Transcript Differences-in-Differences and A (Very) Brief Introduction

Instrumental Variables
Methods of Economic
Investigation
Lecture 15
Last Time
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Introduction to Instrumental Variables
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Correlation with variable of interest
Exclusion restriction
Interpretation of IV with homogeneous
treatment effects
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Gives us a Wald estimate
Nice/well-defined properties of OLS
Today’s Class
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Uses for 2SLS
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Experiments with compliance issues
Omitted Variable Bias
Heterogeneous Treatment Effects
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LATE framework
Interpretation
Review of Instrumental Variables
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Two characteristics
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Instrument (Z) is correlated with (S)
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Must be that S is always increasing (or always
decreasing)
If it changed signs, then the first stage prediction
wouldn’t work
Instrument (Z) is uncorrelated with other
determinants of the outcome (Y)
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This means Z is uncorrelated with unobservables that
affect Y
The only way Z affects Y is through S
Steps to Estimate IV-1
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Step 1: The Structural Equation
Y =ρS + η
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Problems: S correlated with η
OLS estimates won’t recover causal effect of S
on Y
Step 2: Find an Instrument
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Correlated with S
Uncorrelated with η (and so uncorrelated with
the unobservables)
Steps to Estimate IV-2
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Step 3: Estimate the First Stage
S = πZ + ν
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Can estimate this with OLS
Want to test to see if π is significant—will
return to this in the case of weak instruments
where α is close to zero
Step 4: Obtain the fitted values
Sˆ  ˆZ
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This is the component of S that is unrelated to
the error term in the structural equation
Steps to Estimate IV -3
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Step 5: Estimate the Second Stage
Y  Sˆ     (ˆZ )    bZ  
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This is using the fitted value, i.e. the predicted value
of S given the instrument Z
The fitted value captures the component of S that is
uncorrelated with the error
If we want to recover β take the OLS estimate from
the second stage b and divide it by the coefficient
from the first stage α
1
ZY
bˆ N  i i
Cov(YZ )
 
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ˆ 1
Cov( SZ )
Z
S

i i
N
Various uses for IV
Goal: Average Effect of S on Y (ATE)
Omitted Variable
Non-Experimental
Experimental
Perfect
Compliance
Imperfect
Compliance
Matching
IV
Measurement Error
IV
Fixed
Effects
Diff-indiff
Perfect
Compliance
IV
Imperfect
Compliance
IV
Things to worry about
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Is my instrument really uncorrelated with
other determinants of the outcome?
How do I interpret my IV estimate? What
if I think there are heterogeneous
treatment effects?
How strong does my first stage have to
be for this to all work?
We’ll deal with each of these issues
Can we test the exclusion restriction
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This is the assumption of the model and
often cannot be formally tested
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The reduced form gives some information
on the reasonableness of the assumption
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Knowing π and the OLS biased estimated of ρ
we might gut check how reasonable it is for
the only effect of Z to be through S
If we have multiple instruments, we can test
using the overidentification test
Overidentification
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Model is overidentified if we have:
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# instruments variables > # endogenous
variables
Models with exactly same number of instruments
as endogenous variables are just identified
If the model is overidentified we can test
the quality of the fit
Testing Model Fit
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Suppose we have Q instruments and define
(this is our first stage RHS variable)
Z i  [ X i ' z1i ,...,zQi ]'
Wi  [ X i ' sˆi ]'and as before    '  '
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Define
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The residuals from the second stage can be
defined as:
i ()  Yi  'Wi  Yi  [X i  sˆi ]
2SLS Residual Terms
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We assume that η is orthogonal to Z so that
E[Zi η(Γ)]=0
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The sample analog of this is
1
Z i i ()  m N ()

N
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In finite samples, this won’t be exactly zero
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2SLS fits the value of Γ making this closest to
zero
This has an asymptotic distribution of
2
N mN ()  N (0, ) where   E[Zi Zi 'i () ]
The Minimand
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There is an underlying Method of Moments
way to illustrate this but we’ll ignore that
for now.
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Basic idea is to minimize the quadratic
form of the vector mN(Γ)
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The optimal weighting matrix to estimate
this is Λ-1 and then the equation to be
minimized is: J N ( g )  NmN ( gˆ )' 1mN ( gˆ )
The Overidentification Test
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Intuition: is mn(g) close enough to zero
for us to believe that Z uncorrelated with
the error (other unobservable stuff)
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Null hypothesis: E[ηZ]=0 distributed
Χ2(Q-1)
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Can also test this directly
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Estimate the just-identified version for the Q
instruments
Test that the estimate coefficients are
statistically indistinguishable
Next time
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Issues with IV
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Heterogeneity
Weak Instruments