IPSS Selection/ Switching
Review of Selection/Switching Process
Treatment Effects
z = 0,1 Treatment 0,1
(0: non-college 1: college)
Observe y=y1z+y0(1z), P(y1|x, z1), P(y0|x, z0), P(z0|x), P(z1|x)
censored
Treatment: z
Outcome: y
P(y0∈B|x,z0)
z=0
Selection
P(y1∈B | x,z0)
P(y0∈B | x,z1)
z=1
P(y1∈B | x,z1)
1
IPSS Selection/ Switching
Effect of Treatment 1 (improvement over 0):
T(B|x) = P(y1∈B|x) – P(y0∈B|x)
B: “success”
P(y1∈B|x) = P(y1∈B|x,z1) P(z1|x) + P(y1∈B|x,z0) P(z0|x)
-) P(y0∈B|x) = P(y0∈B|x,z1) P(z1|x) + P(y0∈B|x,z0) P(z0|x)
-------------------------------------------------------------------------------P(y1∈B|x,z1) P(z1|x) - P(y0∈B|x,z0) P(z0|x)
+ P(y1∈B|x,z0) P(z0|x) - P(y0∈B|x,z1) P(z1|x)
Width of the interval
P(y1∈B|x,z0) P(z0|x) P(y0∈B|x,z1) P(z1|x)
Max P(z0|x) Min : P(z1|x) => width = P(z0|x)+P(z1|x) = 1
2
IPSS Selection/ Switching
Example : Wage/ Labor Market Participation
z=1 Work z=0 Non-Work
We want to know
y: Market Wage
P(y1|x) = P(y1|x,z0) P(z0|x)
+ P(y1|x,z1) P(z1|x)
Self-selection: If y1> R(x) then z =1, else z=0.
Treatment: z
Outcome: y
P(y0∈B|x,z0)
z=0
P(y1∈B | x,z0)
Selection
P(y0∈B | x,z1)
z=1
P(y1∈B | x,z1)
3
IPSS Selection/ Switching
Assumption: Treatment Independent of Outcomes?
TIO: P(y1|x,z0)= P(y1|x,z1)
No way! It can’t be true.
z=1 work
z=0 non-work
wage
Reservation
wage
4
IPSS Selection/ Switching
Example： Ordered Outcomes (New Drug)
z= 0 Placebo, = 1 New Drug
Event : y∈B:
y≤t
y: life span
We want to know (say something about)
P(y1∈B|x) = P(y1∈B|x,z0) P(z0|x) + P(y1∈B|x,z1) P(z1|x)
No knowledge:
Lower B :
Upper B :
P(y1∈B|x,z1) P(z1|x)
P(z0|x) + P(y1∈B|x,z1) P(z1|x)
Additional Information y1>y0
P(y1∈B|x, z0) < P(y0∈B|x,z0)
Upper B:
P(y0∈B|x,z0)) P(z0|x) + P(y1∈B|x,z1) P(z1|x)
5
IPSS Selection/ Switching
P(y1<t|x,z0) < P(y0<t|x,z0)
y0
y1
t
y
6
IPSS Selection/ Switching
Treatment: z
Outcome: y
P(y0∈B|x,z0)
z=0
P(y1∈B | x, z0)
Selection
z=1
P(y0∈B | x, z1)
P(y1∈B | x, z1)
7
IPSS Selection/ Switching
Example: Roy Model (Wage in sector 0, 1, e.g. 0=Fashion,1= IT)
z=0  y0>y1, z=1  y0<y1,
y0∈B: y≥t
We want to know (say something about)
P(y0∈B|x) = P(y0∈B|x,z0) P(z0|x) + P(y0∈B|x,z1) P(z1|x)
No knowledge:
Lower B :
P(y0∈B|x,z0) P(z0|x)
Upper B :
P(y0∈B|x,z0) P(z0|x) +
P(z1|x)
Additional Information from self-selection
P(y0∈B|x,z1) < P(y1∈B|x,z1)
Upper B:
P(y0∈B|x,z0) P(z0|x) + P(y1∈B|x,z1) P(z1|x)
8
IPSS Selection/ Switching
If z=1 then it has to be true P(y0>t|x,z1) < P(y1> t|x,z1)
y0
y1
t
y
9

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