Sample-selection bound for a selectively observed outcome

Heckman's selection model buys point identification with an exclusion restriction and joint normality. Dropping both leaves the identified set of Manski's worst-case bound: the observed part of the mean is known and the unobserved part can sit anywhere in the support. Stratifying on a discrete X and averaging the within-stratum intervals sharpens the bound whenever selection rates differ by stratum, and reproduces the pooled bound when they do not.

Usage

Bnssel(y, D, X)

Arguments

y

Outcome; entries with D = 0 are unobserved and their recorded value is never used.

D

Selection indicator, 1 when y is observed.

X

Discrete stratum label, one per unit.

Value

List with lower, upper, width, estimate, p_observed, n_strata, n.

Details

Formula, per stratum: \([E(y | D = 1, x) P(D = 1 | x) + y_0 P(D = 0 | x), E(y | D = 1, x) P(D = 1 | x) + y_1 P(D = 0 | x)]\), averaged with weights P(x).

References

Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica 47(1), 153-161. doi:10.2307/1912352 . Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer. Worst-case region as Theorem SIR-2.1, equation (2.2), of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 p. 12).

Examples

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