DGP Dossier: synth_tilt_mild

1. Identity & Status

DGP ID	synth_tilt_mild
Version	1.6.0
Status	experimental
Difficulty	★★☆☆☆ (2/5)
Stress Profile	overlap: mildnoise: gaussianlinearity: lineareffect: lineartarget: both

2. What This DGP Stresses (Intent)

Mild overlap tilt. A warm-up for the severe overlap stress test.

3. Identification Assumptions (Explicit)

• Selection on observables holds.
• Overlap is mildly tilted but still adequate.
• SUTVA holds.

4. Mathematical Specification

Covariates: [ X (0, I_5) ]

Propensity: [ p(X) = (0.45 X_1 - 0.3 X_2 - 0.25 X_4) ]

Including (X_4) induces a mild covariate shift between treated and control units.

Outcome (control): [ _0(X) = 1 + X_1 + 0.5 X_2 ]

Treatment effect: [ (X) = 1 + 0.5 X_1 ]

Noise: [ (0, 0.5^2) ]

5. Oracle Truth Definition

• True ATT is computed from structural_te among treated units.
• True QST is computed on the oracle tau grid (cs_get_oracle_qst()).

6. Visual Diagnostics (n = 5000)

7. Empirical Validation

Mild stress. Most methods pass.

Empirical validation (n=1000, seeds=1:10)

estimator_id	mean_bias	rmse
lm_att	0.008	0.043
oracle_att	0.000	0.000

8. Failure Mode Summary

• Mild tilt mainly increases variance slightly; it should not break estimators.

9. Implementation Reference

• Generator: R/dgp-synth-tilt-mild.R
• Registry: cs_dgp_registry() entry for synth_tilt_mild
• Oracle truth: cs_get_oracle_qst("synth_tilt_mild", version = "1.6.0")

10. Validation Checklist

• Propensity specified as expit(0.45X1 - 0.3X2 - 0.25X4)
• X1 vs X2 titled “Covariate Overlap (Joint Support)”

11. Changelog

• v1.3.0 (2026-01-08): Batch C dossier regenerated with full math specification.
• v1.6.0: Implemented Common Random Numbers (CRN) for Oracle generation. This reduces Monte Carlo variance of the QST contrast; it does not eliminate empirical-quantile sampling uncertainty.

Appendix: Implementation

Source code for: dgp_synth_tilt_mild_v160

function (n, seed = NULL, include_truth = TRUE, oracle_only = FALSE) 
{
    if (!is.null(seed)) {
        cs_set_rng(seed)
    }
    X1 <- stats::rnorm(n, mean = 0, sd = 1)
    X2 <- stats::rnorm(n, mean = 0, sd = 1)
    X4 <- stats::rnorm(n, mean = 0, sd = 1)
    if (!isTRUE(oracle_only)) {
        X3 <- stats::rnorm(n, mean = 0, sd = 1)
        X5 <- stats::rnorm(n, mean = 0, sd = 1)
    }
    mu0 <- 1 + X1 + 0.5 * X2
    tau <- 1 + 0.5 * X1
    p <- stats::plogis(0.45 * X1 - 0.3 * X2 - 0.25 * X4)
    w <- stats::rbinom(n, size = 1L, prob = p)
    eps0 <- stats::rnorm(n, mean = 0, sd = 0.5)
    if (isTRUE(oracle_only)) {
        eps1 <- eps0
    }
    else {
        eps1 <- stats::rnorm(n, mean = 0, sd = 0.5)
    }
    y0 <- mu0 + eps0
    y1 <- mu0 + tau + eps1
    if (isTRUE(oracle_only)) {
        return(list(df = tibble::tibble(w = w, y0 = y0, y1 = y1)))
    }
    y <- ifelse(w == 1L, y1, y0)
    true_att <- cs_true_att(structural_te = tau, w = w)
    true_qst <- if (isTRUE(include_truth)) 
        cs_get_oracle_qst("synth_tilt_mild", version = "1.6.0")
    else NULL
    out <- list(df = tibble::tibble(y = y, w = w, y0 = y0, y1 = y1, 
        p = p, structural_te = tau, X1 = X1, X2 = X2, X3 = X3, 
        X4 = X4, X5 = X5), true_att = true_att, true_qst = true_qst, 
        meta = list(dgp_id = "synth_tilt_mild", version = "1.6.0", 
            type = "synthetic", params = list(n = n, seed = seed), 
            structural_te = tau))
    cs_check_dgp_synthetic(out)
    out
}
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