When cases are assigned to judges at random, the judge's leniency is an
instrument for the decision (detention, a custodial sentence): the
difference in outcomes between lenient and strict judges over the
difference in decisions is the Wald ratio (Imbens and Angrist 1994).
It is the average effect on the compliers, the cases whose decision
depends on the judge, only if no case defies (is detained by the
lenient judge and released by the strict one). With a share of defiers
s whose effect is e, the compliers number the first stage plus s
and their effect is (wald * fs + s * e) / (fs + s); sens gives it
over a grid, so a reader sees how far the conclusion rests on
monotonicity (de Chaisemartin 2017). Code z so that the first stage
is positive (1 for the judge whose decision is more often 1); with a
negative one the complier share is negative and ceff is NA.
Usage
jdgiv(z = NULL, d, y, w = NULL, dsh = c(0, 0.05, 0.1), deff = 0)Arguments
- z
NULL, or the instrument of each case: 0 for the strict judge, 1 for the lenient (orFALSE,TRUE).- d
The decision of each case: 0 or 1; with
z = NULL, two columns (a matrix or data frame), the decisions under the strict and under the lenient judge.- y
The outcome of each case: numbers; with
z = NULL, two columns, the outcomes without and with the decision.- w
NULL, or a weight from 0 for each case.- dsh
Defier shares for the sensitivity grid: shares from 0 to 1.
- deff
Defier effects for the sensitivity grid: numbers.
Value
An object of class jdgiv, a list: type ("obs" or
"pop"); est, a named vector (observed: fs the first stage, itt,
wald; population: itt, fs, wald, ate); shares (observed:
complier, always, never if no one defies; population: also
defier); and sens (observed: dsh, deff, cshare the complier
share, ceff their effect) or effects (population: the mean effect
of each type, NA for an empty one).
Details
Given the decisions and outcomes of a whole population under both
judges (z = NULL, d and y two columns each), the exact
decomposition: the shares and mean effects of compliers, defiers,
always- and never-takers, the intention-to-treat effect and the first
stage, whose ratio is the compliers' effect when there are no defiers.
See proof("jdgiv").
References
Imbens, G. W. and Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62(2), 467-475. doi:10.2307/2951620
de Chaisemartin, C. (2017). Tolerating defiance? Local average treatment effects without monotonicity. Quantitative Economics, 8(2), 367-396. doi:10.3982/QE601
Examples
set.seed(7)
z <- rbinom(1000, 1, 0.5)
d <- rbinom(1000, 1, ifelse(z == 1, 0.3, 0.5))
y <- rbinom(1000, 1, 0.2 + 0.1 * d)
jdgiv(z, d, y)
#> jdgiv: observed
#> fs itt wald
#> -0.17071258 0.01969084 -0.11534496
#> complier always never
#> -0.1707126 0.4723320 0.6983806
#> dsh deff cshare ceff
#> 0.00 0 -0.17071258 NA
#> 0.05 0 -0.12071258 NA
#> 0.10 0 -0.07071258 NA