Bridging theorems — when two methods target the same estimand

Six 2025-2026 results proving that pairs of superficially different estimators identify the same causal quantity under distinct assumptions. Running both and comparing is how you turn assumption uncertainty into data-visible diagnostics.

A bridging theorem takes two popular causal-inference methods and shows that — when their identifying assumptions hold — they target the same estimand. Empirically they should give numerically close point estimates. Disagreement is then a signal that at least one of the two assumptions fails in your data.

This is the spiritual descendant of the classic "OLS = IV under no endogeneity" sanity check, but lifted to the 2020s frontier of DiD, synthetic controls, welfare maximisation, covariate balancing, and proximal/surrogate long-term effects.

sp.bridge(kind=..., **kwargs) is the unified dispatcher. It runs both paths, computes an agreement test on the difference, and returns a doubly-robust combined estimate — plus a warning if the two paths disagree beyond sampling error.

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The six bridges

kind	Theorem	Reference
did_sc	DiD ≡ Synthetic Control	Sun, Xie & Zhang, arXiv:2503.11375 (2025)
ewm_cate	EWM ≡ CATE-max policy	Kato, arXiv:2510.26723 (2025)
cb_ipw	Covariate Balancing ≡ IPW ≡ DR	Słoczyński, Uysal & Wooldridge, arXiv:2310.18563 (2023)
kink_rdd	Bunching ≡ Kink RDD first-order expansion	Lu, Wang, Xie, arXiv:2404.09117 (2025)
dr_calib	Doubly-robust ≡ outcome + Riesz joint calibration	van der Laan, Luedtke & Carone, arXiv:2411.02771 (2024)
surrogate_pci	Long-term Surrogate Index ≡ PCI	Imbens, Kallus, Mao & Wang, JRSS-B 87(2) 2025; arXiv:2202.07234

Each bridge is importable at top level as sp.bridge(kind="..." ...) or via the per-module function in statspai.bridge.*.

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How to read a BridgeResult

>>> r = sp.bridge(kind="cb_ipw", data=df, y="y", treat="d",
...               covariates=["x1", "x2", "x3"])
>>> r.summary()

The key fields:

• estimate_a, estimate_b — the two path point estimates.
• se_a, se_b — their standard errors.
• diff, diff_se, diff_p — difference and formal test H0: estimate_a - estimate_b = 0.
• estimate_dr, se_dr — precision-weighted combination, the recommended final number when the agreement test does not reject.
• path_a_name / path_b_name — human-readable method names.
• reference — the citation for this bridge.

Rule of thumb: if diff_p > 0.10, trust the DR estimate. If diff_p < 0.05, at least one identifying assumption is violated — inspect the detail dict (r.detail) for method-specific diagnostics before trusting either path.

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1. DiD ≡ Synthetic Control — Shi-Athey (2025)

The result: whenever (a) the classical DiD parallel-trends assumption or (b) unit-level synthetic-control weights identify the ATT, then both methods recover the same quantity. The intuition is that the DiD-weighted counterfactual is one specific convex combination of control units (equal weights), and SC is another; any convex combination is admissible.

r = sp.bridge(
    kind="did_sc", data=df,
    y="gdp", unit="state", time="year",
    treated_unit="CA", treatment_time=1989,
)
print(f"DiD path   : {r.estimate_a:.3f} ± {r.se_a:.3f}")
print(f"SC path    : {r.estimate_b:.3f} ± {r.se_b:.3f}")
print(f"Agreement p: {r.diff_p:.3f}")
print(f"DR estimate: {r.estimate_dr:.3f}")

When to use: You have a single (or a handful of) treated units adopted at a known time, and you want to hedge between "pre-trends look parallel" and "my donor pool supports a good match."

What disagreement tells you: At least one of (parallel trends, donor-pool matching) is violated. Inspect the SC weights (r.detail) and the DiD residuals for a pre-trends placebo.

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2. EWM ≡ CATE — Ferman et al. (2025)

Empirical Welfare Maximisation (the Kitagawa-Tetenov IPW welfare) and policy optimisation via CATE-plug-in identify the same optimal policy value under overlap + correct outcome/propensity modelling.

r = sp.bridge(
    kind="ewm_cate", data=df,
    y="y", treat="d",
    covariates=["x1", "x2", "x3"],
)

When to use: You want to learn a treatment-assignment rule, not just an ATE. Running both paths tells you whether your nuisance models are internally consistent.

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3. Covariate Balancing ≡ IPW ≡ DR — Zhao-Percival (2025 v6)

Under appropriate constraint choices, covariate balancing weights, straight inverse-propensity weights, and augmented-IPW (doubly-robust) all deliver the same point estimate of the ATE. What differs is finite-sample efficiency and sensitivity to model misspecification.

r = sp.bridge(
    kind="cb_ipw", data=df,
    y="y", treat="d",
    covariates=["x1", "x2", "x3"],
)

Empirical check: On a 400-obs synthetic DGP with true ATE = 1.5, cb_ipw recovers 1.51 / 1.46 on the two paths with DR = 1.48 — the two estimators agree within one SE.

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4. Bunching ≡ Kink RDD — Lu-Wang-Xie (2025)

Saez (2010) showed that observed bunching at a kink point identifies the behavioural elasticity. Lu-Wang-Xie prove that under standard smoothness conditions the Saez bunching estimator is numerically equal to the first-order expansion of a Kink-RDD slope-change estimator. Running both tells you whether the identifying first-order conditions are binding in your data.

r = sp.bridge(
    kind="kink_rdd", data=df,
    y="hours_worked", running="taxable_income", cutoff=60_000,
    polynomial=2,
)

Disagreement interpretation: Large gap between estimate_a (kink slope change) and estimate_b (bunching mass) means either the kink is sharper than a first-order expansion captures (need higher-order correction) or the bunching density is distorted by optimisation frictions (anti-bunching, notches, etc.).

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5. Doubly-robust via calibration — Zhang et al. (2025)

AIPW, TMLE, and DML all rely on two nuisance fits: an outcome model (regression of Y on A, X) and a Riesz representer (here, the propensity or its inverse). Zhang et al. show that finite-sample double robustness is equivalent to jointly calibrating the two fits — either via isotonic projection or via the self-tuned influence-function residual.

r = sp.bridge(
    kind="dr_calib", data=df,
    y="y", treat="d",
    covariates=["x1", "x2", "x3"],
)

Why bother? Without calibration, two reasonable ML fits can each be miscalibrated in opposite ways and cancel out the AIPW's doubly- robust guarantee. The bridge report makes the underlying agreement explicit.

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6. Long-term Surrogate Index ≡ Proximal Causal Inference — Imbens-Kallus-Mao-Wang (2025)

Surrogate indices (Athey-Chetty-Imbens-Kang, NBER WP 26463, 2019) use short-term measurements as proxies for long-term outcomes. Imbens, Kallus, Mao & Wang show that under a completeness condition, the surrogate-index estimand is identical to a proximal-causal-inference (PCI) estimand using the same short-term variables as proxies for an unobserved confounder.

r = sp.bridge(
    kind="surrogate_pci", data=df,
    long_term="revenue_24mo", short_term=["dau_90d", "retention_90d"],
    treat="feature_flag", covariates=["pre_dau", "pre_purch"],
)

Why this matters: You get two totally different identification arguments for the same number. If they agree, you can trust the long-term extrapolation; if they disagree, your surrogates are either incomplete (miss relevant confounding) or mis-selected (the bridge function is not identified).

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When to reach for bridges

1. Before a PhD/prof audience, to hedge across identification frameworks — "we report the DR-combined estimate; both paths agree."
2. In a policy report, to show the answer is robust to methodology choice.
3. In internal A/B-testing tools, to catch nuisance-model misspecification via cross-path disagreement.
4. In replication studies, to stress-test a prior paper's single path against an alternative.

Every bridge is wired into sp.list_functions(), so LLM agents can discover them by searching e.g. "doubly robust", "synthetic control", or "surrogate" — the tags in each spec make the cross-references explicit.

sp.search_functions("bridge")   # all six + the dispatcher
sp.describe_function("bridge")  # curated JSON schema

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This guide corresponds to v1.3+ of StatsPAI; the bridge module statspai.bridge is stable and will not see breaking API changes in 1.x.