Delta vs simulation vs bootstrap

Three inference paths, one session knob (method=). Picking is a function of curvature, sample structure, and refit cost.

Delta

  • First-order Taylor of g(β) around β̂.

  • Cheap: one Jacobian-vector product per estimand.

  • Accurate when κ is small (see The κ curvature diagnostic).

  • Symmetric Wald CI on the inference scale; can cross natural bounds when back-transformed.

Krinsky–Robb simulation

  • Draw β_s ∼ N(β̂, V̂), evaluate g(β_s), read inference off the empirical distribution.

  • No refits — just S forward evaluations.

  • Reported point estimate stays analytic g(β̂) (matches Stata’s vce(simulation)).

  • Pointwise CIs default to empirical quantiles, so asymmetric CIs for extreme probabilities / ratios are natural.

  • Used as the κ fallback target.

Bootstrap

  • Refit on B resamples, read inference off the bootstrap distribution.

  • Expensive (refit cost × B) but robust to model misspecification and to estimand curvature.

  • Three resampling schemes (pairs, cluster, block); see Cluster and block bootstrap.

  • Required when the asymptotic variance is suspect (small clusters, long-memory time series, mis-specified GLM family).

Decision rule

┌──────────────────────────────────────────────────────┐
│ Curvature high (κ ≥ 0.3) ──────────► simulation      │
│ Cluster correlation ──────────────► cluster bootstrap │
│ Heavy time-series dependence ─────► block bootstrap   │
│ Otherwise ───────────────────────► delta              │
└──────────────────────────────────────────────────────┘

pymargins automates the first row through the κ fallback. The others are user-decisions encoded at session construction.