18 UB
EE α
p = min y :
T
i
⋅ S
i
⋅ 1[Y
i
≥ y
1 − pn
p ]
⋅ 1[Y
i
≤ y]
i=1 n
∑
T
i
⋅ S
i
⋅ 1[Y
i
≥ y
1 − pn
p ]
i=1 n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪ − min y :
1 − T
i
⋅ S
i
⋅ 1[Y
i
≤ y]
i=1 n
∑
1 − T
i
⋅ S
i i=1
n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪ LB
EE α
p = min y :
T
i
⋅ S
i
⋅ 1[Y
i
≤ y
p n
n ]
⋅ 1[Y
i
≤ y]
i=1 n
∑
T
i
⋅ S
i
⋅ 1[Y
i
≤ y
p n
n ]
i=1 n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪ − min y :
1 − T
i
⋅ S
i
⋅ 1[Y
i
≤ y]
i=1 n
∑
1 − T
i
⋅ S
i i=1
n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪ .
A. Tightening Bounds on QTE
EE
α
Using Stochastic Dominance
We tighten the bounds in Equation 17 by strengthening Assumption C to stochastic dominance. Let
F
Y
i
1|EE
⋅ and F
Y
i
1|NE
⋅ denote the cumulative distributions of Y
i
1 for
individuals who belong to the EE and NE strata, respectively: Assumption D. Stochastic Dominance Between the EE and NE
Strata: F
Y
i
1|EE
y ≤ F
Y
i
1|NE
y , for all y.
This assumption directly imposes restrictions on the distribution of potential outcomes under treatment for individuals in the EE stratum, which results in a tighter lower bound
relative to that in Equation 17. Under Assumptions A, B, and D, the resulting sharp bounds are Imai 2008:
LB
EE d
α
≤ QTE
EE α
≤ UB
EE α
, where UB
EE α
is as in Equation 17 and 19
LB
EE d
α
= F
Y
i
|T
i
=1, S
i
=1 −1
α − F
Y
i
|T
i
=0, S
i
=1 −1
α. The estimator of the upper bound is still given
UB
EE α
p by in Equation 18, while the estimator for
LB
EE d
α
is now given by: 20
LB
EE d
α
p = min y :
T
i
⋅ S
i
⋅ 1[Y
i
≤ y]
i=1 n
∑
T
i
⋅ S
i i=1
n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪ − min y :
1 − T
i
⋅ S
i
⋅ 1[Y
i
≤ y]
i=1 n
∑
1 − T
i
⋅ S
i i=1
n
∑
≥ α ⎧
⎨ ⎪
⎩⎪ ⎫
⎬ ⎪
⎭⎪
B. Narrowing Bounds on QTE
EE
α
Using a Covariate
In this section we propose a way to use a pretreatment covariate X taking values on {x
1
, ...., x
J
} to narrow the trimming bounds on
F
Y
i
1|EE −1
α and, thus, the bounds on QTE
EE α
in Equation 17. The idea is similar to that in Lee 2009 described in Section III.C, however, the nonlinear form of the quantile function
F
Y
i
1|EE −1
α prevents us from di- rectly using the law of iterated expectations as in Equation 12. To circumvent this
diffi culty, we fi rst focus on the cumulative distribution function CDF of Y
i
1 for the
stratum EE at a given point y,
F
Y
i
1|EE
y, and write it as the mean of an indicator func- tion, which allows us to use iterated expectations. A similar approach was also used in
Lechner and Melly 2010 to control for selection into treatment based on covariates. Using this insight we can write:
21 F
Y
i
1|EE
y = E[1[Y
i
1 ≤ y] | EE] = E
X
{E[1[Y
i
1 ≤ y] | EE, X
i
= x
j
] | EE}. Note that Equation 21 is similar to Equation 12, except that we now employ
1[Y
i
1 ≤ y] as the outcome instead of
Y
i
1 . Thus, the methods discussed in Section IIIC
and more generally, the trimming bounds in Section IIIA can be used to bound F
Y
i
1|EE
y. As in Section IIIC, let UB
EE y
x
j
and LB
EE y
x
j
denote the upper and lower bounds on
E[1[Y
i
1 ≤ y] | EE, X
i
= x
j
] under Assumptions A and B, which are just the
trimming bounds on E[Y
i
1 | EE] in the fi rst part of Equation 8 within cells with
X
i
= x
j
and employing as outcome the indicator 1[Y
i
1 ≤ y] instead of
Y
i
. After substi- tuting
UB
EE y
x
j
and LB
EE y
x
j
into 21 we obtain the following upper and lower bounds on
F
Y
i
1|EE
y under Assumptions A and B: 22
F
UB
y = E
X
{UB
EE y
x
j
| EE} F
LB
y = E
X
{LB
EE y
x
j
| EE}. Importantly, by the results in Lee 2009 discussed in Section IIIC, these trimming
bounds on F
Y
i
1|EE
y are sharp and tighter than those not employing the covariate X. Given bounds on
F
Y
i
1|EE
y for all y
∈ ℜ, the lower upper bound on the α- quantile
of Y
i
1 for the EE stratum,
F
Y
i
1|EE −1
α, is obtained by inverting the upper lower bound on
F
Y
i
1|EE
y. Using the bounds on F
Y
i
1|EE
y in 22, the lower and upper bounds on F
Y
i
1|EE −1
α are obtained by fi nding the value y
α
such that F
UB
y
α
= α and F
LB
y
α
= α, respectively.
8
Therefore, the bounds on QTE
EE α
under Assumptions A and B are given by: 23
UB
EE α
= F
LB −1
α − F
Y
i
|T
i
=0, S
i
=1 −1
α LB
EE α
= F
UB −1
α − F
Y
i
|T
i
=0, S
i
=1 −1
α. We implement this procedure by estimating the bounds on
F
Y
i
1|EE
y in 22 at M different values of
y spanning the support of the outcome, and then inverting the re- sulting estimated bounds to obtain the estimate of the bounds on the
α- quantile F
Y
i
1|EE −1
α. This last set of estimated bounds are then combined with the estimate of F
Y
i
|T
i
=0, S
i
=1 −1
α to compute estimates of the bounds on QTE
EE α
in 23. The bounds on F
Y
i
1|EE
y
m
at each point y
m
m = 1, . . . , M in 22 are estimated employing the esti- mators of the bounds on
E[Y
i
1 | EE] in the fi rst term of 9 for individuals with
X
i
= x
j
and using as outcome the indicator function 1[Y
i
1 ≤ y
m
] instead of
Y
i
. Finally, just as in the case of the
ATE
EE
in Section III.C, under Assumptions A, B and D, the procedure above is only applied to the upper bound because the lower bound does not
involve trimming.
8. Note that, because we are inverting the CDF, the lower upper bound on the quantile is computed em- ploying the upper lower bound of the CDF.
V. Job Corps and the National Job Corps Study
