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# Instrumental Variables - PowerPoint PPT Presentation

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

• 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

• 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

• 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:

• Instrument Exogeneity:

Two-Stage Least Squares Representation – the First-Stage

• To get bit of X that is correlated with Z, run regression of X on Z

X=ZΠ+v

Two-Stage Least Squares Representation- 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 Representation

Proof of Consistency of IV Estimator Representation

• 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 Representation

• 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 Representation

• 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 Representation

• 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

Testing Over-Identification Representation

• 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 Representation

• In this case (Z’X) is invertible:

• Can write IV estimator as:

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

In one-dimensional case… Representation

• 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 Representation

• 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