Worst-case bounds for a monotonically transformed outcome

Mean bounds are not invariant under a monotone transform: the counterfactual mass is placed at the transformed support, which moves. Quantile bounds are invariant, because both ends are quantiles of the observed distribution and a monotone map commutes with the type-1 quantile. Both facts are reported: the mean bound on t(y) and the equivariance gap of the median bound, exactly zero for increasing t.

Usage

Bndtfm(y, D, X, transform)

Arguments

y

Observed outcome.

D

Binary treatment indicator, coded 0/1.

X

Discrete stratum label, one per unit; the mean bound is computed within stratum and averaged.

transform

t(y_i) already evaluated, one value per unit, non-decreasing in y. Passing values rather than a function keeps the two language arms evaluating the same map.

Value

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

Details

Formula: worst-case bound of Molinari (2021) equation (2.11) applied to t(y); gap = |t(r_y) - r_t| + |t(s_y) - s_t| at the median.

References

Chernozhukov, V., Lee, S. and Rosen, A. M. (2013). Intersection bounds: estimation and inference. Econometrica 81(2), 667-737. doi:10.3982/ECTA8718 . The invariance evaluated here is the quantile bound of Manski (2003) Section 1.3 as printed on pp. 12-13 of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751); the mean bound is equation (2.11) of the same.

Examples

set.seed(1)
y <- rnorm(40)
Bndtfm(y = y, D = rbinom(40, 1, 0.5), X = rbinom(40, 1, 0.5),
       transform = exp(y))
#> $lower
#> [1] -2.304462
#> 
#> $upper
#> [1] 2.516065
#> 
#> $width
#> [1] 4.820527
#> 
#> $estimate
#> [1] 0.1058016
#> 
#> $gap
#> [1] 0
#> 
#> $n_strata
#> [1] 2
#> 
#> $n
#> [1] 40
#> 
#> $method
#> [1] "Bound under outcome transformation"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"