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Econ 140
Simultaneous Equations II
Lecture 23
Lecture 23
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Today’s plan
Econ 140
• More on instrumental variables
• Identification
• The Hausman test
Lecture 23
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Identification
Econ 140
• Our model was:
(D): Hi = a1 + b1Wi + ui
(S): Hi = a2 +b2Wi + c2Xi + d2NKi +vi
– problem: both hours and wages are endogenous
• Instrumental variables and identification to get around this
problem
• If we had a model
(D): Hi = a1 + b1Wi + e1i
(S): Hi = a2 +b2Wi + e2i
– from economic theory we’d expect b1 < 0 and b2 > 0
Lecture 23
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Identification (2)
Econ 140
• If we run a regression of hours on wages, we don’t know
what equation from the graph is estimated:
W
S
D
H
Lecture 23
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Identification (3)
Econ 140
• In order to determine which curve the regression estimates,
we need to identify one of the equations
• Think back to the model and add one more variable to the
supply equation:
(D): Hi = a1 + b1Wi + e1i
(S): Hi = a2 +b2Wi + c2X1 + e2i
– Where X1 is age
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Identification (4)
Econ 140
• With each additional variable that’s not included in the
other equation, we trace out another supply curve:
S1
W
S2
D
H
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Identification (5)
Econ 140
• Each additional variable must be a factor of one equation
but not the other
• Example: if age is a factor of demand such that employers
hire aging workers for less hours, system is no longer
identified
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Order condition
Econ 140
1) “Just identified”: If we have one variable that differentiates
demand from supply
2) “Over-identification”: If we have more than one variable
that differentiates demand and supply
• Must have variables that are exogenous to the system that
don’t count in both equations!
• L23.xls example:
– over-identified: two additional variables
– wrong sign on b1
Lecture 23
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Hausman test
Econ 140
• How do we know the system was identified at all?
• Need to test to see if:
– added variables are in fact exogenous instruments
• Hausman test tests for over-identification
– We have an estimating equation [2nd stage of TSLS]:
Yi  a1  b1 Xˆ i  ui
– First we have: Xi = a2 + b2Zi + vi [1st stage of TSLS]
– Idea of test: look for correlation between the
instruments Zi and the ui from the 2nd stage equation
• if there is correlation, Zi is no longer a valid
instrument
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Steps for the Hausman test
Econ 140
1) Regress errors from the second stage equation on the
instrumental variables:
ui   0   1Z1i  ...   m Z mi  i
2) Calculate R2 *(n-k) where k = m + 1
– test is distributed: [R2*(n-k)] ~ 2(m-1)
– we have m-1 over-identifying variables
– null hypothesis: H0: E(Z, u) = 0
– In this test, R2 is:
ˆ j  j Zu We want numerator to tend
2
R 
towards zero
2
u

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Quick summary
Econ 140
• L23.xls
– Shows how to construct Hausman test
• We’ve learned so far:
– How to estimate given simultaneity (TSLS)
– Working with instrumental variables
– How to test if simultaneity still exists with the selected
instruments (Hausman test)
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