We denote the confidence set generated by Algorithm 1.9.1 asCn,`,x,Bj(1−α,Um,ΘΘΘl). The next theorem confirms that the uniform bootstrap confidence bands for both conditional and overall TEBFs are asymptotically accurate.
Theorem 1.6 Suppose the assumptions of Corollary 1.3 hold, we have
n→∞lim inf
(u,θθθ)∈Um×ΘΘΘlP
ν`,jx (u,θθθ)∈Cn,`,jx,B (1−α,Um,ΘΘΘl)
=1−α,
n→∞lim inf
(u,θθθ)∈Um×ΘΘΘlP ν`,j(u,θθθ)∈Cn,`,jB (1−α,Um,ΘΘΘl)
=1−α,
forx∈X,`∈ {lb,ub}, and j∈ {AT E,DT E,QT E,CHT E}.
is the true propensity score function.
The BGF admits an analytical form whenλT,d(·) =λC,d(·)≡λd(·). Such simplification is handy when checking the coverage of our bootstrap confidence sets. By symmetry, we have thatSYd|X(y|x) =exp
−21/θd∗(x)λd(xγd)y , and SYd,Rd|X(y,1|x) =2−1exp
−21/θd∗(x)λd(xγd)y
, implying a population censoring rate of 50%. Now, from (1.4.5) and by direct calculations,
sTd(t,xγd,θ) =φθ−1
2−1φθ
exp
−21/θd∗(x)λd(xγd)t
,ford∈ {0,1}.
This formula simplifies further when the true copula is Gumbel. In this case,sTd(t,xγd,θ) =exp(−21/θd∗(x)−1/θ· λd(xγd)t), equivalent to an exponential distribution with a rate parameter equal toβd(xγd,θ)≡21/θd∗(x)−1/θλd(xγd).
As a direct consequence,νAT E,lbx (θθθ) =β1(xγ1,θ2)−1−β0(xγ0,θ1)−1, and the overall average effect is also immediately available via taking the expectation with respect toX. In the following, we setγ1=γ0= (0.5,0.5)0,λ1(v) =√
v+v/2, andλ0(v) =√
v. As a result, the true DTE is heterogeneous and uniformly negative across the index set.
To generate variables from the Gumbel copula, we follow the algorithm provided in Section 2.9 in Nelsen (2007), for which purpose, we assume the true copula parameters are θ0∗(·) =1 andθ1∗(·) =1.25. That is, censoring is independent for the treated group, whereasT0andC0are correlated with Kendall’sτequal to 0.2.
In Figure 1.1, we plot unconditional BGFs and DTEs across various levels of the sensitivity parameter, alongside the corresponding Peterson’s bounds. We observe that the worse case bounds (the upper bound in particular) are highly non-informative and the BGFs provide significant improvement over the worst-case bounds under the assumed censoring mechanism. One may question whether the gap between the two can be completely bridged by varying theta. The answer to this question is contingent on the copula under consideration and specifically whether it admits Hoeffding-Frechet bounds as limiting cases. For instance, the Gumbel copula is unable to bridge the gap entirely, due to its inherent incapability to model negative correlation. Additionally, the figure illustrates the stochastic domi- nance relations, as influenced by the concordance ordering within the copula family. Consequently, we can observe a correlation between the size of the identified set and the range of theta values chosen by the researcher.
To assess the performance of TEBF estimators over an index setUm, we adoptaverageandmedian integrated bias,integrated root mean square error(IRMSE), and the coverage rate as the criterion of evaluation.10 Regarding the index setUm, we use an equidistant grid between 0.1 and 1.5 with the interval size of 0.05 for the ATE, DTE and CHTE. For the QTE, an equidistant grid between 0.25 and 0.75 with a step size of 0.05 is adopted. We let both
10Consider a Monte Carlo experiment withSreplications, the average integrated bias is defined byS−1∑Ss=1Ru∈Um
fˆs(u)−f(u)
du, median integrated bias denotes the 50-th percentile ofR
u∈Um
fˆs(u)−f(u) du S
s=1, and the IRMSE, by
S−1∑Ss=1 R
u∈Um
fˆs(u)−f(u)
2du 1/2
, where f(·)is any one of theνj,`(·), for`=lb,ub, andj∈ {AT E,DT E,QT E,CHT E}.
Figure 1.1: BGFs and DTEs with multiple levels ofθ
Notes: The left panel depicts the unconditional BGFs for the control group and the right panel illustrates the overall DTEs. In each plot, the dashed curve depicts the function when the independent censoring mechanism is assumed (equivalently, Gumbel copula withθ=1). The green solid curves represent the true functions (Gumbel copula withθ=1.25). The red solid curves depict the Peterson’s worst-case bounds.
L(·)andK(·)be the Epanechnikov kernel: L(u) =K(u) =0.75(1−u2)1{|u| ≤1}. For treatment groupd∈ {0,1}, the bandwidthbis chosen as the value from the set
2−0.5k(n/2)−0.26 6k=−1, that minimizes the estimated criterion Jˆd(γˆd,ρ), where we letρ(v) =exp(− kvk2/2). We then set the bandwidthhequal tob. To assess the impact of first-step estimation, we provide a set of “oracle” results where the single index parameters take their true values along with “feasible” results where the parameters are estimated according to the procedure from Section 1.4.1.
Table 1.1 reports simulation results based on 1,000 Monte Carlo replications of samples with sizen=1,000. For each type of treatment effect, we show results for two different range ofθ: a narrower one withθθθ = (1,1.5), and a wider one withθθθ= (1,2). Given the one-to-one mapping betweenθ and Kendall’sτ, the twoθθθ choices correspond to Kendall’sτlying between[0,1/3]and[0,1/2], respectively. Results in Table 1.1 suggest that our TEBF estimators exhibit minimal bias, and their confidence intervals generally achieve close-to-nominal-level coverage, irrespective of the choice of copula parameters, the type of treatment effects and whether the effect is conditional. The (C)CHTE perform relatively worse than the other three types, in terms of integrated bias and IRMSE. This is to be expected as the log transformations tend to induce higher bias.
When comparing oracle and feasible results, we find that the oracle results generally exhibit smaller bias and
IRMSE, and they have better coverage properties. The difference is more prominent when conditional treatment effects are considered. This is partially due to the fact that the component in the expansion of the conditional TEBFs, which is associated with the single-index estimation, is of a lower order than the component appearing in the overall TEBFs, even though both are negligible in the first order. To improve the performance of our bootstrap procedure for conditional TEBFs, one may consider adding the influence functions associated with first stage estimation when constructing the bootstrap processes. This is left for future research.
Overall, the results from finite-sample studies align with the theoretical predictions discussed in Sections 1.4 and 1.5.