Econometrics & Partial Identification
Manski Worst-Case and Monotone Causal Bounds Formula
Bound Average Treatment Effects without random assignment on bounded [0, 1] or [K_0, K_1] outcomes using Manski’s (1990) worst-case bounds and MTR/MTS tightening.
See why Manski worst-case bounds on [0, 1] ALWAYS have exact width 1.00
When observational treatment D in {0, 1} is self-selected without valid instruments or unconfoundedness, point identification fails — yet for any bounded outcome Y in [K_min, K_max], Charles Manski (1990) proved sharp assumption-free "No-Assumptions / Worst-Case" bounds on E[Y(1)] - E[Y(0)] by plugging K_min and K_max into the unobserved counterfactuals! Specifically, E[Y(1)] lies in [p_D*mu_1 + (1 - p_D)*K_min, p_D*mu_1 + (1 - p_D)*K_max] and E[Y(0)] lies in [(1 - p_D)*mu_0 + p_D*K_min, (1 - p_D)*mu_0 + p_D*K_max].
Verify on a binary graduation outcome [K_min = 0, K_max = 1] (B5=0, B6=1): p_D = 0.60 treated share (B2), observed treated graduation mu_1 = 0.80 (B3), observed untreated graduation mu_0 = 0.50 (B4). Worst-case E[Y(1)] in [0.48, 0.88] minus E[Y(0)] in [0.20, 0.80] gives ATE in [-0.3200, +0.6800] — exact width K_max - K_min = 1.0000 (so worst-case bounds alone ALWAYS straddle zero!). Adding Manski’s (1997) Monotone Treatment Response (Y(1) >= Y(0)) lifts the lower bound to 0, and adding Positive Monotone Treatment Selection (E[Y(d)|D=1] >= E[Y(d)|D=0]) caps the upper bound at naive observational difference mu_1 - mu_0 = +0.3000 -> tight MTR+MTS bounds [0.0000, 0.3000]!
| Manski Identification Regime | Lower & Upper Bound Formulas | Sample Bounds [L, U] |
|---|---|---|
| Naive Observational Difference | mu_1 - mu_0 (Point assumption) | +0.3000 |
| Manski Worst-Case Lower L_WC | p*mu_1 + (1-p)*K_0 - ((1-p)*mu_0 + p*K_1) | -0.3200 |
| Manski Worst-Case Upper U_WC | p*mu_1 + (1-p)*K_1 - ((1-p)*mu_0 + p*K_0) | +0.6800 (Width = 1.0000) |
| Joint MTR + Positive MTS Bounds | [0, MAX(0, mu_1 - mu_0)] | [0.0000, +0.3000] |
Manski Assumption-Free Worst-Case Lower & Upper ATE Bounds Formula
Check algebraic width U_WC - L_WC = (1 - p_D)*(K_1 - K_0) + p_D*(K_1 - K_0) = K_1 - K_0 regardless of treatment share p_D!
=LET(p_tr, B2, mu_1, B3, mu_0, B4, k_lo, B5, k_hi, B6, IF(OR(p_tr <= 0, p_tr >= 1, k_hi <= k_lo, mu_1 < k_lo, mu_1 > k_hi, mu_0 < k_lo, mu_0 > k_hi), "Invalid Bounded Support", ROUND((p_tr * mu_1 + (1 - p_tr) * k_lo) - ((1 - p_tr) * mu_0 + p_tr * k_hi), 4)))Monotone Treatment Selection (MTS) Upper Bound Formula
Under positive selection (more capable students self-select into tutoring so E[Y(d)|D=1] >= E[Y(d)|D=0]), E[Y(1)] <= mu_1 and E[Y(0)] >= mu_0, tightening the upper bound from U_WC (+0.6800) all the way down to naive difference mu_1 - mu_0 (+0.3000)!
=IF(B6 <= B5, "Invalid Bounded Support", ROUND(MIN(E3, B3 - B4), 4))Try the example
Use the original values or change them to test your understanding.