Test-inversion confidence set for an interval-identified scalar

The confidence set is the sub-level set of the moment-inequality criterion, and the reported interval is its projection onto the parameter line. For a scalar with two inequalities at most one can bind at any parameter value, so the criterion under the null is the square of a single one-sided standard normal and the cutoff is z_(1-alpha)^2 in closed form: no bootstrap, and therefore the same number in both language arms.

Usage

Bndinf(theta, moments, alpha = 0.05)

Arguments

theta

Candidate parameter values to test.

moments

Interval data, an (n, 2) matrix with the lower end in column 1 and the upper end in column 2.

alpha

Miss probability, default 0.05.

Value

List with lower, upper, width, grid_lower, grid_upper, n_in_set, cutoff, criterion_min, n.

Details

Formula: CS = {theta : Q_n(theta) <= z^2} with \(Q_n(theta) = [sqrt(n)(mL - theta)/sL]_+^2 + [sqrt(n)(theta - mU)/sU]_+^2\), whose endpoints are mL - z sL / sqrt(n) and mU + z sU / sqrt(n).

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 . Criterion and level set as equations (4.2) and (4.10) of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 pp. 89, 97).

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
D <- data.frame(x = c(1, 2, 3, 4), y = c(2, 4, 5, 9))
Bndinf(V, D)
#> $lower
#> [1] 1.438252
#> 
#> $upper
#> [1] 7.421159
#> 
#> $width
#> [1] 5.982907
#> 
#> $grid_lower
#> [1] 2
#> 
#> $grid_upper
#> [1] 7
#> 
#> $n_in_set
#> [1] 6
#> 
#> $cutoff
#> [1] 2.705543
#> 
#> $criterion_min
#> [1] 0
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "Inference for partially identified parameters"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"