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Instrumental Variables

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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

- 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

- 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

- Leads to estimates:

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
- Will return to it later

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