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Econometric Approaches to Causal Inference: Difference-in-Differences and Instrumental VariablesPowerPoint Presentation

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

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

Graduate Methods Master Class

Department of Government, Harvard University

February 25, 2005

Overview: diff-in-diffs and IV

Data Randomized experiment Observational data

or natural experiment

Problem We cannot observe the OVB, selection bias,

counterfactual (what if simultaneous causality

treatment group had not

received treatment)

Method Difference-in-differences Instrumental variables

Diff-in-diffs: basic idea

Suppose we randomly assign treatment to some units

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

To estimate the treatment effect, we could just compare the

treated units before and after treatment

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

changed around the time of treatment

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

confounding factors and isolate the treatment effect

Diff-in-diffs: without regression

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

outcome before and after treatment

Treatment group Control group

Before TB CB

After TA CA

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

Treatment effect = (TA -TB ) -(CA -CB )

Diff-in-diffs: with regression

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

allows us to add regression controls, if needed):

yi = β0 + β1 treati +β2 afteri + β3 treati*afteri + ei

where treat = 1 if in treatment group, = 0 if in control group

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

The coefficient on the interaction term (β3 )gives us the

difference-in-differences estimate of the treatment effect

Diff-in-diffs: with regression

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

yi = β0 + β1 treati +β2 afteri + β3 treati*afteri + ei

Treatment Control

Group Group Difference

Before β0 + β1β0β1

After β0 + β1 + β2 + β3β0 + β2β1 + β3

Difference β2 + β3β2 β3

Diff-in-diffs: example

Card and Krueger (1994)

What is the effect of increasing the minimum wage on

employment at fast food restaurants?

Confounding factor: national recession

Treatment group = NJ Before = Feb 92

Control group = PA After = Nov 92

FTEi = β0 + β1 NJi +β2 Nov92i +β3 NJi*Nov92i + ei

Diff-in-diffs: example

FTEi = β0 + β1 NJi +β2 Nov92i +β3 NJi*Nov92i + e

23.33 -2.89 -2.16 2.75

FTE

23.33 Control group (PA)

21.17

20.44 Treatment group (NJ) 21.03

Time

Treatment effect of minimum wage increase = + 2.75 FTE

Diff-in-diff-in-diffs

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

to study the effect of treatment on different groups

If we are concerned that our estimated treatment effect might

be spurious, a common robustness test is to introduce a

comparison group that should not be affected by the treatment

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

affected labor force participation, we can use a DD model

that takes advantage of policy variation across states, and then

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

versus married women

Diff-in-diffs: drawbacks

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

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

to data from natural experiments, raising questions about

whether treatment is truly random

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

data but ignore the resulting inconsistency of standard errors

(see Bertrand, Duflo, and Mullainathan 2004)

IV: basic idea

Suppose we want to estimate a treatment effect using

observational data

The OLS estimator is biased and inconsistent (due to correlation

between regressor and error term) if there is

- omitted variable bias
- selection bias
- simultaneous causality
If a direct solution (e.g. including the omitted variable) is not

available, instrumental variables regression offers an alternative

way to obtain a consistent estimator

IV: basic idea

Consider the following regression model:

yi = β0 + β1 Xi+ ei

Variation in the endogenous regressor Xi has two parts

- the part that is uncorrelated with the error (“good” variation)
- the part that is correlated with the error (“bad” variation)
The basic idea behind instrumental variables regression is to

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

IV: conditions for a valid instrument

The first step is to identify a valid instrument

A variable Zi is a valid instrument for the endogenous regressor

Xi if it satisfies two conditions:

1. Relevance: corr (Zi , Xi) ≠ 0

2. Exogeneity: corr (Zi , ei) = 0

IV: two-stage least squares

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

Stage 1: Decompose Xi into the component that can be

predicted by Zi and the problematic component

Xi = 0 + 1 Zi+ i

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

regression to estimate its effect on Yi

yi = 0 + 1 X-hati+ i

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

single regression, producing the correct standard errors

IV: example

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

on the crime rate?

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

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

signed (positive) coefficient

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

gubernatorial elections as an instrumental variable

Is this instrument valid?

Relevance: police force increases in election years

Exogeneity: election cycles are pre-determined

IV: example

Two-stage least squares:

Stage 1: Decompose police hires into the component that can

be predicted by the electoral cycle and the problematic

component

policei = 0 + 1 electioni+ i

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

regression to estimate its effect on crimei

crimei = 0 + 1 police-hati+ i

Finding: an increased police force reduces violent crime

(but has little effect on property crime)

IV: number of instruments

There must be at least as many instruments as endogenous

regressors

Let k = number of endogenous regressors

m = number of instruments

The regression coefficients are

exactly identified if m=k (OK)

overidentified if m>k (OK)

underidentified if m<k (not OK)

IV: testing instrument relevance

How do we know if our instruments are valid?

Recall our first condition for a valid instrument:

1. Relevance: corr (Zi , Xi) ≠ 0

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

testing the hypothesis that the coefficients on the instruments

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

regressor)

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

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

IV: testing instrument exogeneity

Recall our second condition for a valid instrument:

2. Exogeneity: corr (Zi , ei) = 0

If you have the same number of instruments and endogenous

regressors, it is impossible to test for instrument exogeneity

But if you have more instruments than regressors:

Overidentifying restrictions test – regress the residuals from

the 2SLS regression on the instruments (and any exogenous

control variables) and test whether the coefficients on the

instruments are all zero

IV: drawbacks

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

(not weak) and exogenous

Assessment of instrument exogeneity can be highly subjective

when the coefficients are exactly identified

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

Sources

Stock and Watson, Introduction to Econometrics

Bertrand, Duflo, and Mullainathan, “How Much Should We Trust

Differences-in-Differences Estimates?” Quarterly Journal of Economics

February 2004

Card and Krueger, "Minimum Wages and Employment: A Case Study of

the Fast Food Industry in New Jersey and Pennsylvania," American

Economic Review, September 1994

Angrist and Krueger, “Instrumental Variables and the Search for

Identification: From Supply and Demand to Natural Experiments,”

Journal of Economic Perspectives, Fall 2001

Levitt, “Using Electoral Cycles in Police Hiring to Estimate the Effect of

Police on Crme,” American Economic Review, June 1997

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