Optimal Data-Driven Regression Discontinuity Plots∗ Supplemental Appendix Sebastian Calonico†
Matias D. Cattaneo‡
Rocio Titiunik§
November 25, 2015
Abstract This supplemental appendix contains the proofs of our main theorems, additional methodological and technical results, detailed simulation evidence, and further empirical illustrations not included in the main paper to conserve space.
∗
Financial support from the National Science Foundation (SES 1357561) is gratefully acknowledged. Department of Economics, University of Miami. ‡ Department of Economics and Department of Statistics, University of Michigan. § Department of Political Science, University of Michigan. †
Contents 1 Implied Weights in Optimal WIMSE Approach
2
2 Proofs of Main Theorems
3
2.1
Lemma SA1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Lemma SA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Lemma SA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.5
Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6
Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7
Proof of Remark 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.8
Proof of Theorem 4 and Remark 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Data-Driven Implementations with Arbitrary w(x)
12
3.1
Evenly Spaced RD Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
Quantile Spaced RD Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Other Empirical Applications
18
4.1
U.S. Senate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2
Progresa/Oportunidades Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3
Head Start Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4
Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Complete Simulation Results
28
6 Numerical Comparison of Partitioning Schemes
48
1
1
Implied Weights in Optimal WIMSE Approach
Recall from the main paper that the optimal choices of number of bins based on a WIMSE can be written as JES-ω,−,n = dω− JES-µ,−,n e
and JES-ω,+,n = dω+ JES-µ,+,n e,
where JES-µ,−,n and JES-µ,+,n denote the IMSE-optimal choices and ω− = (ωB,− /ωV ,− )1/3 and ω+ = (ωB,+ /ωV ,+ )1/3 . As discussed in the paper, this result may be used to justify ad-hoc rescalings chosen by the researchers when using the IMSE-optimal choices as a starting point. In particular, given a choice of rescaling factors ω− and ω+ , we have: (ωV ,− , ωB,− ) =
3 ω− 1 , 3 1 + ω3 1 + ω− −
and
(ωV ,+ , ωB,+ ) =
3 ω+ 1 , 3 1 + ω3 1 + ω+ +
,
which are the resulting weights entering the WIMSE objective function that would be compatible with such choices of rescale constants for the IMSE-optimal number of bins. To gain some intuition on the relative weights emerging from manual rescaling of the IMSEoptimal choice, we present the implied weights in the optimal WIMSE approach for different, common choices of rescaling constants ω: ω 0.1 0.2 0.5 1 2 5 10
ωV 0.999 0.992 0.889 0.500 0.111 0.008 0.001
ωB 0.001 0.008 0.111 0.500 0.889 0.992 0.999
As expected, the larger ω the smaller the weight on variance (ωV ) and the larger the weight on bias (ωB ) in the WIMSE objective function. Our software implementations in R and Stata compute this weights explicitly as part of the standard output; see Calonico, Cattaneo and Titiunik (2014a, 2015) for further details.
2
2
Proofs of Main Theorems
We state and prove results only for the treatment group (subindex “+”) because for the control group the results and proofs are analogous. Here we only provide short, self-contained proofs of the main results presented in the paper. To this end, we first state three preliminary technical lemmas. We also offer short proofs of these lemmas, and provide references to the underlying results not reproduced here to conserve space. Recall that the lower and upper end points of P+,j are denoted, respectively, by p+,j−1 and p+,j for j = 1, 2, · · · , J+,n , which are nonrandom under ES partitioning and random under QS partitioning. Let p¯+,j = (p+,j + p+,j−1 )/2 be the middle point of bin P+,j . Throughout the supplemental appendix C denotes an arbitrary positive, bounded constant taking different values in different places.
2.1
Lemma SA1
This lemma holds for any nonrandom partition P+,n satisfying C1 C2 ≤ min |p+,j − p+,j−1 | ≤ max |p+,j − p+,j−1 | ≤ , 1≤j≤J+,n J+,n 1≤j≤J+,n J+,n for fixed positive constants C1 and C2 . In particular, it holds for PES,+,n . Note also that Lemma SA1(i) shows that P(N+,j > 0) → 1 uniformly in j, which guarantees that the estimators for the ES partitioning scheme are well-behaved in large samples. Lemma SA1. Let Assumption 1 hold. For PES,+,n , if J+,n log(J+,n ) →0 n
and
3
J+,n → ∞,
then the following results hold.
(i)
max
1≤j≤J+,n
|1(N+,j > 0) − 1| = oP (1).
−1 |N+,j /n − P[Xi ∈ P+,j ]| = oP (J+,n ). n 1 X X − p ¯ X − p ¯ −1 max 1P+,j (Xi ) i +,j − E 1P+,j (Xi ) i +,j = oP (J+,n ). 1≤j≤J+,n n p+,j − p+,j−1 p+,j − p+,j−1 i=1 Xi − p¯+,j = o(J −1 ). max E 1P+,j (Xi ) +,n 1≤j≤J+,n p+,j − p+,j−1
(ii)
max
1≤j≤J+,n
(iii) (iv)
Proof of Lemma SA1. The proof of this lemma is very similar to the results given in the supplemental appendix of Cattaneo and Farrell (2013). Part (i) follows by properties of the Binomial distribution and simple bounding arguments, under the assumptions imposed. For part (ii), note −1 that E[1(Xi ∈ P+,j )] = P[Xi ∈ P+,j ] = O(J+,n ) and C1 /J+,n ≤ V[1(Xi ∈ P+,j )] ≤ C2 /J+,n ,
uniformly in j = 1, 2, · · · , J+,n . For any ε > 0, and using Bernstein inequality, we have Nj − P[Xi ∈ P+,j ] > ε P J+,n max 1≤j≤J+,n n " n # X ≤ J+,n max P (1(Xi ∈ P+,j ) − P[Xi ∈ P+,j ]) > nε/J+,n 1≤j≤J+,n i=1 ) ( 2 n2 ε2 /J+,n ≤ J+,n max 2 exp − Pn 1≤j≤J+,n 2 i=1 V[1(Xi ∈ P+,j )] + 2nε ) ( Cn Cn ≤ C exp − + log(J ) ≤ C exp − + log(J ) → 0, +,n +,n 2 ε J+,n J+,n + J+,n
provided that J+,n log(J+,n )/n → ∞. Part (iii) follows by similar arguments. Finally, to verify part (iv), using change of variables we obtain Xi − p¯+,j max E 1P+,j (Xi ) 1≤j≤J+,n p+,j − p+,j−1 Z xu x − p ¯ +,j 1P+,j (x) = max f (x)dx 1≤j≤Jn p+,j − p+,j−1 x ¯ Z 1 = max (p+,j − p+,j−1 ) uf (u(p+,j − p+,j−1 ) + p¯+,j )du 1≤j≤J+,n −1 Z xu − x ¯ 1 = max uf (¯ p+,j )du + o(1) , 1≤j≤J+,n J+,n −1 4
and the result follows.
2.2
Lemma SA2
This second lemma characterizes the properties of the random partitioning scheme based on quantile estimates. These results will be used when handling the partitioning scheme PQS,+,n : recall that p+,j = Fˆ+−1 (j/J+,n ) in this case, j = 1, 2, · · · , J+,n , and thus set q+,j = F+−1 (j/J+,n ) with F+−1 (y) = inf{x : F+ (x) ≥ y} with
F+ (x) =
P[Xi ≤ x, Xi ≥ x ¯] = F (x|Xi ≥ x ¯). P[Xi ≥ x ¯]
Lemma SA2. Let Assumption 1 hold. For PQS,+,n , if J+,n log(J+,n ) →0 n
and
J+,n → ∞, log(n)
then the following results hold.
(i) (ii)
max
−1 |N+,j /N+ − 1/J+,n | = oP (J+,n ).
max
−1 |p+,j − p+,j−1 − (q+,j − q+,j−1 )| = oP (J+,n ).
1≤j≤J+,n
1≤j≤J+,n
Proof of Lemma SA2. Because the sample size N+ is random, we employ the following P result: if N+ →as ∞ and Zn →as Z∞ , then ZN+ →as Z∞ . In our case, N+ = ni=1 1(Xi ≥ x ¯) and thus N+ /n →as P+ . Hence, it suffices to assume N+ → ∞ is not random, but we need to prove the statements in an almost sure sense. The rest of the proof takes limits as N+ → ∞. Part (i) now follows from properties of distribution function and quantile processes (e.g., Shorack and Wellner, 2009). Using continuity and boundedness of f (x), we have
N+,j
n X j −1 j − 1 −1 ˆ ˆ ≤ Xi < F+ = 1 F+ J+,n J+,n i=1 j j−1 N+ = N+ Fˆ+ Fˆ+−1 − N+ Fˆ+ Fˆ+−1 {1 + oas (1)} = {1 + oas (1)}, J+,n J+,n J+,n
5
uniformly in j = 1, 2, · · · , J+,n , under the rate restrictions imposed. Similarly, part (ii) follows from properties of the modulus of continuity of the sample quantile process (e.g., Mason (1984) and Shorack and Wellner (2009, Chapter 14)). We have
|p+,j − p+,j−1 − (q+,j − q+,j−1 )| −1 j j −1 −1 j − 1 −1 j − 1 ˆ ˆ = oas (J −1 ), − F+ − F+ − F+ = max F+ +,n 1≤j≤J+,n J+,n J+,n J+,n J+,n max
1≤j≤J+,n
under the rate restrictions imposed.
2.3
Lemma SA3
Our final third technical lemma gives the main convergence results for the spacings estimators used to construct data-driven choices of partition sizes. We employ the notation introduced in Section 5 of the main paper. Lemma SA3. Let Assumption 1 hold, and set ` ∈ Z+ . If Yi (1) is continuously distributed and g : [¯ x, xu ] → R+ is continuous, then the following results hold.
(i)
N+`−1
N+ X
¯ +,(i) ) →P `!P `−1 (X+,(i) − X+,(i−1) )` g(X +
i=2 N+
(ii)
N+`−1
X
Z
xu
f (x)1−` g(x)dx.
x ¯
¯ +,(i) ) →P `!P `−1 2 (X+,(i) − X+,(i−1) ) (Y+,[i] − Y+,[i−1] ) g(X + `
2
i=2
Z
xu
2 f (x)1−` σ+ (x)g(x)dx.
x ¯
Proof of Lemma SA3. We prove the result assuming that N+ is nonrandom, and thus limits are taken as N+ → ∞. Set Ui = F+ (X+,i ) ∼ Uniform(0, 1) and U(i) = F+ (X+,(i) ), i = 1, · · · , N+ . ¯ : i = 2, · · · , N+ }, where {Ei : i = Recall that {N+ (U(i) − U(i−1) ) : i = 2, · · · , N+ } =d {Ei /E ¯ = PN+ Ei /N+ , and where 2, · · · , N+ } i.i.d. random variables with Ei ∼ Exponential(1) and E i=2 Z1 =d Z2 denotes that Z1 and Z2 have the same probability law. Set u ¯i = (i − 1/2)/N+ and recall that max2≤i≤N+ supU(i−1) ≤u≤U(i) |u − u ¯i | →P 0. PN+ ` Ei →P E[Ei` ] = `!, and uniform continuity of g(·) and For part (i), using the above, N+−1 i=2
6
f (·), N+ X ¯ +,(i) ) (X+,(i) − X+,(i−1) )` g(X
N+`−1
i=2 N+ g(F+−1 (un,i )) 1 X = (N+ (U(i) − U(i−1) ))` {1 + oP (1)} N+ f+ (F+−1 (un,i ))` i=2 N+
1 X Ei ` g(F+−1 (un,i )) =d {1 + oP (1)} ¯ N+ E f+ (F+−1 (un,i ))` i=2 N+ −1 1 X ` g(F+ (un,i )) {1 + oP (1)} = E[Ei ] N+ f+ (F+−1 (un,i ))` i=2 Z 1 g(F+−1 (u)) →P `! du, −1 ` 0 f+ (F+ (u))
and the result follows by change of variables and because f+ (x) = f (x)1(x ≥ x ¯)/P+ . This result PN+ ¯ +,(i) ) = OP (N 1−` ). implies, in particular, i=2 (X+,(i) − X+,(i−1) )` g(X + For part (ii), let X(+) = (X+,(1) , X+,(2) , · · · , X+,(N+ ) ). Recall that (Y+,[1] , Y+,[2] , · · · , Y+,[N+ ] ) are independent conditional on X(+) and E[g(Y+,[i] )|X(+) ] = E[g(Y+,[i] )|X+,(i) ] = G(X+,(i) ) with 2 (X 2 G(x) = E[g(Y+,i )|X+,i = x]. Therefore, E[(Y+,[i] − Y+,[i−1] )2 |X(+) ] = σ+ +,(i) ) + σ+ (X+,(i−1) ) + −2 2 (X 2 (E[Y+,[i] |X(+) ] − E[Y+,[i−1] |X(+) ])2 = σ+ +,(i) ) + σ+ (X+,(i−1) ) + OP (N+ ), uniformly in i. This
gives N+`−1
N+ X ¯ +,(i) ) = T1 + T2 , (X+,(i) − X+,(i−1) )` (Y+,[i] − Y+,[i−1] )2 g(X i=2
with T1 =
N+`−1
N+ X 2 2 ¯ +,(i) ) + oP (1), (X+,(i) − X+,(i−1) )` (σ+ (X+,[i] ) + σ+ (X+,[i−1] ))g(X i=2
T2 =
N+`−1
N+ X
¯ +,(i) ). (X+,(i) − X+,(i−1) )` (Y+,[i] − Y+,[i−1] )2 − E[(Y+,[i] − Y+,[i−1] )2 |X[+] ] g(X
i=2 2 (X 2 2 ¯ Noting that σ+ +,(i) ) + σ+ (X+,(i−1) ) = 2σ+ (X+,(i) ){1 + oP (1)}, uniformly in i, it follows that Rx 2 (x)g(x)dx, as in part (i). Thus, it remains to show that T → T1 →P `!P+`−1 2 x¯ u f (x)1−` σ+ 2 P
0. To this end, first define Y˜i = (Y+,[i] − Y+,[i−1] )2 − E[(Y+,[i] − Y+,[i−1] )2 |X(+) ], and note that
7
E[Y˜i , Y˜i−s |X(+) ] = 0 whenever s ≥ 2, which implies
2(`−1)
V[T2 |X(+) ] ≤ N+
N+ X
¯ +,(i) )2 (X+,(i) − X+,(i−1) )2` V[Y˜i |X(+) ]g(X
i=2 N+ 2(`−1)
X ¯ +,(i) )g(X ¯ +,(i−1) ) (X+,(i) − X+,(i−1) )` (X+,(i−1) − X+,(i−2) )` E[Y˜i Y˜i−1 |X(+) ]g(X
+ 2N+
i=2
≤ CN+−1 ,
and the result follows by the dominated convergence theorem. P The random sample size case (N+ = ni=1 1(Xi ≥ x ¯)) can be handled, for example, using the approach described in Aras et al. (1989) and references therein.
2.4
Proof of Theorem 1
For the variance part, we have n X 1(N+,j > 0)1P+,j (x) X V[ˆ µ+ (x; J+,n )|Xn ] = 1P+,j (Xi )σ+2 (Xi ), 2 N +,j j=1 i=1 J+,n
2 (·) on [¯ x, xu ] and Lemma SA1, we obtain and using uniform continuity of w(·) and σ+
Z
xu
V[ˆ µ+ (x; J+,n )|Xn ]w(x)dx x ¯ J+,n
=
X n X 1(N+,j > 0) Z xu (x)w(x)dx 1 1P+,j (Xi )σ+2 (Xi ) P+,j 2 N x ¯ +,j j=1 i=1
J+,n
=
X 1(N+,j > 0) 2 (p+,j − p+,j−1 )σ+ (¯ p+,j )w(¯ p+,j ){1 + oP (1)} N+,j j=1
J+,n 2 (¯ p+,j )w(¯ p+,j ) 1 X σ+ {1 + oP (1)}, = n f (¯ p+,j ) j=1
8
because P[Xi ∈ P+,j ] =
R p+,j p+,j−1
f (x)dx = (p+,j − p+,j−1 )f (¯ p+,j ){1 + o(1)} uniformly in j. Using
properties of the Riemann integral it then follows that Z
xu
V[ˆ µES,+ (x; J+,n )|Xn ]w(x)dx x ¯ J+,n X σ 2 (¯ p+,j )w(¯ p+,j ) J+,n 1 = (p+,j − p+,j−1 ) + {1 + oP (1)} n xu − x ¯ f (¯ p+,j ) j=1 Z xu 2 σ+ (x) J+,n 1 w(x)dx{1 + oP (1)} = n xu − x ¯ x¯ f (x) J+,n = VES,+ {1 + oP (1)}, n
because p+,j+1 − p+,j = (xu − x ¯)/J+,n for the evenly spaced partition. Rx µ+ (x; Jn )|Xn ] − µ+ (x))2 w(x)dx = T1 + T2 + T3 with Next, for the bias term, note that x¯ u (E[ˆ xu
Z
Z
2
T1 (x) w(x)dx,
T1 =
xu
2
T2 (x) w(x)dx,
T2 =
xu
T1 (x)T2 (x)w(x)dx,
T3 = 2 x ¯
x ¯
x ¯
Z
J+,n
T1 (x) =
X
1P+,j (x)(1(N+,j > 0)µ+ (¯p+,j ) − µ+ (x)),
j=1 J+,n
T2 (x) =
X
1P+,j (x)
j=1
n 1(N+,j > 0) X
N+,j
!
1P+,j (Xi )(µ+ (Xi ) − µ+ (¯p+,j )) .
i=1
Using uniform continuity of µ+ (·) and w(·) on [¯ x, xu ] and Lemma SA1, we obtain J+,n Z xu 2 1 X (1) T1 = µ+ (¯ p+,j ) w(¯ p+,j ) 1P+,j (x)(¯p+,j − x)2 dx{1 + oP (1)} 12 x ¯ j=1
J+,n
2 1 X (1) p+,j ){1 + oP (1)} (p+,j − p+,j−1 )3 µ+ (¯ p+,j ) w(¯ 12 j=1 Z 1 (xu − x ¯)2 xu (1) 2 −2 = 2 µ+ (x) w(x)dx{1 + oP (1)} = J+,n BES,+ {1 + oP (1)}, 12 J+,n x ¯ =
because
Rb
2 a ((a + b)/2 − x) dx
= (b − a)3 /12 and p+,j+1 − p+,j = (xu − x ¯)/J+,n for the evenly spaced
−2 −2 partition. This implies that T1 = OP (J+,n ). Thus, to finish the proof, we show that T2 = oP (J+,n ) −2 and T3 = oP (J+,n ). For T2 , using uniform continuity of µ+ (·) and w(·) on [¯ x, xu ] and Lemma SA1
9
we have J+,n
|T2 | ≤ C
X
1(N+,j > 0)
j=1
2 N 2 /n2 J+,n +,j
n
X − p¯ 1X 1P+,j (Xi ) i +,j n p+,j − p+,j−1
−2 {1 + oP (1)} = oP (J+,n ),
i=1
while, for T3 , Cauchy-Swartz inequality implies |T3 | ≤
2.5
!2
√
√ −1 −1 −2 T1 T2 = OP (J+,n )oP (J+,n ) = oP (J+,n ).
Proof of Theorem 2
Recall that p+,j = Fˆ+−1 (j/J+,n ) and q+,j = F+−1 (j/J+,n ). If J+,n < N+ , then
1(N+,j > 0) = 1,
but now the partitioning scheme PQS,+,n is random. For the variance part, letting q¯+,j = (q+,j + q+,j−1 )/2, we have xu
Z
V[ˆ µQS,+ (x; J+,n )|Xn ]w(x)dx x ¯ J+,n
=
X j=1
1 2 N+,j
Z x ¯
xu
1P+,j (x)w(x)dx
X n
1P+,j (Xi )σ+2 (Xi )
i=1
J+,n
=
J+,n X 2 (p+,j − p+,j−1 )σ+ (¯ p+,j )w(¯ p+,j ){1 + oP (1)} N+ j=1
J+,n J+,n X 2 (q+,j − q+,j−1 )σ+ (¯ q+,j )w(¯ q+,j ){1 + oP (1)} N+ j=1 Z xu J+,n J+,n 1 2 σ+ (x)w(x)dx{1 + oP (1)} = VQS,+ {1 + oP (1)}, = n P+ x¯ n
=
using Lemma SA2 and properties of the Riemann integral. For the bias part, using the previous results and proceeding as in the proof of Theorem 1, Z
xu
(E[ˆ µQS,+ (x; Jn )|Xn ] − µ+ (x))2 w(x)dx
x ¯ J+,n 2 1 X (1) = (p+,j − p+,j−1 )3 µ+ (¯ p+,j ) w(¯ p+,j ){1 + oP (1)} 12 j=1
J+,n 2 1 X (1) (q+,j − q+,j−1 )3 µ+ (¯ q+,j ){1 + oP (1)} q+,j ) w(¯ 12 j=1 !2 Z (1) 1 P+2 xu µ+ (x) −2 = 2 w(x)dx{1 + oP (1)} = J+,n BQS,+ {1 + oP (1)}, f (x) J+,n 12 x¯
=
10
because, for quantile spaced partitions, expanding F+−1 (u) around u ¯ = F+ (¯ q+,j ) ∈ [(j−1)/J+,n , j/J+,n ]), q+,j − q+,j−1 = F+−1
j J+,n
− F+−1
j−1 J+,n
=
1 1 {1 + oP (1)}, f+ (¯ q+,j ) J+,n
uniformly in j = 1, 2, · · · , J+,n , where f+ (x) = ∂F+ (x)/∂x = f (x)1(x ≥ x ¯)/P+ .
2.6
Proof of Theorem 3
Using Lemma SA3 with ` = 1 and g(x) = 1, N+
VˆES,+ =
1 1X 1 (X+,(i) − X+,(i−1) )(Y+,[i] − Y+,[i−1] )2 = xu − x ¯2 xu − x ¯ i=2
Z x ¯
xu
2 σ+ (x)dx + oP (1),
which gives VˆES,+ →P VES,+ . Next, note that for power series estimators, Newey (1997, Theorem 4) gives (1)
(1)
sup |ˆ µ+,kn (x) − µ+ (x)|2 = OP (kn7 /n + kn−2S+8 ) = oP (1).
x∈[¯ x,xu ]
Using this uniform consistency result we have n n 2 (x − x 2 (xu − x ¯)2 X ¯ )2 1 X (1) u BˆES,+ = 1(Xi < x¯) µˆ(1) (X ) 1 (X < x ¯ ) µ (X ) = + oP (1) i i i + +,kn 12n 12 n i=1 i=1 Z (xu − x ¯)2 xu (1) 2 µ+ (x) w(x)dx + oP (1), = 12 x ¯
which gives BˆES,+ →P BES,+ . Putting the above together, consistency of all the data-driven selectors follows.
2.7
Proof of Remark 1
Note that for power series estimators, Newey (1997, Theorem 4) gives sup |ˆ µ+,kn ,p (x) − E[Y (1)p |Xi = x]|2 = OP (kn3 /n + kn−2S+2 ) = oP (1)
x∈[¯ x,xu ]
11
for p = 1, 2, under the assumptions imposed, which implies 2 2 2 sup |ˆ σ+ (x) − σ+ | = OP (kn3 /n + kn−2S+2 ) = oP (1).
x∈[¯ x,xu ]
2 (x), Using this result, and Lemma SA3 with ` = 1 and g(x) = σ+ N+
VˇES,+ =
1 X 2 ¯ +,(i) ) (X+,(i) − X+,(i−1) )ˆ σ+,k (X xu − x ¯ i=2 N+
=
1 1 X 2 ¯ +,(i) ) + oP (1) →P (X+,(i) − X+,(i−1) )σ+,k (X xu − x ¯ xu − x ¯ i=2
Z
xu
2 σ+ (x)dx = VES,+ .
x ¯
Combining this with Theorem SA1, the different consistency results follow.
2.8
Proof of Theorem 4 and Remark 2
Proceeding as in the proofs of Theorem 3 and Remark 1, the results are established using Lemma SA3, N+ /n →P P+ , and uniform consistency of power series estimators, as appropriate for each case.
3
Data-Driven Implementations with Arbitrary w(x)
In this section we provide data-driven implementations for all of our number of bins selectors when w(x) is taken as given. As discussed in the main text, we estimate the unknown constants using ideas related to spacings estimators whenever possible, but we also discuss series (polynomial) nonparametric regression estimates for completeness (to handle the non-continuous outcome case). Recall the notation introduced in the main paper related to order statistics and concomitants. For a collection of continuous random variables {(Zi , Wi ) : i = 1, 2, · · · , n} we let W(i) be the i-th order statistic of Wi and Z[i] its corresponding concomitant. That is, W(1) < W(2) < · · · < W(n) and (Z[i] , W(i) ) = (Zi , W(i) ) for all i = 1, 2, · · · , n. Letting {(Y−,i , X−,i ) : i = 1, 2, · · · , N− } and {(Y+,i , X+,i ) : i = 1, 2, · · · , N+ } be the subsamples of control (Xi < x ¯) and treatment (Xi ≥ x ¯)
12
units, respectively. We also have: ¯ −,(i) = X
X−,(i) + X−,(i−1) , 2
i = 2, 3, · · · , N− ,
ˆ , µ ˆ−,k (x) = rk (x)0 β −,k
¯ +,(i) = X
X+,(i) + X+,(i−1) , 2
i = 2, 3, · · · , N+ ,
ˆ , µ ˆ+,k (x) = rk (x)0 β +,k
(1)
(1)
(1)
(1)
(1)
and rk (x) = ∂rk (x)/∂x = (0, 1, 2x, 3x2 , · · · , kxk−1 )0 .
3.1
Evenly Spaced RD Plots
For the case of ES RD Plots with generic w(x) weighting scheme, we propose the following estimators:
N−
VˆES,− =
1 nX ¯ −,(i) ), (X−,(i) − X−,(i−1) )2 (Y−,[i] − Y−,[i−1] )2 w(X x ¯ − xl 4
(SA-1)
i=2
N− 2 (¯ x − xl )2 X (1) ¯ ¯ −,(i) ), BˆES,− = (X−,(i) − X−,(i−1) ) µ ˆ−,k (X ) w(X −,[i] 12
(SA-2)
i=2
and
N+
VˆES,+ =
1 nX ¯ +,(i) ), (X+,(i) − X+,(i−1) )2 (Y+,[i] − Y+,[i−1] )2 w(X xu − x ¯4
(SA-3)
i=2
N+ 2 (xu − x ¯)2 X (1) ¯ ˆ ¯ +,(i) ). (X+,(i) − X+,(i−1) ) µ ˆ+,k (X BES,+ = w(X +,[i] ) 12
(SA-4)
i=2
Thus, our proposed data-driven selectors for ES RD Plots take the form:
JˆES-µ,−,n
2BˆES,− = VˆES,−
JˆES-ω,−,n = ω−
2BˆES,− VˆES,− &
JˆES-ϑ,−,n =
!1/3
n1/3
!1/3
and JˆES-µ,+,n
n1/3
Vˆ− n ˆ VES,− log(n)2
2BˆES,+ = VˆES,+
and JˆES-ω,+,n = ω+ '
& and JˆES-ϑ,+,n =
!1/3
2BˆES,+ VˆES,+
n1/3 ,
(SA-5)
!1/3
n1/3 ,
' Vˆ+ n , VˆES,+ log(n)2
(SA-6)
(SA-7)
using the estimators in (SA-1)–(SA-4), and where Vˆ− and Vˆ+ are consistent estimators of their population counterparts V− and V+ . The following theorem shows that, when the polynomial 13
fits are viewed as nonparametric approximations with k = kn → ∞, the different number of bins selectors are nonparametric consistent. Theorem SA1. Suppose Assumption 1 holds with S ≥ 5, w : [xl , xu ] 7→ R+ is continuous, and Yi (0) and Yi (1) are continuously distributed. If kn7 /n → 0 and kn → ∞, then JˆES-ϑ,−,n →P 1, JES-ϑ,−,n
JˆES-ω,−,n →P 1, JES-ω,−,n
JˆES-ω,+,n →P 1, JES-ω,+,n
JˆES-ϑ,+,n →P 1, JES-ϑ,+,n
provided that Vˆ− →P V− and Vˆ+ →P V+ . Proof of Theorem SA1. Using Lemma A3 with k = 2 and N+ /n →P P+ , N+
VˆES,+ =
1 nX ¯ +,(i) ) (X+,(i) − X+,(i−1) )2 (Y+,[i] − Y+,[i−1] )2 w(X xu − x ¯4 i=2 N+ N+ X
1 ¯ +,(i) ) + oP (1) (X+,(i) − X+,(i−1) )2 (Y+,[i] − Y+,[i−1] )2 w(X xu − x ¯ 4P+ i=2 Z xu 2 σ+ (x) 1 w(x)dx + oP (1), = xu − x ¯ x¯ f+ (x) =
which gives VˆES,+ →P VES,+ . Similarly, VˆES,− →P VES,− . (1)
(1)
Next, recall that for power series estimators supx∈[¯x,xu ] |ˆ µ+,kn (x) − µ+ (x)|2 = OP (kn7 /n + kn−2S+8 ) = oP (1). Using this uniform consistency result, and Lemma A3 with k = 1, we have N+ 2 (xu − x ¯ )2 X (1) ¯ +,(i) ) w(X ¯ +,(i) ) BˆES,+ = (X+,(i) − X+,(i−1) ) µ ˆ+,kn (X 12
= =
(xu − 12 (xu − 12
i=2 N+ 2 x ¯) X
x ¯)2
2 (1) ¯ ¯ +,(i) ) + oP (1) w(X (X+,(i) − X+,(i−1) ) µ+ (X ) +,(i)
i=2 xu
Z
x ¯
2 (1) µ+ (x) w(x)dx + oP (1),
which gives BˆES,+ →P BES,+ . Similarly, BˆES,− →P BES,− .
Recall that the special case ωV ,− = ωV ,+ = 1/2 gives JˆES-µ,−,n = JˆES-ω,−,n and JˆES-µ,+,n = JˆES-ω,+,n . Theorem SA1 therefore gives a formal justification for employing any of the selectors introduced in our paper for the number of bins in ES RD Plots constructed with a known, arbitrary 14
weight function w(x); a particular choice being w(x) = 1. As discussed in the main text, when Yi (0) and Yi (1) are not continuously distributed, the concomitant-based estimation method becomes invalid. In this case, we need to employ other more standard nonparametric techniques. For example, assuming that E[Yi (t)2 |Xi = x], t = 0, 1, are twice continuously differentiable, we can use the following estimators: N−
VˇES,− =
1 nX 2 ¯ −,(i) )w(X ¯ −,(i) ), (X−,(i) − X−,(i−1) )2 σ ˆ−,k (X x ¯ − xl 2 i=2
N+
VˇES,+ =
1 nX 2 ¯ +,(i) )w(X ¯ +,(i) ), (X+,(i) − X+,(i−1) )2 σ ˆ+,k (X xu − x ¯2 i=2
2 (x) = µ ˆ+,k,2 (x) − (ˆ µ+,k,1 (x))2 , σ ˆ+,k
2 (x) = µ ˆ−,k,2 (x) − (ˆ µ−,k,1 (x))2 , σ ˆ−,k
where, for k ∈ Z+ and p ∈ Z++ , ˆ µ ˆ−,k,p (x) = rk (x)0 β −,k,p ,
ˆ µ ˆ+,k,p (x) = rk (x)0 β +,k,p ,
ˆ β −,k,p = arg min
β∈Rk+1
ˆ β +,k,p = arg min
β∈Rk+1
n X
1(Xi < x¯)(Yip − rk (Xi )0 β)2 ,
i=1 n X
1(Xi ≥ x¯)(Yip − rk (Xi )0 β)2 ,
i=1
and note that µ ˆ−,k (x) = µ ˆ−,k,1 (x) and µ ˆ+,k (x) = µ ˆ+,k,1 (x) with our notation. From results for power series estimators, sup |ˆ µ+,kn ,p (x) − E[Y (1)p |Xi = x]|2 = OP (kn3 /n + kn−2S+2 ) = oP (1)
x∈[¯ x,xu ]
for p = 1, 2, under the assumptions imposed, which implies 2 2 2 sup |ˆ σ+ (x) − σ+ | = OP (kn3 /n + kn−2S+2 ) = oP (1).
x∈[¯ x,xu ]
15
Therefore, Lemma A3 with k = 2 and N+ /n →P P+ , N+
VˇES,+ =
1 nX 2 ¯ ¯ +,(i) ) (X+,(i) − X+,(i−1) )2 σ ˆ+ (X+,(i) )w(X xu − x ¯2 i=2 N+ N+ X
1 2 ¯ ¯ +,(i) ) + oP (1) (X+,(i) − X+,(i−1) )2 σ+ (X+,(i) )w(X xu − x ¯ 2P+ i=2 Z xu 2 σ+ (x) 1 = w(x)dx + oP (1), xu − x ¯ x¯ f+ (x) =
which gives VˇES,+ →P VES,+ . Similarly, VˇES,− →P VES,− . Combining these results with Theorem SA1, it can easily be shown that the following selectors are consistent for any continuous, arbitrary choice of w(x):
JˇES-µ,−,n
2BˆES,− = Vˇ ES,−
JˇES-ω,−,n = ω−
2BˆES,− VˇES,− &
JˇES-ϑ,−,n =
!1/3
n1/3
and JˇES-µ,+,n
!1/3
n1/3
n Vˆ− ˇ VES,− log(n)2
2BˆES,+ = Vˇ ES,+
and JˇES-ω,+,n = ω+ '
& and JˇES-ϑ,+,n =
!1/3
2BˆES,+ VˇES,+
n1/3 ,
(SA-8)
!1/3
n1/3 ,
' n Vˆ+ , VˇES,+ log(n)2
(SA-9)
(SA-10)
provided that Vˆ− →P V− and Vˆ+ →P V+ .
3.2
Quantile Spaced RD Plots
We discuss generic estimators for QS RD Plots employing an arbitrary, known weighting function w(x), paralleling the results given above for ES RD Plots. The underlying estimators are: N− n X ¯ −,(i) ), (X−,(i) − X−,(i−1) )(Y−,[i] − Y−,[i−1] )2 w(X 2N−
(SA-11)
N− 2 N−2 X (1) ¯ ˆ ¯ −,(i) ), BQS,− = (X−,(i) − X−,(i−1) )3 µ ˆ−,k (X ) w(X −,(i) 72
(SA-12)
VˆQS,− =
i=2
i=2
16
and
N+ n X ¯ +,(i) ), (X+,(i) − X+,(i−1) )(Y+,[i] − Y+,[i−1] )2 w(X 2N+
(SA-13)
N+ 2 N+2 X (1) ¯ ˆ ¯ +,(i) ). BQS,+ = w(X (X+,(i) − X+,(i−1) )3 µ ˆ+,k (X +,(i) ) 72
(SA-14)
VˆQS,+ =
i=2
i=2
Therefore, the resulting selectors for QS partitions take the form:
JˆQS-µ,−,n
2BˆQS,− = VˆQS,−
JˆQS-ω,−,n = ω−
2BˆQS,− VˆQS,−
!1/3
n1/3
!1/3
& JˆQS-ϑ,−,n =
and JˆQS-µ,+,n
n1/3
Vˆ− n ˆ VQS,− log(n)2
2BˆQS,+ = VˆQS,+
and JˆQS-ω,+,n = ω+ '
& and JˆQS-ϑ,+,n =
!1/3
2BˆQS,+ VˆQS,+
n1/3 ,
(SA-15)
!1/3
n1/3 ,
' Vˆ+ n , VˆQS,+ log(n)2
(SA-16)
(SA-17)
using the estimators in (SA-11)–(SA-14), and appropriate consistent estimators Vˆ− and Vˆ+ . As in the case of Theorem SA1 for ES RD plots, the following theorem shows that these automatic partition-size selectors are nonparametric consistent if the polynomial fits are viewed as nonparametric approximations with k = kn → ∞. Theorem SA2. Suppose Assumption 1 holds with S ≥ 5, w : [xl , xu ] 7→ R+ is continuous, and Yi (0) and Yi (1) are continuously distributed. If kn7 /n → 0 and kn → ∞, then JˆQS-ω,−,n →P 1, JQS-ω,−,n
JˆQS-ϑ,−,n →P 1, JQS-ϑ,−,n
JˆQS-ω,+,n →P 1, JQS-ω,+,n
JˆQS-ϑ,+,n →P 1, JQS-ϑ,+,n
provided that Vˆ− →P V− and Vˆ+ →P V+ . In practice, the choice w(x) = 1 is arguably the simplest one, but our results permit any continuous function w(x). The proof of Theorem SA2 is very similar to the proof of Theorem SA1 given above, and hence omitted here to conserve space. Next, for the case of non-continuous potential outcomes Yi (0) and Yi (1), we use the series polynomial estimation approach already introduced. Assuming that E[Yi (t)2 |Xi = x], t = 0, 1, are 17
twice continuously differentiable, we may use the following estimators: N− n X 2 ˇ ¯ −,(i) )w(X ¯ −,(i) ), VQS,− = (X−,(i) − X−,(i−1) )ˆ σ−,k (X N− i=2
N+ n X 2 ¯ +,(i) )w(X ¯ +,(i) ), VˇQS,+ = (X+,(i) − X+,(i−1) )ˆ σ+,k (X N+ i=2
2 (x) and σ 2 (x) are the polynomial approximations already discussed. The associated where σ ˆ−,k ˆ+,k
data-driven partition-size selectors are
JˇQS-µ,−,n
2BˆQS,− = Vˇ QS,−
JˇQS-ω,−,n = ω−
2BˆQS,− VˇQS,− &
JˇQS-ϑ,−,n =
!1/3
n1/3
!1/3
and JˇQS-µ,+,n
n1/3
n Vˆ− ˇ VQS,− log(n)2
2BˆQS,+ = Vˇ QS,+
and JˇQS-ω,+,n = ω+ '
& and JˇQS-ϑ,+,n =
!1/3
2BˆQS,+ VˇQS,+
n1/3 ,
(SA-18)
!1/3
n1/3 ,
' n Vˆ+ , VˇQS,+ log(n)2
(SA-19)
(SA-20)
which are easily shown to be consistent in the sense of Theorem SA2, provided the conditions in that theorem hold.
4
Other Empirical Applications
In this section we include three additional empirical applications to illustrate the performance of our proposed methods when applied to different real datasets. Software packages in R and STATA are described in Calonico et al. (2015, 2014a).
4.1
U.S. Senate Data
We employ an extract of the dataset constructed by Cattaneo et al. (2015), who study several measures of incumbency advantage in U.S. Senate elections for the period 1914–2010. In particular, we focus here on the RD effect of the Democratic party winning a U.S. Senate seat on the vote share obtained in the following election for that same seat. This empirical illustration is analogous
18
to the one presented by Lee (2008) for U.S. House elections: the running variable is the state-level margin of victory of the Democratic party in an election for a Senate seat, the threshold is x ¯=0 and the outcome is the vote share of the Democratic party in the following election for the same Senate seat in the state, which occurs six years later. The unit of observation is the state, and the data set has a total of n = 1, 297 state-year complete observations. Results are presented in Figures SA-1 and SA-2.
4.2
Progresa/Oportunidades Data
We illustrate the performance of our methods employing household data from Oportunidades (formerly known as Progresa), a well-known large-scale anti-poverty conditional cash transfer program in Mexico. This conditional cash transfer program targeted poor households in rural and urban areas in Mexico. The program started in 1998 under the name of Progresa in rural areas. The most important elements of the program are the nutrition, health and education components. The nutrition component consists of a cash grant for all treated households and an additional supplement for households with young children and pregnant or lactating mothers. The educational grant is linked to regular attendance in school and starts on the third grade of primary school and continues until the last grade of secondary school. The transfer constituted a significant contribution to the income of eligible families. This social program is best known for its experimental design: treatment was initially randomly assigned at the locality level in rural areas. Progresa was expanded to urban areas urban in 2003. Unlike the rural program, the allocation across treatment and control areas was not random. Instead, it was first offered in blocks with the highest density of poor households. In order to accurately target the program to poor households, in both rural and urban areas Mexican officials constructed a pre-intervention (at baseline) household poverty-index that determined each household’s eligibility. Thus, Progresa/Oportunidades’ eligibility assignment rule naturally leads to sharp (intention-to-treat) regression-discontinuity designs. For additional details for data construction, empirical analysis and related literature, see Calonico et al. (2014b, Section S.4). Our empirical exercise investigates the program treatment effect on household non-food consumption expenditures two years after its implementation. In this application, Xi denotes the 19
household’s poverty-index, x ¯ = 0 denotes the centered cutoff for each RD design, and Yi denotes per capita non-food consumption. Our final database contains 691 control households (Xi < 0) and 2, 118 intention-to-treat households (Xi ≥ 0) in the urban RD design (n = 2, 809, Xi ∈ [−2.25 , 4.11]). Results are presented in Figures SA-3 and SA-4.
4.3
Head Start Data
Head Start is a program of the United States Department of Health and Human Services that provides early childhood education, health, nutrition, and parent involvement services to lowincome children and their families. It was established in 1965 as part of the War on Poverty, in order to foster stable family relationships, enhance children’s physical and emotional well-being, and establish an environment to develop cognitive skills. For each county, eligibility is based on the county’s poverty rate, inducing a natural RD design. Ludwig and Miller (2007) uses this to identify the program’s effects on health and schooling. For each county i = 1, 2, ..., n, the forcing variable is the county’s 1960 poverty rate with treatment assignment given by Ti = 1(Xi ≥ x ¯), where Xi represents the county’s poverty rate in 1960 and x ¯ is the fixed threshold level. The cutoff is set to the poverty rate value of the 300th poorest county in 1960, which in this dataset is given by x ¯ = 59.198. Here we consider as outcome variable the mortality rates per 100, 000 for children between 5–9 years old, with Head Start-related causes, for 1973 − 1983 (see Panel A, Figure IV in Ludwig and Miller (2007)). Results are presented in Figures SA-5 and SA-6.
4.4
Summary of Results
In all the empirical applications we considered, the data-driven selectors introduced in the main paper seemed to perform very well. The mimicking variance selector for the number of bins consistently delivered a disciplined “cloud of points”, which appears to be substantially more useful than the scatter plot of the raw data. In addition, the IMSE-optimal choice of number of bins also performed well, in all cases “tracing out” the estimated smooth polynomial regression fits. As for the implementations, spacings estimators perform on par with polynomial estimators in all the 20
applications considered. Finally, it is worth noting that ES and QS RD plots do not necessarily deliver different number of bins. For example, in the Head Start data set, the mimicking variance choices are essentially identical for both types of RD plots.
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N− = 595 ; N+ = 702
(d) Scatter Plot of Raw Data
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(a) Scatter Plot of Raw Data
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(b) Mimicking Variance, Spacings JˆES-ϑ,−,n = 15 ; JˆES-ϑ,+,n = 35
Vote Share in Election at time t+1
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(e) Mimicking Variance, Series JˇES-ϑ,−,n = 21 ; JˇES-ϑ,+,n = 36
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(f) IMSE-optimal, Series ˇ JES-µ,−,n = 8 ; JˇES-µ,+,n = 9
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(c) IMSE-optimal, Spacings JˆES-µ,−,n = 8 ; JˆES-µ,+,n = 9
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Figure SA-1: Scatter Plot and Automatic Data-driven ES RD Plots for Senate Elections Data
100
100
Notes: (i) sample size is n = 1, 297; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
Vote Share in Election at time t
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Vote Share in Election at time t Vote Share in Election at time t
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N− = 595 ; N+ = 702
(d) Scatter Plot of Raw Data
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N− = 595 ; N+ = 702
(a) Scatter Plot of Raw Data
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(e) Mimicking Variance, Series JˇQS-ϑ,−,n = 29 ; JˇQS-ϑ,+,n = 50
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(f) IMSE-optimal, Series ˇ JQS-µ,−,n = 21 ; JˇQS-µ,+,n = 16
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(c) IMSE-optimal, Spacings ˆ JQS-µ,−,n = 21 ; JˆQS-µ,+,n = 16
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Figure SA-2: Scatter Plot and Automatic Data-driven QS RD Plots for Senate Elections Data
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Notes: (i) sample size is n = 1, 297; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
Vote Share in Election at time t
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(d) Scatter Plot of Raw Data
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N− = 691 ; N+ = 2, 118
(a) Scatter Plot of Raw Data
−2
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(e) Mimicking Variance, Series JˇES-ϑ,−,n = 77 ; JˇES-ϑ,+,n = 67
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(b) Mimicking Variance, Spacings JˆES-ϑ,−,n = 69 ; JˆES-ϑ,+,n = 59
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(f) IMSE-optimal, Series ˇ JES-µ,−,n = 7 ; JˇES-µ,+,n = 9
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(c) IMSE-optimal, Spacings JˆES-µ,−,n = 7 ; JˆES-µ,+,n = 9
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Notes: (i) sample size is n = 2, 809; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
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Figure SA-3: Scatter Plot and Automatic Data-driven ES RD Plots for Progresa/Oportunidades (Urban Localities).
Per Capita Food Consumption
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Per Capita Food Consumption Per Capita Food Consumption
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−1
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0
1 Poverty Index
2
3
N− = 691 ; N+ = 2, 118
(d) Scatter Plot of Raw Data
−2
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(a) Scatter Plot of Raw Data
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(e) Mimicking Variance, Series JˇQS-ϑ,−,n = 46 ; JˇQS-ϑ,+,n = 47
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(b) Mimicking Variance, Spacings JˆQS-ϑ,−,n = 45 ; JˆQS-ϑ,+,n = 46
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(f) IMSE-optimal, Series ˇ JQS-µ,−,n = 36 ; JˇQS-µ,+,n = 14
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(c) IMSE-optimal, Spacings ˆ JQS-µ,−,n = 36 ; JˆQS-µ,+,n = 14
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Notes: (i) sample size is n = 2, 809; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
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Figure SA-4: Scatter Plot and Automatic Data-driven QS RD Plots for Progresa/Oportunidades (Urban Localities).
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Ages 5−9, Head Start−related causes, 1973−1983
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●
●
●
● ● ● ● ●●● ●
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●
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●
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● ● ● ●● ●
●
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●
●
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●
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●
●
●
●
●
●
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●
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40
40 Poverty Rate
60
60
80
●●●●
●
N− = 2, 810 ; N+ = 294
(d) Scatter Plot of Raw Data
20 20
● ● ●●● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●●● ● ●●● ● ● ●●●●● ●● ●● ●●● ●● ● ●● ● ● ●●● ●●●●● ●●● ●● ● ●
●
●
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● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ●●● ● ● ●● ●● ● ● ● ● ●● ● ●●● ● ● ●● ● ● ●●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ●● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●●●●● ●● ● ● ● ● ●●●● ●● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ●●●●●● ●● ● ● ● ●● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ●●●● ● ●● ● ● ● ● ●● ●●● ● ●●● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ● ●●● ● ●●● ● ●● ●● ● ● ●● ●●● ● ● ● ●● ● ● ● ●●●●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ●● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ●●●●● ● ●●● ● ● ●● ● ●● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ●●●●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ● ● ● ● ● ● ● ●● ● ●●●●● ●● ●●● ● ● ● ● ● ●● ● ●● ●● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ●●●● ●●●●● ● ●● ●●●●●● ●● ● ● ● ● ● ●●●● ● ● ● ●●●●●● ● ●●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ●●● ●● ● ● ● ● ● ● ● ●●● ● ● ●
●
●●
●
N− = 2, 810 ; N+ = 294
(a) Scatter Plot of Raw Data
20 20
● ● ●●● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●●● ● ●●● ● ● ●●●●● ●● ●● ●●● ●● ● ●● ● ● ●●● ●●●●● ●●● ●● ● ●
●
●
●
●
●
●
●
80
80
0
0
●
● ● ●
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●
● ● ● ●
20
●
●
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●
●
●
●
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● ●
●
●
●
40
●
● ●
●
Poverty Rate
●
●
● ●
●
●
●
● ● ● ●
●
●
60
●●
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●
●
●
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●
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●
●
●
●
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●
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●
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●
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80
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●
●
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●
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●
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20
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●
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● ●
● ●
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40
● ●
●
Poverty Rate
●
●
●
●
●
● ●
● ● ●
●
60
●
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●
●
● ●
●
●
●
●
● ●
● ●
●
●
●
●
●
●
●
●
●
●
●
●● ● ●
●
●
●●●
●
● ● ●
●●
●
●
●
(e) Mimicking Variance, Series JˇES-ϑ,−,n = 39 ; JˇES-ϑ,+,n = 52
●
80
● ●●
●
(b) Mimicking Variance, Spacings JˆES-ϑ,−,n = 45 ; JˆES-ϑ,+,n = 42
●
●
0
0
40 Poverty Rate
60
40 Poverty Rate
60
(f) IMSE-optimal, Series ˇ JES-µ,−,n = 6 ; JˇES-µ,+,n = 6
20
(c) IMSE-optimal, Spacings JˆES-µ,−,n = 6 ; JˆES-µ,+,n = 6
20
Figure SA-5: Scatter Plot and Automatic Data-driven ES RD Plots for Head Start Assistance
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15 10 5
15 10 5
80
80
Notes: (i) sample size is n = 3, 104; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
Ages 5−9, Head Start−related causes, 1973−1983
−5
15
10
5
0
−5
Ages 5−9, Head Start−related causes, 1973−1983 Ages 5−9, Head Start−related causes, 1973−1983
0 −5 15 10 5 0 −5
Ages 5−9, Head Start−related causes, 1973−1983 Ages 5−9, Head Start−related causes, 1973−1983
0 −5 15 10 5 0 −5
26
Ages 5−9, Head Start−related causes, 1973−1983
15
10
5
0
0 0
0 0
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40
40 Poverty Rate
60
60
80
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40
40 Poverty Rate
60
60
80
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N− = 2, 810 ; N+ = 294
(d) Scatter Plot of Raw Data
20 20
● ● ●●● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●●● ● ●●● ● ● ●●●●● ●● ●● ●●● ●● ● ●● ● ● ●●● ●●●●● ●●● ●● ● ●
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●●
●
N− = 2, 810 ; N+ = 294
(a) Scatter Plot of Raw Data
20 20
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(f) IMSE-optimal, Series ˇ JQS-µ,−,n = 31 ; JˇQS-µ,+,n = 6
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(c) IMSE-optimal, Spacings JˆQS-µ,−,n = 31 ; JˆQS-µ,+,n = 6
20
Figure SA-6: Scatter Plot and Automatic Data-driven QS RD Plots for Head Start Assistance
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Notes: (i) sample size is n = 3, 104; (ii) N− and N+ denote the sample sizes for control and treatment units, respectively; (iii) solid blue lines depict 4th order polynomial fits using control and treated units separately.
Ages 5−9, Head Start−related causes, 1973−1983
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Ages 5−9, Head Start−related causes, 1973−1983 Ages 5−9, Head Start−related causes, 1973−1983
0 −5 15 10 5 0 −5
Ages 5−9, Head Start−related causes, 1973−1983 Ages 5−9, Head Start−related causes, 1973−1983
0 −5 15 10 5 0 −5
27
5
Complete Simulation Results
We report the results from a Monte Carlo experiment to study the finite-sample behavior of our proposed methods. We consider several data generating processes, which vary in the distribution of the running variable, the conditional variance, and the distribution of the unobserved error term in the regression function. Specifically, the data is generated as i.i.d. draws, {(Yi , Xi )0 : i = 1, 2, ..., n} following
Yi = µ(Xi ) + εi ,
where µ(x) =
Xi ∼ Fx ,
εi ∼ σ(Xi )Fε ,
0.48 + 1.27x + 7.18x2 + 20.21x3 + 21.54x4 + 7.33x5
if x < 0
0.52 + 0.84x − 3.00x2 + 7.99x3 − 9.01x4 + 3.56x5
if x ≥ 0
,
and Fx equals either (2B(p1 , p2 ) − 1), with B(p1 , p2 ) denoting a Beta distribution with parameters p1 and p2 , or equals a mixture of two normal distributions with means µ1 and µ2 , respectively, same standard deviations set to 1/4 and mixing weights ω1 and ω2 , respectively. In addition, σ(x) is either equal to 1 (homoskedasticity) or equal to exp(−|x|/2) (heteroskedasticity), and Fε is either √ N (0, 1) or (χ4 − 4)/ 8. The functional form of µ(x) is obtained by fitting a 5-th order global polynomial with different coefficients for control and treatment units separately using the original data of Lee (2008), after discarding observations with past margin of victory greater than 99 and less than −99 percentage points. Figure SA-7 plots the regression function µ(x) and the different choices for the density of Xi . Notice that some of these densities take on “low” values in some regions of the support of Xi , in same cases near the RD cutoff. Our Monte Carlo experiment considers 16 models that combine different choices of Fx , σ(x) and Fε , as described in Table SA-1. For each model in Table SA-1, we set n = 5, 000 and generate 5, 000 simulations to compute the IMSE of both ES and QS partitioning schemes for different possible number of bins, as well as for the IMSE-optimal data-driven selector proposed. In each case considered, we also computed the mimicking variance selectors introduced in the paper, both infeasible and data-driven versions. All tables include results for both ES and QS RD plots organized in two distinct panels. Panel 28
A focuses attention on the IMSE of different partitioning schemes in finite samples, as well as the performance of the associated IMSE-optimal data-driven selectors. All IMSEs are normalized relative to the IMSE evaluated at the optimal partition-size choice to avoid any scaling issue. Panel B reports several features of the empirical (finite-sample) distribution of the different data-driven number of bins selectors introduced in this paper: (i) spacings-based selectors for ES RD plots, (ii) polynomial-based selectors for ES RD plots, (iii) spacings-based selectors for QS RD plots, and (iv) polynomial-based selectors for QS RD plots. Therefore, our Monte Carlo experiment is designed to capture the finite-sample performance of Theorems 1 and 2 in terms of providing a good approximation to the IMSE (Panel A), and the finite-sample performance of Theorems 3 and 4 as well as the other consistency results discussed in the remarks in the paper (Panel B). The results of our simulation experiment are very encouraging. First, in all cases the IMSE is minimized at the corresponding IMSE-optimal number of bins choice derived in the paper, suggesting that Theorems 1 and 2 provide a good finite-sample approximation. The theoretical IMSE-optimal number of bins almost always exactly coincides with the simulated IMSE-optimal number of bins. Second, in all models we find that our proposed data-driven implementations of the different number of bins selectors perform quite well, exhibiting a concentrated finite-sample distribution centered at the target population (optimal) choice introduced in this paper. That is, the summary statistics in Panel B of each table show that our data-driven implementations of the population selectors choices have a finite sample distribution well centered and concentrated around their population targets, when using either spacings estimators or polynomial estimators. In sum, our extensive simulation study indicates that the different data-driven number of bins selectors underlying the construction of the RD plots perform well in finite samples.
29
0.0
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µ(x)
0.6
0.8
1.0
Figure SA-7: Data Generating Processes
−1.0
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0.5
1.0
x
1.0
(a) Regression function, µ(x).
0.0
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0.4
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0.6
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p1 = 1, p2 = 1 p1 = 1 2, p2 = 1 2 p1 = 4 5, p2 = 1 5 p1 = 1 5, p2 = 4 5
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(b) Xi ’s distribution, B(p1 , p2 ).
0.0
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µ = (−1/4 , 1/4), ω = (1/2 , 1/2) µ = (−1/2 , 1/2), ω = (1/2 , 1/2) µ = (−1/2 , 1/2), ω = (4/5 , 1/5) µ = (−1/2 , 1/2), ω = (1/5 , 4/5)
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(c) Xi ’s distribution, Mixture of Normals
30
1.0
Table SA-1: Data Generating Processes Panel A: Models 1 to 8 Model 1 2 3 4 5 6 7 8
p1 1 0.5 0.2 0.8 1 0.5 0.2 0.8
p2 1 0.5 0.8 0.2 1 0.5 0.8 0.2
σ 2 (x) 1 1 exp(−|x|/2) exp(−|x|/2) 1 1 exp(−|x|/2) exp(−|x|/2)
Fε N (0, 1) N (0, 1) N (0, 1) N (0, 1) √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8
Panel B: Models 9 to 16 Model 9 10 11 12 13 14 15 16
µ1 -0.25 -0.5 -0.5 -0.5 -0.25 -0.5 -0.5 -0.5
µ2 0.25 0.5 0.5 0.5 0.25 0.5 0.5 0.5
ω1 0.5 0.5 0.8 0.2 0.5 0.5 0.8 0.2
ω2 0.5 0.5 0.2 0.8 0.5 0.5 0.2 0.8
31
σ 2 (x) 1 1 exp(−|x|/2) exp(−|x|/2) 1 1 exp(−|x|/2) exp(−|x|/2)
Fε N (0, 1) N (0, 1) N (0, 1) N (0, 1) √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8 √ (χ4 − 4)/ 8
Table SA-2: Simulations Results for Model 1 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
20 21 22 23 24 25 26 27 28 29 30
1.047 1.027 1.013 1.005 1.000 1.000 1.003 1.008 1.016 1.025 1.036
11 12 13 14 15 16 17 18 19 20 21
1.148 1.081 1.039 1.014 1.002 1.000 1.006 1.017 1.033 1.053 1.076
20 21 22 23 24 25 26 27 28 29 30
1.047 1.027 1.013 1.005 1.000 1.000 1.003 1.008 1.016 1.025 1.036
11 12 13 14 15 16 17 18 19 20 21
1.148 1.081 1.039 1.014 1.002 1.000 1.006 1.017 1.033 1.053 1.076
JˆES-µ,−,n JˇES-µ,−,n
1.033 1.034
JˆES-µ,+,n JˇES-µ,+,n
0.9435 0.9428
JˆQS-µ,−,n JˇQS-µ,−,n
1.072 1.073
JˆQS-µ,+,n JˇQS-µ,+,n
0.9351 0.9347
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 25 JES-ϑ,−,n = 118
JES-µ,+,n = 16 JES-ϑ,+,n = 116
JQS-µ,−,n = 25 JQS-ϑ,−,n = 118
JQS-µ,+,n = 16 JQS-ϑ,+,n = 116
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
22 23 105 110
25 25 116 117
26 26 120 119
25.95 25.93 119.6 119.3
27 26 123 121
29 29 139 131
0.93 0.87 5.05 2.72
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
14 14 103 107
15 15 113 115
15 15 117 117
15.34 15.34 116.7 116.7
16 16 120 118
17 17 139 128
0.57 0.55 4.71 2.65
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
23 23 108 110
26 26 117 117
27 27 120 119
26.91 26.89 119.6 119.3
27 27 122 121
30 30 134 131
0.92 0.90 3.66 2.71
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
14 14 106 107
15 15 114 115
15 15 117 117
15.21 15.21 116.6 116.7
15 15 119 118
17 17 130 128
0.51 0.50 3.50 2.65
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
32
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-3: Simulations Results for Model 2 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
26 27 28 29 30 31 32 33 34 35 36
1.032 1.019 1.010 1.004 1.001 1.000 1.001 1.004 1.009 1.015 1.022
11 12 13 14 15 16 17 18 19 20 21
1.157 1.088 1.043 1.017 1.003 1.000 1.004 1.015 1.030 1.050 1.072
19 20 21 22 23 24 25 26 27 28 29
1.047 1.026 1.012 1.004 1.000 1.000 1.004 1.010 1.019 1.029 1.042
13 14 15 16 17 18 19 20 21 22 23
1.086 1.045 1.018 1.004 0.998 1.000 1.007 1.019 1.035 1.054 1.075
JˆES-µ,−,n JˇES-µ,−,n
1.086 1.088
JˆES-µ,+,n JˇES-µ,+,n
0.9009 0.9005
JˆQS-µ,−,n JˇQS-µ,−,n
0.9271 0.9292
JˆQS-µ,+,n JˇQS-µ,+,n
0.9399 0.9394
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 31 JES-ϑ,−,n = 114
JES-µ,+,n = 16 JES-ϑ,+,n = 118
JQS-µ,−,n = 24 JQS-ϑ,−,n = 114
JQS-µ,+,n = 18 JQS-ϑ,+,n = 118
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
30 31 98 104
33 33 112 112
34 34 115 114
34.13 34.08 115.1 114.5
35 35 118.2 117
39 38 134 126
1.09 1.01 5.18 3.05
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
13 13 102 110
14 14 116 118
15 15 120 120
14.84 14.83 120.3 120.2
15 15 124 122
18 17 145 133
0.72 0.70 5.63 3.22
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
21 21 104 106
22 22 112 113
22 22 115 114
22.24 22.2 114.8 114.4
23 22 117 116
24 24 128 124
0.53 0.50 3.46 2.56
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
15 15 108 109
16 16 117 118
17 17 120 120
16.71 16.72 119.9 119.9
17 17 122 122
20 20 134 132
0.65 0.65 3.66 2.81
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
33
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-4: Simulations Results for Model 3 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
49 50 51 52 53 54 55 56 57 58 59
1.008 1.005 1.002 1.001 1.000 1.000 1.001 1.002 1.004 1.006 1.009
8 9 10 11 12 13 14 15 16 17 18
1.279 1.149 1.071 1.027 1.005 1.000 1.007 1.022 1.044 1.071 1.102
40 41 42 43 44 45 46 47 48 49 50
1.010 1.006 1.002 1.000 1.000 1.000 1.001 1.003 1.006 1.010 1.014
8 9 10 11 12 13 14 15 16 17 18
1.265 1.139 1.064 1.023 1.003 1.000 1.008 1.025 1.048 1.076 1.108
JˆES-µ,−,n JˇES-µ,−,n
1.09 1.097
JˆES-µ,+,n JˇES-µ,+,n
0.9534 0.9504
JˆQS-µ,−,n JˇQS-µ,−,n
0.869 0.872
JˆQS-µ,+,n JˇQS-µ,+,n
0.9628 0.9609
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 54 JES-ϑ,−,n = 112
JES-µ,+,n = 13 JES-ϑ,+,n = 149
JQS-µ,−,n = 45 JQS-ϑ,−,n = 155
JQS-µ,+,n = 13 JQS-ϑ,+,n = 149
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
54 54 90 99
58 58 108 108
59 59 112 111
59.05 58.85 112.1 110.9
60 60 116 114
65 64 138 127
1.59 1.28 6.65 4.08
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
11 11 111 125
12 12 140 143
13 13 147 148
12.79 12.8 147.6 147.8
13 13 155 152
16 16 193 174
0.73 0.68 10.94 6.47
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
36 36 140 142
38 38 151 151
39 39 154 153
38.8 38.72 154.2 153.3
39 39 157 155
42 42 168 165
0.82 0.78 4.07 3.12
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
11 11 119 125
12 12 142 143
13 13 147 147
12.74 12.76 147.5 147.8
13 13 153 152
15 15 182 174
0.61 0.59 8.29 6.47
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
34
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-5: Simulations Results for Model 4 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
16 17 18 19 20 21 22 23 24 25 26
1.080 1.047 1.024 1.010 1.002 1.000 1.002 1.009 1.018 1.030 1.044
19 20 21 22 23 24 25 26 27 28 29
1.059 1.035 1.018 1.008 1.002 1.000 1.002 1.006 1.014 1.023 1.034
15 16 17 18 19 20 21 22 23 24 25
1.072 1.039 1.017 1.005 1.000 1.000 1.005 1.014 1.027 1.042 1.059
30 31 32 33 34 35 36 37 38 39 40
1.025 1.015 1.008 1.003 1.001 1.000 1.001 1.003 1.007 1.011 1.017
JˆES-µ,−,n JˇES-µ,−,n
1.065 1.067
JˆES-µ,+,n JˇES-µ,+,n
0.8511 0.8504
JˆQS-µ,−,n JˇQS-µ,−,n
0.9663 0.9679
JˆQS-µ,+,n JˇQS-µ,+,n
0.9004 0.9003
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 21 JES-ϑ,−,n = 145
JES-µ,+,n = 24 JES-ϑ,+,n = 102
JQS-µ,−,n = 20 JQS-ϑ,−,n = 145
JQS-µ,+,n = 35 JQS-ϑ,+,n = 141
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
19 19 106 125
22 22 141 143
23 23 148 147
22.86 22.83 148.3 147.6
24 23 156 152
28 26 201 179
1.04 0.91 11.48 6.59
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
17 17 82 90
20 20 99 101
21 21 103 103
20.91 20.91 103.6 103.5
22 22 108 106
27 27 130 119
1.33 1.30 6.29 3.95
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
17 17 120 126
19 19 144 145
19 19 149 149
19.44 19.43 149.6 149.1
20 20 155 153
23 22 187 181
0.74 0.70 8.59 6.60
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
28 28 130 131
31 31 140 141
32 32 143 143
31.91 31.92 142.9 142.9
33 33 146 145
40 40 159 155
1.61 1.61 3.97 3.25
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
35
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-6: Simulations Results for Model 5 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
41 42 43 44 45 46 47 48 49 50 51
1.013 1.008 1.004 1.002 1.000 1.000 1.001 1.002 1.004 1.007 1.011
7 8 9 10 11 12 13 14 15 16 17
1.247 1.113 1.039 1.004 0.994 1.000 1.018 1.045 1.078 1.116 1.158
30 31 32 33 34 35 36 37 38 39 40
1.016 1.008 1.003 1.000 0.999 1.000 1.002 1.006 1.011 1.017 1.024
6 7 8 9 10 11 12 13 14 15 16
1.472 1.240 1.110 1.041 1.008 1.000 1.008 1.028 1.057 1.092 1.131
JˆES-µ,−,n JˇES-µ,−,n
1.095 1.099
JˆES-µ,+,n JˇES-µ,+,n
0.9544 0.9521
JˆQS-µ,−,n JˇQS-µ,−,n
0.8966 0.8977
JˆQS-µ,+,n JˇQS-µ,+,n
0.9651 0.9629
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 46 JES-ϑ,−,n = 109
JES-µ,+,n = 12 JES-ϑ,+,n = 119
JQS-µ,−,n = 35 JQS-ϑ,−,n = 109
JQS-µ,+,n = 11 JQS-ϑ,+,n = 119
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
44 45 77 92
50 50 104 105
51 51 110 109
50.93 50.82 109.9 109.1
52 52 115 113
58 57 139 130
1.83 1.61 8.11 5.60
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
9 9 82 102
11 11 113 116
11 11 120 120
11.17 11.17 120.4 120.4
12 12 127 124
15 14 161 141
0.74 0.69 10.36 5.89
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
28 28 99 101
30 30 107 108
31 31 109 109
31.02 31 109.4 109.1
32 31 111 111
35 35 120 117
0.86 0.84 2.75 2.17
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
9 9 99 101
11 11 115 116
11 11 119 120
11.06 11.06 119.8 120.1
11 11 124 124
13 13 149 140
0.59 0.57 6.86 5.55
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
36
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-7: Simulations Results for Model 6 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
13 14 15 16 17 18 19 20 21 22 23
1.119 1.068 1.035 1.014 1.003 1.000 1.003 1.011 1.023 1.039 1.057
16 17 18 19 20 21 22 23 24 25 26
1.069 1.039 1.018 1.006 1.000 1.000 1.004 1.012 1.022 1.035 1.051
12 13 14 15 16 17 18 19 20 21 22
1.121 1.066 1.031 1.011 1.001 1.000 1.006 1.017 1.032 1.050 1.072
22 23 24 25 26 27 28 29 30 31 32
1.044 1.026 1.014 1.005 1.001 1.000 1.002 1.005 1.011 1.019 1.029
JˆES-µ,−,n JˇES-µ,−,n
1.065 1.065
JˆES-µ,+,n JˇES-µ,+,n
0.8495 0.8493
JˆQS-µ,−,n JˇQS-µ,−,n
1.008 1.008
JˆQS-µ,+,n JˇQS-µ,+,n
0.9261 0.9264
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 18 JES-ϑ,−,n = 119
JES-µ,+,n = 21 JES-ϑ,+,n = 102
JQS-µ,−,n = 17 JQS-ϑ,−,n = 119
JQS-µ,+,n = 27 JQS-ϑ,+,n = 102
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
16 16 87 102
19 19 113 116
20 20 120 120
19.71 19.69 120.2 119.7
20 20 127 124
24 24 165 145
1.23 1.17 9.88 5.92
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
13 14 75 82
17 17 97 98
18 18 102 102
18.14 18.13 102.4 102.2
19 19 108 106
25 26 137 124
1.71 1.69 7.90 5.77
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
15 15 97 104
17 17 115 116
17 17 120 119
17.31 17.31 119.8 119.6
18 18 124 123
20 20 146 142
0.94 0.92 6.81 5.43
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
22 22 94 96
25 25 100 100
25 25 101 101
25.42 25.42 101.3 101.2
26 26 103 102
31 31 109 109
1.32 1.31 2.43 1.85
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
37
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-8: Simulations Results for Model 7 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
49 50 51 52 53 54 55 56 57 58 59
1.008 1.005 1.002 1.001 1.000 1.000 1.001 1.002 1.004 1.006 1.009
8 9 10 11 12 13 14 15 16 17 18
1.279 1.149 1.071 1.027 1.005 1.000 1.007 1.022 1.044 1.071 1.102
40 41 42 43 44 45 46 47 48 49 50
1.010 1.006 1.002 1.000 1.000 1.000 1.001 1.003 1.006 1.010 1.014
8 9 10 11 12 13 14 15 16 17 18
1.265 1.139 1.064 1.023 1.003 1.000 1.008 1.025 1.048 1.076 1.108
JˆES-µ,−,n JˇES-µ,−,n
1.097 1.104
JˆES-µ,+,n JˇES-µ,+,n
0.9335 0.9308
JˆQS-µ,−,n JˇQS-µ,−,n
0.9043 0.9079
JˆQS-µ,+,n JˇQS-µ,+,n
0.9649 0.9629
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 54 JES-ϑ,−,n = 113
JES-µ,+,n = 13 JES-ϑ,+,n = 144
JQS-µ,−,n = 45 JQS-ϑ,−,n = 156
JQS-µ,+,n = 13 JQS-ϑ,+,n = 145
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
53 54 82 94
58 58 108 108
59 59 114 113
59.38 59.15 114.1 112.7
61 60 120 117
66 65 149 137
1.98 1.60 8.89 6.08
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
10 10 105 117
12 12 142 146
13 13 152 152
12.57 12.59 152.6 152.5
13 13 162 159
17 16 227 188
0.82 0.76 15.09 9.37
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
38 38 138 142
40 40 153 153
40 40 156 156
40.33 40.24 156.6 155.6
41 41 160 158
44 44 177 170
0.84 0.82 4.77 3.95
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
11 11 112 115
12 12 143 144
13 13 150 151
12.7 12.71 150.8 151.1
13 13 158 157
16 16 208 188
0.69 0.67 11.11 9.56
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
38
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-9: Simulations Results for Model 8 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
16 17 18 19 20 21 22 23 24 25 26
1.080 1.047 1.024 1.010 1.002 1.000 1.002 1.009 1.018 1.030 1.044
19 20 21 22 23 24 25 26 27 28 29
1.059 1.035 1.018 1.008 1.002 1.000 1.002 1.006 1.014 1.023 1.034
15 16 17 18 19 20 21 22 23 24 25
1.072 1.039 1.017 1.005 1.000 1.000 1.005 1.014 1.027 1.042 1.059
30 31 32 33 34 35 36 37 38 39 40
1.025 1.015 1.008 1.003 1.001 1.000 1.001 1.003 1.007 1.011 1.017
JˆES-µ,−,n JˇES-µ,−,n
1.039 1.042
JˆES-µ,+,n JˇES-µ,+,n
0.8473 0.8474
JˆQS-µ,−,n JˇQS-µ,−,n
1.019 1.021
JˆQS-µ,+,n JˇQS-µ,+,n
0.9442 0.9443
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 21 JES-ϑ,−,n = 150
JES-µ,+,n = 24 JES-ϑ,+,n = 102
JQS-µ,−,n = 20 JQS-ϑ,−,n = 151
JQS-µ,+,n = 35 JQS-ϑ,+,n = 142
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
18 18 103 121
22 22 139 142
22 22 147 146
22.36 22.32 147.9 146.7
23 23 156 151.2
26 26 207 175
1.16 1.04 12.86 7.34
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
16 16 68 77
20 20 94 95
21 21 100 100
20.85 20.84 100.5 100.1
22 22 107 105
26 26 140 128
1.47 1.40 9.47 6.87
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
17 17 119 123
20 20 143 144
21 20 149 148
20.53 20.5 149 148.4
21 21 155 153
24 24 191 176
0.93 0.89 9.17 7.31
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
28 29 122 123
32 32 136 136
34 34 139 139
33.72 33.72 139.2 139.2
35 35 142 142
43 43 157 154
1.85 1.84 4.79 4.18
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
39
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-10: Simulations Results for Model 9 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
15 16 17 18 19 20 21 22 23 24 25
1.088 1.051 1.026 1.010 1.002 1.000 1.003 1.010 1.020 1.034 1.049
12 13 14 15 16 17 18 19 20 21 22
1.133 1.075 1.037 1.014 1.003 1.000 1.004 1.014 1.027 1.045 1.065
61 62 63 64 65 66 67 68 69 70 71
1.006 1.004 1.002 1.001 1.000 1.000 1.000 1.001 1.002 1.004 1.006
23 24 25 26 27 28 29 30 31 32 33
1.028 1.015 1.006 1.001 0.999 1.000 1.003 1.009 1.016 1.025 1.035
JˆES-µ,−,n JˇES-µ,−,n
0.9429 0.9447
JˆES-µ,+,n JˇES-µ,+,n
0.9666 0.9633
JˆQS-µ,−,n JˇQS-µ,−,n
1.026 1.027
JˆQS-µ,+,n JˇQS-µ,+,n
0.71 0.7095
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 20 JES-ϑ,−,n = 103
JES-µ,+,n = 17 JES-ϑ,+,n = 96
JQS-µ,−,n = 66 JQS-ϑ,−,n = 103
JQS-µ,+,n = 28 JQS-ϑ,+,n = 96
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
16 17 71 83
19 19 97 99
19 19 103 102
19.22 19.19 103.1 102.4
20 20 109 106
23 23 132 123
0.94 0.86 8.83 5.77
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
14 15 69 77
16 16 92 93
17 17 96 97
16.81 16.83 96.25 96.48
17 17 101 100
20 20 120 114
0.82 0.77 7.06 4.64
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
45 45 93 95
64 64 101 101
68 68 103 103
68.02 68.01 102.7 102.6
72 72 105 104
89 89 114 112
6.29 6.26 3.02 2.18
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
14 14 86 89
18 18 94 95
19 19 96 96
19.77 19.77 95.83 95.91
21 21 98 97
41 41 107 103
3.08 3.08 2.66 1.86
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
40
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-11: Simulations Results for Model 10 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
17 18 19 20 21 22 23 24 25 26 27
1.064 1.036 1.018 1.006 1.001 1.000 1.003 1.010 1.020 1.032 1.046
11 12 13 14 15 16 17 18 19 20 21
1.129 1.068 1.030 1.008 0.999 1.000 1.008 1.021 1.039 1.061 1.086
28 29 30 31 32 33 34 35 36 37 38
1.018 1.009 1.003 1.000 0.999 1.000 1.003 1.007 1.012 1.019 1.027
13 14 15 16 17 18 19 20 21 22 23
1.110 1.062 1.030 1.011 1.002 1.000 1.004 1.013 1.027 1.043 1.062
JˆES-µ,−,n JˇES-µ,−,n
1.047 1.049
JˆES-µ,+,n JˇES-µ,+,n
0.9967 0.995
JˆQS-µ,−,n JˇQS-µ,−,n
1.044 1.045
JˆQS-µ,+,n JˇQS-µ,+,n
0.8817 0.8811
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 22 JES-ϑ,−,n = 121
JES-µ,+,n = 16 JES-ϑ,+,n = 111
JQS-µ,−,n = 33 JQS-ϑ,−,n = 121
JQS-µ,+,n = 18 JQS-ϑ,+,n = 111
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
20 20 91 99
22 22 106 107
23 23 110 109
22.89 22.86 109.8 109.3
23 23 113 111
26 26 131 120
0.81 0.75 5.35 2.97
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
14 14 78 89
15 15 94 96
16 16 97 98
15.66 15.68 97.45 97.57
16 16 101 99
18 17 116 107
0.54 0.51 4.68 2.62
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
27 28 97 101
32 32 107 107
33 33 109 109
33.45 33.43 109.4 109.2
35 35 111 111
41 41 121 120
1.69 1.67 3.33 2.46
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
13 13 88 90
15 15 95 96
16 16 97 97
15.93 15.93 97.35 97.4
17 17 99 99
22 22 108 105
1.21 1.20 2.82 1.98
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
41
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-12: Simulations Results for Model 11 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
25 26 27 28 29 30 31 32 33 34 35
1.026 1.014 1.006 1.002 1.000 1.000 1.003 1.007 1.013 1.021 1.029
9 10 11 12 13 14 15 16 17 18 19
1.224 1.122 1.059 1.022 1.004 1.000 1.006 1.019 1.039 1.063 1.091
40 41 42 43 44 45 46 47 48 49 50
1.008 1.004 1.001 1.000 0.999 1.000 1.002 1.004 1.007 1.011 1.016
10 11 12 13 14 15 16 17 18 19 20
1.169 1.091 1.042 1.014 1.001 1.000 1.007 1.021 1.040 1.062 1.089
JˆES-µ,−,n JˇES-µ,−,n
1.036 1.041
JˆES-µ,+,n JˇES-µ,+,n
0.9962 0.9944
JˆQS-µ,−,n JˇQS-µ,−,n
1.083 1.085
JˆQS-µ,+,n JˇQS-µ,+,n
0.9214 0.9201
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 30 JES-ϑ,−,n = 150
JES-µ,+,n = 14 JES-ϑ,+,n = 147
JQS-µ,−,n = 45 JQS-ϑ,−,n = 153
JQS-µ,+,n = 15 JQS-ϑ,+,n = 144
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
28 29 112 119
30 30 128 129
31 30 132 131
30.57 30.49 132 130.9
31 31 136 133
33 32 155 144
0.73 0.63 5.48 3.28
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
12 12 99 108
14 14 121 124
14 14 127 127
14.1 14.12 127 127
14 14 133 130
17 16 165 148
0.68 0.63 8.76 5.06
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
42 42 120 123
46 46 130 130
47 47 133 132
47.28 47.22 132.9 132.3
48 48 135 134
52 52 146 143
1.45 1.42 3.52 2.72
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
11 11 103 106
13 13 119 120
14 14 123 123
13.75 13.75 123.5 123.7
14 14 127 127
18 18 147 144
0.93 0.92 6.05 4.64
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
42
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-13: Simulations Results for Model 12 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
15 16 17 18 19 20 21 22 23 24 25
1.061 1.030 1.011 1.001 0.998 1.000 1.007 1.017 1.031 1.047 1.066
16 17 18 19 20 21 22 23 24 25 26
1.075 1.043 1.021 1.008 1.001 1.000 1.003 1.010 1.020 1.033 1.047
24 25 26 27 28 29 30 31 32 33 34
1.024 1.012 1.004 1.000 0.999 1.000 1.003 1.009 1.016 1.025 1.034
21 22 23 24 25 26 27 28 29 30 31
1.034 1.018 1.007 1.001 0.999 1.000 1.004 1.010 1.018 1.028 1.040
JˆES-µ,−,n JˇES-µ,−,n
1.014 1.015
JˆES-µ,+,n JˇES-µ,+,n
0.9924 0.9926
JˆQS-µ,−,n JˇQS-µ,−,n
1.097 1.098
JˆQS-µ,+,n JˇQS-µ,+,n
0.8544 0.8545
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 20 JES-ϑ,−,n = 157
JES-µ,+,n = 21 JES-ϑ,+,n = 134
JQS-µ,−,n = 29 JQS-ϑ,−,n = 153
JQS-µ,+,n = 26 JQS-ϑ,+,n = 135
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
17 17 108 124
19 19 136 138
19 19 143 142
19.48 19.46 143.3 142.5
20 20 150 147
23 22 189 164
0.86 0.77 10.43 5.94
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
19 20 94 100
20 21 108 109
21 21 111 111
20.81 20.81 111 110.8
21 21 114 113
22 22 130 120
0.54 0.47 4.85 2.80
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
25 25 118 122
30 30 135 136
31 31 140 139
30.67 30.65 139.9 139.7
32 32 144 143
37 37 169 160
1.77 1.73 7.19 5.49
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
18 18 103 106
21 21 111 111
22 22 113 113
21.8 21.8 113 113
23 23 115 114
29 29 125 122
1.68 1.66 2.90 2.19
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
43
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-14: Simulations Results for Model 13 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
15 16 17 18 19 20 21 22 23 24 25
1.088 1.051 1.026 1.010 1.002 1.000 1.003 1.010 1.020 1.034 1.049
12 13 14 15 16 17 18 19 20 21 22
1.133 1.075 1.037 1.014 1.003 1.000 1.004 1.014 1.028 1.045 1.066
61 62 63 64 65 66 67 68 69 70 71
1.006 1.003 1.002 1.001 1.000 1.000 1.000 1.001 1.002 1.004 1.006
23 24 25 26 27 28 29 30 31 32 33
1.028 1.014 1.006 1.001 0.999 1.000 1.003 1.009 1.016 1.025 1.035
JˆES-µ,−,n JˇES-µ,−,n
0.95 0.9532
JˆES-µ,+,n JˇES-µ,+,n
0.9652 0.9578
JˆQS-µ,−,n JˇQS-µ,−,n
1.092 1.093
JˆQS-µ,+,n JˇQS-µ,+,n
0.8257 0.8247
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 20 JES-ϑ,−,n = 104
JES-µ,+,n = 17 JES-ϑ,+,n = 96
JQS-µ,−,n = 66 JQS-ϑ,−,n = 104
JQS-µ,+,n = 28 JQS-ϑ,+,n = 96
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
15 16 55 64
19 19 97 98
19 19 106 104
19.4 19.36 104.8 103.9
20 20 113 110
24 23 143 135
1.02 0.92 11.59 8.93
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
13 14 46 57
16 16 88 90
17 17 96 96
16.84 16.87 94.64 95.13
17 17 102 100
21 20 126 117
1.00 0.88 10.72 7.54
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
49 49 93 96
68 68 102 102
72 72 104 104
72.34 72.33 103.8 103.7
77 77 106 105
104 105 118 114
6.56 6.53 3.22 2.47
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
13 13 85 88
19 19 92 92
22 22 93 93
22.89 22.9 93.37 93.49
25 25 95 95
51 51 103 102
4.86 4.86 2.67 1.96
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
44
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-15: Simulations Results for Model 14 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
17 18 19 20 21 22 23 24 25 26 27
1.064 1.036 1.017 1.006 1.001 1.000 1.003 1.010 1.020 1.032 1.046
11 12 13 14 15 16 17 18 19 20 21
1.129 1.068 1.030 1.008 0.999 1.000 1.008 1.021 1.040 1.061 1.086
28 29 30 31 32 33 34 35 36 37 38
1.018 1.009 1.003 1.000 0.999 1.000 1.003 1.007 1.012 1.019 1.027
13 14 15 16 17 18 19 20 21 22 23
1.110 1.061 1.030 1.011 1.002 1.000 1.004 1.014 1.027 1.043 1.062
JˆES-µ,−,n JˇES-µ,−,n
1.059 1.061
JˆES-µ,+,n JˇES-µ,+,n
0.9816 0.98
JˆQS-µ,−,n JˇQS-µ,−,n
1.172 1.173
JˆQS-µ,+,n JˇQS-µ,+,n
0.8606 0.8602
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 22 JES-ϑ,−,n = 121
JES-µ,+,n = 16 JES-ϑ,+,n = 111
JQS-µ,−,n = 33 JQS-ϑ,−,n = 121
JQS-µ,+,n = 18 JQS-ϑ,+,n = 111
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
20 20 77 92
23 23 105 106
23 23 109 108
23.06 23.03 109 108.5
24 24 113 111
26 26 131 125
0.86 0.77 6.57 3.58
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
14 14 75 85
15 15 95 97
15 15 99 99
15.43 15.43 98.67 98.73
16 16 102 101
17 17 119 110
0.59 0.54 5.67 3.31
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
30 30 97 98
36 36 106 107
37 37 109 109
37.42 37.4 108.8 108.6
39 39 111 110
45 44 121 119
1.94 1.92 3.57 2.77
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
13 13 88 91
15 15 96 97
15 15 98 98
15.56 15.56 98.56 98.59
16 16 101 100
21 21 111 108
1.14 1.13 3.05 2.30
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
45
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-16: Simulations Results for Model 15 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
25 26 27 28 29 30 31 32 33 34 35
1.026 1.014 1.006 1.001 0.999 1.000 1.003 1.007 1.013 1.021 1.030
9 10 11 12 13 14 15 16 17 18 19
1.223 1.121 1.058 1.022 1.004 1.000 1.006 1.020 1.039 1.063 1.091
40 41 42 43 44 45 46 47 48 49 50
1.008 1.004 1.001 1.000 0.999 1.000 1.002 1.004 1.007 1.011 1.016
10 11 12 13 14 15 16 17 18 19 20
1.168 1.090 1.041 1.014 1.001 1.000 1.007 1.021 1.040 1.063 1.089
JˆES-µ,−,n JˇES-µ,−,n
1.041 1.046
JˆES-µ,+,n JˇES-µ,+,n
0.9839 0.9816
JˆQS-µ,−,n JˇQS-µ,−,n
1.158 1.161
JˆQS-µ,+,n JˇQS-µ,+,n
0.9187 0.9176
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 30 JES-ϑ,−,n = 149
JES-µ,+,n = 14 JES-ϑ,+,n = 140
JQS-µ,−,n = 45 JQS-ϑ,−,n = 151
JQS-µ,+,n = 15 JQS-ϑ,+,n = 137
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
28 28 102 111
30 30 126 127
31 30 130 129
30.51 30.42 130 129
31 31 134 132
33 33 158 141
0.78 0.67 6.56 3.77
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
12 12 87 101
13 14 117 119
14 14 124 124
13.93 13.94 124.4 124.3
14 14 131 129
17 16 168 155
0.75 0.68 11.00 6.83
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
44 44 120 120
49 49 129 129
50 50 131 131
50.34 50.28 131.3 130.8
51 51 134 133
56 56 144 142
1.63 1.59 3.61 2.84
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
11 11 98 103
13 13 115 116
14 14 120 120
13.65 13.66 120.4 120.5
14 14 125 124
19 19 152 143
1.10 1.11 7.05 5.95
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
46
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
Table SA-17: Simulations Results for Model 16 Panel A: IMSE for Grid of Number of Bins and Estimated Choices J−,n
IMSEES,− (J−,n ) IMSE∗ ES,−
J+,n
IMSEES,+ (J+,n ) IMSE∗ ES,+
J−,n
IMSEQS,− (J−,n ) IMSE∗ QS,−
J+,n
IMSEQS,+ (J+,n ) IMSE∗ QS,+
15 16 17 18 19 20 21 22 23 24 25
1.059 1.030 1.011 1.001 0.998 1.000 1.007 1.018 1.031 1.048 1.066
16 17 18 19 20 21 22 23 24 25 26
1.073 1.042 1.021 1.007 1.001 1.000 1.003 1.010 1.021 1.033 1.048
24 25 26 27 28 29 30 31 32 33 34
1.023 1.011 1.004 1.000 0.999 1.000 1.004 1.009 1.016 1.025 1.035
21 22 23 24 25 26 27 28 29 30 31
1.033 1.017 1.007 1.001 0.999 1.000 1.004 1.010 1.019 1.029 1.041
JˆES-µ,−,n JˇES-µ,−,n
1.048 1.05
JˆES-µ,+,n JˇES-µ,+,n
0.9941 0.9938
JˆQS-µ,−,n JˇQS-µ,−,n
1.071 1.072
JˆQS-µ,+,n JˇQS-µ,+,n
0.8365 0.8365
Panel B: Summary Statistics for the Estimated Number of Bins Pop. Par. JES-µ,−,n = 20 JES-ϑ,−,n = 155
JES-µ,+,n = 21 JES-ϑ,+,n = 134
JQS-µ,−,n = 29 JQS-ϑ,−,n = 151
JQS-µ,+,n = 26 JQS-ϑ,+,n = 136
Min.
1st Qu.
Median
Mean
3rd Qu.
Max.
Std. Dev.
JˆES-µ,−,n JˇES-µ,−,n JˆES-ϑ,−,n JˇES-ϑ,−,n
17 17 94 116
20 20 132 134
20 20 139 138
20.09 20.05 138.9 138
21 21 146 142
24 23 179 164
0.92 0.80 11.15 6.18
JˆES-µ,+,n JˇES-µ,+,n JˆES-ϑ,+,n JˇES-ϑ,+,n
19 19 85 98
20 20 108 109
21 21 112 112
20.74 20.74 112 111.9
21 21 116 114.2
23 23 132 127
0.66 0.58 6.23 4.09
JˆQS-µ,−,n JˇQS-µ,−,n JˆQS-ϑ,−,n JˇQS-ϑ,−,n
24 24 113 117
29 29 132 132
30 30 136 136
30.13 30.11 136.6 136.3
31 31 141 140
37 37 163 161
1.73 1.70 7.10 5.74
JˆQS-µ,+,n JˇQS-µ,+,n JˆQS-ϑ,+,n JˇQS-ϑ,+,n
17 17 102 104
20 20 111 111
21 21 113 113
21.08 21.07 113 113
22 22 115 115
28 28 125 125
1.42 1.41 3.35 2.74
Notes: (i) Population quantities: JES-µ,·,n = IMSE-optimal partition size for ES RD Plot. JES-ϑ,·,n = Mimicking variance partition size for ES RD Plot. JQS-µ,·,n = IMSE-optimal partition size for QS RD Plot. JQS-ϑ,·,n = Mimicking variance partition size for QS RD Plot. IMSE∗ES,· = IMSEES,· (JES-µ,·,n ) = ES IMSE function evaluated at optimal choice. IMSE∗QS,· = IMSEQS,· (JQS-µ,·,n ) = QS IMSE function evaluated at optimal choice. (ii) Estimators: JˆES-µ,·,n = spacings JˆES-ϑ,·,n = spacings JˆQS-µ,·,n = spacings JˆQS-ϑ,·,n = spacings
estimator estimator estimator estimator
of of of of
JES-µ,·,n ; JES-ϑ,·,n ; JQS-µ,·,n ; JQS-ϑ,·,n ;
JˇES-µ,·,n JˇES-ϑ,·,n JˇQS-µ,·,n JˇQS-ϑ,·,n
= = = =
polynomial polynomial polynomial polynomial
47
estimator estimator estimator estimator
of of of of
JES-µ,·,n . JES-ϑ,·,n . JQS-µ,·,n . JQS-ϑ,·,n .
6
Numerical Comparison of Partitioning Schemes
We proposed two alternative ways of constructing RD plots, one employing ES partitioning and the other employing QS partitioning. While developing a general theory for optimal partitioning scheme selection is beyond the scope of this paper, we can employ our IMSE expansions to compare the two partitioning schemes theoretically in order to assess their relative IMSE-optimality properties. Without loss of generality we focus on the IMSE for the treatment group (“+” subindex). Assuming the regularity conditions imposed in the paper hold, we obtain (up to the ceiling operator for selecting the optimal partition sizes): √ 3 IMSEES,+ (JES,+,n ) =
3 CES,+ n−2/3 {1 + oP (1)}, 4
√ 3 IMSEQS,+ (JQS,+,n ) =
3 CQS,+ n−2/3 {1 + oP (1)}, 4
where xu
Z CES,+ =
2 (1) µ+ (x) w(x)dx
1/3 Z
x ¯
CQS,+
Z = x ¯
xu
xu
x ¯ (1) µ+ (x)
f (x)
1/3
!2
Z
w(x)dx
2 (x) σ+ w(x)dx f (x)
xu
2/3
2 σ+ (x)w(x)dx
, 2/3 .
x ¯
Thus, in order to compare the performance of the partition-size selectors for ES and QS RD plots we need to compare the two DGP constants CES,+ and CQS,+ . It follows that when f (x) ∝ κ (i.e., the running variable is uniformly distributed), then CES,+ = CQS,+ and therefore both partitioning schemes have equal (asymptotic) IMSE when the corresponding optimal partition size is used. Unfortunately, when the density f (x) is not constant on the support [xl , xu ], it is not possible to obtain a unique ranking between IMSEES,+ (JES,+,n ) and IMSEQS,+ (JQS,+,n ). Heuristically, the QS RD plots should perform better in cases where the data is sparse because the estimated quantile spaced partition should adapt to this situation better, but we have been unable to provide a formal ranking along these lines. Nonetheless, in Table SA-18 we explore the ranking between the two partitioning schemes using the 16 data generating processes discussed in our simulation study (Table SA-1). As expected, this
48
Table SA-18: Comparison of Partitioning Schemes
Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
BES,− BQS,−
VES,− VQS,−
IMSEES,− (JES-µ,−,n ) IMSEQS,− (JQS-µ,−,n )
BES,+ BQS,+
VES,+ VQS,+
IMSEES,+ (JES-µ,+,n ) IMSEQS,+ (JQS-µ,+,n )
1.000 2.290 2.466 1.258 2.466 1.258 2.466 1.258 0.028 0.309 0.301 0.309 0.028 0.309 0.301 0.309
1.000 1.000 1.389 1.004 1.000 1.000 1.389 1.004 1.000 1.000 1.015 0.977 1.000 1.000 1.015 0.977
1.000 1.319 1.682 1.084 1.352 1.081 1.682 1.084 0.303 0.677 0.677 0.666 0.303 0.677 0.677 0.666
1.000 0.784 1.038 0.447 1.038 0.447 1.038 0.447 0.241 0.655 0.831 0.570 0.241 0.655 0.831 0.570
1.000 1.000 1.004 1.389 1.000 1.000 1.004 1.389 1.000 1.000 0.977 1.015 1.000 1.000 0.977 1.015
1.000 0.925 1.016 0.953 1.004 0.765 1.016 0.953 0.624 0.867 0.928 0.839 0.624 0.867 0.928 0.839
table shows that when f (x) is uniform both IMSE are equal, while when f (x) is not uniform either IMSE may dominate the other. This depends on the shape of the regression function (different for control and treatment sides) and conditional heteroskedasticity in the underlying true data generating process.
49
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