Simultaneous Equations II

1
Simultaneous Equations II

• Lecture 24

2
Todays plan

• More on instrumental variables
• Identification
• The Hausman test
• LATE and random variation in the number of kids
  you may have.

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Identification

• 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 wed expect b1 lt 0 and b2 gt 0

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Identification (2)

• If we run a regression of hours on wages, we
  dont know what equation from the graph is
  estimated

W
S
D
H
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Identification (3)

• 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)

• With each additional variable thats not included
  in the other equation, we trace out another
  supply curve

S1
W
S2
D
H
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Identification (5)

• 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

• 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 dont count in both equations!
• L23.xls example we identified the demand
  equation
• over-identified two additional variables in
  demand equation
• wrong sign on b1

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Hausman test

• 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
• 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

• 1) Regress errors from the second stage equation
  on the instrumental variables
• 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

We want numerator to tend towards zero
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Quick summary

• L24.xls
• Shows how to construct Hausman test
• Weve 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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LATE and Angrist Evans

• Instrumental variables are important in economics
  because of local average treatment effects (LATE)
• Angrist and Evans
• tested for the effect of having children on hours
  worked
• looked at random instruments for a treatment and
  control group
• Number of kids and propensity to have another
  child
• testing to see if sex of the first two affects
  the likelihood of having a third

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LATE and Angrist Evans (2)

• Random assignment of treatment and control groups
• cant determine sex of children
• assumption number of kids influences hours of
  work because the more kids you have, the less
  time time you have to work
• Identification comes off small part of sample
  need very large sample size

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LATE and Angrist Evans (3)

• Estimated the following equationHi a1 b1Xi
  b2NKi ?I
• here they identifying on the number of kids (NKi)
• Why can we expect NKi to be endogenous?
• Might choose combination of number of kids and
  hours of work, but cant choose sex of children
• So Angrist and Evans used the same sex variable
  as the identifying variable NKi g1 g2Xi
  g3samesexi vi
• samesexi is a dummy variable equal to 1 if the
  children are the same sex and 0 otherwise
• Negative result more kids leads to less hours of
  work
• L24.xls has a worked example