The resulting upper UB
EE
and lower LB
EE
bounds for ATE
EE
are Lee 2009; Zhang, Rubin, and Mealli 2008:
8 UB
EE
= E[Y
i
| T
i
= 1, S
i
= 1, Y
i
≥ y
1 − p
1|0 p1|1 11
] − E[Y
i
| T
i
= 0, S
i
= 1] LB
EE
= E[Y
i
| T
i
= 1, S
i
= 1, Y
i
≤ y
p1|0 p1|1 11
] − E[Y
i
| T
i
= 0, S
i
= 1], where
y
1 − p
1 | 0
p
1 |1
11
and y
p
1 | 0
p
1 |1
11
denote the 1 −
p
1|0
p
1|1
and the p
1|0
p
1|1
quantiles of Y
i
conditional on T
i
= 1 and S
i
= 1, respectively. Lee 2009 shows that these bounds are sharp that is, there are no shorter bounds possible under the current assump-
tions. The bounds in Equation 8 can be estimated with sample analogs:
9 UB
EE
p =
Y
i
⋅ T
i
⋅ S
i
⋅ 1[Y
i
≥ y
1 − ˆp
p ]
i=1 n
∑
T
i
⋅ S
i
⋅ 1[Y
i
≥ y
1 − ˆp
p ]
i=1 n
∑
− Y
i
⋅ 1 − T
i
⋅ S
i i=1
n
∑
1 − T
i
⋅ S
i i=1
n
∑
LB
EE
p =
Y
i
⋅ T
i
⋅ S
i
⋅ 1[Y
i
≤ y
ˆp
n ]
i=1 n
∑
T
i
⋅ S
i
⋅ 1[Y
i
≤ y
ˆp
n ]
i=1 n
∑
− Y
i
⋅ 1 − T
i
⋅ S
i i=1
n
∑
1 − T
i
⋅ S
i i=1
n
∑
, where
y
1 − ˆp
p and y
ˆp
n are the sample analogs of the quantities y
1 − p
1|0
p
1|1
11
and y
p
1|0
p
1|1
11
in 8, respectively, and
ˆp, the sample analog of p
1|0
p
1|1
, is calculated as follows: 10
ˆp = 1
− T
i
⋅ S
i i=1
n
∑
1 − T
i i=1
n
∑
T
i
⋅ S
i i=1
n
∑
T
i i=1
n
∑
. Lee 2009 shows that these estimators are asymptotically normal.
B. Tightening the Bounds by Adding Weak Monotonicity of Mean Potential Outcomes Across Strata
We present a weak monotonicity assumption of mean potential outcomes across the EE
and NE strata that tightens the bounds in Equation 8. This assumption was origi- nally proposed by Zhang and Rubin 2003 and employed in Zhang, Rubin, and Mealli
2008: Assumption C. Weak Monotonicity of Mean Potential
Outcomes Across the EE and NE Strata: E[Y 1 | EE]
≥ E[Y1 | NE]. Intuitively, in the context of JC, this assumption formalizes the notion that the EE
stratum is likely to be comprised of more “able” individuals than those belonging to the NE stratum. Because “ability” is positively correlated with labor market outcomes,
one would expect wages for the individuals who are employed regardless of treatment status the EE stratum to weakly dominate on average the wages of those individuals
who are employed only if they receive training the NE stratum. Hence, Assumption C requires a positive correlation between employment and wages. Although Assump-
tion C is not directly testable, one can indirectly gauge its plausibility by comparing
the average of pretreatment covariates that are highly correlated with wages between the EE and NE strata, as we illustrate in Section VIB below.
Employing Assumptions A, B, and C results in tighter bounds. To see this, recall that the average outcome in the observed group with
T
i
, S
i
= 1, 1 contains units from two strata, EE and NE, and can be written as the weighted average shown in
Equation 7. By replacing E[Y 1 | NE]
with E[Y 1 | EE]
in Equation 7 and using the inequality in Assumption C, we have that
E[Y
i
| T
i
= 1, S
i
= 1] ≤ E[Y
i
1 | EE] , and thus
that E[Y 1 | EE]
is bounded from below by E[Y
i
| T
i
= 1, S
i
= 1]. Therefore, the lower bound for
ATE
EE
becomes: E[Y
i
| T
i
= 1, S
i
= 1] − E[Y
i
| T
i
= 0, S
i
= 1]. Imai 2008 shows that these bounds are sharp.
To estimate the bounds under Assumptions A, B, and C, note that the upper bound estimator of Equation 8 remains
UB
EE
p from Equation 9, although the estima- tor of the lower bound is the corresponding sample analog of
E[Y
i
| T
i
= 1, S
i
= 1] − E[Y
i
| T
i
= 0, S
i
= 1]: 11
LB
EE C
p =
Y
i
⋅ T
i
⋅ S
i i=1
n
∑
T
i
⋅ S
i i=1
n
∑
− Y
i
⋅ 1 − T
i
⋅ S
i i=1
n
∑
1 − T
i
⋅ S
i i=1
n
∑
.
C. Narrowing Bounds on ATE
