ECON0019: Quant Econ and Econometrics

Instrumental Variables and LATE

Prof. Áureo de Paula

: @PaulaAureo

Department of Economics
University College London

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Outline: Instrumental Variables (IV)

Part I: The basics of IV

§ Motivation and basic idea
§ IV assumptions and estimator
‚ Example
Part II: Issues in IV estimation
§ Inference and weak instruments
§ 2SLS
‚ Multiple explanatory variables
‚ Multiple instruments
§ Testing for endogeneity
§ Overidentification
Part III: IV and LATE

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Introduction

Remember the classical linear regression model:

y “ β0 ` β1 x ` u p” β J x ` u using matrix algebraq.

For the OLS estimator to be consistent (i.e. to get “close” to β

as the sample increases) we assume

covpx, uq “ 0.

If covpx, uq ‰ 0 we say that x is (econometrically) endogenous.

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What does covpx, uq ‰ 0 mean?

Let’s think about this in the context of an example:

lnpwageq “ β0 ` β1 educ ` u.

We need to consider what factors are captured by u.

cov pu, educq “ 0 implies no (linear) association between educ

and u, which may incorporate variables such as ability. But
more educated individuals will tend to be more “able”.

Remember that if covpeduc, uq ‰ 0, OLS estimators for β0 and

β1 are not consistent.

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When could covpx, uq ‰ 0 arise?

Common sources of endogeneity are:

1. Ommited variables
2. Simultaneity (Reverse Causality) (y Ñ x or y Ø x)
3. Measurement error (i.e., errors-in-variables) (Practical
Session)

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You’ve seen some of this before . . .

§ Ommited variables:

“A natural disaster like Hurricane Katrina in the US state of Louisiana in

2005 could have triggered an increase in both stress and
unemployment. This is an example where a third factor – in this case,
the weather – might account for the positive correlation between
searches for antidepressants and unemployment. It warns us to be
careful in concluding that an observed correlation implies a causal
relationship between variables.” CORE’s The Economy 1.0 (Unit 13)

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You’ve seen some of this before . . .

§ Simultaneity

“Using the US stimulus program that was implemented in the wake of

the financial crisis (the American Recovery and Reinvestment Act of
2009, a $787 billion fiscal stimulus), we would expect that US states
that were more severely hit by the financial crisis would have had higher
unemployment and attracted more stimulus spending by the
government. So unemployment causes more spending in those states.
This makes it difficult to estimate the effect of higher spending on output
and unemployment, which is what we want to do if we want to know the
size of the multiplier.” CORE’s The Economy 1.0 (Unit 14.7)

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And maybe you heard of this . . .

§ Reverse Causality

“In the South Seas there is a Cargo Cult of people. During the war they
saw airplanes land with lots of good materials, and they want the same
thing to happen now. So they’ve arranged to make things like runways,
to put fires along the sides of the runways, to make a wooden hut for a
man to sit in, with two wooden pices on his head like headphones and
bars of bamboo sticking out like antennas – he’s the controller – and
they wait for the airplanes to land. They’re doing everything right. The
form is perfect. It looks exactly the way it looked before. But it doesn’t
work. No airplanes land.” Cargo Cult Science, by R. Feynman

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If you build it, will they come?

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Small detour: unbiasedness ‰ consistency!

§ Unbiasedness: small sample; Consistency: large sample.

§ Conditions are different:

Epu|xq “ 0 ñ cov px, uq “ 0

§ It is not true that cov px, uq “ 0 implies Epu|xq “ 0.

§ Intuitively, these two conditions highlight that x and u
should be unrelated. Technically, they are nevertheless
different!
§ If you confuse the two in the exam, you will be penalized.

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IV: The Basic Idea
OLS: Exogenous regressors

x Ñ y
Õ
u

OLS: Omitted variables bias:

x Ñ y
Ù Õ
u

IV: Suppose there exists a variable z such that

z Ñ x Ñ y
Ù Õ
u

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IV Assumptions
Consider the simple regression model

yi “ β0 ` β1 xi ` ui p2SLS.1q

where cov pu, xq ‰ 0

Assume now that we have an instrumental variable zi that

satisfies the following two conditions
§ cov pzi , ui q “ 0 (exogeneity or validity)
(`Epui q “ 0 ñ 2SLS.4)
§ cov pzi , xi q ‰ 0 (relevance) (ð 2SLS.3)

The exogeneity assumption requires that:

§ zi affects yi only through xi
§ zi is unrelated to ui
and is not testable (since cov pzi , ui q involves ui ).
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IV Assumptions

Instrument relevance requires that zi affect xi

§ It can be verified by estimating the following regression

xi “ π0 ` π1 zi ` vi

Since π1 “ cov pzi , xi q{var pzi q, we can (and must!) test

relevance:
§ H0 : π1 “ 0: instrument irrelevant
§ H1 : π1 ‰ 0: instrument relevant
‚ As usual, perform t-test

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IV Estimator
Now we can identify β1 by (i) noting that

cov pzi , yi q “ β1 cov pzi , xi q ` cov pzi , ui q

and ii) using the IV assumptions to write

cov pzi , yi q cov pzi , yi q{var pzi q
β1 “ “
cov pzi , xi q cov pzi , xi q{var pzi q
which is the slope coefficient estimator from the reduced form
divided by the slope coefficient estimator from the first stage.

Reduced form: cov pzi , yi q{var pzi q

§ slope coefficient from a regression of yi on zi and an
intercept
First stage: cov pzi , xi q{var pzi q
§ slope coefficient from a regression of xi on zi and an
intercept
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IV Estimator

The sample analog is the instrumental variable estimator of β1 :

řn
pzi ´ zqpyi ´ y q
β̂1 “ ři“1
n
i“1 pzi ´ zqpxi ´ xq

The IV estimator of β0 is:

βˆ0 “ y ´ βˆ1 x

Note that βˆ1 is the OLS estimator of β1 when zi “ xi .

2SLS.1, 3-4 + Random Sampling (2SLS.2) ñ IV estimator

(2SLS) is consistent.

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Special Case of IV: Wald Estimator
A common (and simple) example of IV is one where the
instrument is binary
zi P t0, 1u
Note that

Eryi |zi “ 1s “ β0 ` β1 Erxi |zi “ 1s

Eryi |zi “ 0s “ β0 ` β1 Erxi |zi “ 0s

so that

Eryi |zi “ 1s ´ Eryi |zi “ 0s “ β1 pErxi |zi “ 1s ´ Erxi |zi “ 0sq

which after rearranging and taking sample analogues gives the

Wald estimator:
y pz“1q ´ y pz“0q
βˆ1 “
x pz“1q ´ x pz“0q
ř
where x pz“1q ” p|ti : zi “ 1u|q´1 ti:zi “1u xi , etc.
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Example
Consider our example of returns to schooling

lnpwagei q “ β0 ` β1 educi ` ui

Angrist and Krueger (1991) came up with an instrument for

education in the US: quarter of birth.

Arguments for instrument:

§ Education is compulsory by law until your 16th birthday
§ School start in the year you turn 6:
‚ children born early in the year begin school at an older age
‚ and may therefore leave school with somewhat less
education
§ Quarter of birth is arguably uncorrelated with
unobservables affecting wages (though see Bound, Jaeger
and Baker (1995) for arguments otherwise).

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Mean Years of Completed Education, by Quarter of
Birth

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Mean Log Weekly Earnings, by Quarter of Birth

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OLS and IV estimates

Quarter of birth Difference

1st 4th (2)-(1)
(1) (2) (3)
ln(weekly wage) 5.892 5.905 0.0135
(0.0034)
Years of education 12.688 12.839 0.151
(0.016)

Wald estimate of 0.089

return to education (0.021)

OLS estimate of 0.070

return to education (0.0005)

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Part I: The basics of IV
§ Motivation and basic idea
§ IV assumptions and estimator
‚ Example
Part II: Issues in IV estimation
§ Inference and weak instruments
§ 2SLS
‚ Multiple explanatory variables
‚ Multiple instruments
§ Testing for endogeneity
§ Overidentification
Part III: IV and LATE

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Inference
Assuming homoscedasticity Erui2 |zi s “ σ 2 , it can be shown that
the estimated (asymptotic) variance of the IV estimator is:

σ̂ 2
vy
ar pβ̂IV q “ 2
SSTx Rx,z

§ SSTx “ ni“1 pxi ´ xq2

ř
2 : the R-squared from a regression of x on z and an
§ Rx,z i i
intercept
§ σ̂ 2 “ pn ´ 2q´1 ni“1 ûi2 where ûi “ yi ´ β̂0 ´ β̂1 xi
ř

As before, we compute the standard error and use it to perform

tests and derive confidence intervals, but we might need large
sample sizes.

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IV vs. OLS
Advantage of IV estimator: Consistent even if u and x are
correlated, in which case the OLS estimator is biased and
inconsistent.

Disadvantage of IV estimator: less efficient if u and x are

uncorrelated.

Assume that u and x are uncorrelated. Then:

σ̂ 2 σ̂ 2
vy
ar pβ̂IV q “ 2
ą “ vy
ar pβ̂OLS q
SSTx Rx,z SSTx

and we can see the variance of the IV estimator

§ is always larger than the variance of the OLS estimator and
§ depends crucially on the correlation between z and x.

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Weak Instruments and Bias
Weak instrument means that z and x are only weakly
correlated:
§ not only lead to imprecise IV estimates,
§ but can also give large bias.

Recall that:
řn řn
i“1 pzi ´ zqpyi ´ y q pzi ´ zqpui ´ uq
β̂1 “ řn “ β1 ` ři“1
n
i“1 pzi ´ zqpxi ´ xq i“1 pzi ´ zqpxi ´ xq

Even if cov pz, uq is smaller than cov px, uq, it is not necessarily
better when the denominator is small.

Even if cov pz, uq “ 0, β̂1 becomes very unstable and unreliable

if z and x are only weakly correlated, so that the denominator is
close to zero

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Weak Instruments and Bias

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Weak Instruments and Bias

§ If the IVs are weak, the sampling distribution of the IV

(TSLS) estimator (and its t-statistic) is not well
approximated by its large n normal approximation.

§ IV estimation thus requires fairly strong instruments

§ Rule of thumb: F-statistic above 10 (same as t-statistic

?
above 10q for the instrument in the first stage (though
see recent research by Lee et al. (2022)).

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Example - Weak instrument: Birth weight and smoking

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IV in the MLR model

We can add an additional explanatory variables x2 to the model:

yi “ β0 ` β1 xi1 ` β2 xi2 ` ui

Assume that x2 is uncorrelated with u, while x1 is correlated

with u
§ xi2 : exogenous explanatory variable
§ xi1 : endogenous explanatory variable

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IV in the MLR model

To consistently estimate all the β’s we use the sample

analogues of the moment conditions
n
ÿ
Epui q “ 0 ñ pyi ´ β̂0 ´ β̂1 xi1 ´ β̂2 xi2 q “ 0
i“1
ÿn
cov pui , zi q “ 0 ñ pyi ´ β̂0 ´ β̂1 xi1 ´ β̂2 xi2 qzi “ 0
i“1
ÿn
cov pui , xi2 q “ 0 ñ pyi ´ β̂0 ´ β̂1 xi1 ´ β̂2 xi2 qxi2 “ 0
i“1

3 equations with 3 unknowns: can be solved as we did for OLS

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IV in the MLR model
As before we need z to be correlated with x1 , but now over and
above x2 .

We can test this by estimating the following regression

xi1 “ π0 ` π1 zi ` π2 xi2 ` vi

and instrument relevance is tested as:

H0 : π1 “ 0 vs H1 : π1 ‰ 0

In other words, IV in the MLR model is just as IV in the SLR

model except the exogeneity assumption is now:

cov pzi , ui | xi2 q “ 0

(Note that zi and xi2 can be correlated!)

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Two-stage least squares (2SLS)

We still consider the following model

yi “ β0 ` β1 xi1 ` β2 xi2 ` ui

but now with M ą 1

cov pui , zim | xi2 q “ 0 m “ 1, . . . , M

We only need to modify the first stage such that:

xi1 “ π0 ` π1 zi1 ` . . . ` πM ziM ` πM`1 xi2 ` vi

Instrument relevance is tested using an F statistic for

H0 : π1 “ . . . “ πM “ 0

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2SLS: Step-by-step
1. Estimate the first-stage regression:

xi1 “ π0 ` π1 zi1 ` . . . ` πM ziM ` πM`1 xi2 ` vi

- regressing the endogenous explanatory variable on the

instruments and all the other exogenous explanatory
variable
2. Compute the predicted value of x1 :

x̂i1 “ π̂0 ` π̂1 zi1 ` . . . ` π̂M ziM ` π̂M`1 xi2

3. Estimate the second-stage regression:

yi “ β0 ` β1 x̂i1 ` β2 xi2 ` ei

- regressing the outcome variable on x̂i1 and all the other

exogenous explanatory variable
Note that using more than one instrument is not necessary, but
it can give you more efficient IV estimates.
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Multiple Endogenous Variables
If there is more than one endogenous variable, e.g.

y1 “ β0 ` β1 x1 ` β2 x2 ` β3 x3 ` u

where x1 and x2 are potentially correlated with the error u, we

need at least two exogenous variables z1 and z2 that do not
appear in the equation above. (This is known as the order
condition.)

Both of these variables still need to be relevant though. If z1

does not correlate with either endogenous variable or if both z1
and z2 correlate with only one of the endogenous variables we
would not be able to identify the desired parameters. The
sufficient condition for identification is called the rank condition.
(We will revisit that when we discuss simultaneous equations.)

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Weak IV Revisited

§ Weak IV are also a problem with many instruments.

§ Adding instruments with low predictive power in the first

stage lowers the F -statistic and exacerbates the bias in the
2SLS estimator.

§ Bound, Jaeger and Baker (1995) illustrate this using

Angrist and Krueger (1991). AK present results using
different sets of IVs (plus other covariates):
- quarter of birth dummies: M “ 3 instruments.
- quarter of birth + (quarter of birth) x (year of birth)
dummies: M “ 30 instruments.
- quarter of birth + (quarter of birth) x (year of birth) +
(quarter of birth) x (state of birth): M “ 180 instruments.

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Weak IV Revisited

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Weak IV Revisited

With more than one endogenous variable the F -statistic in the

first stage may not suffice for detection of weak IVs either.

One alternative is to perform a test based on the Cragg-Donald

Eigenvalue statistic (see Stock, Wright and Yogo (2002)).

(Alternatively, see Sanderson and Windmeijer, Journal of

Econometrics, 2016.)

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Testing for endogeneity: Hausman test
Consider the simple regression model
yi “ β0 ` β1 xi1 ` ui
Test for endogeneity:
§ H0 : cov pxi1 , uq “ 0, both OLS and IV are consistent
§ H1 : cov pxi1 , uq ‰ 0, only IV is consistent and xi1
We perform a Hausman test by
1. Calculating the first-stage residual v̂i (this contains the
endogenous part of xi1 )
xi1 ´ x̂i1 “ v̂i
2. Adding v̂i to the regression model, and estimate by OLS:
yi “ β0 ` β1 xi1 ` θv̂i ` ei
3. Using a t-test to check if θ is significantly different from
zero ñ reject H0 : cov pxi1 , ui q “ 0
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Testing Overidentification Restrictions

§ When there are as many IVs as endogenous variables,

exogeneity of the instruments is not testable.

§ However, when there are more IVs than endogenous

variables, we can test whether some of them are
uncorrelated with the u.

§ With two IVs and one endogenous variable, for example,

we could compute alternative 2SLS estimates using each
of the IVs. If the IVs are both exogenous, the 2SLS will
converge to the same parameter and they will differ only by
sampling error.

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Testing Overidentification Restrictions

§ If the two estimates are statistically different, we would not

be able to reject the hypothesis that at least one of the IVs
is invalid.

§ But we would not be able to ascertain which one!

§ Moreover, if they are similar and pass the test it could

because both IVs fail the exogeneity requirement.

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Testing Overidentification Restrictions
In practice, under homoskedasticity (i.e., Epu 2 |zq “ σ 2 ),

1. Estimate coefficients by 2SLS and obtain residuals ûi . For

a linear regression with one endogenous regressor this is

ûi “ yi ´ β̂0 ´ β̂1 xi .

Note: These are not the residulas from the second stage
OLS regression in TSLS: it uses xi , not x̂i .

2. Regress ûi on all exogenous variables. Record the R 2 .

3. Under the null hypothesis that all IVs are exogenous,

nR 2 „ χ2M´1 .

This test can be made robust to heteroskedasticity.

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Part I: The basics of IV
§ Motivation and basic idea
§ IV assumptions and estimator
‚ Example
Part II: Issues in IV estimation
§ Inference and weak instruments
§ 2SLS
‚ Multiple explanatory variables
‚ Multiple instruments
§ Testing for endogeneity
§ Overidentification
Part III: IV and LATE

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Example: Job Training Programme

Out of 600 workers, 300 were invited to participated in a job

training programme, and 200 of them chose to do so. The table
shows average wages:
Participated
Invited No Yes Mean
No 11 - 11
r300s
Yes 9 15 13
r100s r200s
Mean 10.5 15

What is the average causal effect of the program?

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 Participated
Invited No Yes Mean
No 11 - 11
r300s
Yes 9 15 13
r100s r200s
Mean 10.5 15

§ 15 ´ 9 “ 6 : compares workers who chose to participate

and those who chose not to. They may not be similar on
Y0 -selection bias.
§ 15 ´ 10.5 “ 4.5 : compares workers who chose to
participate and those who either chose not to or weren’t
given an option. The participant group may still be
self-selected.

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 Participated
Invited No Yes Mean
No 11 11
r300s -
Yes 9 15 13
r100s r200s
Mean 10.5 15

§ 13 ´ 11 “ 2 : compares workers invited to participate with

those not invited (by chance). This is a causal effect but of
a different treatment: getting invited – the Intention-to-Treat
(ITT) effect.
§ ITT averages the outcome for 2{3 of invited workers who
participated and for 1{3 of invited workers who did not.

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 Participated
Invited No Yes Mean
No 11 11
r300s -
Yes 9 15 13
r100s r200s
Mean 10.5 15

§ Another approach: by randomness of invitations,

11 “ 13 ˆ 9 ` 23 ˆ?, where ? “ 12 is the average untreated
potential outcome for the group who would participate if
given an option
§ Thus, the average causal effect of participation on this
group is 15 ´ 12 “ 3.

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This is a Special Case of IV
§ X “ participated in the program
§ Z “ invited to participate
§ Relevance: the first-stage coefficient is the compliance
rate:

2 2
ErX | Z “ 1s ´ ErX | Z “ 0s “ ´0“
3 3
§ Exogeneity: the invitation does not affect wages, other
than through participation (a reasonable but non-trivial
assumption!)
§ The reduced-form coefficient is the ITT:

ErY | Z “ 1s ´ ErY | Z “ 0s “ 13 ´ 11 “ 2
2
§ IV (Wald) estimator: 2{3 “3
§ To understand what causal effect this is, we need an IV
framework with heterogeneous effects.
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IV with Heterogeneous Effects

§ Outcome equation: Yi “ β0 ` β1i Xi ` Ui

§ Exclusion: Zi does not have a direct causal effect
§ Potential outcomes: Y0i “ β0 ` Ui and Y1i “ β0 ` β1i ` Ui
and βi1 “ Y1i ´ Y0i .
§ First-stage equation: Xi “ π0 ` π1i Zi ` Vi
§ Potential values: Xi “ Zi X1i ` p1 ´ Zi q X0i where
X0i “ π0 ` Vi and X1i “ π0 ` π1i ` Vi .
§ Independence: Zi K
K pUi , β1i , Vi , π1i q
§ “Monotonicity”: π1i ě 0 (or π1i ď 0) for all i (ñ sign
switches across i are ruled out), plus E rπ1i s ‰ 0

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Local Average Treatment Effects (LATE)

The IV estimator identifies “LATE,” a weighted average of β1i

with weights proportional to how much xi is affected by zi (π1i ):
„ 
E rβ1i π1i s π1i
βIV ” plim β̂IV “ “ E β1i
E rπ1i s E rπ1i s

Proof

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When LATE is Average Treatment Effect (ATE)
Note that π1i is unobserved, and ATE is generally not identified
Erβ1i π1 s
When would ATE ” E rβ1i s equal to βIV “ Erπ1i s ?
§ When causal effects are homogenoeous, β1i ” β1
§ When the instrument affects all individuals equally, π1i ” π1
§ When the heterogeneity in the treatment effect and the
heterogeneity in the effect of the instrument are
uncorrelated, E rβ1i π1i s “ E rβ1i s E rπ1i s.

Otherwise, two researchers with two valid instruments can find

different estimates, even in large samples! (Overidentifying
restriction tests can reject even if both instruments are valid, but
treatment effects are heterogeneous.)

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These slides covered:

Wooldridge 15.1-15.6, Stock and Watson 12 + SW 13.6 and

Appendix 13.2.

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Appendix

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IV ñ LATE
We know that βIV “ Cov pYi , Zi q { Cov pXi , Zi q. Denote
µz “ E rZi s. Then:

Cov pXi , Zi q “ E rXi pZi ´ µz qs “ E rpπ0 ` π1i Zi ` Vi q pZi ´ µz qs

“ E rπ1i Zi pZi ´ µz qs “ E rπ1i s Var rZi s
” ı
since Zi KK pUi , β1i , Vi , π1i q and E rZi pZi ´ µz qs “ E pZi ´ µz q2 .
Furthermore,
Cov rYi , Zi s “ E rYi pZi ´ µz qs “ E rpβ0 ` β1i Xi ` Ui q pZi ´ µz qs “
“ E rβ1i Xi pZi ´ µz qs “ E rβ1i pπ0 ` π1i Zi ` Vi q pZi ´ µz qs
“ E rβ1i π1i Zi pZi ´ µz qs “ E rβ1i π1i s Var rZi s

since Zi KK pUi , β1i , Vi , π1i q.

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