Research Method
Lecture 12 (Ch16)
Simultaneous Equations
Models (SEMs)
©
1
Introducdtion
We have learned two “sources” of
endogeneity.
1. Omitted variables
2. Errors in variables
In this handout, we will learn another
source of endogeneity: Simultaneity.
2
In econometrics, “endogeneity” usually means
that an explanatory variable is correlated with
the error term.
In simultaneous equation models, endogeneity
means that the observed variable is determined
by the equilibrium. For example, an observed
quantity is determined by the equilibrium
between demand and supply.
When a variable is endogenous in ‘simultaneous
equation’ sense, it is usually endogenous in
econometric sense (i.e., correlated with the error
term). We will see this soon.
3
The nature of simultaneous
equation.
Consider the following model describing
equilibrium quantity of labor (in hours) in
agricultural sector in a country.
Labor supply : hs=α1w+β1z1+u1
Labor demand: hd=α2w+β2z2+u2
hs is the hours of labor supplied, and hd is the hours of
labor demanded. These quantities depends on the
wage rate, w, and other factors, z1 and z2.
4
z1 would be the wage rate of the
manufacturing sector. If the
manufacturing wage increases, people
would move to manufacturing sector,
reducing hours worked in agricultural
sector. z1 is called the observed demand
shifter. u1 is called the unobserved
demand shifter.
z2 would be agricultural land area. The
more land available, more demand for
labor. z2 is the observed supply shifter. u1
is the unobserved supply shifter.
5
Demand and supply describes entirely
different relationships.
The observed labor quantity and wage
rate are determined by the equilibirum
between these two equations.
The equilibrium: hs=hd
6
Consider you have country level data. Then,
for each country, we observe only the
equilibirum labor supply and wage rate.
Demand: hi=α1wi+β1zi1+ui1
Supply: hi=α2wi+β2zi2+ui2
where i is the country subscript.
These two equations constitute a simultaneous
equations model (SEM). These two equations
are called the structural equations. α1,β1, α2,
7
β2 are called the structural parameters.
In SEM framework, hi and wi are
endogenous variables because they are
determined by the equilibrium between
the two equations.
In the same way, zi1 and zi2 are exogenous
variables because they are determined
outside of the model.
u1 and u2 are called the structural errors.
One more important point: Without z1 or
z2, there is no way to distinguish whether
one equation is demand or supply.
8
Simultaneous equation bias
Consider the following simultaneous
equation model.
y1=α1y2+β1z1+u1…………….(1)
y2=α2y1+β2z2+u2…………….(2)
In this model, y1 and y2 are endogenous
variables since they are determined by the
equilibrium between the two equations.
z1 z2 are exogenous variables.
9
Since z1 and z2 are determined outside of
the model, we assume that z1 and z2 are
uncorrelated with both of the structural
errors.
Thus, by definition, the exgoneous
variables in SEM are exogenous in
‘econometric sense’ as well.
In addition, the two structural errors, u1 &
u2, are assumed to be uncorrelated with
each other.
10
Now, solve the equations (1) and (2) for y1
and y2, then you get the following
reduced form equations.
y1=п11z1+п12z2+v1
y2=п21z1+п22z2+v2
where
п11= β1/(1- α1 α2)
п112= α1 β2/(1- α1 α2)
v1 =(u1+ α1 u2)/(1- α1 α2)
п21 =α2β1/(1- α2 α1)
п22 = β2/(1- α2 α1)
v2=(α2u1+u2)/(1- α2 α1)
These parameters are called the reduced
form parameters.
11
You can check that v1 and v2 are
uncorrelated with z1 and z2. Therefore,
you can estimate these reduced form
parameters by OLS (Just apply OLS
separately for each equation).
12
However, you cannot estimate the
structural equations with OLS. For
example, consider the first structural
equation.
y1=α1y2+β1z1+u1
Notice that
Cov(y2, u1) =[α2/(1-α2α1)]E(u12)
=[α2/(1-α2α1)]σ21 ≠0
Thus, y2 is correlated with u1 (assuming that
α2 ≠0.) In other words, y2 is endogenous in
‘econometric sense’.
13
Thus, endogenous variables in SEM are usually
endogenous in ‘econometric sense’ as well.
Thus, you cannot apply OLS to the structural
equations.
Cov(y2, u1) =[α2/(1-α2α1)]σ21 can be used to
predict the direction of bias. If this is positive,
OLS estimate of α1 will be biased upward. If it is
negative, it will be biased downward.
The formula above does not carry over to more
general models. But we can use this as a guide
to check the direction of the bias.
14
An example
Suppose that you are interested in
estimating the effect of police size on the
city murder rate.
Notice that the ‘supply’ of murder would
be a function of police size. But the
‘demand’ for police is a function of
murder rates.
15
Thus, the observed murder rate and the police size are
determined simultaneously by the following model.
(Murder)=α1(police)+β10+β1(Income per capita)+u1..(3)
(Police)=α2(Murder)+ β20+β2(other vars)+u2………..(4)
Allthe variables are the city-level variables. (Murder) is
the number of murders per capita. (Police) is the
number of police officers per capita.
We are interested in estimating the effect of police on the
murder rate: equation (3).
16
However, since murder rate and police
force are determined simultaneously,
(police) is endogenous in equation (3).
Thus OLS estimate for α1 is biased.
Question: What would be the direction of
the bias?
17
Identifying and estimating a
structural equation:
2 equations case
When we learned OLS, a parameter was
said to be identified when the explanatory
variable is not correlated with the error. In
2SLS chapter, we learned how to identify
(i.e., eliminate the bias) by apply IV
method.
In SEM, the term ‘identification’ is used
slightly differently.
18
Suppose the following model describing
the supply and demand.
Supply: q =α1p+β1z1+u1
Demand: q =α2p+u1
Note that supply curve has an observed
supply shifter z1, but demand has no
obsedved supply shifter.
Given the data on q, p and z1, which
equation can be estimated? That is, which
is an identified equation?
19
Demand
Supply: location is
different depending on
the value of z1.
These are the data
points.
Notice: data points trace the demand curve.
Thus, it is the demand equation that can be
estimated.
20
Because there is observed supply shifter
z1 which is not contained in demand
equation, we can identify the demand
equation.
It is the presence of an exogenous variable
in the supply equation that allows us to
estimate the demand equation.
In SEM, identification is used to mean
which equation can be estimated.
21
Now turn to a more general case.
y1  10  1 y2  11z11    1k z1k  u1
y2   20   2 y1   21z21     2l z2l  u2
(z11~z1k) and (z21~ z2l ) may contain the same
variables, but may contain different
variables as well.
When one equation contains exogenous
variables not contained in the other
equation, this means that we have
imposed exclusion restrictions.
22
The condition for identification is the
following.
The condition for identification: The first
equation is identified if and only if the
second equation contains at least one
exogenous variable (non zero coefficient)
that is excluded from the first equation.
23
The above condition have two
components. First, at least one exogenous
variable should be excluded from the first
equation (order condition). Second, the
excluded variable should have non zero
coefficients in the second equation (rank
condition).
The identification condition for the
second equation is just a mirror image of
the statement.
24
Example
Labor supply of married working women.
Labor supply equation:
hours  1lwage  10  11educ  12age  13kids6
 14 ( NonWifeIncome)  u1
Wage offer equation:
lwage   2 hours   30   21educ   22 exp   23 exp 2  u2
In the model, hours and lwage are endogenous
variables. All other variables are exogenous.
(Thus, we are ignoring the endogeneity of educ
25
arising from omitted ability.)
Suppose that you are interested in
estimating the first equation.
Since exp and exp2 are excluded from the
first equation, the order condition is
satisfied for the first equation. The rank
condition is that, at least one of exp and
exp2 has a non zero coefficient in the
second equation. Assuming that the rank
condition is satisfied, the first equation is
identified.
In a similar way, you can see that the
second equation is also identified.
26
Estimating SEM using 2SLS
Once we have determined that an
equation is identified, we can estimate it
by two stage least square.
27
Consider the labor supply equation
example again. You are interested in
estimating the first equation.
hours  1lwage  10  11educ  12age  13kids6
 14 ( NonWifeIncome)  u1
lwage   2 hours   30   21educ   22 exp   23 exp 2  u2
Suppose that the first equation is
identified (both order and rank conditions
are satisfied).
lwage is correlated with u1. Thus, OLS
cannot be used.
28
However, exp and exp2 can be used as
instruments for lwage in the first
equation.
Why? First, exp and exp2 are uncorrelated
with u1 by assumption of the model
(instrument exogeneity satisfied). Second
exp and exp2 are correlated with lwage by
the rank condition (instrument relevance
satisfied).
29
In general, you can use the excluded
exogenous variables as the instruments.
30
Exercise
Consider the following simultaneous
equation model.
hours  1lwage  10  11educ  12age  13kids6
 14 ( NonWifeInc )  u1
lwage   2 hours   30   21educ   22 exp   23 exp 2  u2
Q1: Which equation(s) is/are identified?
Q2: Estimate the identified equation(s).
31
Answer
. reg hours lwage educ age kidslt6 nwifeinc, robust
Linear regression
Number of obs
F( 5, 422)
Prob > F
R-squared
Root MSE
hours
Coef.
lwage
educ
age
kidslt6
nwifeinc
_cons
-2.046796
-6.62187
.5622541
-328.8584
-5.918459
1523.775
Robust
Std. Err.
82.02275
18.43784
5.360839
126.681
3.385146
309.4226
t
P>|t|
-0.02
-0.36
0.10
-2.60
-1.75
4.92
0.980
0.720
0.917
0.010
0.081
0.000
=
428
=
2.46
= 0.0324
= 0.0361
= 766.63
[95% Conf. Interval]
-163.2708
-42.86331
-9.975019
-577.8629
-12.57231
915.5734
OLS
159.1772
29.61957
11.09953
-79.85399
.7353893
2131.976
. ivregress 2sls hours educ age kidslt6 nwifeinc (lwage=exper expersq), robust
Instrumental variables (2SLS) regression
hours
Coef.
lwage
educ
age
kidslt6
nwifeinc
_cons
1639.556
-183.7513
-7.806092
-198.1543
-10.16959
2225.662
Instrumented:
Instruments:
Robust
Std. Err.
593.3108
67.78742
10.48746
208.4247
5.287486
603.0964
Number of obs
Wald chi2(5)
Prob > chi2
R-squared
Root MSE
z
2.76
-2.71
-0.74
-0.95
-1.92
3.69
P>|z|
0.006
0.007
0.457
0.342
0.054
0.000
lwage
educ age kidslt6 nwifeinc exper expersq
=
=
=
=
=
428
12.60
0.0274
.
1344.7
[95% Conf. Interval]
476.6879
-316.6122
-28.36114
-606.6592
-20.53287
1043.615
2SLS
2802.423
-50.89039
12.74896
210.3506
.1936911
3407.709
32
. reg lwage hours educ exper expersq, robust
Linear regression
Number of obs
F( 4, 423)
Prob > F
R-squared
Root MSE
lwage
Coef.
hours
educ
exper
expersq
_cons
-.0000565
.1062139
.0447035
-.0008585
-.4619955
Robust
Std. Err.
.0000654
.0133269
.0152503
.0004166
.2113449
t
-0.86
7.97
2.93
-2.06
-2.19
P>|t|
0.388
0.000
0.004
0.040
0.029
=
428
= 20.24
= 0.0000
= 0.1601
= .6659
[95% Conf. Interval]
-.0001852
.0800187
.0147277
-.0016773
-.8774124
.0000721
.1324091
.0746793
-.0000397
-.0465786
. ivregress 2sls lwage (hours=age kidslt6 nwifeinc) educ exper expersq, robust
Instrumental variables (2SLS) regression
lwage
Coef.
hours
educ
exper
expersq
_cons
.0001259
.11033
.0345824
-.0007058
-.6557254
Robust
Std. Err.
.0002924
.0148178
.0185052
.0004265
.4097655
Number of obs
Wald chi2(4)
Prob > chi2
R-squared
Root MSE
z
0.43
7.45
1.87
-1.65
-1.60
P>|z|
0.667
0.000
0.062
0.098
0.110
=
428
= 83.56
= 0.0000
= 0.1257
= .67545
[95% Conf. Interval]
-.0004472
.0812877
-.0016872
-.0015418
-1.458851
.000699
.1393723
.0708519
.0001302
.1474001
Instrumented: hours
Instruments: educ exper expersq age kidslt6 nwifeinc
33
Note on the terminology
In the previous slides, the exogenous
variables excluded from the equation
were called the instruments.
In SEM (and in usual IV method too),
people often refer to all the exogenous
variables (regardless of whether they are
included or excluded) as the instruments.
The instruments that are excluded from
the equation is called specifically as the
‘excluded instruments’.
34
Simultaneous equations
models with panel data.
Consider the following SEM.
yit1  1 yit 2  zit11  (ai1  uit1 )
yit 2   2 yit1  zit 2  2  ( ai 2  uit 2 )
The notation zit11 is a short hand notation
for 11z1t1   1k z1tk . The same for zit 2  2.
Due to the fixed effect term ai1 and ai 2 , zvariables are correlated with the composite
error terms. Therefore, the excluded
exogenous variables cannot be used as
35
instruments unless we do something.
To apply 2SLS, we should first (i) firstdifference, or (i) demean the equations.
First-differenced version
yit1  1yit 2  zit11  uit1
yit 2   2 yit1  zit 2  2  uit 2
Time demeaned (fixed effect) version
yit1  1 yit 2  zit11  u
it1
yit 2   2 yit1  zit 2  2  u
it 2
36
Then zit1 , zit 2 or zit1 , yit1 are not correlated
with the error term. Thus we can apply
the 2SLS method.
Estimation procedure is the same. First,
determine which equation is identified.
Then, use the excluded exogenous
variable as the instruments in the 2SLS
method.
37
An application
The effect of prison population on the
violent crime rate (Levitte 1996).
This paper answers to the following
question: To what extent an increase in
prison population would decrease the
violent crime?
38
Consider the following model.
log( crime)it  t  1 log( prison )  zit11  (ai1  uit1 )......( 4)
(Crime): the number of violent crimes per capita.
(Prison) prison population per capita.
 t : intercepts (different at each year: just include
year dummies.)
z1: police per capita, log of income per capita,
unemployment rate, proportions of black and
those living in metropolitan areas, and age
distributions.
39
First-differece the equation to eliminate
the fixed effect ai.
 log( crime)it  t  1 log( prison )  zit11  uit1......(5)
Even after eliminating the fixed effect,
there still is the simultaneous equation
bias, because the prison population is
determined by the crime rate as well.
40
The simultaneity can be expressed in the
SEM framework as:
 log( crime)it  1t  1 log( prison )it  zit11  uit1......(6)
 log( prison ) it   2t   2  log( crime)
 (Exogenous Vars) it  2  uit 2 .............................................(7)
(Exogenous vars) in equation (7) could
contain zit1 . However, in order to identify
the crime equation (6), (exogenous vars)
should contain variables that are not
included the crime equation. What can be
the variable?
41
Levitte (1996) used the overcrowding
litigation as the excluded instruments.
In the US, prisoner’s right groups have
filed law suits to mitigate the
overcrowding of the prisons.
When the law suit is successful, the court
orders the prisons to mitigate the
overcrowding of the prisons. It usually
takes the form of population caps.
42
Thus, overcrowding litigation, if victories
are achieved, will affect the change in the
prison population. At the same time, it is
reasonable to assume that the
overcrowding litigation affect crime rate
only through prison population.
Thus, the model is now:
 log( crime)it  1t  1 log( prison )it  zit11  uit1......(8)
 log( prison )it   2t   2  log( crime)
 (Final) it  2  (Other Factors)  uit 2 ..............(9)
Whether the final decisions about the
overcrowiding litigation is reached.
43
The results
. reg
gcriv gpris gpolpc gincpc cunem cblack cmetro cag0_14 cag15_17 cag18_24 cag25_34
>
y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93, robust
Linear regression
Number of obs
F( 23,
690)
Prob > F
R-squared
Root MSE
gcriv
Coef.
gpris
gpolpc
gincpc
cunem
cblack
cmetro
cag0_14
cag15_17
cag18_24
cag25_34
y81
y82
y83
y84
y85
y86
y87
y88
y89
y90
y91
y92
y93
_cons
-.1808974
.0514239
.7383676
.41126
-.0147435
.5383056
.989306
4.98384
2.412758
2.879946
-.0686258
-.0407726
-.0421775
-.0136596
.0094042
.0440948
-.0239597
.0347581
.0253571
.0871704
.038884
.0081502
.0087141
-.0056706
Robust
Std. Err.
.0557664
.0553029
.2294963
.399978
.0336211
1.417525
2.38922
4.998959
2.937888
2.549853
.0203006
.0225609
.024122
.0240967
.0229413
.0272085
.0237496
.0221166
.0235964
.0226556
.0235126
.0245474
.0271283
.0282869
t
-3.24
0.93
3.22
1.03
-0.44
0.38
0.41
1.00
0.82
1.13
-3.38
-1.81
-1.75
-0.57
0.41
1.62
-1.01
1.57
1.07
3.85
1.65
0.33
0.32
-0.20
P>|t|
0.001
0.353
0.001
0.304
0.661
0.704
0.679
0.319
0.412
0.259
0.001
0.071
0.081
0.571
0.682
0.106
0.313
0.117
0.283
0.000
0.099
0.740
0.748
0.841
=
=
=
=
=
714
10.66
0.0000
0.2311
.07893
[95% Conf. Interval]
-.2903896
-.0571583
.2877727
-.3740601
-.0807554
-2.244874
-3.701708
-4.831156
-3.355514
-2.126456
-.1084842
-.0850688
-.0895389
-.0609713
-.0356389
-.0093267
-.0705898
-.0086658
-.0209723
.0426883
-.0072807
-.0400464
-.0445498
-.0612093
-.0714052
.1600061
1.188963
1.19658
.0512685
3.321485
5.68032
14.79884
8.18103
7.886347
-.0287675
.0035237
.0051838
.033652
.0544473
.0975162
.0226704
.0781819
.0716865
.1316525
.0850488
.0563468
.0619781
.0498682
Simple first
differenced
model. The
coefficient
would be
biased.
44
. ivregress 2sls
gcriv (gpris= final1 final2) gpolpc gincpc cunem cblack cmetro cag0_1
> 4 cag15_17 cag18_24 cag25_34 y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93, robust
Instrumental variables (2SLS) regression
gcriv
Coef.
gpris
gpolpc
gincpc
cunem
cblack
cmetro
cag0_14
cag15_17
cag18_24
cag25_34
y81
y82
y83
y84
y85
y86
y87
y88
y89
y90
y91
y92
y93
_cons
-1.031956
.035315
.9101992
.5236958
-.0158476
-.591517
3.379384
3.549945
3.358348
2.319993
-.0560732
.0284616
.024703
.0128703
.0354026
.0921857
.004771
.0532706
.0430862
.1442652
.0618481
.0266574
.0222739
.0148377
Instrumented:
Instruments:
Robust
Std. Err.
.3314008
.0602729
.3257538
.4809308
.0403465
1.603944
2.938685
5.910124
3.321347
2.87316
.0251341
.0390173
.0388549
.0320581
.0308006
.0385744
.0320625
.0284254
.0292807
.036031
.0303677
.0300795
.0340501
.0367185
Number of obs
Wald chi2(23)
Prob > chi2
R-squared
Root MSE
z
-3.11
0.59
2.79
1.09
-0.39
-0.37
1.15
0.60
1.01
0.81
-2.23
0.73
0.64
0.40
1.15
2.39
0.15
1.87
1.47
4.00
2.04
0.89
0.65
0.40
P>|z|
0.002
0.558
0.005
0.276
0.694
0.712
0.250
0.548
0.312
0.419
0.026
0.466
0.525
0.688
0.250
0.017
0.882
0.061
0.141
0.000
0.042
0.375
0.513
0.686
=
=
=
=
=
714
145.87
0.0000
.
.09385
[95% Conf. Interval]
-1.68149
-.0828178
.2717334
-.4189114
-.0949254
-3.73519
-2.380332
-8.033685
-3.151374
-3.311297
-.105335
-.0480108
-.0514511
-.0499624
-.0249655
.0165812
-.0580704
-.0024422
-.0143029
.0736458
.0023285
-.0322974
-.044463
-.0571292
-.3824227
.1534478
1.548665
1.466303
.0632302
2.552156
9.1391
15.13358
9.868069
7.951282
-.0068113
.104934
.1008572
.0757031
.0957707
.1677902
.0676124
.1089833
.1004754
.2148846
.1213676
.0856122
.0890109
.0868047
First-difference
plus 2SLS to
eliminate the
simultaneous
equation bias.
gpris
gpolpc gincpc cunem cblack cmetro cag0_14 cag15_17 cag18_24
cag25_34 y81 y82 y83 y84 y85 y86 y87 y88 y89 y90 y91 y92 y93
final1 final2
45
The results of the overidentifying restriction test and
endogeneity test.
. estat overid
Test of overidentifying restrictions:
Score chi2(1)
=
.012929
(p = 0.9095)
. estat endog
Tests of endogeneity
Ho: variables are exogenous
Robust score chi2(1)
Robust regression F(1,689)
=
=
6.14067
9.09532
(p = 0.0132)
(p = 0.0027)
46
The first stage regression
First-stage regressions
Number of obs
F( 24,
689)
Prob > F
R-squared
Adj R-squared
Root MSE
gpris
Coef.
gpolpc
gincpc
cunem
cblack
cmetro
cag0_14
cag15_17
cag18_24
cag25_34
y81
y82
y83
y84
y85
y86
y87
y88
y89
y90
y91
y92
y93
final1
final2
_cons
-.0286921
.2095521
.1616595
-.0044763
-1.418389
2.617307
-1.608738
.9533678
-1.031684
.0124113
.0773503
.0767785
.0289763
.0279051
.0541489
.0312716
.019245
.0184651
.0635926
.0263719
.0190481
.0134109
-.077488
-.0529558
.0272013
Robust
Std. Err.
.033455
.1941286
.3106113
.0259464
.8617595
1.665526
3.739701
1.630762
1.813735
.0156429
.0174195
.0170559
.0190171
.0175523
.0212384
.0178416
.0171349
.0174176
.0175546
.0176008
.0175037
.0201717
.0162845
.0195862
.0224239
t
-0.86
1.08
0.52
-0.17
-1.65
1.57
-0.43
0.58
-0.57
0.79
4.44
4.50
1.52
1.59
2.55
1.75
1.12
1.06
3.62
1.50
1.09
0.66
-4.76
-2.70
1.21
P>|t|
0.391
0.281
0.603
0.863
0.100
0.117
0.667
0.559
0.570
0.428
0.000
0.000
0.128
0.112
0.011
0.080
0.262
0.289
0.000
0.135
0.277
0.506
0.000
0.007
0.226
=
=
=
=
=
=
714
5.64
0.0000
0.1522
0.1226
0.0624
[95% Conf. Interval]
-.0943781
-.1716025
-.4481988
-.0554198
-3.110379
-.6528087
-8.951316
-2.248492
-4.592794
-.0183022
.0431486
.0432907
-.008362
-.0065574
.0124491
-.0037587
-.0143979
-.0157328
.0291257
-.0081856
-.0153188
-.0261945
-.1094613
-.0914117
-.016826
.0369938
.5907068
.7715178
.0464671
.2736011
5.887423
5.73384
4.155228
2.529427
.0431248
.111552
.1102663
.0663146
.0623676
.0958488
.066302
.0528878
.052663
.0980595
.0609295
.053415
.0530164
-.0455148
-.0144999
.0712287
Overcrowding
litigation
reduces the
prison
population
growth.
47