# Differences in Differences and Instrumental Variables

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

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

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 + β1             β0            β1

After          β0 + β1 + β2 + β3     β0 + β2       β1 + β3

Difference         β2 + β3             β2            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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