Instrumental Variables

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Instrumental Variables
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General Use

• For getting a consistent estimate of ß in
• YXße
• when X is correlated with e
• Will see it working with omitted variable bias,
  endogeneity, measurement error
• Intuition variation in X can be divided into two
  bits
• Bit correlated with e this causes the problems
• Bit uncorrelated with e
• Want to use the second bit this is what IV does

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Some Terminology

• Denote set of instruments by Z.
• Dimension of X is (Nxk), dimension of Z is (Nxm).
• If km this is just-identified case
• If kltm this is over-identified case
• If kgtm this is under-identified case (go home)
• Some variables in X may also be in Z these are
  the exogenous variables
• Variables in X but not in Z are the endogenous
  variables
• Variables in Z but not in X are the instruments

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Conditions for a Valid Instrument

• Instrument Relevance
• Cov(Zi,Xi)?0
• Instrument Exogeneity
• Cov(Zi,ei)0
• These conditions ensure that the part of X that
  is correlated with Z only contains the good
  variation
• Instrument relevance is testable
• Instrument exogeneity is not fully testable (can
  test over-identifying restrictions) need to
  argue plausibility

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Instrument Relevance and Exogeneity Alternative
Representation

• Instrument Relevance

• Instrument Exogeneity

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Two-Stage Least Squares the First-Stage

• To get bit of X that is correlated with Z, run
  regression of X on Z
• XZ?v
• Leads to estimates

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Two-Stage Least Squares- the Second Stage

• Need to ensure the predicted value of X is of
  rank k this is why cant have mltk
• Run regression of y on predicted value of X
• IV (2SLS) estimate of ß is

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Use formula for X-hat
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Proof of Consistency of IV Estimator

• Substitute yXße to give

• Take plims
• Second term is zero when can invert first inverse
• Can do this when instrument relevance satisfied
• Note IV estimator is not unbiased, just
  consistent
• Estimate should be independent of instrument used

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The Asymptotic Variance of the IV estimator

• Class exercise
• Need to get estimate of s2
• Use estimated residual to do this (as in OLS)
• To estimate residual must use X not X-hat i.e.

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Implication

• Never do 2SLS in two stages standard errors in
  second stage will be wrong as STATA will compute
  residuals as

• Easier to do it in one line if x1 endogenous, x2
  exogenous, z instruments
• . reg y x1 x2 (x2 z)
• . ivreg y x2 (x1z)

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The Finite Sample Distribution

• Results on IV estimator are asymptotic
• Small sample distribution may be very different
• Especially when instruments are weak not much
  correlation between X and Z
• Instruments should not be weak in experimental
  context
• Will return to it later

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Testing Over-Identification

• If mgtk then over-identified and can test
  instrument validity for (m-k) instruments
• Basic idea is

• If instruments valid then E(eZ)0 so Z should
  not matter when X-hat included
• Can test this but not for all Zs as X-hat a
  linear combination of Zs

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Some Special Cases The Just-Identified Case

• In this case (ZX) is invertible
• Can write IV estimator as

(using (AB)-1B-1A-1
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In one-dimensional case

• Can write this as

• i.e. ratio of coefficient on Z in regression of y
  on Z to coefficient on Z in regression of X on Z

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Binary Instrument No other covariates

• Where Instrument is binary should recognise the
  previous as sample equivalent to

• This is called the Wald estimator
• Simple intuition take effect of Z on y and
  divide by effect of Z on X