Instrumental Variables

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Instrumental Variables

• Methods of Economic Investigation
• Lecture 15

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Last Time

• Introduction to Instrumental Variables
• Correlation with variable of interest
• Exclusion restriction
• Interpretation of IV with homogeneous treatment
  effects
• Gives us a Wald estimate
• Nice/well-defined properties of OLS

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Todays Class

• Uses for 2SLS
• Experiments with compliance issues
• Omitted Variable Bias
• Heterogeneous Treatment Effects
• LATE framework
• Interpretation

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Review of Instrumental Variables

• Two characteristics
• Instrument (Z) is correlated with (S)
• Must be that S is always increasing (or always
  decreasing)
• If it changed signs, then the first stage
  prediction wouldnt work
• Instrument (Z) is uncorrelated with other
  determinants of the outcome (Y)
• This means Z is uncorrelated with unobservables
  that affect Y
• The only way Z affects Y is through S

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Steps to Estimate IV-1

• Step 1 The Structural Equation
• Y ?S ?
• Problems S correlated with ?
• OLS estimates wont recover causal effect of S on
  Y
• Step 2 Find an Instrument
• Correlated with S
• Uncorrelated with ? (and so uncorrelated with the
  unobservables)

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Steps to Estimate IV-2

• Step 3 Estimate the First Stage
• S pZ ?
• Can estimate this with OLS
• Want to test to see if p is significantwill
  return to this in the case of weak instruments
  where a is close to zero
• Step 4 Obtain the fitted values
• This is the component of S that is unrelated to
  the error term in the structural equation

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Steps to Estimate IV -3

• Step 5 Estimate the Second Stage
• 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 ß take the OLS estimate
  from the second stage b and divide it by the
  coefficient from the first stage a

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Various uses for IV
Goal Average Effect of S on Y (ATE)
Omitted Variable
Non-Experimental
Experimental
Perfect Compliance
Imperfect Compliance
Matching
Diff-in-diff
IV
Fixed Effects
IV
Perfect Compliance
Imperfect Compliance
Measurement Error
IV
IV
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Things to worry about

• 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?
• Well deal with each of these issues

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Can we test the exclusion restriction

• This is the assumption of the model and often
  cannot be formally tested
• The reduced form gives some information on the
  reasonableness of the assumption
• Knowing p and the OLS biased estimated of ? 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

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Overidentification

• Model is overidentified if we have
• instruments variables gt 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

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Testing Model Fit

• Suppose we have Q instruments and define
• (this is our first stage RHS variable)
• Define and as before
• The residuals from the second stage can be
  defined as

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2SLS Residual Terms

• We assume that ? is orthogonal to Z so that EZi
  ?(G)0
• The sample analog of this is
• In finite samples, this wont be exactly zero
• 2SLS fits the value of G making this closest to
  zero
• This has an asymptotic distribution of
• where

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The Minimand

• There is an underlying Method of Moments way to
  illustrate this but well ignore that for now.
• Basic idea is to minimize the quadratic form of
  the vector mN(G)
• The optimal weighting matrix to estimate this is
  ?-1 and then the equation to be minimized is

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The Overidentification Test

• Intuition is mn(g) close enough to zero for us
  to believe that Z uncorrelated with the error
  (other unobservable stuff)
• Null hypothesis E?Z0 distributed ?2(Q-1)
• Can also test this directly
• Estimate the just-identified version for the Q
  instruments
• Test that the estimate coefficients are
  statistically indistinguishable

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Next time

• Issues with IV
• Heterogeneity
• Weak Instruments