Imbens-Manski confidence interval for a partially identified parameter

Given estimated bounds \([\hat L, \hat U]\) with standard errors, the interval \([\hat L - c\,\sigma_L,\; \hat U + c\,\sigma_U]\) covers every point of the identification region with probability \(1-\alpha\) (uniformly over the region's width) when \(c\) solves \(\Phi(c + \Delta) - \Phi(-c) = 1 - \alpha\) with \(\Delta = (\hat U - \hat L)/\max(\sigma_L, \sigma_U)\) (Imbens & Manski 2004). The cutoff is non-increasing in \(\Delta\) (Research.P11.im_cutoff_antitone), lies between the one-sided and the two-sided normal quantiles (im_cutoff_between), and the two-sided quantile over-covers whenever the region has positive width (two_sided_overcovers). An interval built to cover the whole region is at least as conservative for the parameter (region_coverage_le); the function reports both. The cutoff the root finder returns exists and is unique: the coverage is continuous and strictly increasing in \(c\) with limits \(-1\) and \(1\) (im_cutoff_exists, im_cutoff_unique).

Usage

morie_bounds_confidence(lower, upper, se_lower, se_upper, level = 0.95)

Arguments

lower, upper: Estimated bounds (e.g. from morie_sentence_effect_bounds with bootstrap standard errors).
se_lower, se_upper: Their standard errors, positive.
level: Confidence level.

Value

A list with cutoff (\(c\)), delta, interval (the Imbens-Manski interval), region_interval (the interval that covers the whole region, using the two-sided quantile), z_one_sided, z_two_sided, coverage_two_sided (what the two-sided cutoff would deliver for a point at one end, above level), and theorems.

Examples

morie_bounds_confidence(lower = 0.10, upper = 0.35, se_lower = 0.03, se_upper = 0.04)
#> $cutoff
#> [1] 1.644854
#> 
#> $delta
#> [1] 6.25
#> 
#> $level
#> [1] 0.95
#> 
#> $interval
#>      lower      upper 
#> 0.05065439 0.41579415 
#> 
#> $region_interval
#>      lower      upper 
#> 0.04120108 0.42839856 
#> 
#> $z_one_sided
#> [1] 1.644854
#> 
#> $z_two_sided
#> [1] 1.959964
#> 
#> $coverage_two_sided
#> [1] 0.975
#> 
#> $theorems
#> [1] "Research.P11.region_coverage_le"    "Research.P11.coverage_strictMono_c"
#> [3] "Research.P11.im_cutoff_antitone"    "Research.P11.im_cutoff_between"    
#> [5] "Research.P11.two_sided_overcovers"  "Research.P11.im_cutoff_exists"     
#> [7] "Research.P11.im_cutoff_unique"     
#>