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Ec1123 Section 7Instrumental Variables

Andrea Passalacqua

Harvard [email protected]

November 16th, 2017

Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 1 / 28

Outline

1 Simultaneous Causality

2 Instrumental Variable Regression: Introduction

Conditions

3 IV: Examples

4 Two-Stage Least Squares

5 Testing the Validity of Instruments


Simultaneous causality: What is that?

Yi = β0 + β1Xi + β2W1i + β3W2i + ui

Simultaneous causality arises when

X −→ Y and X ←− Y

OLS will fail because conditional mean independence (CMI) is violated.

I Recall CMI: E [u|X ,W1,W2] = E [u|W1,W2] – conditional on Z1 and Z2, X isas good as randomly assigned

However, it is not as “fixable” as OVB (ie. when E [u|X ] 6= 0) since addingcontrols won’t address the fact that X ←− Y


Examples of simultaneous causality

QUANTITYi = β0 + β1PRICEi + ui

Changes in price affect quantities supplied and demanded

Changes in quantity supplied and demanded affect price

Price and quantity are jointly determined by a set ofsimultaneous equations (ie. the demand and supply curve)

Other examples:

Democracy and growth

Parental involvement and child’s performance in school

Police and crime


Outline



Conditions

3 IV: Examples




Instrumental Variables

Instrumental Variables (IV) are useful for estimating models

with measurement error

with simultaneous causalityor

with omitted variable bias

I IV especially useful when we cannot plausibly control for all omitted variables

More generally, whenever conditional mean independence on X fails

I Our estimated coefficient is biased and cannot be interpreted causally

I IV relies on using a valid instrumental variable Z to recover as-if randomassignment of X



Yi = β0 + β1Xi + ui

We want to know the causal effect of X on Y

Xi Yi

S1i

W1iW2i

but are confounded by OVB and simultaneous causality−→ β2 is biased


Conditions for IV: Intuition


Z is an instrumental variable for X if:

Condition 1: Relevance

Z is related to X

Condition 2: Exogeneity of Z

Two ways of saying the Exogeneity condition:Z is as-if randomly assignedThe only relationship between Z and Y goes through X

after conditioning on any control variable W s.




Xi Yi

S1i

W1iW2i

Z1i




Xi Yi

S1i

W1iW2i

Z1i

Z2i

There can be multiple instruments Z1 and Z2 for the same X


IV - Instrument Relevance

In the single-variable case:


Suppose the model fails CMI (Recall CMI:E [u|X ,W1,W2] = E [u|W1,W2]), but we have a valid instrument Z :

βIV =Cov(Y ,Z )

Cov(X ,Z )

Notice that clearly we need Cov(X ,Z ) 6= 0. In fact, in our dataset, wereally want Cov(X ,Z ) to be far away from zero.

An instrument where Cov(X ,Z ) (ie. the “relevance” of Z ) is close to zerois known as a weak instrument.


IV - Exogeneity Condition


All of these would violate Condition 2 (exogeneity):

Zi ←→ Yi

Zi ←→ S1i

Zi ←→W1i and Z ←→W2i

I But we can control for W1i or W2i

In other words, if Z is related to Y (causally or not) through anyrelationship other than through X .

In the figure, this would be any path from Z to Y that does not go throughX .


Instrumental Variables – RECAP

ConsiderYi = β0 + β1Xi + ui

where OLS yields a biased β1

Conditions for IV

Z is an instrumental variable for X in this model if:1 Relevance: Z is related to X

Corr(Z ,X ) 6= 0

2 Exogeneity: The only relationship between Z and Y goes through X

Corr(Z , u) = 0


Outline



Conditions

3 IV: Examples




Instrumental Variables Example: Roommate Assignment

Suppose we analyzing among first semester college freshmen:

GPAi = β0 + β1 × (Hours Studying)i + ui

E [u|X ] 6= 0 because of omitted variable bias.

Proposed instrumental variable:

Z = whether randomly assigned roommatebrought a video game to college

Relevant?

Exogeneity?





Z is related to X . i.e. Corr(Z ,X ) 6= 0

How to check? Examine the first-stage relationship between Z and X :

First-stage: (Hours Studying)i = γ0 + γ1Zi + viregress hours videogame

From lecture, γ1 = −0.668∗∗ (large and significant)

(later) Conduct a F -test to confirm




Condition 2: Exogeneity

The only relationship between Z and Y goes through X : Corr(Z , u) = 0

Condition 2 holds if roommates having video games (Z ) only affects GPA(Y ) through affecting hours studying (X )

Plausible? Are there other channels through which Z is related to Y ?

What if being assigned a roommate with video games made students sleepless?

What if male students are more likely to bring video games and males havelower grades on average? (for argument’s sake)



The Z to Y alternate channel does NOT have to be causal to violateCondition 2

More mathematically: hours of sleep and gender lie in the error term u, sowe have Corr(Z , u) 6= 0

So control for hours of sleep (W1i ) and gender (W2i ):

GPAi = β0 + β1 × (Hours Studying)i + β2W1i + β3W2i + ui

Similar to OVB, we can never directly test whether the exogeneity conditionis violated or satisfied since u is unobserved. We can only provide arguments(using theory or institutional knowledge)


Examples of Instrumental Variables

Z X YHow does prenatal health affect a child’s long-run development?

In wombPrenatal health

Adult health &

during Ramadan income

What effect does serving in the military have on future wages?

Military draft lottery # Military service Income

What is the effect of rioting on community development?

Rainfall on day Number of Long-run

of MLK assassination riots property values

Each of these examples also requires some control variables W s for theexogeneity condition to hold. In general, arguing the exogeneity conditioncan be very difficult.


Outline



Conditions

3 IV: Examples




IV in STATA

IV regression of Y on X using instrument Z :

ivregress 2sls y (x = z), robust

IV regression of Y on X using instrument Z and controls W1 and W2:

ivregress 2sls y w1 w2 (x = z), robust

IV regression of Y on X using instruments Z1 and Z2 and controls W1 andW2:

ivregress 2sls y w1 w2 (x = z1 z2), robust

where 2sls stands for Two-Stage Least Squares


Two-Stage Least Squares (TSLS or 2SLS)

Goal of IV: estimate the causal relationship of X on Y using instrument Z


2SLS estimates β1 in “two stages”:

Stage 1: Regress X on Z and calculate predicted values Xi

Xi = γ0 + γ1Zi

Stage 2: Regress Y on X to get β1 = β2SLS = βIV


2SLS – Intuition


We cannot causally interpret β1 since X is not “randomly assigned”

Think of the variation (not variance) in X as coming from two separatesources:

Variation in X = As-if random part + Non-random part

The non-random part is giving us problems (OVB or simultaneouscausality)

Stage 1 of 2SLS isolates the as-if random part of X , which is X .

I Since Z is a valid instrument for X in this model, Z is as-if randomlyassigned by Condition 2

I The variation in X related to Z is also as-if randomly assigned

Stage 2 uses the as-if random part to estimate the causal relationship of Xon Y . Regressing Y on X is like regressing Y on the random variation in X .


Outline



Conditions

3 IV: Examples




Testing the Validity of Instruments


Conditions for IV

Z is an instrumental variable for X in this model if controlling for W s:1 Relevance: Corr(Z ,X ) 6= 0

2 Exogeneity: Corr(Z , u) = 0

Testing Condition 1 is straightforward, since we have data on both Z and X

Testing Condition 2 is trickier, because we never observe u. In fact, we canonly test Condition 2 when we have more instruments Z s than X s


Testing Condition 1: Relevance


Z must be related to X . i.e. Corr(Z ,X ) 6= 0

We need the relationship between X and Z to be “meaningfully large”

How to check?

Run first-stage regression with OLS

Xi = α0 + α1Z1i + α2Z2i + α3W1i + α4W2i + · · ·+ vi

Check the F-test on all the coefficients on the instrumentsH0 : α1 = α2 = 0

If F > 10, we claim that Z is a strong instrument

If F ≤ 10, we have a weak instruments problem


Testing Condition 2: ExogeneityCondition 2: Exogeneity of Z

Z is as-if randomly assigned. i.e. Corr(Z , u) = 0

To check exogeneity, we need more instruments Z s than endogenous X s(ie., our model is overidentified)

Suppose there is one treatment variable of interest X , multiple Z s,potentially, multiple control variables W s.


Use Z1 to estimate β1 and predict ui . If Z1 and Z2 are both validinstruments, then:

Corr(Z2, u) = 0

This is the basic idea of a J-test

for a model with just one instrument and one endogenous variable there isno formal test


Testing Condition 2: Exogeneity

J-test for overidentifying restrictions:

H0 : Both Z1 and Z2 satisfy the exogeneity condition

Ha : Either Z1,Z2, or both are invalid instruments

In STATA:

ivregress 2sls y w1 w2 (x = z1 z2), robust

estat overid

display "J-test = " r(score) " p-value = " r(p score)

If the p-value < 0.05, then we reject the null hypothesis that all ourinstruments are valid

But just like an F -test, rejecting the test does not reveal which instrument isinvalid, only that at least one fails the exogeneity condition


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