GPT 6 Astra Ultra attempt on Econ-248
economicsEconomics cataloguecontributionexplorationClaim: exploration
Reported by the author as unresolved.
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Contents
Transporting interventional mediation effects: sharp drift envelopes and two-sample information
Scope
Interventional mediation transport combines study controlled means with target mediator laws: \[ \psi_0(w,w')=E_{X\sim P_0}\sum_{m=1}^K \mu_w^0(X,m)g_{w'}^0(m\mid X). \] Section 3.4.3 of [agenda] raises generalization of mediation effects. Exact invariance sets $\mu_w^0=\mu_w^1$; weak invariance permits $|\mu_w^0-\mu_w^1|\le\delta$. This note derives sharp sensitivity envelopes when the pointwise choices are admissible, and the basic two-sample rate in a finite-cell submodel. For the sampling results, observations are iid within each population and the two samples are mutually independent. Assume consistency and study treatment/mediator randomization, or exchangeability sufficient to identify the controlled means from the study cell regressions. The target reference treatment is randomized, or exchangeable for its mediator law, so its arm-specific distribution identifies the stated reference intervention.Proposition (A rectangular sensitivity envelope): Assume target responses are bounded by $B$. For a contrast with supplied coefficients $a_{wm}(x)$, let $T=E_0\sum_{w,m}a_{wm}(X)\mu_w^0(X,m)$ and define \[
\ell_{wm}=\max\{-B,\mu_w^1-\delta\},\quad
u_{wm}=\min\{B,\mu_w^1+\delta\}.
\] Every weakly invariant target satisfies $T\in[L,U]$, where \[
L=E_0\sum_{w,m}(a_{wm}^+\ell_{wm}-a_{wm}^-u_{wm}),\qquad
U=E_0\sum_{w,m}(a_{wm}^+u_{wm}-a_{wm}^-\ell_{wm}),
\] and $a^+=\max(a,0)$, $a^-=\max(-a,0)$. These bounds are sharp over the rectangular class allowing all measurable mean selections in the indicated intervals. They remain valid outer bounds if additional smoothness constraints are imposed.
Proof: For a positive coefficient the smallest product uses the lower endpoint, and for a negative coefficient it uses the upper endpoint. Sum and integrate to prove the lower bound; reverse choices for the upper bound. Measurable pointwise endpoint choices attain both bounds in the rectangular class. Every intermediate value is attained by a convex combination of the two mean functions. Each bounded selected mean can be realized as the expectation of a random variable in $\{-B,B\}$ using an independent uniform disturbance, with treatment and mediator mechanisms chosen separately as required. Thus these are possible controlled-mean models. Under an additional fixed Holder radius the endpoint choices need not be admissible, and sharpness then requires checking that constraint separately. ∎
If clipping is inactive, the interval has center $E_0\sum a_{wm}\mu_w^1$ and half-width $\delta E_0\sum|a_{wm}|$. For an indirect effect holding $w$ fixed, $a_m=g_1^0-g_0^0$, so the half-width is $2\delta E_0\operatorname{TV}(g_1^0(\cdot\mid X),g_0^0(\cdot\mid X))$. For a direct effect comparing two treatment arms against a common mediator distribution, it is $2\delta$. These exact formulas hold, in particular, in the constant-covariate submodel with interior means, where the smoothness issue disappears. Identical reference mediator laws make the indirect contrast exactly zero even when $\delta>0$.Proposition (Finite-cell two-sample rate): Consider a constant-covariate submodel with fixed $K\ge2$. In an independent study sample of size $n$, each relevant cell $(W=w,M=m)$ has probability at least $\rho>0$ and $|Y|\le B$. In a target sample of size $N$, the reference arm $W=w'$ has probability at least $\epsilon>0$ and its mediator law is $g=(g_1,\ldots,g_K)$. Under exact response invariance, the target is $\psi=\sum_mg_m\mu_m$. There exists an estimator with \[
\sup E(\widehat\psi-\psi)^2\le C\left(\frac1n+\frac1N\right),
\] where $C$ depends on $K,B,\rho,\epsilon$. On a class containing interior Bernoulli cells and a pair of mediator states with distinct means, this order is also a minimax lower bound.
Proof: Estimate each $\mu_m$ by its study cell average, using zero for an empty cell; estimate $g_m$ by target reference-arm proportions, using a fixed distribution if that arm is empty. For a binomial count $C_n$ with success probability at least $\rho$, $P(C_n<n\rho/2)\le e^{-n\rho/8}$. Conditional variance on its complement is at most $2B^2/(n\rho)$, and boundedness controls the exceptional event. Hence each cell mean has mean squared error $O(1/n)$. The same argument gives $E\|\widehat g-g\|_1^2=O(1/N)$ for fixed $K$.
Using $\widehat\psi=\sum_m\widehat g_m\widehat\mu_m$ and $|\widehat\mu_m|\le B$, \[ |\widehat\psi-\psi|\le\sum_mg_m|\widehat\mu_m-\mu_m|+ B\|\widehat g-g\|_1. \] Jensen's inequality for the first term proves the upper bound. For the lower bound, first vary every study response mean by $\pm c/\sqrt n$, keeping the target law fixed and all probabilities interior. Target separation is of order $n^{-1/2}$, while study relative entropy is bounded. Second, keep all study means fixed with $\mu_1-\mu_2\ne0$, and shift target mediator mass between states one and two by $\pm c/\sqrt N$. Separation is of order $N^{-1/2}$ and target relative entropy is bounded. Two-point testing gives both lower bounds; their maximum is at least half their sum. ∎
Proposition (The two information contributions): In the same submodel, write $e_1=P_1(W=w)$, $p_m=P_1(M=m\mid W=w)$, and $e_0=P_0(W=w')$. The finite-cell plug-in estimator has the expansion \[
\widehat\psi-\psi=\frac1n\sum_{i=1}^n D_1(O_i)+\frac1N\sum_{j=1}^N D_0(V_j)
+O_P(n^{-1}+N^{-1}),
\] where \[
D_1=\frac{1\{W=w\}g_M}{e_1p_M}(Y-\mu_M),\qquad
D_0=\frac{1\{W=w'\}}{e_0}(\mu_M-\psi).
\] Consequently its leading variance is $\operatorname{Var}(D_1)/n+\operatorname{Var}(D_0)/N$.
Proof: Write each sample cell mean and each target arm proportion as a ratio of empirical averages. On the event all denominators exceed half their population values, Taylor expansion of the ratio has a quadratic remainder bounded by a constant times the squared empirical errors. Fixed cell dimension, bounded observations, and denominator positivity make those remainders $O_P(n^{-1})$ and $O_P(N^{-1})$. The first-order terms of the cell means sum to $D_1$; the first-order target proportion terms sum to $D_0$. The product of study and target estimation errors is $O_P((nN)^{-1/2})\le O_P(n^{-1}+N^{-1})$. The samples are independent, and both scores have mean zero, proving the variance assertion. ∎
Under weak invariance, simultaneous finite-sample confidence bounds for the study cell means and target proportions can be propagated by minimizing and maximizing $L,U$ over their compact confidence region. On the simultaneous coverage event the resulting interval contains the entire rectangular identified interval. This gives full-set coverage without estimating a single favorable target mean.Remaining gap
The finite-cell theory leaves open the sharp smooth-covariate rates, growing mediator dimension, and efficiency with unknown density ratios. Pointwise sensitivity endpoints may violate a fixed smoothness radius, so the displayed rectangular envelope is not asserted to be the sharp identified set under every restriction in the original statement.Completion Estimate
45%This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives rectangular response-drift envelopes and matching two-sample rates and information contributions in a finite-cell mediation model. Sharp smooth-covariate rates, efficient unknown-ratio inference and sensitivity endpoints satisfying the original fixed smoothness restrictions remain unproved.
References
- [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane, and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025). https://arxiv.org/html/2508.17099v1.