Under no assumptions about selection, the ATE is only partially
identified (Manski 1990): with the outcome in \([a, b]\),
\(E[Y_1] \in [p \bar y_1 + a(1-p), p \bar y_1 + b(1-p)]\) and
\(E[Y_0] \in [(1-p) \bar y_0 + a p, (1-p) \bar y_0 + b p]\), so the
bounds always have width \(b - a\). Returns a named list with
lower_bound, upper_bound, point_estimate, width.
Usage
manski_bounds(
outcome_treated,
outcome_control,
p_treated,
outcome_range = NULL
)
Arguments
- outcome_treated
Outcomes for treated units.
- outcome_control
Outcomes for control units.
- p_treated
Proportion treated.
- outcome_range
c(min, max) on the outcome. Default c(0, 1).
Examples
set.seed(1)
res <- manski_bounds(runif(50), runif(50), p_treated = 0.5)
c(res$lower_bound, res$upper_bound)
#> [1] -0.4852542 0.5147458
