allows to estimate consistently Eq. (3) as
a binary choice model without imposing distributional constraints to the error terms. In
particular, this estimator involves choosing γ to maximize the following objective
function:
n
Qn = ∑ [Li ⋅1 ∗ ( z 'i γˆ ≥ 0 ) + (1 − Li ) ⋅1 ⋅ ( z 'i γˆ < 0 )]
(15)
i =1
The estimator seeks to maximize the number of correct predictions3. However this
procedure is often computationally cumbersome, especially if a large number of
variables are included as regressors. To solve this problem Khan (2005) shows how an
3
As an alternative to the Manski's Maximum Score estimator Horowitz (1992) proposed a version that
includes a smooth function. This approach is computationally simpler than the Mansky's estimator
11
alternative NLLS (Non Linear Least Squares) procedure under a conditional median
restriction is observationally equivalent to both Mansky and Horowitz estimators.
This procedure, known as a Sieve-NLLS estimator, is obtained by the minimization of
the following function:
1 n
Qn = ∑ Li − Φ z 'i γˆ * exp(l ( zi ))
n i =1
[
(
2
)]
(16)
where Li is a binary response variable (0;1), Φ () is the standard normal cumulative
distribution function, γ* is a normalized vector (1, γ) of parameters, and exp(l(zi)) is a
“sieve” function4. The Monte Carlo results reported in the study of Khan confirm
consistency and finite-sample inferential properties of Sieve-NLLS estimator5.
Then a Sieve-NLLS procedure can be applied to estimate, in a reduced form, the
selection (participation) function and consequently to select the observations to include
at the second stage in the median quantile regression. Regarding the present model, we
can use the Sieve-NLLS estimation results to select, at a second stage, the observations
with a "positive" index. In particular, if the number of observations conditional on L = 1
is higher than the number of observations conditional on L = 0, Eq. (1) (wage equation)
can be estimated using the CLAD estimator at a second stage. In particular, using
CLAD, we obtain the value of coefficients α̂ which minimizes:
1 n

ˆ
α = arg min ∑ Wì − max(0, x'1 α ) 
α
 n i =1

(17)
This procedure, based upon the conditional median of Wi, is analogous to the median
quantile regression (Powell, 1986):
1 n

ρθ [Wì − max(0, x'1 α )]
∑
 n i =1

αˆ = arg min 
α
where ρθ (
)
(with θ = 0.5)
(18)
is the "check function" introduced by Koenker and Basset (1978) that
simplifies the following equivalent equation:
4
Consider a simple model with two regressors, z1 and z2; the “Sieve” function is given by the following
polynomial terms:
exp(l0 + l1 z1 + l2 z2 + l3 z12+ l4z22 + l5z1 z2)
5
However, we must consider that the estimated coefficients of the selection criterion using Sieve-NLLS
procedure, given by the normalized vector γ*, are generally not comparable with the estimated
coefficients obtained adopting parametric methods.
12


αˆ = arg min θ
α

∑ αW − x' α + (1 − θ ) ∑ αW − x' α 
1
ì
Wì ≥ x '1
1
ì
Wì < x '1
(19)

In the next paragraph we compare on simulated data the results of the Two-Stage
Heckman, FIML and Two-Stage CLAD (Khan-Powell) estimation procedures.
4. Monte Carlo Results
Investigation by a Monte Carlo experiment is motivated by a desire to compare finite
sample performance and inferential properties of parametric and semiparametric
approach to the model estimation. All designs here considered are generated by the
model equations (1), (2) and (3), characterized by the inclusion in their right side of a
single regressor variable. Each regressor (x1, x2, and z) has mean and variance equal to
one. Slope and intercept coefficients and the elements of the error terms covariance
matrix are set as follows:
α = 20
α = 0
;
 α1 = 5 
σ u2 = σ v2 = σ ε2 = 1;
 β = 20
β = 0
;
 β1 = 5 
γ = −0.2
γ = 0
;
 γ1 = 1 
and:
σ uε = σ vε = 0.95
We impose a high correlation (95%) between the error terms of, respectively, Eq. (1)
and Eq. (3) and Eq. (2) and Eq. (3) to simulate the presence of strong endogeneity.
Three different error distributions are considered: Normal(0;1); Student-t (0;1) (d.o.f. =
3); Log-Normal (0;1). Moreover experiments conducted under these distributional
assumptions are replicated for the three distinct cases in which, respectively,
homoskedasticity, heteroskedasticity and "strong" heteroskedasticity in the error terms
of all the three equations are simulated. Heteroskedasticity is produced imposing to
the error term of each equation a variance proportional to the absolute values, |x1|, |x2|
and |z|, of the regressor; for strong heteroskedasticity, the variance is proportional to an
exponential function of the regressor: exp( ).
The simulation experiments consists of 10000 replications of the model estimation for
random generated samples of size equal to 1000. We utilize mean bias and root mean-
13
square error (RMSE) to compare the performance of each estimator: T-S Heckman,
FIML
and T-S CLAD (Khan-Powell). Censoring for the Eq. (1) estimation is
reproduced by imposing equal to 30% the percentage of cases in which the selection
response variable is equal to zero (L = 0). Simmetrically, the censored observations in
Eq. (2) estimation are equal to 70% . In semiparametric estimation of the coefficients of
Eq. (3) we impose γ1 = 1 as normalization criterion, in this way normalized vector of
parameters is equal to the vector of the "true" parameters, and Sieve-NLLS estimation
of the coefficient γ0 results comparable with the parametric procedures results.
Simulation results are reported extensively in the Appendix. In the first experiment,
homoskedasticity condition is assumed (Table A1). If the error distribution is normal,
we can observe how the bias in the estimated coefficients of Eq. (1) and Eq. (2) is
substantially negligible for all the three estimation methods, but FIML
procedure
performs better than T-S Heckman and T-S CLAD in terms of relative efficiency.
If we consider error terms distributed like a Student-t (3 d.o.f.), the bias of T-S Heckman
and FIML estimates of Eq. (1) and in Eq. (2) varies from 0.1% to 1% (in absolute
value), and bias of TS-CLAD estimates of Eq. (1) results negligible. However Eq. (3)
estimates are generally biased in each procedure. Analogously, if error terms
distribution is assumed to be log-normal (0;1), all the procedures are affected by a
small bias in Eq. (1) and in Eq. (2) estimates, and large bias in estimated coefficients of
Eq. (3). Variability of TS-CLAD
estimates seems to be smaller than parametric
methods.
A second Monte Carlo experiment takes into account heteroskedasticity in the error
terms distributions (Table A2). In this case, semiparametric estimators with normal or
Student-t errors distribution seems to have a negligible bias in the Eq. (1) coefficients.
Parametric procedures, for normal and Student-t distribution of error terms, in Eq. 1 and
in Eq. (2) are affected by a relative bias varying between 0.3% and 3% (in absolute
value). With log-normal errors, the relative bias of the Eq. (1) and the Eq. (2)
coefficients is between 0.24% and 2.37%. The estimation of Eq. (3) coefficients shows
a strong bias in all parametric and semiparametric procedures.
In the third Monte Carlo experiment strong heteroskedasticity in the error terms is
assumed (Table A3). In this case, the bias of parametric procedures markedly increases;
14
and especially if the errors are normally or Student-t distributed, T-S CLAD procedure
performs better than parametric methods.
Generally, comparing simulation results reported in tables A1, A2 and A3, we may
observe that the bias in estimated coefficients seems to depend particularly on the
presence of heteroskedasticity rather than distributional form of the error terms. The
estimated coefficients of Eq. 3 only show a higher sensitivity across different errors
distributions.
In conclusion, semiparametric procedure generally performs better than parametric
methods. Comparing parametric methods only, FIML estimator seems to be more
efficient than T-S Heckman in the estimation of the coefficients of Eq. 1 (regression
with a smaller percentage of censoring). RMSE of estimated coefficients of FIML
procedure are particularly lower than T-S Heckman when errors are normally or
Student-t distributed.
5. Empirical application: Italian graduates' labour income and reservation wage
estimation
We propose also an empirical application in which the performance of the previously
discussed parametric and semiparametric procedures are compared estimating both
graduate’s labour income and reservation wage on a sample selected from the ISTAT6
Survey on Italian Graduates in the year 2001 interviewed in the year 2004, three years
after their degree. The survey sample is composed by 14126 individuals; 12109 of them
are employed and 2017 are unemployed and looking for a job. Monthly labour income
is observed on the 12109 employed graduates only, and logarithm of monthly labour
income is here assumed as dependent variable in Eq. (1). In the survey, unemployed
graduates declare their own desired monthly salary, here considered as their desired
reservation wage and assumed as dependent variable of Eq. (2).
For wage equation (Eq. (1)) a specification similar to the well-known function
developed by Mincer7 is here adopted. But, with respect to the Mincer's function, it is
difficult to find here explanatory variables proxy of the difference across the subjects in
their working experience and education. In fact, cross-sectional survey sample is
6
ISTAT: Italian National Institute of Statistics
7
Cfr., inter alia, Mincer (1974) and Willis (1986)
15
composed by graduates interviewed only three years after their degree and with a
consequent low experience in the labour market.
We try to model individual difference in human capital endowment and ability by
introducing as regressors in Eq. (1) both high school test score (a positive coefficient is
expected) and
duration of the period of attendance of university beyond regular
completion time (a negative coefficient is expected). Furthermore, gender gap in wage
(generally unfavourable to women) is modeled by the introduction of sex as a dummy
regressor.
As a result of a preliminary reduced-form regression, reduced-form
estimated weekly working hours are also introduced to explain the wage difference due
to the choice of the subject to work full-time or part-time8. Dimension of firm or
administration in which the subject works is also a determinant of wage level, in the
sense that the individual wage is expected to be higher in large dimension firms.
Moreover, we consider career satisfaction and perspectives of the subject directly
related to individual labour income. For this reason, a dummy variable regarding career
perspectives of the subject is introduced as regressor.
The reservation wage equation (Eq. (2)) is specified taking into account the individual
factors influencing the desired wage of unemployed subjects. These factors and related
to the job search activity and to the expected job characteristics. In particular, we
consider as an important explanatory variable the decision to search for a part-time or a
full-time job.
To better identify the selection equation (Eq. (3)), we include as regressor the month in
which the subject took the degree, here assumed as a variable influencing working
status of the subject, but considered also to be independent with respect to both
individual wage and reservation wage. Specification of Eq. (3) involves also the use of
information about previous labour experience of the subject and the introduction of
regional fixed effects as proxy of economic activity influence. Another factor
potentially influencing the individual participation to the labour market is the decision
to attend training courses and, consequently, to postpone the access to labour market.
Then we estimate the previous model of three simultaneous equations, in which logwage and reservation wage (in Eq. (1) and in Eq. (2)) are considered as limited
8
The preliminary estimation, by a reduced-form OLS regression, of weekly hours serves to correct the
influence of potential endogeneity of the time devoted by the subject to paid work and his/her perceived
wage. The results of the estimation are not reported here for the sake of brevity.
16
dependent variables whose selection criterion for censoring is specified (symmetrically)
in the Eq, (3) represented by a reduced-form participation function with a binary
dependent variable.
Preliminary estimates show how the residuals of a Probit participation equation (Eq.
(3)) are correlated with both the residuals of the OLS estimates (run without any
selectivity correction) of Wage Equation (Eq. (1)) and reservation Wage Equation (Eq.
(2)), and the empirical values of correlation are 9% and 13%, respectively. We may
consider these statistics as gross measures of
endogeneity potentially affecting
estimates.
Estimation results are reported in Appendix (Tables A4, A5 and A6). We can observe
that the estimated coefficients of Eq. (1) (log-Wage equation) are similar if we apply
T-S Heckman or FIML as parametric procedures. T-S CLAD estimation results are
different, especially considering the influence of the sex and the satisfaction of job on
the individual wage. Furthermore, estimated standard errors of the semiparametric
procedure are lower than both parametric methods.
In Reservation Wage estimation (Eq. 2) we can compare the performance of the
parametric methods only. In this circumstance, estimation results show non-negligible
differences in estimated coefficients. Moreover standard errors of FIML estimates are
lower than T-S Heckman.
Estimation results of Eq. (3) (selection criterion equation) show small differences in
estimated coefficients if we compare the two parametric procedures. FIML estimates are
characterized by lower standard errors. We cannot consider a comparison with
semiparametric estimates because of the normalization of coefficients used adopting
Sieve-NLLS estimator. However we can also observe how the latter semiparametric
procedure run with nine regressors, intercept included (see Table A6). In this specific
case, the intercept is constrained to a value equal to one as a normalization (or
identification) restriction.
17
6. Final remarks
The results on simulated data show that, with respect to a parametric Two-Stage
procedure, FIML estimator performs better in terms of finite sample properties and
relative efficiency if estimates are affected by endogeneity.
However, semiparametric T-S CLAD estimator reports a smaller bias and lower
variability in estimates with respect to the FIML procedure; of course, the
semiparametric estimator can be utilized only in the estimates of the equation in which
the number of uncensored observations is larger than censored ones.
Still Monte Carlo experiments show that estimates are generally more sensitive to
heteroskedasticity than to the distribution form of the error terms. Semiparametric T-S
CLAD
simulation
results
seem
to
be
comparatively
less
influenced
by
heteroskedasticity.
The results of empirical application using the Italian graduates dataset show how FIML
and T-S Heckman estimates differ markedly in the regression on Reservation Wage
function, where uncensored observations are less than censored ones.
18
References
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Treatment Effects with an Application to the Returns to Schooling and Measurement
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19
Moon C. G. (1989) "A Monte Carlo Comparison of Semiparametric Tobit Estimators "
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Labor Economics, Amsterdam: North-Holland.
20
Appendix - Monte Carlo experiments and empirical application results
Table A1 - Simulation results for parametric and semiparametric estimators with homoskedastic
disturbances (percentage of cases with L = 0 index: 30%)
A1a -Error Terms Distribution:Normal (0:1); sample: 1000; NREP: 10000
Two-Stage Heckman
α0=20
α1=5
β0=20
β1=5
γ0= −0.2
γ1= 1
mean
mean
bias
20.0000
5.0002
19.9989
5.0001
-0.2017
1.0045
0.0000
0.0002
-0.0011
0.0001
-0.0017
0.0045
FIML
% mean
RMSE
bias
0.00%
0.00%
-0.01%
0.00%
0.84%
0.45%
0.0587
0.0305
0.1091
0.0406
0.0623
0.0639
mean
mean
bias
19.9997
4.9995
20.0003
5.0006
-0.1997
1.0009
-0.0003
-0.0005
0.0003
0.0006
0.0003
0.0009
Two-Stage CLAD
% mean
RMSE
bias
0.00%
-0.01%
0.00%
0.01%
-0.13%
0.09%
0.0446
0.0267
0.0710
0.0356
0.0517
0.0479
mean
mean
bias
20.0000
4.9999
---------0.1904
1.0000
0.0000
-0.0001
--------0.0097
0.0000
% mean
RMSE
bias
0.00%
0.00%
---------4.83%
0.00%
0.0739
0.0532
--------0.0772
0.0000
A1b - Error Terms Distribution: t-Student (3 d.o.f.); sample 1000; NREP: 10000
Two-Stage Heckman
mean
bias
mean
α0=20
α1=5
β0=20
β1=5
γ0= −0.2
γ1= 1
20.0166
4.9994
20.1250
4.9971
-0.2334
1.2118
% mean
RMSE
bias
0.0166 0.08% 0.0541
-0.0006 -0.01% 0.0294
0.1250 0.63% 0.1476
-0.0029 -0.06% 0.0538
-0.0334 16.70% 0.0648
0.2118 21.18% 0.0827
FIML
mean
19.9763
4.9997
20.2112
5.0003
-0.1154
0.9600
mean
bias
% mean
RMSE
bias
-0.0237 -0.12% 0.0390
-0.0003 -0.01% 0.0233
0.2112 1.06% 0.1604
0.0003 0.01% 0.0389
0.0846 -42.29% 0.0625
-0.0400 -4.00% 0.1126
Two-Stage CLAD
mean
mean
bias
19.9999
4.9991
---------0.1930
1.0000
-0.0001
-0.0009
--------0.0070
0.0000
% mean
RMSE
bias
0.00%
-0.02%
---------3.51%
0.00%
0.0441
0.0299
--------0.0578
0.0000
A1c - Error Terms Distribution: Log-Normal (0;1); sample 1000; NREP: 10000
Two-Stage Heckman
FIML
Two-Stage CLAD
mean
α0=20
α1=5
β0=20
β1=5
γ0= −0.2
γ1= 1
19.9477
5.0010
19.7867
5.0010
-0.5746
1.8368
mean
bias
% mean
RMSE
bias
-0.0523 -0.26% 0.0619
0.0010 0.02% 0.0396
-0.2133 -1.07% 0.0442
0.0010 0.02% 0.0190
-0.3746 187.30% 0.0969
0.8368 83.68% 0.1311
mean
19.9018
4.9997
19.8087
5.0007
-0.4236
1.4727
mean
bias
% mean
RMSE
bias
-0.0982 -0.49% 0.0528
-0.0003 -0.01% 0.0342
-0.1913 -0.96% 0.0481
0.0007 0.01% 0.0191
-0.2236 111.80% 0.1126
0.4727 47.27% 0.2127
21
mean
19.7014
4.9999
---------0.4804
1.0000
mean
bias
% mean
RMSE
bias
-0.2986 -1.49% 0.0339
-0.0001 0.00% 0.0243
-------------------------0.2804 140.21% 0.0490
0.0000 0.00% 0.0000
Table A2 - Simulation results for parametric and semiparametric estimators with heteroskedastic
disturbances (percentage of cases with L = 0 index: 30%)
A2a -Error Terms Distribution:Normal (0:1); sample: 1000; NREP: 10000
Two-Stage Heckman
mean
α0=20
mean
bias
FIML
% mean
RMSE
bias
mean
mean
bias
% mean
RMSE
bias
Two-Stage CLAD
%
mean
mean
mean RMSE
bias
bias
19.8858 -0.1142
-0.57%
0.0658 19.9404 -0.0596
-0.30%
0.0415 19.9993 -0.0007 0.00% 0.0407
α1=5
5.1191
0.1191
2.38%
0.0401 5.0531
0.0531
1.06%
0.0426 4.9994 -0.0006 -0.01% 0.0546
β0=20
20.3189 0.3189
1.59%
0.1022 20.2276 0.2276
1.14%
0.0678
-----
-----
-----
-----
β1=5
4.7565 -0.2435
-4.87%
0.0533 4.7898 -0.2102
-4.20%
0.0512
-----
-----
-----
-----
γ0=
−0.2 -0.2876 -0.0876 43.82% 0.0620 -0.2710 -0.0710 35.49% 0.0542 -0.2618 -0.0618 30.91% 0.0521
γ1= 1
0.8883 -0.1117 -11.17% 0.6367 0.8841 -0.1159 -11.59% 0.0600 1.0000 0.0000 0.00% 0.0000
A2b - Error Terms Distribution: t-Student (3 d.o.f.); sample 1000; NREP: 10000
Two-Stage Heckman
mean
α0=20
mean
bias
FIML
% mean
RMSE
bias
mean
mean
bias
% mean
RMSE
bias
Two-Stage CLAD
%
mean
mean
mean RMSE
bias
bias
19.9309 -0.0691
-0.35%
0.0618 19.9327 -0.0673
-0.34%
0.0423 19.9993 -0.0007 0.00% 0.0236
α1=5
5.0815
0.0815
1.63%
0.0392 5.0056
0.0056
0.11%
0.0349 4.9982 -0.0018 -0.04% 0.0319
β0=20
20.3252 0.3252
1.63%
0.1291 20.4162 0.4162
2.08%
0.2037
-----
-----
-----
-----
β1=5
4.7996 -0.2004
-4.01%
0.0699 4.8510 -0.1490
-2.98%
0.0568
-----
-----
-----
-----
γ0=
−0.2 -0.3354 -0.1354 67.71% 0.0612 -0.1749 0.0251 -12.53% 0.0741 -0.2265 -0.0265 13.23% 0.0332
γ1= 1
1.1510 0.1510 15.10% 0.0874 0.8970 -0.1030 -10.30% 0.1305 1.0000 0.0000 0.00% 0.0000
A2c - Error Terms Distribution: Log-Normal (0;1); sample 1000; NREP: 10000
Two-Stage Heckman
mean
α0=20
mean
bias
FIML
% mean
RMSE
bias
mean
mean
bias
% mean
RMSE
bias
Two-Stage CLAD
%
mean
mean
mean RMSE
bias
bias
19.8993 -0.1007
-0.50%
0.0637 19.8698 -0.1302
-0.65%
0.0608 19.8423 -0.1577 -0.79% 0.0217
α1=5
5.0457
0.0457
0.91%
0.0525 5.0302
0.0302
0.60%
0.0456 4.8813 -0.1187 -2.37% 0.0283
β0=20
19.9514 -0.0486
-0.24%
0.0423 19.9424 -0.0576
-0.29%
0.0445
-----
-----
-----
-----
β1=5
4.8853 -0.1148
-2.30%
0.0307 4.8894 -0.1106
-2.21%
0.0312
-----
-----
-----
-----
γ0=
−0.2 -0.6556 -0.4556 227.82% 0.1116 -0.5039 -0.3039 151.97% 0.1344 -0.3896 -0.1896 94.81% 0.0425
γ1= 1
2.0921 1.0921 109.21% 0.1833 1.7288 0.7288 72.88% 0.2874 1.0000 0.0000 0.00% 0.0000
22
Table A3 - Simulation results for parametric and semiparametric estimators with strong heteroskedastic
disturbances (percentage of cases with L = 0 index: 30%)
A3a -Error Terms Distribution:Normal (0:1); sample: 1000; NREP: 10000
Two-Stage Heckman
mean
mean
bias
FIML
% mean
RMSE
bias
mean
α0=20
19.5801 -0.4199
-2.10%
α1=5
5.5234
10.47% 0.0932 5.2407
β0=20
20.8650 0.8650
β1=5
0.5234
Two-Stage CLAD
RMSE
mean
mean
bias
0.2289 19.6864 -0.3136 -1.57% 0.0969 19.9894 -0.0106
% mean
RMSE
bias
-0.05%
0.0694
0.2407
4.81%
0.0886 4.9972 -0.0028
-0.06%
0.0891
0.2520 20.5980 0.5980
2.99%
0.1657
-----
-----
-----
-----
4.2902 -0.7098 -14.20% 0.1138 4.4939 -0.5061 -10.12% 0.0927
-----
-----
-----
-----
-0.60%
0.0806
0.00%
0.0000
γ0= −0.2 -0.2121 -0.0121
γ1= 1
mean
bias
%
mean
bias
4.32%
6.06%
0.0537 -0.1821 0.0179
-8.94% 0.0485 -0.1988 0.0012
0.4922 -0.5078 -50.78% 0.0458 0.4603 -0.5397 -53.97% 0.0433 1.0000
0.0000
A3b - Error Terms Distribution: t-Student (3 d.o.f.); sample 1000; NREP: 10000
Two-Stage Heckman
mean
mean
bias
FIML
% mean
RMSE
bias
mean
mean
bias
%
mean
bias
Two-Stage CLAD
RMSE
mean
mean
bias
% mean
RMSE
bias
α0=20
19.6866 -0.3134
-1.57%
0.2133 19.7317 -0.2683 -1.34% 0.1128 19.9912 -0.0088
-0.04%
0.0416
α1=5
5.3631
0.3631
7.26%
0.0960 5.0128
0.0128
0.26%
0.0703 4.9938 -0.0062
-0.12%
0.0521
β0=20
20.7357 0.7357
3.68%
0.2469 20.9519 0.9519
4.76%
0.3448
-----
-----
-----
-----
β1=5
4.3944 -0.6056 -12.11% 0.1409 4.6865 -0.3135 -6.27% 0.0959
-----
-----
-----
-----
-0.43%
0.0622
0.00%
0.0000
γ0= −0.2 -0.2266 -0.0266 13.28% 0.0539 -0.0712 0.1288 -64.38% 0.0683 -0.1991 0.0009
γ1= 1
0.6548 -0.3452 -34.52% 0.0517 0.4182 -0.5818 -58.18% 0.0744 1.0000
0.0000
A3c - Error Terms Distribution: Log-Normal (0;1); sample 1000; NREP: 10000
Two-Stage Heckman
mean
mean
bias
FIML
% mean
RMSE
bias
mean
mean
bias
%
mean
bias
Two-Stage CLAD
RMSE
mean
mean
bias
% mean
RMSE
bias
α0=20
19.2328 -0.7672
-3.84%
0.3164 19.1930 -0.8070 -4.03% 0.2382 19.6780 -0.3220
-1.61%
0.0374
α1=5
5.3088
0.3088
6.18%
0.1548 4.8691 -0.1309 -2.62% 0.0801 4.7683 -0.2317
-4.63%
0.0487
β0=20
20.0630 0.0630
0.31%
0.0764 20.0120 0.0120
0.0983
-----
-----
-----
-----
β1=5
4.5953 -0.4047
-8.09%
0.5441 4.6192 -0.3808 -7.62% 0.0544
-----
-----
-----
-----
0.06%
γ0= −0.2 -0.5106 -0.3106 155.29% 0.6250 -0.3141 -0.1141 57.05% 0.0543 -0.5542 -0.3542 177.10% 0.0569
γ1= 1
0.7790 -0.2210 -22.10% 0.0586 0.4952 -0.5048 -50.48% 0.0778 1.0000
23
0.0000
0.00%
0.0000
Table A4 - Estimation of Eq 1 - dependent variable: log of monthly wage
T-S HECKMAN
FIML
T-S ClAD
SE
SE.
SE
coeff
coeff
coeff
0.0280
0.0039
intercept
5.9340 0.0281
5.8860
6.0180
Dummy Sex (man=0)
-0.1088 0.0065 -0.1223 0.0060 -0.0441 0.0008
Years beyond regular
-0.0046 0.0020 -0.0082 0.0020 -0.0012 0.0003
completion time
High School final exam
0.0004
0.0001
0.0016 0.0004
0.0016
0.0002
test score
Dummy satisfaction of
job (unsatisfied = 0)
0.0883
0.0086
0.0876
0.0087
-0.0316
0.0009
dummy Career
(unsatisfied = 0)
0.0446
0.0062
0.0437
0.0062
0.0101
0.0009
0.0274
0.0005
0.0283
0.0005
0.0271
0.0001
0.0003
0.0000
0.0003
0.0000
0.0003
0.0000
-0.2416
0.0206
instrumental: weekly
working hours
estimated in R.F.
No of employees in the
firm or in the
Administation
lambda Heckman
Table A5 - Estimation of Equation 2
Dependent Variable: RW (monthly reservation wage)
T-S
FIML
SE
SE
Coeff.
Coeff.
Intercept
909.3394 27.3661 532.6730 17.7061
5.6354
Dummy Sex (man=0)
-95.5861 12.2727 -14.7290
Dummy employee =1 self8.1394
23.7694 10.6078 17.6415
employee=0
Dummy Msc after degree
8.6825
35.4462 12.3481 16.0356
(No =0)
Dummy he/she prefers
126.6278
part-time =0; full-time=1
Dummy he/she could
look for a job in foreign
state (No =0)
Dummy he/she attends
program-training course
(No =0)
lambda Heckman
11.9903
132.6950
9.0556
75.1360
11.1506
29.9026
11.2068
72.9295
14.8289
42.6367
8.5750
86.8783
14.4559
24
Table A6 - Estimation of Eq. 3 (Participation function)
Dependent variable : L =0 (employed) L =1 (unemployed)
Estimator:
Probit
FIML
SIEVE NLLS
SE
SE
SE
Coeff.
Coeff.
Coeff. (*)
0.1570
0.1193 1.0000
--Intercept
-2.3958
-0.9908
Dummy Sex (man=0)
-0.2963
0.0289
-0.2789
0.0227
-7.2148
1.6596
Years beyond regular
completion time
-0.0800
0.0099
-0.0453
0.0068
-0.2902
0.0719
Month of the Degree
-0.0110
0.0292
-0.0123
0.0027
0.0101
0.0229
0.3606
0.0291
0.2543
0.0210
1064.95
0.0000
-0.3251
0.0281
-0.2563
0.0208
0.4303
0.1716
Dummy: if he/she
worked before the
degree (No = 0)
Dummy: if he/she
attends stage
program (No = 0)
Fixed effect: Regional
6.880E-05 3.000E-06 3.246E-05 2.456E-06 1.712E-04 3.940E-05
GDP pro capite
Fixed effect:
percentage in the
college of graduates
who found a fixed job
0.0100
0.0145
0.0009
0.0012
(*) Coefficients normalized imposing constant term equal to one
N. obs: 14126. Censored obs. : 2017
25
0.0110
0.0054