We consider a simultaneous equation model with two endogenous limited dependent variables (individual wage and reservation wage) characterized by a selection mechanism determining a two-regimes endogenous-switching. We extend the FIML procedure proposed by Poirier-Ruud (1981) for a single equation switching model providing a stochastic specification for both equations and for the selection criterion. An accurate Monte Carlo experiment shows that the relative efficiency of the FIML estimator over to the Two-Stage procedure is remarkably high in presence of a high degree of endogeneity in the selection equation.Selection bias; endogenous switching

Calzolari, Giorgio

Di Pino, Antonino

English

Research Papers in Economics

Individual Wage and Reservation Wage: Efficient 
Estimation of a Simultaneous Equation Model  with 
Endogenous Limited Dependent Variables
(1) 
 
Giorgio Calzolari 
Università di Firenze, Department of Statistics “G. Parenti”, calzolar@ds.unifi.it 
Antonino Di Pino 
Università di Messina, Department “V. Pareto”, dipino@unime.it 
 
Abstract: We consider a simultaneous equation model  with two endogenous limited 
dependent variables (individual wage and reservation wage) characterized by a selection 
mechanism determining a two-regimes  endogenous-switching. We extend the FIML 
procedure  proposed  by  Poirier-Ruud  (1981)  for  a  single  equation  switching  model 
providing a stochastic specification for both equations and for the selection criterion. An 
accurate  Monte  Carlo  experiment  shows  that  the  relative  efficiency  of  the  FIML 
estimator over to the Two-Stage procedure is remarkably high in presence of a high 
degree of endogeneity in the selection equation.  
 
Keywords: Selection Bias, Endogenous Switching 
 
1. Estimation of Regression Models with Two-Regimes Specification 
 
We study a simultaneous equation model with two equations, Eq. (1) and Eq. (2), whose 
dependent variables (individual wage and reservation wage) are both partially observed 
or  "limited"  as  a  consequence  of  a  selection  mechanism  that  doesn’t  permit  us  to 
observe  the  two  dependent  variables  together.  The  selection  mechanism  may  be 
specified by a third equation, Eq. (3), whose dependent variable is a binary dummy 
“indicator”  that  produces,  alternatively,  two  different  regimes  where  the  dependent 
variables of Eq. (1) and Eq. (2) are alternatively observable. Furthermore, we assume 
that  the  binary  dummy  indicator  is  not  exogenous  with  respect  to  the  two  limited 
dependent variables. The binary indicator involving  the two different regimes is given 
by the working status of the subject: its value is one if the subject works, and it is zero if 
the subject does not work. 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. 
One may adopt several ways to specify and to estimate simultaneously both wage and 
reservation  wage  equations.  The  two-regimes  characteristic  suggests  to  specify  the 
model as an endogenous two-regimes “switching” regression model (Poirier and Ruud, 
                                                           
(1) A. Di Pino has benefited from financial support of MIUR - National Research Project  (PRIN 2008-
2010) 1981, inter alia).  In this context, we may adopt two alternative approaches to estimate 
the model: i) a Two-Stage procedure generally utilized to estimate regression models 
with limited dependent variable (Heckman, 1976); ii) a FIML approach  utilized  to 
estimate regression models with endogenous switching (Poirier and Ruud, 1981). The 
Two-Stage  is  the  most  widely  used  by  applied  econometricians;  it  is  simpler  and 
provides  a  consistent  parameters  estimate.  FIML  is  less  popular  in  applied  works, 
presumably  for  its  computational  complexity  being  a  maximum  likelihood  method; 
besides  consistency  it  also  attains  asymptotic  efficiency,  if  the  model  is  correctly 
specified. What is the loss of efficiency implied by the widely used Two-Stage method? 
In the next section, we provide a brief discussion on the stochastic specification of the 
model and on the FIML estimator characteristics. Then we show the performances of 
both Two-Stage and FIML estimators in a detailed set of Monte Carlo experiments. 
Moreover, the results of an empirical application will be briefly resumed. 
 
2. The Model and the Estimation Procedures 
   
Poirier-Ruud (1981) shows 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 both regimes. If applied to our model, it 
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 
condition. Analogously, working income or wage depends on several exogenous factors 
which determine the working status, but does not depend on the employment condition 
of the subject. In this context, we utilize an estimation procedure based on a likelihood 
function given by the product of marginal probabilities of the subjects of perceiving a 
wage or, alternatively, of desiring a reservation wage.   Let’s start considering a three 
equations model. The purpose is to estimate simultaneously the individual wage, the 
reservation wage and the individual participation propensity in the labour market. 
 
  u x W + = α 1 '         if L = 1; 0 otherwise        (1) 
v x RW + = β 2 '       if L = 0; 0 otherwise        (2)  
ε γ + = ' * z L        



=
> =
otherwise L
L if L
0
0 1
*
      (3)   
Moreover, if L = 1,   γ ε ' z − > ; if L = 0,   γ ε ' z − ≤  
The variable W is a vector of n1 elements, the number of pays perceived  by the n1 
employed  people.  The  variable  RW  is  a  vector  of  n2  elements,  the  number  of  the 
reservation wages desired by the n2 subjects unemployed and looking for a job. The 
binary variable L is a vector of  n= n1 + n2 elements composed by n1 unitary elements 
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:










=
1
0
0
2
2
ε ε
ε
ε
σ σ
σ σ
σ σ
v u
v v
u u
Σ . Taking into account  the well-known 
properties of density functions and of conditional density functions: ( ) ( ) ( ) u u u ε ϕ ϕ ε ϕ = ,  
and  ( ) ( ) ( ) v v v ε ϕ ϕ ε ϕ = , , the probability marginal distributions are: 
( ) ( ) ( ) ∫ ∫
∞ −
+∞
−
= =
γ
γ
ε ε ϕ ε ε ϕ
'
'
, ,
z
z
d u d u u P    and     ( ) ( ) ( ) ∫ ∫
+∞ −
∞ −
= =
γ
γ
ε ε ϕ ε ε ϕ
'
'
, ,
z
z
d v d v v P     (4) 
Furthermore,  by  substituting  L  =  1  or  L  =  0  into  the  third  equation,  we  have, 
respectively: 
( ) ( ) ( ) ( ) γ γ γ ε ' ' 1 ' 0 z z z P u u P Φ = − Φ − = − > = >    if L = 1      (5a) 
( ) ( ) ( ) ( ) γ γ γ ε ' 1 ' ' 0 z z z P v v P Φ − = − Φ = − ≤ = ≤   if L = 0      (5b)   
If we consider the error terms u and v normally distributed, the p.d.f. of u if  0 > u u  and 
of v if  0 ≤ v v  are, respectively:  
( ) ( ) γ
σ
α
ϕ
σ
ϕ
'
'
1
1
z
x W
u
u
u Φ
 


 

 −
=   if L = 1,     and  ( ) ( ) γ
σ
β
ϕ
σ
ϕ
' 1
'
1
2
z
x RW
v
v
v Φ −
 


 

 −
=   if L = 0  (6)   
 
Furthermore, utilizing the conditional density of a bivariate normal distribution  and 
assuming  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∏ ∫ ∏ ∫ ∏ ∏
= = = =
≤ ≥ = =
0 1 0 1
0 0 , , , ,
L L L L
v v u u v u v u ε ϕ ϕ ε ϕ ϕ ε ϕ ε ϕ ε L   the 
Likelihood function to be maximized is:  
( )
( )
∏ ∏
= =
⋅



 


 

 −
⋅




























−
 


 


− +
Φ  


 

 −
0
2
1 2
1
2 2
1 2
1 ' 1
' '
' 1
L v v L
u u
u
u
u u
x RW
x W z
x W
σ
β
ϕ
σ σ σ
α
σ
σ
γ
σ
α
ϕ
σ
ε
ε
   
( )
( )



































−
 


 


− +
Φ − ⋅
2
1
2 2
2 2 ' '
1
ε
ε
σ σ
β
σ
σ
γ
v v
v
v x RW z
            (7) 
The results of  a Monte Carlo experiment show that both estimated coefficients  and  
standard  errors  are  asymptotically  equal  for  FIML  and  T-S  estimator  when  the covariances  of  the  error  terms,  ε σ u and  ε σ v ,  are  imposed  equal  to  zero  (simulating 
absence of endogeneity in the switching-model).  Instead, we can observe that relative 
efficiency of FIML  procedure is significant when the error terms correlation is close to 
50% or more (cfr. Table 1).   
 
Table 1  -  Montecarlo results of FIML and T-S estimations   
Error terms 
correlation (in 
percent) 
Differential (in percent)  of T-S 
estimator variance with respect to 
the ML estimator 
10%  close to 0 
50%    + 10-20% 
95%    + 30-100% 
 
For an empirical comparison, we apply both FIML and T-S estimators to the  ISTAT  
Survey on Italian Graduates in 2001 dataset. In this context, wage equation,  reservation 
wage equation, and  participation equation are simultaneously estimated by applying 
both estimation procedures. The estimation results, not reported here for the sake of 
brevity,  show  that  estimated  coefficients  and  standard  errors  are  very  similar  by 
utilizing both FIML and T-S procedures, and that the residual-based estimates of the 
error terms covariances,  ε σ u and  ε σ v , are close to zero (endogeneity seems to be absent 
in the model). 
 
 
References 
 
Heckman  J.  J.  (1978)  “Dummy  Endogenous  Variables  in  a  Simultaneous  Equation 
System”, Econometrica, 46,  931-959. 
Heckman  J.  J.,Tobias  J.,  Vytlacil  E.  (2003)  “Simple  Estimators  for  Treatment 
Parameters in a Latent Variable framework”, Review of Economics and Statistics, 85 
Issue 3, 748-755. 
ISTAT (2004) Survey on Italian Graduates in 2001 
Poirier D. J., Ruud P. A. (1981) “On The Appropriateness of Endogenous Switching”, 
Journal of Econometrics, 16, 249-256.  

Individual wage and reservation wage: efficient estimation of a simultaneous equation model with endogenous limited dependent variables

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