Econometric Approaches to Causal Econometric Approaches to Causal Inference:Inference:Difference-in-Differences and Difference-in-Differences and Instrumental VariablesInstrumental Variables

Graduate Methods Master ClassGraduate Methods Master Class

Department of Government, Harvard University Department of Government, Harvard University

February 25, 2005February 25, 2005

Overview: diff-in-diffs and IVOverview: diff-in-diffs and IV

Data Data Randomized experiment Randomized experiment Observational dataObservational data or natural experimentor natural experiment

Problem We cannot observe theProblem We cannot observe the OVB, selection bias, OVB, selection bias, counterfactual (what ifcounterfactual (what if simultaneous simultaneous

causalitycausality treatment group had nottreatment group had not received treatment)received treatment)

MethodMethod Difference-in-differences Difference-in-differences Instrumental Instrumental variablesvariables

Diff-in-diffs: basic ideaDiff-in-diffs: basic idea

Suppose we randomly assign treatment to some units Suppose we randomly assign treatment to some units

(or nature assigns treatment “as if” by random assignment) (or nature assigns treatment “as if” by random assignment)

To estimate the treatment effect, we could just compare the To estimate the treatment effect, we could just compare the

treated units before and after treatment treated units before and after treatment

However, we might pick up the effects of other factors that However, we might pick up the effects of other factors that

changed around the time of treatmentchanged around the time of treatment

Therefore, we use a control group to “difference out” theseTherefore, we use a control group to “difference out” these

confounding factors and isolate the treatment effectconfounding factors and isolate the treatment effect

Diff-in-diffs: without regressionDiff-in-diffs: without regression

One approach is simply to take the mean value of each One approach is simply to take the mean value of each group’s group’s

outcome before and after treatmentoutcome before and after treatment

Treatment groupTreatment group Control groupControl group

BeforeBefore T TBB C CBB

AfterAfter T TAA C CAA

and then calculate the “difference-in-differences” of the and then calculate the “difference-in-differences” of the means:means:

Treatment effect = (TTreatment effect = (TA A -- TTB B ) -) - (( CCA A -- CCB B ) )

Diff-in-diffs: with regressionDiff-in-diffs: with regression

We can get the same result in a regression framework We can get the same result in a regression framework (which (which

allows us to add regression controls, if needed):allows us to add regression controls, if needed):

yyii = = ββ00 + + ββ11 treat treatii + + ββ22 after afterii + + β β33 treat treatii*after*afterii + e + eii

wherewhere treat = 1 if in treatment group, = 0 if in control treat = 1 if in treatment group, = 0 if in control groupgroup

after = 1 if after treatment, = 0 if before after = 1 if after treatment, = 0 if before treatmenttreatment

The coefficient on the interaction term (The coefficient on the interaction term (ββ3 3 )) gives us the gives us the

difference-in-differences estimate of the treatment effectdifference-in-differences estimate of the treatment effect

Diff-in-diffs: with regressionDiff-in-diffs: with regression

To see this, plug zeros and ones into the regression To see this, plug zeros and ones into the regression equation:equation:

yyii = = ββ00 + + ββ11 treat treatii + + ββ22 after afterii + + β β33 treat treatii*after*afterii + e + eii

TreatmentTreatment Control Control GroupGroup Group Difference Group Difference

BeforeBefore ββ0 0 + + ββ11 ββ00 ββ11

AfterAfter ββ0 0 + + ββ1 1 + + ββ2 2 + + ββ33 ββ00 + + ββ22 ββ11 + + ββ33

DifferenceDifference ββ2 2 + + ββ33 ββ22 ββ33

Diff-in-diffs: exampleDiff-in-diffs: example

Card and Krueger (1994)Card and Krueger (1994)

What is the effect of increasing the minimum wage on What is the effect of increasing the minimum wage on employment at fast food restaurants?employment at fast food restaurants?

Confounding factor: national recessionConfounding factor: national recession

Treatment group = NJ Treatment group = NJ Before = Feb 92Before = Feb 92

Control group = PAControl group = PA After = Nov 92After = Nov 92

FTEFTEii = = ββ00 + + ββ11 NJ NJii + + ββ22 Nov92 Nov92ii + + ββ33 NJ NJii*Nov92*Nov92ii + e + eii

Diff-in-diffs: exampleDiff-in-diffs: example

FTEFTEii = = ββ00 + + ββ11 NJ NJii + + ββ22 Nov92 Nov92ii + + ββ33 NJ NJii*Nov92*Nov92ii + e + e

23.33 -2.89 -2.1623.33 -2.89 -2.16 2.75 2.75

FTEFTE

23.3323.33 Control group (PA)Control group (PA)

21.1721.17

20.4420.44 Treatment group (NJ)Treatment group (NJ) 21.03 21.03

TimeTime

Treatment effect of minimum wage increase = + 2.75 FTETreatment effect of minimum wage increase = + 2.75 FTE

Diff-in-diff-in-diffsDiff-in-diff-in-diffs

A difference-in-difference-in-differences (DDD) model allows us A difference-in-difference-in-differences (DDD) model allows us

to study the effect of treatment on different groupsto study the effect of treatment on different groups

If we are concerned that our estimated treatment effect might If we are concerned that our estimated treatment effect might

be spurious, a common robustness test is to introduce a be spurious, a common robustness test is to introduce a

comparison group that should not be affected by the treatmentcomparison group that should not be affected by the treatment

For example, if we want to know how welfare reform has For example, if we want to know how welfare reform has

affected labor force participation, we can use a DD modelaffected labor force participation, we can use a DD model

that takes advantage of policy variation across states, and then that takes advantage of policy variation across states, and then

use a DDD model to study how the policy has affected single use a DDD model to study how the policy has affected single

versus married womenversus married women

Diff-in-diffs: drawbacksDiff-in-diffs: drawbacks

Diff-in-diff estimation is only appropriate if treatment is Diff-in-diff estimation is only appropriate if treatment is randomrandom

- however, in the social sciences this method is usually - however, in the social sciences this method is usually applied applied

to data from natural experiments, raising questions to data from natural experiments, raising questions about about

whether treatment is truly random whether treatment is truly random

Also, diff-in-diffs typically use several years of serially-Also, diff-in-diffs typically use several years of serially-correlated correlated

data but ignore the resulting inconsistency of standard data but ignore the resulting inconsistency of standard errors errors

(see Bertrand, Duflo, and Mullainathan 2004)(see Bertrand, Duflo, and Mullainathan 2004)

IV: basic ideaIV: basic idea

Suppose we want to estimate a treatment effect using Suppose we want to estimate a treatment effect using

observational data observational data

The OLS estimator is biased and inconsistent (due to The OLS estimator is biased and inconsistent (due to correlation correlation

between regressor and error term) if there isbetween regressor and error term) if there is

- omitted variable biasomitted variable bias- selection biasselection bias- simultaneous causalitysimultaneous causality

If a direct solution (e.g. including the omitted variable) is not If a direct solution (e.g. including the omitted variable) is not

available, instrumental variables regression offers an available, instrumental variables regression offers an alternative alternative

way to obtain a consistent estimatorway to obtain a consistent estimator

IV: basic ideaIV: basic idea

Consider the following regression model:Consider the following regression model:

yyii = = ββ00 + + ββ11 X Xii + e+ eii

Variation in the endogenous regressor XVariation in the endogenous regressor X ii has two parts has two parts

- the part that is uncorrelated with the error (“good” the part that is uncorrelated with the error (“good” variation)variation)

- the part that is correlated with the error (“bad” variation)the part that is correlated with the error (“bad” variation)

The basic idea behind instrumental variables regression is The basic idea behind instrumental variables regression is to to

isolate the “good” variation and disregard the “bad” isolate the “good” variation and disregard the “bad” variationvariation

IV: conditions for a valid instrumentIV: conditions for a valid instrument

The first step is to identify a valid instrumentThe first step is to identify a valid instrument

A variable ZA variable Zii is a valid instrument for the endogenous is a valid instrument for the endogenous regressor regressor

XXii if it satisfies two conditions: if it satisfies two conditions:

1. Relevance:1. Relevance: corr (Zcorr (Zi i , X, Xii) ≠ 0) ≠ 0

2. Exogeneity:2. Exogeneity: corr (Zcorr (Zi i , e, eii) = 0) = 0

IV: two-stage least squaresIV: two-stage least squares

The most common IV method is two-stage least squares The most common IV method is two-stage least squares (2SLS)(2SLS)

Stage 1: Decompose XStage 1: Decompose Xii into the component that can be into the component that can be

predicted by Zpredicted by Zii and the problematic component and the problematic component

XXii = = 00 + + 11 Z Zii + + ii

Stage 2: Use the predicted value of XStage 2: Use the predicted value of Xii from the first-stage from the first-stage

regression to estimate its effect on Yregression to estimate its effect on Y ii

yyii = = 00 + + 11 X-hat X-hatii + + ii

Note: software packages like Stata perform the two stages in Note: software packages like Stata perform the two stages in a a

single regression, producing the correct standard errorssingle regression, producing the correct standard errors

IV: exampleIV: example

Levitt (1997): what is the effect of increasing the police forceLevitt (1997): what is the effect of increasing the police force

on the crime rate?on the crime rate?

This is a classic case of simultaneous causality (high crime This is a classic case of simultaneous causality (high crime areas areas

tend to need large police forces) resulting in an incorrectly-tend to need large police forces) resulting in an incorrectly-

signed (positive) coefficientsigned (positive) coefficient

To address this problem, Levitt uses the timing of mayoral To address this problem, Levitt uses the timing of mayoral and and

gubernatorial elections as an instrumental variablegubernatorial elections as an instrumental variable

Is this instrument valid?Is this instrument valid?

Relevance: police force increases in election yearsRelevance: police force increases in election years

Exogeneity: election cycles are pre-determinedExogeneity: election cycles are pre-determined

IV: exampleIV: example

Two-stage least squares:Two-stage least squares:

Stage 1: Decompose police hires into the component that can Stage 1: Decompose police hires into the component that can be predicted by the electoral cycle and the be predicted by the electoral cycle and the

problematic problematic componentcomponent

policepoliceii = = 00 + + 11 election electionii + + ii

Stage 2: Use the predicted value of policeStage 2: Use the predicted value of police ii from the first-stage from the first-stage regression to estimate its effect on crimeregression to estimate its effect on crime ii

crimecrimeii = = 00 + + 11 police-hat police-hatii + + ii

Finding:Finding: an increased police force reduces violent crime an increased police force reduces violent crime (but has little effect on property crime)(but has little effect on property crime)

IV: number of instrumentsIV: number of instruments

There must be at least as many instruments as There must be at least as many instruments as endogenous endogenous

regressorsregressors

Let k = number of endogenous regressorsLet k = number of endogenous regressors

m = number of instrumentsm = number of instruments

The regression coefficients areThe regression coefficients are

exactly identified if m=k exactly identified if m=k (OK)(OK)

overidentified if m>k overidentified if m>k (OK) (OK)

underidentified if m<k underidentified if m<k (not OK)(not OK)

IV: testing instrument relevanceIV: testing instrument relevance

How do we know if our instruments are valid? How do we know if our instruments are valid?

Recall our first condition for a valid instrument: Recall our first condition for a valid instrument:

1. Relevance: corr (Z1. Relevance: corr (Z i i , X, Xii) ≠ 0) ≠ 0

Stock and Watson’s rule of thumbStock and Watson’s rule of thumb: the first-stage F-statistic : the first-stage F-statistic

testing the hypothesis that the coefficients on the instruments testing the hypothesis that the coefficients on the instruments

are jointly zero should be at least 10 (for a single endogenous are jointly zero should be at least 10 (for a single endogenous

regressor)regressor)

A small F-statistic means the instruments are “weak” (they A small F-statistic means the instruments are “weak” (they

explain little of the variation in X) and the estimator is biased explain little of the variation in X) and the estimator is biased

IV: testing instrument exogeneityIV: testing instrument exogeneity

Recall our second condition for a valid instrument:Recall our second condition for a valid instrument:

2. Exogeneity: corr (Z2. Exogeneity: corr (Zi i , e, eii) = 0) = 0

If you have the same number of instruments and endogenous If you have the same number of instruments and endogenous

regressors, it is impossible to test for instrument exogeneityregressors, it is impossible to test for instrument exogeneity

But if you have more instruments than regressors:But if you have more instruments than regressors:

Overidentifying restrictions testOveridentifying restrictions test – regress the residuals from – regress the residuals from

the 2SLS regression on the instruments (and any exogenous the 2SLS regression on the instruments (and any exogenous

control variables) and test whether the coefficients on the control variables) and test whether the coefficients on the

instruments are all zeroinstruments are all zero

IV: drawbacksIV: drawbacks

It can be difficult to find an instrument that is both It can be difficult to find an instrument that is both relevant relevant

(not weak) and exogenous(not weak) and exogenous

Assessment of instrument exogeneity can be highly Assessment of instrument exogeneity can be highly subjectivesubjective

when the coefficients are exactly identifiedwhen the coefficients are exactly identified

IV can be difficult to explain to those who are unfamiliar IV can be difficult to explain to those who are unfamiliar with itwith it

SourcesSources

Stock and Watson, Stock and Watson, Introduction to EconometricsIntroduction to Econometrics

Bertrand, Duflo, and Mullainathan, “How Much Should We Trust Bertrand, Duflo, and Mullainathan, “How Much Should We Trust Differences-in-Differences Estimates?” Differences-in-Differences Estimates?” Quarterly Journal of EconomicsQuarterly Journal of Economics February 2004February 2004

Card and Krueger, "Minimum Wages and Employment: A Case Study of Card and Krueger, "Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania," the Fast Food Industry in New Jersey and Pennsylvania," American American Economic ReviewEconomic Review, September 1994, September 1994

Angrist and Krueger, “Instrumental Variables and the Search for Angrist and Krueger, “Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments,”Identification: From Supply and Demand to Natural Experiments,”Journal of Economic PerspectivesJournal of Economic Perspectives, Fall 2001, Fall 2001

Levitt, “Using Electoral Cycles in Police Hiring to Estimate the Effect ofLevitt, “Using Electoral Cycles in Police Hiring to Estimate the Effect ofPolice on Crme,” Police on Crme,” American Economic ReviewAmerican Economic Review, June 1997, June 1997