Identified set as the zero-level set of the moment-inequality criterion

The set estimate and the confidence set are different objects and the difference is visible here: this function reports the level set at criterion zero, with no critical value inflating it. Every candidate inside the interval satisfies both inequalities in sample and scores exactly zero; every candidate outside scores strictly positive.

Usage

Bnsiii(y, X, moments)

Arguments

y

Lower end of each observation's interval.

X

Upper end of each observation's interval, same length as y.

moments

Candidate parameter values at which the criterion is evaluated.

Value

List with lower, upper, width, n_in_set, q_min, q_max_stat, n.

Details

Formula: H = {theta : q(theta) = 0} with q the sum of squared positive parts, Molinari (2021) equations (4.2) and (4.4).

References

Chernozhukov, V., Hong, H. and Tamer, E. (2007). Estimation and confidence regions for parameter sets in econometric models. Econometrica 75(5), 1243-1284. doi:10.1111/j.1468-0262.2007.00794.x . Criterion and level set as equations (4.2), (4.3) and (4.4) of Molinari, F. (2021), Handbook of Econometrics 7A (arXiv:2004.11751 pp. 88-89).

Examples

Bnsiii(y = c(1, 2, 3, 4, 5, 6, 7, 8), X = c(1, 2, 3, 4, 5, 6, 7, 8),
  moments = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $lower
#> [1] 4.5
#> 
#> $upper
#> [1] 4.5
#> 
#> $width
#> [1] 0
#> 
#> $n_in_set
#> [1] 0
#> 
#> $q_min
#> [1] 0.3333333
#> 
#> $q_max_stat
#> [1] 4.041452
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Identification by intersection of inequalities"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"