Instrumental variables
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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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Instrumental Variables

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Instrumental variables

Instrumental Variables


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:


Use formula for x hat

Use formula for X-hat


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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