Bound the complier effect when defiers are not ruled out

The first stage identifies only the NET complier share, pi_c - pi_d; monotonicity is what turns that into pi_c. Without it the Wald ratio is a difference of two effects, and the data bound the defier share only through pi_c <= min(P(D = 1 | Z = 1), P(D = 0 | Z = 0)). The interval widens with the admissible defier share, so the union over admissible pi_c is attained at that maximum, and collapses to the Wald ratio exactly when the maximum equals the net share.

Usage

Bndnmt(y, D, Z)

Arguments

y

Observed outcome.

D

Binary treatment, coded 0/1.

Z

Binary instrument, coded 0/1.

Value

List with lower, upper, width, estimate, wald, pi_net, pi_c_max, pi_d_max, itt_y, n. pi_d_max, itt_y, n.

Details

Derivation: ITT_y = pi_c E[D | c] - pi_d E[D | d] with |E[D | d]| <= y_1 - y_0, so with pi_d = pi_c - ITT_D, LATE in [(ITT_y - pi_d R) / pi_c, (ITT_y + pi_d R) / pi_c] at pi_c = pi_c_max.

References

de Chaisemartin, C. (2017). Tolerating defiance? Local average treatment effects without monotonicity. Quantitative Economics 8(2), 367-396. doi:10.3982/QE601 , for the problem; the interval above is the elementary mixture bound, derived in Details.

Examples

set.seed(1)
n <- 500
Z <- rbinom(n, 1, 0.5)
D <- rbinom(n, 1, 0.2 + 0.5 * Z)
y <- 1 + D + rnorm(n)
Bndnmt(y, D, Z)[c("lower", "upper", "wald")]
#> $lower
#> [1] -1.164068
#> 
#> $upper
#> [1] 2.81464
#> 
#> $wald
#> [1] 1.149536
#>