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Imbens-Manski confidence interval for a partially identified parameter

Source: R/bnd_native.R
morie_bnd_imbens_manski.Rd

[lower - c s_l/sqrt(n), upper + c s_u/sqrt(n)] with c solving pnorm(c + sqrt(n) delta / max(s_l, s_u)) - pnorm(-c) = 1 - alpha.

Usage

morie_bnd_imbens_manski(
  lower_hat,
  upper_hat,
  se_lower,
  se_upper,
  n,
  alpha = 0.05
)

Arguments

lower_hat, upper_hat

estimated bounds.

se_lower, se_upper

their standard deviations on the sqrt(n) scale, i.e. sqrt(n) * se(bound).

n

sample size.

alpha

miss probability.

Value

list: ci, c, z_one_sided, z_two_sided, delta, covers, stoye_caveat, n, method.

Details

The construction exists because covering the identified SET and covering the true PARAMETER are different targets. The parameter can only be missed at one end at a time, so a wide identified set needs only the one-sided z; as the set collapses to a point the problem becomes point identification and the two-sided z is what comes out. c interpolates and attains both limits exactly. Stoye (2009) showed the interpolation needs delta's own sampling error negligible or the bound superefficient; the output says so.

References

Imbens and Manski (2004), Econometrica 72:1845-1857, Eq. (6); Stoye (2009), Econometrica 77:1299-1315.

Examples

morie_bnd_imbens_manski(0.2, 0.8, 1, 1, 400)$c
#> [1] 1.644854