Positive-only treatment bound (increasing monotone treatment response)

Assuming treatment never hurts, y(1) >= y(0) for every unit, pins the lower bound at exactly zero and leaves only the upper bound to be estimated. Unlike the worst-case bound, the arm bounds use every (y, D) pair rather than only the pairs in the arm of interest, so they stay informative even when one arm is nearly empty.

Usage

Bndpos(y, D, y_max)

Arguments

y

Observed outcome.

D

Binary treatment indicator, coded 0/1.

y_max

Upper end of the logically possible support; at least max(y).

Value

List with lower, upper, width, estimate, p_treated, n.

Details

Formula: \(upper = [E(y | D = 1) P(D = 1) + y_max P(D = 0)] - [E(y | D = 0) P(D = 0) + y_min P(D = 1)]\), lower = 0.

References

Manski, C. F. (1997). Monotone treatment response. Econometrica 65(6), 1311-1334. doi:10.2307/2171738 . Binary-treatment form as equation (2.13) and the paragraph beneath it in Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 pp. 18-19).

Examples

set.seed(1)
Bndpos(y = rnorm(40), D = rbinom(40, 1, 0.5), y_max = 3)
#> $lower
#> [1] 0
#> 
#> $upper
#> [1] 2.643278
#> 
#> $width
#> [1] 2.643278
#> 
#> $estimate
#> [1] 1.321639
#> 
#> $p_treated
#> [1] 0.45
#> 
#> $n
#> [1] 40
#> 
#> $method
#> [1] "Positive-only treatment bound"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"