DGP Dossier: synth_nonlinear_heteroskedastic

1. Identity & Status

DGP ID synth_nonlinear_heteroskedastic
Version 1.6.0
Status experimental
Difficulty ★★★☆☆ (3/5)
Stress Profile  overlap: moderatenoise: heteroskedasticlinearity: smootheffect: constanttarget: both

2. What This DGP Stresses (Intent)

Monotone nonlinear confounding + severe heteroskedasticity. OLS fails because it fits a plane to a cubic signal and cannot adapt to variance that explodes in the tails of the covariate distribution.

3. Identification Assumptions (Explicit)

  • Selection on observables holds.
  • Overlap is moderate (restored in v1.5.0, carried forward to v1.6.0).
  • SUTVA holds.

4. Mathematical Specification

Covariates: [ X_1 (0, 1^2), X_2 (0, 1^2), X_3 [-2, 2], X_4 (0.4) ]

Outcome (control): [ _0(X) = 1 + 0.5 X_1^3 + 1.5 X_2^2 - 1.0 X_4 ]

Treatment Effect: [ (X) ]

Propensity: [ p(X) = (0.5 X_1 - 0.5 X_2) ]

Heteroskedastic noise: [ !(0, ^2(X)),(X) = 0.1 + 1.0(0.5 X_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

Under the v1.6.0 cubic/exponential regime (with overlap restored), OLS (lm_att) exhibits significant structural bias. While it avoids the variance explosion of severe overlap violations, the linear model cannot adapt to the cubic outcome surface. In quick validation (n = 1000, seeds = 1:10), lm_att shows mean bias around 0.32 (32% error), while oracle_att remains exact.

estimator_id mean_bias rmse
lm_att 0.321 0.404
oracle_att 0.000 0.000

8. Failure Mode Summary

  • Structural underfitting of nonlinear signal (linear model misspecification) drives ATT bias.
  • Severe heteroskedasticity creates variance instability even under moderate overlap.

9. Implementation Reference

  • Generator: R/dgp-synth-nonlinear-heteroskedastic.R
  • Registry: cs_dgp_registry() entry for synth_nonlinear_heteroskedastic
  • Oracle truth: cs_get_oracle_qst("synth_nonlinear_heteroskedastic", version = "1.6.0")

10. Validation Checklist

11. Changelog

  • v1.4.0: Parameter hardening (Deprecated v1.3.0).
  • v1.5.0: Relaxed propensity coefficients to 0.5. v1.4.0 deprecated due to unintended overlap violation.
  • 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_nonlinear_heteroskedastic_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)
    X3 <- stats::runif(n, min = -2, max = 2)
    X4 <- stats::rbinom(n, size = 1L, prob = 0.4)
    mu0 <- 1 + 0.5 * X1^3 + 1.5 * X2^2 - 1 * X4
    tau <- rep(1, n)
    sigma <- 0.1 + 1 * exp(0.5 * X2)
    eps0 <- stats::rnorm(n, mean = 0, sd = sigma)
    if (isTRUE(oracle_only)) {
        eps1 <- eps0
    }
    else {
        eps1 <- stats::rnorm(n, mean = 0, sd = sigma)
    }
    p <- stats::plogis(0.5 * X1 - 0.5 * X2)
    w <- stats::rbinom(n, size = 1L, prob = p)
    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_nonlinear_heteroskedastic", 
            version = "1.6.0")
    }
    else {
        tibble::tibble(tau = cs_tau_oracle, value = rep(NA_real_, 
            length(cs_tau_oracle)))
    }
    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), true_att = true_att, true_qst = true_qst, meta = list(dgp_id = "synth_nonlinear_heteroskedastic", 
        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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