Economics 20 - Prof,Anderson 1
Simultaneous Equations
y1 = a1y2 + b1z1 + u1
y2 = a2y1 + b2z2 + u2
Economics 20 - Prof,Anderson 2
Simultaneity
Simultaneity is a specific type of
endogeneity problem in which the
explanatory variable is jointly determined
with the dependent variable
As with other types of endogeneity,IV
estimation can solve the problem
Some special issues to consider with
simultaneous equations models (SEM)
Economics 20 - Prof,Anderson 3
Supply and Demand Example
Start with an equation you’d like to
estimate,say a labor supply function
hs = a1w + b1z + u1,where
w is the wage and z is a supply shifter
Call this a structural equation – it’s derived
from economic theory and has a causal
interpretation where w directly affects hs
Economics 20 - Prof,Anderson 4
Example (cont)
Problem that can’t just regress observed
hours on wage,since observed hours are
determined by the equilibrium of supply
and demand
Consider a second structural equation,in
this case the labor demand function
hd = a2w + u2
So hours are determined by a SEM
Economics 20 - Prof,Anderson 5
Example (cont)
Both h and w are endogenous because they
are both determined by the equilibrium of
supply and demand
z is exogenous,and it’s the availability of
this exogenous supply shifter that allows us
to identify the structural demand equation
With no observed demand shifters,supply
is not identified and cannot be estimated
Economics 20 - Prof,Anderson 6
Identification of Demand Equation
w
h
D S (z=z1) S (z=z2)
S (z=z3)
Economics 20 - Prof,Anderson 7
Using IV to Estimate Demand
So,we can estimate the structural demand
equation,using z as an instrument for w
First stage equation is w = p0 + p1z + v2
Second stage equation is h = a2? + u2
Thus,2SLS provides a consistent estimator
of a2,the slope of the demand curve
We cannot estimate a1,the slope of the
supply curve
Economics 20 - Prof,Anderson 8
The General SEM
Suppose you want to estimate the structural
equation,y1 = a1y2 + b1z1 + u1
where,y2 = a2y1 + b2z2 + u2
Thus,y2 = a2(a1y2 + b1z1 + u1) + b2z2 + u2
So,(1 – a2a1)y2 = a2 b1z1 + b2z2 + a2 u1 +
u2,which can be rewritten as
y2 = p1z1 + p2z2 + v2
Economics 20 - Prof,Anderson 9
The General SEM (continued)
By substituting this reduced form in for y2,
we can see that since v2 is a linear function
of u1,y2 is correlated with the error term and
a1 is biased – call it simultaneity bias
The sign of the bias is complicated,but can
use the simple regression as a rule of thumb
In the simple regression case,the bias is the
same sign as a2/(1 – a2a1)
Economics 20 - Prof,Anderson 10
Identification of General SEM
Let z1 be all the exogenous variables in the
first equation,and z2 be all the exogenous
variables in the second equation
It’s okay for there to be overlap in z1 and z2
To identify equation 1,there must be some
variables in z2 that are not in z1
To identify equation 2,there must be some variables in z
1 that are not in z2
Economics 20 - Prof,Anderson 11
Rank and Order Conditions
We refer to this as the rank condition
Note that the exogenous variable excluded
from the first equation must have a non-zero
coefficient in the second equation for the
rank condition to hold
Note that the order condition clearly holds
if the rank condition does – there will be an
exogenous variable for the endogenous one
Economics 20 - Prof,Anderson 12
Estimation of the General SEM
Estimation of SEM is straightforward
The instruments for 2SLS are the
exogenous variables from both equations
Can extend the idea to systems with more
than 2 equations
For a given identified equation,the
instruments are all of the exogenous
variables in the whole system
Simultaneous Equations
y1 = a1y2 + b1z1 + u1
y2 = a2y1 + b2z2 + u2
Economics 20 - Prof,Anderson 2
Simultaneity
Simultaneity is a specific type of
endogeneity problem in which the
explanatory variable is jointly determined
with the dependent variable
As with other types of endogeneity,IV
estimation can solve the problem
Some special issues to consider with
simultaneous equations models (SEM)
Economics 20 - Prof,Anderson 3
Supply and Demand Example
Start with an equation you’d like to
estimate,say a labor supply function
hs = a1w + b1z + u1,where
w is the wage and z is a supply shifter
Call this a structural equation – it’s derived
from economic theory and has a causal
interpretation where w directly affects hs
Economics 20 - Prof,Anderson 4
Example (cont)
Problem that can’t just regress observed
hours on wage,since observed hours are
determined by the equilibrium of supply
and demand
Consider a second structural equation,in
this case the labor demand function
hd = a2w + u2
So hours are determined by a SEM
Economics 20 - Prof,Anderson 5
Example (cont)
Both h and w are endogenous because they
are both determined by the equilibrium of
supply and demand
z is exogenous,and it’s the availability of
this exogenous supply shifter that allows us
to identify the structural demand equation
With no observed demand shifters,supply
is not identified and cannot be estimated
Economics 20 - Prof,Anderson 6
Identification of Demand Equation
w
h
D S (z=z1) S (z=z2)
S (z=z3)
Economics 20 - Prof,Anderson 7
Using IV to Estimate Demand
So,we can estimate the structural demand
equation,using z as an instrument for w
First stage equation is w = p0 + p1z + v2
Second stage equation is h = a2? + u2
Thus,2SLS provides a consistent estimator
of a2,the slope of the demand curve
We cannot estimate a1,the slope of the
supply curve
Economics 20 - Prof,Anderson 8
The General SEM
Suppose you want to estimate the structural
equation,y1 = a1y2 + b1z1 + u1
where,y2 = a2y1 + b2z2 + u2
Thus,y2 = a2(a1y2 + b1z1 + u1) + b2z2 + u2
So,(1 – a2a1)y2 = a2 b1z1 + b2z2 + a2 u1 +
u2,which can be rewritten as
y2 = p1z1 + p2z2 + v2
Economics 20 - Prof,Anderson 9
The General SEM (continued)
By substituting this reduced form in for y2,
we can see that since v2 is a linear function
of u1,y2 is correlated with the error term and
a1 is biased – call it simultaneity bias
The sign of the bias is complicated,but can
use the simple regression as a rule of thumb
In the simple regression case,the bias is the
same sign as a2/(1 – a2a1)
Economics 20 - Prof,Anderson 10
Identification of General SEM
Let z1 be all the exogenous variables in the
first equation,and z2 be all the exogenous
variables in the second equation
It’s okay for there to be overlap in z1 and z2
To identify equation 1,there must be some
variables in z2 that are not in z1
To identify equation 2,there must be some variables in z
1 that are not in z2
Economics 20 - Prof,Anderson 11
Rank and Order Conditions
We refer to this as the rank condition
Note that the exogenous variable excluded
from the first equation must have a non-zero
coefficient in the second equation for the
rank condition to hold
Note that the order condition clearly holds
if the rank condition does – there will be an
exogenous variable for the endogenous one
Economics 20 - Prof,Anderson 12
Estimation of the General SEM
Estimation of SEM is straightforward
The instruments for 2SLS are the
exogenous variables from both equations
Can extend the idea to systems with more
than 2 equations
For a given identified equation,the
instruments are all of the exogenous
variables in the whole system
