Intersection bounds from many candidate lower and upper bounds

When several inequalities each bound the same parameter, the binding one is the largest lower and the smallest upper. Taking a min of noisy estimates biases it downward, so the plug-in upper bound is too tight; the precision correction pushes each candidate out by its own standard error before the min is taken. The correction reported here is the Bonferroni one, exact at a single candidate (the multiplier is then zero) and conservative beyond it.

Usage

Bndlmm(theta, moments)

Arguments

theta

An (n, K) matrix of observation-level estimates of K candidate lower bounds.

moments

An (n, J) matrix of observation-level estimates of J candidate upper bounds.

Value

List with lower, upper, width, lower_pc, upper_pc, width_pc, K, J, n.

Details

Formula: [max_k m_k, min_j m_j] plug-in, and [max_k (m_k - z_K s_k / sqrt(n)), min_j (m_j + z_J s_j / sqrt(n))] with z_J = qnorm(1 - 1 / (2 J)) for the half-median-unbiased version.

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 half-median-unbiased criterion the correction targets is equation (4.9) of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 p. 96); the multiplier used here is Bonferroni rather than the paper's bootstrap, and is labelled as such.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Bndlmm(V, V)
#> $lower
#> [1] 4.5
#> 
#> $upper
#> [1] 4.5
#> 
#> $width
#> [1] 0
#> 
#> $lower_pc
#> [1] 4.5
#> 
#> $upper_pc
#> [1] 4.5
#> 
#> $width_pc
#> [1] 0
#> 
#> $K
#> [1] 1
#> 
#> $J
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Linear min-max bound"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"