1
Synthetic controls
Suppose one has panel data but very limited evidence on the treated and/or
untreated subpopulations (for example, perhaps only one individual is treated).
Then, standard matching strategies (general or propensity score) are likely of
limited utility. On the other hand, perhaps one can construct a match for the
treated based on a composite of the controls (untreated). This is an extension
of dierence-in-dierences (D-I-D) in which unobservable factors (perhaps correlated with covariates, Xit ) are allowed to vary cross-sectionally whereas such
unobservables are a cross-sectional constant for the D-I-D strategy. This is
commonly referred to as a factor model since the unobservables can be thought
of as factor scores, t , (which vary through time) weighted by factor loadings,
µi , (which vary cross-sectionally) both of which are unobservable in a factor
analysis.
Yit = dit +  T Xit + t µi + it
where Yit is outcome for individual i during period t,  is an unknown vector
of coecients common across individuals, it is unobservable (random) noise,
dit is an indicator of treatment (1) or nontreatment (0), and  is the treatment
eect of interest.
Synthetic controls are constructed based on a convex combination (with
T0


weights wi  0,
wi = 1) of the pre-intervention outcomes, Yik =
kt Yit ,
t=1
and covariates, Xi , from the control subpopulation such that
T
arg min (H1  [H2 · · · Hn ] W ) V (H1  [H2 · · · Hn ] W )
W

T
where Hi = Yik1 , Yik2 , . . . Yikm , XiT , i = 1 denotes treated while i = 2, . . . , n
denotes control, and V is a diagonal matrix controlling potentially dierent
weights applied to the components of Hi . Abadie et al [2010, 2011] propose
choosing V to minimize the expected mean square error
T
arg min (Y1  Y0 W  (V )) (Y1  Y0 W  (V ))
V
during the pre-intervention period where Y1 refers to outcomes for the treated
and Y0 outcomes for the control group.
Next, we explore several variations via some highly stylized examples. First,
we visit examples with constant factors such that both D-I-D and synthetic
controls identify the treatment eect. Then, we visit settings with heterogeneous factors in which D-I-D fails but synthetic controls eectively identify the
treatment eect. Finally, we consider settings where neither D-I-D nor synthetic variables eectively identify the treatment eect. The first failure is due
to violation of conditional mean independence of potential outcome without
treatment while the second failure is due to the treated individuals’ H lying
outside the convex hull of the controls. In the latter case, synthetic controls can
produce greater selection bias than D-I-D.
1
1.1
Examples
Suppose there are three individuals who are untreated during the first two periods but the third individual adopts treatment in the third period. The data
generating process (DGP ) is
Y = I1 + I2 + I3 + T + 2I3  T + µ
where Y is observed outcome, Ij denotes and indicator variable equal to 1 for
individual j and 0 otherwise, and T is an indicator equal to one for intervention
period three and zero otherwise. In the examples below, Y0 represents potential
outcome without treatment and only µ (and, of course, outcome) varies.
Example 1 (D-I-D setting) Suppose the DGP is
Y0
2
2
3
2
2
3
2
2
3
Y
2
2
3
2
2
3
2
2
5
I1
1
1
1
0
0
0
0
0
0
I2
0
0
0
1
1
1
0
0
0
I3
0
0
0
0
0
0
1
1
1
T
0
0
1
0
0
1
0
0
1
d = I3  T
0
0
0
0
0
0
0
0
1
µ
1
1
1
1
1
1
1
1
1
 = 5  3 = 2. D-I-D identifies
E [Y | X] = 2I1 + 2I2 + 2I3 + T + 2I3  T
That is, even though the nonzero error term biases the coecients on the individual mean eects, the treatment eect is eectively identified. Any weights V
and W produce synthetic controls that eectively identify the treatment eect.
Example 2 (D-I-D setting with time variation) Suppose the DGP is
Y0
2
2
4
2
2
4
2
2
4
Y
2
2
4
2
2
4
2
2
6
I1
1
1
1
0
0
0
0
0
0
I2
0
0
0
1
1
1
0
0
0
I3
0
0
0
0
0
0
1
1
1
T
0
0
1
0
0
1
0
0
1
d = I3  T
0
0
0
0
0
0
0
0
1
µ
3
3
2
3
3
2
3
3
2
 = 6  4 = 2. D-I-D identifies
E [Y | X] = 2I1  2I2  2I3 + 6T + 2I3  T
2
That is, even though the nonzero error term biases the coecients on the individual and time mean eects, the treatment eect is eectively identified. Again,
any weights V and W produce synthetic controls that eectively identify the
treatment eect.


Example 3 (synthetic control setting W T = 0.1 0.9 ) Suppose the DGP
is
Y0
Y
I1 I2 I3 T d = I3  T
µ
2 2 1
0
0
0
0
3
2 2 1
0
0
0
0
3
4
4
1
0
0
1
0
2
4
4
0
1
0
0
0
3
4
4
0
1
0
0
0
3
0
0
0
1
0
1
0
2
3.4 3.4 0
0
1
0
0
2.4
3.4 3.4 0
0
1
0
0
2.4
0.4 2.4 0
0
1
1
1
1.6
 = 2.4  0.4 = 2. D-I-D identifies
1
1
2
E [Y | X] =  I1 + 2 I2 + 3 I3 + T  2I3  T
3
3
5
Cross-sectional variation in the unobservable factors, µ, produces substantial
bias in the D-I-D estimate of the treatment eect. That is, it is so underestimated the sign is reversed. On the other hand, since the test subject lies in
a convex hull of the control subjects’ predicted outcomes, the synthetic control
strategy eectively identifies the treatment eect.


I1
 I2 

V
 Yit=1 
Y t=2
i
WT
 = Y33  W T
where Yit=1 =

2

t=1
1 Identification
Y13
Y23
V fixed

0 0 0
0
 0 0 0
0 


 0 0 0.5 0 
0 0 0 0.5


0.1 0.9



0.335
 0

 0
0
2
kt Yit = Yi1 and Yit=2 =
is also eective with Yit=1 =
2

V estimated
0
0
0
0.037
0
0
0
0.332
0
0
0
0.296


0.1 0.9
2
kt Yit = Yi2 .1
t=1
2

kt Yit =
1
2
(Yi1 + Yi2 ) (or, for that matter,
t=1
any convex combination of Yi1 and Yi2 ) in place of both Yit=1 and Yit=2 .
3






Example 4 (synthetic control setting W T = 0.9 0.1 ) Suppose the DGP
is
Y0
Y
I1 I2 I3 T d = I3  T
µ
2
2
1
0
0
0
0
3
2
2
1
0
0
0
0
3
4
4
1
0
0
1
0
2
4
4
0
1
0
0
0
3
4
4
0
1
0
0
0
3
0
0
0
1
0
1
0
2
1.4 1.4 0
0
1
0
0
2.4
1.4 1.4 0
0
1
0
0
2.4
3.6
5.6
0
0
1
1
1
1.6
 = 5.6  3.6 = 2. D-I-D identifies
1
1
2
E [Y | X] = 1 I1 + 1 I2  2 I3 + T + 6I3  T
3
3
5
Cross-sectional variation in the unobservable factors, µ, produces substantial
(overestimation) bias in the D-I-D estimate of the treatment eect. On the other
hand, since the test subject lies in a convex hull of the control subjects’ predicted
outcomes, the synthetic control strategy eectively identifies the treatment eect.


I1
 I2 

V
 Yit=1 
Y t=2

0
 0

 0
0
i
WT
 = Y33  W
where Yit=1 =
T

2

t=1
Y13
Y23

V fixed

0 0
0
0 0
0 

0 0.5 0 
0 0 0.5


0.9 0.1

0.037
 0

 0
0
2
kt Yit = Yi1 and Yit=2 =
2

V estimated
0
0
0
0.335
0
0
0
0.332
0
0
0
0.296


0.9 0.1
2
kt Yit = Yi2 .
t=1
Example 5 (both d-i-d and synthetic controls fail) Suppose the DGP is
almost identical to example 1 except that potential outcome without treatment is
aected by the intervention (not conditionally mean independent).
Y0
2
2
3
2
2
3
2
2
5
Y
2
2
3
2
2
3
2
2
5
I1
1
1
1
0
0
0
0
0
0
I2
0
0
0
1
1
1
0
0
0
I3
0
0
0
0
0
0
1
1
1
T
0
0
1
0
0
1
0
0
1
4
d = I3  T
0
0
0
0
0
0
0
0
1
µ
1
1
1
1
1
1
1
1
1




 = 5  5 = 0. As before, D-I-D identifies
E [Y | X] = 2I1 + 2I2 + 2I3 + T + 2I3  T
but the treatment eect is not 2 but rather zero. Further, since mean conditional
independence fails, synthetic controls cannot eectively identify the treatment
eect either (also identifies 2 rather than zero).
Example 6 (treated lie outside convex hull of controls) Suppose the DGP
is
as example 3 for individuals 1 and 2 but the weights are W T =
 the same

1 1 (not a convex combination).
Y0
2
2
4
4
4
0
2
2
4
Y
2
2
4
4
4
0
2
2
6
I1
1
1
1
0
0
0
0
0
0
I2
0
0
0
1
1
1
0
0
0
I3
0
0
0
0
0
0
1
1
1
T
0
0
1
0
0
1
0
0
1
d = I3  T
0
0
0
0
0
0
0
0
1
µ
3
3
2
3
3
2
0
0
0
 = 6  4 = 2. D-I-D identifies
1
1
E [Y | X] =  I1 + 2 I2 + 2I3 + T + 3I3  T
3
3
but the treatment eect is 2 not 3. Since the test subject lies outside the convex
hull of the control subjects’ predicted outcomes, the synthetic control strategy
ineectively identifies the treatment eect.


I1
V  I2 
Yi
WT
 = Y33  W T
where Yi =
2


Y13
Y23
kt Yit =

1
2
 1V fixed 
0 0
3
 0 1 0 
3
0 0 13

 
0.435 0.565 = 1
V estimated

0.44
0
0
 0
0.22
0 
0
0
0.34

 1 2  
= 1 1
3
3

1
4.26 = 2

4 32 = 2
(Yi1 + Yi2 ). Hence, the bias in the treatment eect is
t=1
greater with synthetic controls than the bias for D-I-D.
5