Supplementary material to: Tolerating deance?
Identication of treatment eects without monotonicity.
∗
Clément de Chaisemartin
April 13, 2016
Abstract
This paper gathers the supplementary material to de Chaisemartin (2016). I show that
one can estimate quantile treatment eects in the comvivors subpopulation, that one can
test the CD condition, and that my results extend to multivariate treatment and instrument,
to fuzzy regression discontinuity designs, and to the bounds in Lee (2009) for survey nonresponse.
∗
Department of Economics, University of Warwick, clementdechaisemartin@gmail.com
1 Testability and quantile treatment eects.
In this section, I show that random instrument, exclusion restriction, and CD have a testable
implication.
Then, I discuss how the testable implications of random instrument, exclusion
restriction, and ND studied in Huber & Mellace (2012), Kitagawa (2013), and Mourie & Wan
(2014) relate to the CD condition.
R, let S(R) denote the support of R. Let Udz be a random variable
uniformly distributed on [0, 1] denoting the rank of an observation in the distribution of Y |D =
d, Z = z . If the distribution of Y |D = d, Z = z is continuous, Udz = FY |D=d,Z=z (Y ). If the
distribution of Y |D = d, Z = z is discrete, one can randomly allocate a rank to tied observations
For any random variable
by setting
Udz = FY |D=d,Z=z (Y − ) + V (FY |D=d,Z=z (Y ) − FY |D=d,Z=z (Y − )),
V is uniformly distributed on [0, 1] and independent of (Y, D, Z),
FS
.
S(Y |D = d, Z = z) : y < Y }.1 For every d ∈ {0, 1}, let pd = P (D=d|Z=d)
and p1 are included between 0 and 1. Finally, let
where
Y − = sup{y ∈
Notice that both p0
and
L = E(Y |D = 1, Z = 1, U11 ≤ p1 ) − E(Y |D = 0, Z = 0, U00 ≥ 1 − p0 )
L = E(Y |D = 1, Z = 1, U11 ≥ 1 − p1 ) − E(Y |D = 0, Z = 0, U00 ≤ p0 ).
Theorem S1
If Assumptions 1, 2, and 5 are satised,
L ≤ W ≤ L.
(29)
The intuition of this result goes as follows. Under random instrument, exclusion restriction, and
E(Y1 − Y0 |CV ) is point identied: following Theorem 2.1 in de Chaisemartin (2016), it is
equal to W . It is also partially identied. CV is included in {D = 0, Z = 0}, and it accounts for
p0 % of this population. E(Y0 |CV ) can therefore not be larger than the mean of Y0 of the p0 %
of this population with largest Y0 . It can also not be smaller than the mean of Y0 of the p0 %
with lowest Y0 (see Horowitz & Manski, 1995 and Lee (2009)). Combining this with a similar
reasoning for E(Y1 |CV ) yields worst case bounds for E(Y1 − Y0 |CV ), L and L. The estimand of
E(Y1 − Y0 |CV ) should lie within its bounds, hence the testable implication.
CD,
To implement this test, one can use results from Andrews & Soares (2010). Let
two following moment inequality conditions:
θ
satisfy the
0
0
YZ
Y (1 − Z)
Y DZ1{U11 ≤ p1 }
Y (1 − D)(1 − Z)1{U00 ≥ 1 − p0 }
−
− FS θ +
−
P (Z = 1)
P (Z = 0)
P (D = 1, Z = 1, U11 ≤ p1 )
P (D = 0, Z = 0, U00 ≥ 1 − p0 )
Y DZ1{U11 ≥ 1 − p1 }
Y (1 − D)(1 − Z)1{U00 ≤ p0 }
YZ
Y (1 − Z)
≤ E FS θ +
−
−
−
.
P (D = 1, Z = 1, U11 ≥ 1 − p1 )
P (D = 0, Z = 0, U00 ≤ p0 )
P (Z = 1)
P (Z = 0)
≤E
This denes a moment inequality model with preliminary estimated parameters.
satises all the technical conditions required in Andrews & Soares (2010) if
1Y −
should be understood as
−∞
if
Y = inf{S(Y |D = d, Z = z)}.
1
This model
2+δ
E(|Y |
) < +∞
δ , and if P (Z = 0), P (Z = 1), P (D = 1, Z = 1, U11 ≤ p1 ), P (D =
1, Z = 1, U11 ≥ 1 − p1 ), P (D = 0, Z = 0, U00 ≤ p0 ), and P (D = 0, Z = 0, U00 ≥ 1 − p0 ) are all
bounded by below by some strictly positive constant . Under these two mild restrictions, their
Theorem 1 applies, and one can use it to construct a uniformly valid condence interval for θ .
(29) is rejected when 0 does not belong to this condence interval.
for some strictly positive
As pointed out in Balke & Pearl (1997) and Heckman & Vytlacil (2005), random instrument,
exclusion restriction, and ND also have testable implications. Huber & Mellace (2011), Kitagawa
(2013), Machado et al. (2013), and Mourie & Wan (2014) have developed statistical tests of
these implications. I now discuss how these tests relate to the CD condition.
The test suggested in Huber & Mellace (2011) is not a test of CD. Huber & Mellace (2011) use
the fact that
E(Y0 |N T )
and
E(Y1 |AT )
are both point and partially identied under random in-
E(Y |D = 0, Z = 1) is equal to E(Y0 |N T )
(1 − p0 )E(Y0 |N T ) + p0 E(Y0 |C). This implies
strument, exclusion restriction, and ND. For instance,
and
E(Y |D = 0, Z = 0)
is equal to
E(Y |D = 0, Z = 0, U00 ≤ 1 − p0 ) ≤ E(Y |D = 0, Z = 1) ≤ E(Y |D = 0, Z = 0, U00 ≥ p0 ).
But under CD,
E(Y |D = 0, Z = 1)
is equal to
P (N T )
E(Y0 |N T )
P (N T )+P (F )
+
P (F )
E(Y0 |F ), so
P (N T )+P (F )
the previous implication need not be true anymore.
The tests suggested in Kitagawa (2013), Machado et al. (2013), and Mourie & Wan (2014) are
not tests of CD, but they are tests of the CDM condition introduced hereafter.
(Compliers-deers for marginals: CDM)
There is a subpopulation of C denoted CF which satises:
Assumption S1
P (CF ) = P (F )
Y0 |CF ∼ Y0 |F
Y1 |CF ∼ Y1 |F.
The CDM condition requires that a subgroup of compliers have the same size and the same
marginal distributions of
Y0
and
Y1
as deers. CDM is stronger than CD. On the other hand, it
is invariant to the scaling of the outcome, while CD is not. With non-binary outcomes, working
under assumptions invariant to scaling might be desirable so one might then prefer to invoke
CDM than CD, despite the fact it is a stronger assumption.
Following the logic of Theorem
2.2 in de Chaisemartin (2016), a sucient condition for CDM to hold is that there be more
compliers than deers in each subgroup with the same value of
P (F |Y0 , Y1 ) ≤ P (C|Y0 , Y1 ).
2
(Y0 , Y1 ):
One can show that under CDM, the marginal distributions of
Y0
and
Y1
are identied for com-
vivors. The estimands are the same as in Imbens & Rubin (1997):
P (D = 0|Z = 0)fY |D=0,Z=0 (y0 ) − P (D
P (D = 0|Z = 0) − P (D
P (D = 1|Z = 1)fY |D=1,Z=1 (y1 ) − P (D
fY1 |CV (y1 ) =
P (D = 1|Z = 1) − P (D
fY0 |CV (y0 ) =
= 0|Z
= 0|Z
= 1|Z
= 1|Z
= 1)fY |D=0,Z=1 (y0 )
= 1)
= 0)fY |D=1,Z=0 (y1 )
.
= 0)
The testing procedures in Kitagawa (2013) and Mourie & Wan (2014) test whether the rhs of
the two previous equations are positive everywhere. Under CDM, these quantities are densities
so they must be positive. These procedures are therefore tests of CDM. Machado et al. (2013)
focus on binary outcomes. Their procedure tests inequalities equivalent to those considered in
Kitagawa (2013). Therefore, their procedure is also a test of the CDM condition.
2 Multivariate treatment and instrument
Results presented in the main paper extend to applications where treatment is multivariate.
z ∈ {0; 1}, Dz ∈ {0, 1, 2, ..., J} for some integer J . Let (Yj )j∈{0,1,2,...,J}
j
denote the corresponding potential outcomes. Let C = {D1 ≥ j > D0 } denote the subpopulation of compliers induced to go from a treatment below to above j by the instrument. Let
F j = {D0 ≥ j > D1 } denote the subpopulation of deers induced to go from above to below j by
the instrument. As in Angrist & Imbens (1995), I assume that the cdf of D|Z = 0 stochastically
dominates that of D|Z = 1. This implies that P (Cj ) ≥ P (Fj ). Consider the following CD
Assume that for every
condition:
(Compliers-deers for multivariate treatment: CDMU)
For every j ∈ {1, 2, ..., J}, there is a subpopulation of C j denoted CFj which satises:
Assumption S2
P (CFj ) = P (F j )
E(Yj − Yj−1 |CFj ) = E(Yj − Yj−1 |F j ).
If CDMU is satised, one can show that
J
X
W
is equal to the following average causal response:
wj E(Yj − Yj−1 |CVj ),
j=1
where
CVj
is a subset of
Cj
of size
P (Cj ) − P (Fj ),
and
wj
are positive real numbers.
This
generalizes Theorem 1 in Angrist & Imbens (1995).
Their Theorem 2 considers the case where the instrument is multivariate:
Let
β
denote the coecient of
D
in a 2SLS regression of
3
Y
on
D
Z ∈ {0, 1, 2, ..., K}.
using the set of dummies
(1{Z = k})1≤k≤K as instruments. Angrist & Imbens (1995) show that if Dz0 ≥ Dz for every
z 0 ≥ z , β is equal to a weighted average of average causal responses. Let C jz = {Dz ≥ j > Dz−1 }
jz
(resp. F
= {Dz ≥ j > Dz−1 }) denote the subpopulation of compliers induced to go from a
treatment below to above j (resp. above to below j ) when the instrument moves from z − 1 to
z . Consider the following condition:
(Compliers-deers for multivariate treatment and instrument: CDMU2)
For every j ∈ {1, 2, ..., J} and z ∈ {0, 1, 2, ..., K}, there is a subpopulation of C jz denoted CFjz
which satises:
Assumption S3
P (CFjz ) = P (F jz )
E(Yj − Yj−1 |CFjz ) = E(Yj − Yj−1 |F jz ).
Under CDMU2, one can show that
β
is equal to a weighted average of average causal responses
for comvivors.
3 Fuzzy regression discontinuity designs and survey non response
My results also extend to many treatment eect models relying on ND type of conditions.
An important example are fuzzy regression discontinuity (RD) designs studied in Hahn et al.
(2001). Their Theorem 3 still holds if their Assumption A.3, a ND at the threshold assumption,
S denote the forcing variable, and let s
F = {F, S = s} respectively denote compliers
is replaced by a CD at the threshold assumption. Let
denote the threshold. Let
s
C = {C, S = s}
and
s
and deers at the threshold.
(Compliers-deers at the threshold: CDT)
There is a subpopulation of C s denoted CFs which satises:
Assumption S4
P (CFs |S = s) = P (F s |S = s)
E(Y1 − Y0 |CFs , S = s) = E(Y1 − Y0 |F s , S = s).
Finally, my results extend to the sample selection and survey non-response literatures. ND type
of conditions have also been used in the sample selection and survey non-response literatures,
R0 and R1 denote a subject's potential
responses to a survey without and with the treatment. Let AR, RC , and RF respectively denote
always respondents (subjects who satisfy R0 = R1 = 1), reponse-compliers (those who satisfy
R0 = 0 and R1 = 1), and response-deers (those who satisfy R0 = 1 and R1 = 0). Under
the assumption that there are no response-deers (R1 ≥ R0 ), Lee (2009) derives bounds for
E(Y1 − Y0 |AR). Under the weaker assumption that a subpopulation RCF of RC has the same
size and the same average treatment eect as RF , the bounds derived in Lee (2009) are valid
bounds for E(Y1 − Y0 |AR ∪ RCF ).
for instance in Lee (2009) or Behaghel et al. (2012). Let
4
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using generalized moment selection',
Econometrica 78(1), 119157.
Angrist, J. D. & Imbens, G. W. (1995), `Two-stage least squares estimation of average causal
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Journal of the American Statistical Asso-
ciation 90(430), pp. 431442.
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the Study of Labor (IZA).
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identication of treatment eects without
monotonicity.'.
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5
Machado, C., Shaikh, A. & Vytlacil, E. (2013), Instrumental variables, and the sign of the
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6
A Proofs
Proof of Theorem S1
Proving that
L
and
L
are valid bounds for
E(Y1 − Y0 |CV )
relies on a similar argument as the
proof of Proposition 1.a in Lee (2009). Due to a concern for brevity, I refer the reader to this
paper for this part of the proof. The testable implication directly follows.
QED.
7