ATE bound under monotone treatment selection

Monotone treatment selection says the treated would have done at least as well as the untreated under either treatment, E[y(t) | D = 1] >= E[y(t) | D = 0]. The consequence is one line of algebra: E[y(1)] = E[y | D = 1] P(D = 1) + E[y(1) | D = 0] P(D = 0) <= E[y | D = 1], and symmetrically E[y(0)] >= E[y | D = 0], so the naive observed difference, which without the assumption bounds nothing, becomes an exact upper bound on the ATE. The other end is left at the worst case.

Usage

Bndsdo(y, D, skew = 1)

Arguments

y

Observed outcome.

D

Binary treatment indicator, coded 0/1.

skew

Direction of the selection: positive for selection on gains (the default), negative for the reverse.

Value

List with lower, upper, width, estimate, naive, wc_lower, wc_upper, p_treated, n.

References

Manski, C. F. and Pepper, J. V. (2000). Monotone instrumental variables, with an application to the returns to schooling. Econometrica 68(4), 997-1010. doi:10.1111/1468-0262.00144 . The worst-case end is equation (2.11) of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 p. 17). The derivation above is written out because the paper was not accessible: what is cited to Manski and Pepper is the assumption, not a copied formula.

Examples

set.seed(1)
r <- Bndsdo(y = rnorm(10), D = rbinom(10, 1, 0.5)); TRUE
#> [1] TRUE