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Instrumental Variables. General Use. For getting a consistent estimate of β in Y=X β + ε when X is correlated with ε Will see it working with omitted variable bias, endogeneity, measurement error Intuition: variation in X can be divided into two bits:

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general use
General Use
  • For getting a consistent estimate of β in

Y=Xβ+ε

when X is correlated with ε

  • Will see it working with omitted variable bias, endogeneity, measurement error
  • Intuition: variation in X can be divided into two bits:
    • Bit correlated with ε – this causes the problems
    • Bit uncorrelated with ε
  • Want to use the second bit – this is what IV does
some terminology
Some Terminology
  • Denote set of instruments by Z.
  • Dimension of X is (Nxk), dimension of Z is (Nxm).
  • If k=m this is just-identified case
  • If k<m this is over-identified case
  • If k>m 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
conditions for a valid instrument
Conditions for a Valid Instrument
  • Instrument Relevance

Cov(Zi,Xi)≠0

  • Instrument Exogeneity

Cov(Zi,εi)=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’
instrument relevance and exogeneity alternative representation
Instrument Relevance and Exogeneity: Alternative Representation
  • Instrument Relevance:
  • Instrument Exogeneity:
two stage least squares the first stage
Two-Stage Least Squares – the First-Stage
  • To get bit of X that is correlated with Z, run regression of X on Z

X=ZΠ+v

  • Leads to estimates:
two stage least squares the second stage
Two-Stage Least Squares- the Second Stage
  • Need to ensure the predicted value of X is of rank k – this is why can’t have m<k
  • Run regression of y on predicted value of X
  • IV (2SLS) estimate of βis:
proof of consistency of iv estimator
Proof of Consistency of IV Estimator
  • Substitute y=Xβ+ε 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
the asymptotic variance of the iv estimator
The Asymptotic Variance of the IV estimator
  • Class exercise
  • Need to get estimate of σ2
  • Use estimated residual to do this (as in OLS)
  • To estimate residual must use X not X-hat i.e.
implication
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 (x1=z)

the finite sample distribution
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
testing over identification
Testing Over-Identification
  • If m>k then over-identified and can test instrument validity for (m-k) instruments
  • Basic idea is:
  • If instruments valid then E(ε|Z)=0 so Z should not matter when X-hat included
  • Can test this – but not for all Z’s as X-hat a linear combination of Z’s
some special cases the just identified case
Some Special Cases: The Just-Identified Case
  • In this case (Z’X) is invertible:
  • Can write IV estimator as:

(using (AB)-1=B-1A-1

in one dimensional case
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
binary instrument no other covariates
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
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