Lecture 2: Diff-in-Diff and Instrumental Variables 2.pdf · Difference-in-Differences Instrumental Variables Example For a little variety, I will do different examples today

Lecture 2: Diff-in-Diff and Instrumental Variables

January 14, 2016

Difference-in-Differences Instrumental Variables

Causal Inference

There are 5 basic empirical methods to obtain causal inference:

1 Controls (includes matching/fixed-effects)2 Randomized Experiments3 Difference-in-Differences4 Instrumental Variables5 Regression Discontinuity


Difference-in-Differences

Difference-in-differences (DD) extends the idea behind individualfixed-effects

Instead of just comparing before and after, it compares before andafter among a ‘treated’ and a ‘control’ group

In fact, fixed-effects is a difference; DD just adds another layerSome people add even another layer, making it adifference-in-difference-in-differences


Example

For a little variety, I will do different examples today than my usualeducation cases

Consider the question of immigration


A View on Immigration

Ted Cruz (Nov 10, 2015 GOP debate): “I will say, the politics of it[immigration] would be very, very different if a bunch of lawyers orbankers were crossing the Rio Grande. Or if a bunch of people withjournalism degrees were coming over and driving down the wages inthe press, then we would see stories about the economic calamity thatis befalling our nation.”


Immigration and Wages

This is one of the major arguments against immigration: thatimmigrants will drive down the wages of locals

Especially in the low-skilled sector (depending on the type ofimmigrant)

We would like to look at this empirically


Framing the Research Question

Our question is: Does immigration drive down the wages (or theemployment rate) of locals?

1 What is the unit of analysis?

2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?6 How could we mimic this? Can we do a randomized experiment?




1 What is the unit of analysis?2 What is the treatment?

3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?6 How could we mimic this? Can we do a randomized experiment?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?

4 What are the counterfactual outcomes?5 What is the causal link?6 How could we mimic this? Can we do a randomized experiment?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?

5 What is the causal link?6 How could we mimic this? Can we do a randomized experiment?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?

6 How could we mimic this? Can we do a randomized experiment?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?6 How could we mimic this? Can we do a randomized experiment?


The Mariel Boatlift

What was the Mariel Boatlift?


DD

The DD term refers to two levels of differenceThe first difference is (almost) always timeThe second difference can be: schools, grades, geography, individuals,etc.


First-Difference

Let’s look at the first-difference of timeNotation: Y1,city ,t is treated city at time tY0,city ,t is control city at time t

What outcomes do we observe?

Write out the treatment effect and the selection bias:


Second-Difference I

What is a reasonable second layer of difference?Two possible:

David Card (1990) v. George Borjas (2015)


Second-Difference II

We now incorporate the second layer of difference

What outcomes do we observe:

What idea can we use to mimic the counterfactual?


Estimating Equations

Y1,miami ,1981 − Y0,atlanta,1981 − [Y0,miami ,1979 − Y0,atlanta,1979]

In regression form:

Open up STATA!


Interpretation of DD

How do we interpret the DD estimate?

We call this a Average Treatment Effect (ATE)


Internal Validity I

What is our key assumption?

Visual Representation:


Internal Validity II

How do we check our identifying assumption? Possible failures?Ashenfelter dipPolitical Endogeneity (i.e. Reverse Causality)

How can we weaken the key assumption?

http://www.motherjones.com/kevin-drum/2015/09/another-shot-fired-great-immigration-vs-wages-war




External Validity

Generalizability?

General Equilibrium Effects?


Pros and Cons of DD

Pros:

Cons:


Causal Inference

There are 5 basic empirical methods to obtain causal inference:

1 Controls (includes matching/fixed-effects)2 Randomized Experiments3 Difference-in-Differences4 Instrumental Variables5 Regression Discontinuity


Basic Intuition

Basic idea: In a randomized experiment, we randomized people to‘treatment’ and ‘control’

What happens if some external ‘thing’ (geography, mass layoff,storms, etc) strikes some people and not others (at random!)

Well the external ‘thing’ has randomized for us! We call this anInstrumental Variable (IV)Randomized experiments can thus be treated as IVs


Example I

Does earlier colonization improve economic outcomes?

Problem: European empires likely colonized the ‘best’ places (mostfertile, etc) first

Solution: Ships had to sail with the wind


Example II

Thought Experiment:

Compare two islands: Guam and Fefan (in Micronesia)

Guam was directly on the East-West route across the pacific (used byMagellan)

Fefan was not on this route (was on the much more difficultWest-East route)

https://www.google.ca/maps/dir/Guam/Fefan,+Federated+States+of+Micronesia/@10.3941378,130.3823886,4z/data=!4m13!4m12!1m5!1m1!1s0x671f76ff930f24ef:0x5571ae91c5b3e5a6!2m2!1d144.793731!2d13.444304!1m5!1m1!1s0x6667a4d6a8ce100d:0xc47882f565ab012a!2m2!1d151.8379961!2d7.3487617








Example III

Therefore, Guam was colonized before Fefan

‘Wind’ randomized the colonization for us!


IV Basics I

The (basic) math:

What outcomes do we observe?

What is our treatment and selection effect?


IVs Basics II

The (more complex) math:

In IVs, randomization is not perfect

Fefan could have been discovered ‘by luck’ earlierFor example, Pitcairn was discovered earlier even though it was not onthe main wind routeThis was because of the mutiny on the ship “HMS Bounty”

We account for this by dividing by the probability randomizationaffected your treatment status:


IV Notation

Notation:

We call the instrument (here wind patterns) Z

We call the ‘endogenous regressor’ (here colonization date) X


IV Assumptions

For an IV to be valid it must be both:

Relevant =⇒ Corr(X ,Z ) 6= 0All this says is that there was randomization (i.e. Islands on favorablewind routes were colonized first)We can check this through the equation:ColonizationDateisland,t = α + βFavorableWindisland,t + εit

Exclusion =⇒ Corr(Z , ε) = 0This says that randomization was ‘proper’ (i.e. Islands on favorablewind routes had no other advantage (more fertile soil, etc.))This assumption cannot be tested as ε is unobserved

Do these assumptions seem likely to hold in this case?


IV as a Regression

We have two terms to estimate: E[D1 − D0] (first-stage) andE[Y1 − Y0|Z = 1] (reduced-form)

First-stage is the effect of Z on X (i.e. how much more likely was it forislands on favorable wind routes to be colonized first)Reduced-form is the effect of Z on Y (i.e. how much better off areislands on favorable wind routes?)

First-stage:ColonizationDateisland ,t = α + β1FavorableWindisland ,t + εit

In general: Xi = α + β1Zi + εi

Reduced-form:EconomicOutcomesisland ,t = α + β2FavorableWindisland ,t + εit

In general: Yi = α + β2Zi + εi

The IV estimate is then: β2β1


For the Keeners

In the last slide, we got the Wald estimator

However, when implementing IVs we use the IV estimatorIntuition is identical; the IV estimator is just more efficient

Only difference is that in the reduced-form regression we plug in a‘predicted’ X rather than Z

First-stage: X̂i = α + β1Zi + εi

Reduced-form: Yi = α + β2X̂i + εi

In STATA you do all this by simply typing: “ ivregress 2sls y (x=z),vce(robust) first”


Another IV

Our question is: Does having another child affect a mother’s laboursupply?

OLS: HasWorkedi = α + β#Kidsi + εi

Possible failures of OLS in this instance?

Idea: Use the fact parents like to have a boy AND a girl in theirfamily (Angrist and Evans, 1998)



Our question is: Does having another child affect the mother’s laboursupply?

1 What is the unit of analysis?

2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?




1 What is the unit of analysis?2 What is the treatment?

3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?

4 What are the counterfactual outcomes?5 What is the causal link?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?

5 What is the causal link?




1 What is the unit of analysis?2 What is the treatment?3 What outcome are we interested in?4 What are the counterfactual outcomes?5 What is the causal link?


To STATA

How do we mimic this? (think of the IV as having your first 2 kids bethe same gender)

Open up STATA


Interpretation of IVs

How do we interpret the IV estimate? (i.e. who complies withtreatment?)

We call this a Local Average Treatment Effect (LATE)

This is (in my view) the biggest weakness of IVs


Internal Validity

Relevance?

Exclusion?


External Validity

Generalizability?

Mechanisms?


Pros and Cons of IV

Pros:

Cons:

Documents

Lecture 2: Diff-in-Diff and Instrumental Variables 2.pdf · Difference-in-Differences Instrumental Variables Example For a little variety, I will do different examples today