Sharp quantile bounds for a selectively observed outcome

Quantiles behave better than means under selection. The mean bound is informative only when the outcome support is bounded on the relevant side; the quantile bound is informative whenever the observed fraction exceeds 1 - alpha (lower) or alpha (upper), regardless of the range of y. Because both ends are quantiles of the observed distribution, the bound is equivariant under any increasing transform.

Usage

Bndmoq(y, D, X, quantile)

Arguments

y

Outcome; entries with D = 0 are unobserved.

D

Observation indicator, coded 0/1.

X

Discrete stratum label, one per unit; used to report the widest within-stratum interval alongside the pooled one.

quantile

Quantile level alpha in (0, 1).

Value

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

Details

Formula, with q(p) the type-1 quantile among the observed and p_1 = P(D = 1): lower = q(1 - (1 - alpha) / p_1) if p_1 > 1 - alpha and y_0 otherwise; upper = q(alpha / p_1) if p_1 >= alpha and y_1 otherwise.

Note

this construction is sometimes attributed to Chernozhukov and Hansen (2005), which is the instrumental-variable quantile regression model, a different object. The bound evaluated here is Manski's and is cited as such.

References

Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer, Section 1.3. The two expressions are r(alpha, x) and s(alpha, x) on pp. 12-13 of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751).

Examples

set.seed(1)
Bndmoq(y = rnorm(40), D = rbinom(40, 1, 0.5), X = rnorm(40), quantile = 0.5)
#> $lower
#> [1] -2.2147
#> 
#> $upper
#> [1] 1.595281
#> 
#> $width
#> [1] 3.809981
#> 
#> $estimate
#> [1] -0.3097095
#> 
#> $max_width
#> [1] 3.809981
#> 
#> $n_strata
#> [1] 40
#> 
#> $p_observed
#> [1] 0.45
#> 
#> $n
#> [1] 40
#> 
#> $method
#> [1] "Quantile-equivariant bound"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"