Manski-Pepper MTR-MTS bounds (the returns-to-schooling application)

Joint monotone-treatment-response / monotone-treatment-selection bounds on \(E[y(t)]\). Under MTR (each unit response weakly increasing in the ordered treatment) and MTS (units selecting higher treatment have weakly higher mean response functions), Proposition 3 Corollary 2 of Manski and Pepper (2000), eq. (36), gives sharp bounds that need no outcome-support assumption: $$\sum_{u<t} E[y|z=u] P(z=u) + E[y|z=t] P(z \ge t) \le E[y(t)] \le \sum_{u>t} E[y|z=u] P(z=u) + E[y|z=t] P(z \le t).$$ For a contrast \(t_1 > t_0\) the upper bound is \(U(t_1) - L(t_0)\); the lower bound is 0 because MTR alone already implies \(y(t_1) \ge y(t_0)\) unit by unit (Manski 1997, Proposition M2).

Usage

Bndapp(y, z, t1 = NULL, t0 = NULL)

Arguments

y

Observed outcome.

z

Realized treatment on an ordered scale; every distinct value is a level.

t1

Upper contrast level; default the largest realized level.

t0

Lower contrast level; default the smallest realized level.

Value

List with levels, lower, upper (per-level, parallel vectors), ate_lower (0), ate_upper, t1, t0, n, method.

References

Manski, C. F. and Pepper, J. V. (2000), Monotone Instrumental Variables: With an Application to the Returns to Schooling, Econometrica 68(4):997-1010, eq. (36) (Proposition 3, Corollary 2) and eq. (34). Manski, C. F. (1997), Monotone Treatment Response, Econometrica 65(6):1311-1334, Proposition M2, as printed in Molinari, F. (2021), Handbook of Econometrics 7A, eq. (2.13) (arXiv:2004.11751).

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Bndapp(V, V)
#> $levels
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $lower
#> [1] 1.000 1.875 2.625 3.250 3.750 4.125 4.375 4.500
#> 
#> $upper
#> [1] 4.500 4.625 4.875 5.250 5.750 6.375 7.125 8.000
#> 
#> $ate_lower
#> [1] 0
#> 
#> $ate_upper
#> [1] 7
#> 
#> $t1
#> [1] 8
#> 
#> $t0
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Manski-Pepper (2000) MTR-MTS bounds, eq. (36)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"