1.
Difference-in-Difference
Methods
Day 2, Lecture 1
By Ragui Assaad
Training on Applied Micro-Econometrics
and Public Policy Evaluation
July 25-27, 2016
Economic Research Forum

2.
Type I Methods Identification
Assumptions
“conditional exogeneity of placement”
or
“conditional exogeneity of placement to changes in outcomes”
• “Conditional exogeneity of placement” was discussed in the context
of regular regression and propensity score matching and weighting
models.
• It simply means that selection into treatment is assumed not to depends
on any unobservables that affect outcomes
• “conditional exogeneity of placement to changes in outcomes” is a
much weaker assumption.
• Here we distinguish between two kinds of unobservables, time invariant
unobservables and time varying unobservables.
• Under this assumption, selection can depend on time invariant
unobservables, but is assumed not to depend on time-varying
unobservables

3.
First Difference and Difference-in
Difference Approaches
• Under the stronger “conditional exogeneity of
placement”, all we needed to do is compare
outcomes for a treatment and control group at one
point in time controlling for observables X
– This is called a first difference approach (see D(X) estimator)
• Under the weaker “exogeneity of placement to
changes in outcomes” we need to compare the
difference from before and after the program for a
treatment group to the same difference for a control
group.
– This is called difference-in-difference or second difference
approach

4.
The Identification Assumption
Graphically

5.
The Difference-in-Difference Estimator
DD(X ) = E(Y1
T
-Y0
T
| X ,T1
=1)- E(Y1
C
-Y0
C
| X,T1
= 0)
If identifying assumption is met, then
DD(X) = E(G1
| X ,T1
=1) = ATT(X )
If the counterfactual expectations are not expected to
change over time, then the term on the right is zero
E(Y1
C
-Y0
C
| X,T1
=1) = 0
Under such an assumption, the DiD estimator
reduces to the Before and After (BA) estimator on
the treated group, making measurement on a
comparison group unnecessary
BA(X ) = E(Y1
T
-Y0
T
| X,T1
=1)
Such an assumption is unlikely to hold in practice.. Gives

6.
Data Structures for DiD Estimation
When we have multiple observations on the same units, we
effectively have panel data
• Panel data can come in a number of different shapes
• Let’s talk about shapes with a person-time period
structure (say, before and after)
• “Long” data: an individual shows up twice, once in time period 0
and once in time period 1. An observation is an individual in a time
period
• “Wide” data: an observation is a person, and there are different
variables for each time period

7.
Notation for data shapes
(in STATA)
• i is a unique ‘logical observation’ (like a person, who may
then be observed over time)
• j is a ‘sub-observation’ (like a time period—people are
observed in time periods)
• x_ij are variables that can change over i and j (people and
time), like employment status
• Some variables are ‘time-varying,’ and can change over
time (i.e. income or employment status)
• Some variables are ‘time invariant’ and cannot change
over time (i.e. sex)
• Data entry errors or other issues may make time invariant variables
time-varying in actual data
7

8.
Long Data: An Example
• An observation is a
person (id=i) in a
year(=j)
• A person has multiple
observations (multiple
years)
• inc (income) is time
varying (x_ij)
• Sex is time invariant
8

9.
Wide data: an example
• An observation is a
person (id=i)
• A person has only one
observation
• There is still only one
variable for sex (time
invariant)
• Now there are three
variables for income
• One for each year
• There is no more year
variable—it is a suffix
on income
9

10.
Reshaping Data from Long to Wide
and vice versa
• You can use the “reshape” command in STATA
to reshape data from long to wide and vice
versa.
• However before you do that, you need to know
what form your data is in.
• The “duplicate report” command on the unique
individual identifier can allow you to do that.

11.
Estimating the DD Estimator in Practice:
The Regression Approach
• Shape the data up as “long”, with each observation being an individual i time
period t
• Run a regression of the outcome (Yit) on the treatment status (Ti1), a time
dummy (t) (t=0, before, t=1, after) (Ti0 = 0, by definition)
Yit
=a +bTi1
t +gTi1
+dt +eit
Calculate the following expectations:
E(Yi0
|Ti1
= 0) =a
E(Yi1
|Ti1
= 0) =a +d
E(Yi0
|Ti1
=1) =a +g
Note: We can ignore the X’s since this regression does not et have
covariates. I also dropped the T and C superscripts since we only
observed treatment for treated observations and non-treatment for
comparison observations
E(Yi1
|Ti1
=1) =a +b +g +d

12.
SD1 = E(Yi1
|Ti1
=1)- E(Yi1
|Ti1
= 0) = (a +b +g +d)-(a +d) = b +g
SD0
= E(Yi0
|Ti1
=1)- E(Yi0
|Ti1
= 0) = (a +g)-(a) =g
Estimating the DD Estimator in Practice:
The Regression Approach
The Single Difference Estimator after Treatment is given by:
The Single Difference Estimator before Treatment is given by:
You can think of this estimator as measuring the effect of
selection – the difference between treatment and
comparisons observations before any treatment is applied.

13.
• The Difference-in-Difference Estimator or the Double
Difference estimators is given by:
Estimating the DD Estimator in Practice:
The Regression Approach
DD = SD1
-SD0
= b
Thus, under the weaker Type I identification assumptions,
the effect of the treatment on the treated (ATT) is given by
the regression coefficient β, which is the coefficient of the
interaction term between the treatment dummy and the
time dummy.
SD1 is an unbiased measured of ATT if and only if ϒ=0,
which means there is no selection.

14.
Introducing Covariates
• It is easy to introduce covariates in DiD estimation
Yit
=a +bTi1
t +gTi1
+dt + qj
Xitj
j
å +eit
All the expectations can now be written given X.
ATT is still equal to β.
To obtain heterogeneous effects, all you ned to
do is interact β & ϒ with the Xs.

15.
An Example: Wahba and Assaad (2014). Flexible
Labor Regulations and Informality in Egypt
• Studies the effects of the 2003/04 labor law on informality
of employment in Egypt
• Research Design
• Two observation periods
• 1998-2002 pre-intervention P=0 (P=post)
• 2004-2008 post-intervention P=1
• Two groups of private non-agricultural wage workers who would be
differentially affected are observed over a 5-yer period, either pre-
or post-intervention
• Workers initially non-contracted working for formal/semiformal
enterprises (F) (T=1)
• Workers initially non-contracted working in informal enterprises (I) (T=0)
• Contention is that second group of workers unlikely to be affected by new
labor law

16.
Wahba and Assaad (2014)
• Estimation Equation
Where:
Cit is a dummy indicating the contract status of the worker
Tit is a dummy indicating belonging to F (versus i)
Pit is a dummy indicating post
Xit is a vector of observables including individual characteristics,
such as gender, education and region, GDP growth rates, annual
unemployment rates, a time trend variable capturing the time
since the job started
We posit that δ captures he effect of the law.
Cit
=a +bTit
+g Pit
+dTit
Pit
+jXit
+eit

17.
Difference-in-Difference Estimates of the Determinants of
Acquiring a Job Contract
α α+β β α+ϒ α+β+ϒ+δ β+δ δ

18.
Falsification Tests
• The identifying assumption for DiD means that the trend
in the outcome variable between treated and control
observations (conditional on observables) should not be
different absent treatment.
• To check on this, one can check whether the trend is the
same in the pre-treatment period.
• To do this, you will need two points in time prior to treatment
begins.
• The same regression is carried out for that pre-treatment period.
The coefficient β needs to be statistically insignificant in this case to
pass the falsification test.

19.
Cross-Sectional Difference-in-
Difference
• DiD estimators do not have to involve measurements
across time.
• Can be done across groups of individuals, with one group
not expected to be affected by treatment.
• E.g.
• Target and non-target villages for micro-credit programs
• Participating and non-participating households
• Non-targeted villages are not expected to be affected, so
difference between participating and non-participating HH
in such villages could be a measure of selection into
participation
• DiD: difference between participants and non-participants in
targeted villages, minus the same difference in non-targeted
villages

20.
Difference-in-Difference and PSM
• One can use DiD in the context of PSM as well
• Matching is still done along pre-treatment characteristics
ATT =(1/NT
) [(Y1 j
T
-Y0 j
t
)
j=1
NT
å - wij
iÎC
å (Y1ij
C
-Y0ij
C
)]
Using “wide” data, calculate Y1j - Y0j at the individual
level and then carry out PSM on that outcome
variable

21.
Wahba and Assaad (2014)
PSM Robustness check:
• Treatment: (as before) is belonging to a formal
firm or informal firm at the beginning of the perio
T=1 in formal firm
T=0 in informal firm
• Outcome: whether individual with no contract in
2002-2004 moved to a contract job in 2004-2008.
Y=1 obtained contract
Y=0 did not obtain contract
• Falsification test
• Do it again for period 1996-98 to 1998-2002

22.
Wahba and Assaad (2014)
Coefficient
(DiD)
Bootstrapped Std Err
Post Policy
(2004-2008)
0.0330** 0.0174
Pre Policy
(1998-2002)
0.0043 0.0145
Covariates used in matching:
Gender, Age, Education, Region of Residence
Conclusion:
• The proportion who moved from no contract to a
contracts in Formal Firms exceeds that of in Informal
Firms by 3.3 percentage points
• No such difference can be seen in the pre-policy period