Coordinate projection of a d-dimensional identified set

Subset inference is a projection: what a joint identified set says about one coordinate is the shadow the set casts on that axis. The projection is always at least as wide as any conditional slice, which is exactly why projection-based inference is conservative; the reported per-coordinate widths make that visible.

Usage

Bndsbs(theta_full, subset_idx)

Arguments

theta_full

Points of the identified set, one per row of an (m, d) matrix.

subset_idx

Zero-based coordinate indices to project onto.

Value

List with lower, upper, width for the first selected coordinate, plus total_width, max_width, d_subset, m, d.

Details

Formula: proj_k(H) = [min h_k, max h_k] over the points of H, for each selected coordinate k.

References

Romano, J. P. and Shaikh, A. M. (2008). Inference for identifiable parameters in partially identified econometric models. Journal of Statistical Planning and Inference 138(9), 2786-2807. doi:10.1016/j.jspi.2008.03.015 . The coverage-of-a-component distinction is Section 4.3.3 of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 p. 101).

Examples

Bndsbs(theta_full = matrix(c(1, 2, 3, 4, 5, 6), nrow = 2), subset_idx = list(a = 1, b = 2))
#> $lower
#> [1] 3
#> 
#> $upper
#> [1] 4
#> 
#> $width
#> [1] 1
#> 
#> $total_width
#> [1] 2
#> 
#> $max_width
#> [1] 1
#> 
#> $d_subset
#> [1] 2
#> 
#> $m
#> [1] 2
#> 
#> $d
#> [1] 3
#> 
#> $method
#> [1] "Subset-inference bound"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"