1. Econometric Approaches to Causal Inference: Difference-in-Differences and Instrumental Variables
Graduate Methods Master Class
Department of Government, Harvard University
February 25, 2005

2. 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

3. 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

4. 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 )

5. 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

6. 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 + β β0 β1
After β0 + β1 + β2 + β β0 + β β1 + β3
Difference β2 + β β2 β3

7. 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

8. Diff-in-diffs: example
FTEi = β0 + β1 NJi + β2 Nov92i + β3 NJi*Nov92i + e
FTE
Control group (PA)
21.17
Treatment group (NJ)
Time
Treatment effect of minimum wage increase = FTE

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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)

17. 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)

18. 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

19. 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

20. 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

21. 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