RECITATION 2
FIXED
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RECRE ION
DI CONTINUITY
RECITATION 2
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2
RECITATION 2
FIXED
EFFECT
RECRE ION
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UC Berkeley gender case.
Berkeley sued for bias against women in 1973.
Evidence:
44% men admitted
35% women admitted
Convincing?
SIMPSON'S
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RECITATION 2
FIXED
EFFECT
RECRE ION
DI CONTINUITY
Breakdown by department:
Dept
A
B
C
D
E
F
Male Female
App Admit App Admit
825 62% 108 82%
560 63% 25 68%
325 37% 593 34%
417 33% 375 35%
191 28% 393 24%
272 6% 341 7%
SIMPSON'S
3

RECITATION 2
FIXED
EFFECT
RECRE ION
DI CONTINUITY
Breakdown by department:
Dept
A
B
C
D
E
F
Male Female
App Admit App Admit
825 62% 108 82%
560 63% 25 68%
325 37% 593 34%
417 33% 375 35%
191 28% 393 24%
272 6% 341 7%
SIMPSON'S
4

RECITATION 2
FIXED
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RECRE ION
DI CONTINUITY
FIXED EFFECTS
JORE JOTIVATION
5

RECITATION 2
FIXED EFFECTS
FIXED
EFFECT
RECRE ION
DI CONTINUITY
Bottom line, we might care about within variation.
We have a panel data set, which consists of n individuals observed for
T periods.
y it
= α i
+ γ t
+ x it
β
0
+ it
.
Interpretations:
What is
α i
?
What is
γ t
?
What is
β
0
?
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RECITATION 2
FIXED EFFECTS
FIXED
EFFECT
RECRE ION
DI CONTINUITY y it
= α i
+ x it
β
0
+ it
.
Let us take an average.
n
− 1 y it
= n
− 1
α i
+ n
− 1 x it t t t
� � � � � � � � � y i
:=??
i
β
0
+ n
− 1
� � t
:=¯ i it
�
.
s o we have
¯ i
= α i x i
β
0
¯ i
.
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RECITATION 2
FIXED EFFECTS
FIXED
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RECRE ION
DI CONTINUITY
Target equation: y it
= α i
+ x it
β
0
+ it
.
Average equation:
¯ i
= α i
+ ¯ i
β
0
+ ¯ i
.
Let us subtract the second from the frst: y it
− y i
= α i
− α i
+ x it
β
0
− x i
β
0
+ it
− ¯ i
.
� {z } interp ??
| {z
=??
} | {z interp ??
� | {z � interp ??
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RECITATION 2
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RECRE ION
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FIXED
Bottom line: y it
− y i
= ( x it
− x i
) β
0
+ ( it
− ¯ i
) .
EFFECTS
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RECITATION 2
REMINDER - POTENTIAL OUTCOMES
FIXED
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RECRE ION
DI CONTINUITY
Two universes:
Y i
(0)
: outcome for person i without treatment
Y i
(1)
: outcome for person i with treatment
W hat do we want to study?
Y i
(1) − Y i
(0) .
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RECITATION 2
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POTENTIAL OUTCOMES
Let W i
∈ { 0 , 1 } be whether the treatment was received.
What is the observed outcome?
Y i
= (1 − W i
) · Y i
(0) + W i
· Y i
(1) .
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RECITATION 2
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COVARIATES
We also have access to variables
X i been afected by the treatment.
and
Z i which have not
In particular,
X i will be important for the RD design.
We observe:
( Y i
, W i
, X i
, Z i
) .
Basic idea is that the assignment to the treatment is going to be determined fully or partially by the value of a predictor
(the covariate
X i
).
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RECITATION 2
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SHARP RD
We call
X the forcing variable or the treatment-determining variable:
W i
= 1 { X i
≥ c } .
Interpret
τ
SRD
= E [ Y i
(1) − Y i
(0) | X i
= c ]
We can estimate lim x ↓ c
E [ Y i
| X i
= x ] − lim x ↑ c
E [ Y i
| X i
= x ] .
13
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RECITATION 2
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RECRE ION
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SHARP RD - ASSUMPTIONS
(1)
Y i
(0) , Y i
(1) ⊥ W i
| X i
- does it hold?
Trivially, but
X fully determines
W and there is no variation.
(2) Continuity of conditional expectations in x
E [ Y (0) | X = x ] and
E [ Y (1) | X = x ] .
Notice, this method requires extrapolation.
Why?
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RECITATION 2
FIXED
EFFECT
RECRE ION
DI CONTINUITY
SHARP RD - ASSUMPTIONS
Observe that lim x ↓ c
E [ Y (0) | X = x ] = E [ Y (0) | X = c ] = E [ Y (1) | X = x ] = lim x ↑ c
E [ Y (1) | X = x ]
Therefore the average treatment efect at c
,
τ
SRD
, is
τ
SRD
= lim x ↓ c
E [ Y | X = x ] − lim x ↑ c
E [ Y | X = x ] .
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RECITATION 2
FIXED
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FUzzy RD
Imagine the probability of treatment changes as we cross c
.
Examples?
Idea:
Variation in
X = ⇒ variation in
W = ⇒ variation in
Y
.
Looks familiar?
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RECITATION 2
FIXED
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RECRE ION
DI CONTINUITY
FUzzy RD
We then estimate
τ
F RD
= lim x ↓ c lim x ↓ c
E [ Y | X = x ] − lim x ↑ c
E [ W | X = x ] − lim x ↑ c
E [ Y | X = x ]
E [ W | X = x ]
Assumptions:
W i
( x ) is non-increasing in x at x = c
.
Compliers:
Nevertakers: lim W i x ↓ X i
( x ) = 0 and lim W i x ↑ X i
( x ) = 1 lim W i x ↓ X i
( x ) = 0 and lim W i x ↑ X i
( x ) = 0
Alwaystakers: lim W i x ↓ X i
( x ) = 1 and lim W i x ↑ X i
( x ) = 1
Then
τ
F RD
= E [ Y i
(1) − Y i
17
(0) | i is a complier and
X i
= c ] .

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14.
7 5 Political Economy and Economic Development
Fall 201 2
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