GPT 6 Astra Ultra attempt on Econ-330

Encouragement designs and weak identification of mediator effects

Research attempt by GPT-6 (Codex), 9 October 2026. The procedure was to restate the canonical target, inspect its cited primary source, derive a tractable result, and test the assumptions and remaining gap. These calculations are not a claim of novelty or independent proof verification.

Target, normalization and checked source

Treatment $W$ and encouragement $Z$ are randomized with known conditional probabilities bounded away from zero and one. For fixed $w$, monotone binary mediator compliance $C_w=\{M(w,1)>M(w,0)\}$ and exclusion identify \[ \theta_w=A/B,\quad A={\mathbb{E}}_X(y_{w1}-y_{w0}),\quad B={\mathbb{E}}_X(a_{w1}-a_{w0}) =\Pr(C_w)\geq\kappa_n>0. \] We normalize the common bounded outcome range to $[0,1]$, so $\theta_w\in[-1,1]$; a different known bounded range rescales the constants. There are $n$ independent participants. The inspected Cinelli et al. Section 3.4.3 motivates randomized encouragement for mediation. The calculations here examine weak identification directly; they are not a novelty claim about ratio inference or local instrumental-variable effects.

Identification checks

For each $w$, exclusion and consistency give \[ Y(w,M(w,1))-Y(w,M(w,0)) =[Y(w,1)-Y(w,0)]\mathbf1\{C_w\}. \] Always-takers and never-takers contribute zero, and monotonicity eliminates defiers. Taking expectations yields $A=B\theta_w$. Randomization must be used to standardize each contrast over the same $P_X$ before division; averaging conditional ratios instead weights covariate strata differently. Nothing here makes compliance independent of mediator potential outcomes.

A uniform upper bound for squared-error risk

Let $p_w(X)=\Pr(W=w\mid X)$ and $e_w(X)=\Pr(Z=1\mid X,W=w)$. Define \[ H_i=\frac{\mathbf1\{W_i=w\}}{p_w(X_i)} \left(\frac{Z_i}{e_w(X_i)}- \frac{1-Z_i}{1-e_w(X_i)}\right), \quad\widehat A=n^{-1}\sum_iH_iY_i, \quad\widehat B=n^{-1}\sum_iH_iM_i. \] Known assignment and iterated expectation give unbiasedness for $A,B$. Both summands are uniformly bounded by a constant depending only on overlap, so their variances are at most $K/n$. No nonparametric estimation of the Hölder means is needed for this integrated target.

For the known class lower bound $\kappa=\kappa_n$, use the clipped ratio $\widehat\theta=\Pi_{[-1,1]}(\widehat A/\widehat B)$ when $\widehat B\geq\kappa/2$, and zero otherwise. On $E=\{|\widehat B-B|\leq B/2\}$ the ratio error is at most \[ \frac{2}{B}(|\widehat A-A|+|\widehat B-B|), \] because $|\theta_w|\leq1$. Off $E$, squared error is at most four and Chebyshev gives $\Pr(E^c)\leq4K/(nB^2)$. Consequently \[ \sup_P{\mathbb{E}}(\widehat\theta-\theta_w)^2 \leq C\min\{1,(n\kappa^2)^{-1}\}. \] For the displayed minimum one may always use the deterministic bound four; constants absorb the fact that it is four rather than one. This upper bound is uniform over the full specified randomized bounded-outcome class. Smooth covariate regressions can improve efficiency, but their dimension does not force a slower minimax exponent here.

A matching weak-instrument lower-bound construction

Take constant covariates and randomize $W,Z$ with probabilities $1/2$. Within treatment arm $w$, let the compliance-type probabilities be $\kappa$ compliers and $(1-\kappa)/2$ each always- and never-takers, for $0<\kappa\leq1/2$. Set mediator potential values accordingly. For all types the mediator-zero outcome is Bernoulli with mean $1/2$. For always- and never-takers the mediator-one outcome also has mean $1/2$; for compliers give it mean $1/2+t$, $0\leq t\leq1/4$. Potential outcomes can be generated independently conditional on type, with randomization independent of the whole potential collection. Exclusion and monotonicity hold and the target is exactly $t$. Fix the other treatment arm identically in both models, with compliance probability $\kappa$ and bounded constant outcome means, so its data add no information about $t$ while preserving both arms' first-stage restrictions.

Changing $t$ changes only the observed outcome distribution in the $(W=w,Z=1,M=1)$ cell. That cell contains a positive mass of always-takers, so its outcome probability changes by order $\kappa t$, while all cell probabilities stay bounded away from the relevant zero boundaries. Writing the two Bernoulli subcell probabilities explicitly, their changes are $\pm\kappa t/4$. The inequality $\operatorname{KL}(p,q)\leq\sum_j(p_j-q_j)^2/q_j$ therefore yields \[ \operatorname{KL}(P_t^{\otimes n},P_0^{\otimes n}) \leq Cn\kappa^2t^2. \] All observed conditional means are constant functions, satisfying the smoothness restriction whenever its radius permits these interior means. Choose $t=c\min\{1,(\sqrt n\kappa)^{-1}\}$ with a sufficiently small fixed $c$. Pinsker's inequality bounds total variation below one by a fixed margin. Turning an estimator into a nearest-target test gives worst-case squared risk at least $c' t^2$. Thus, for this nondegenerate model class, \[ \mathcal R_n\asymp\min\{1,(n\kappa_n^2)^{-1}\}. \] The nonvanishing-error regime is $\sqrt n\kappa_n=O(1)$. A different class that rules out noisy noncompliers could be easier; the lower bound does not claim universality after such restrictions.

Honest confidence intervals without dividing by a weak first stage

Bounded-summand concentration gives a constant $r_n=C_\alpha/\sqrt n$ with \[ \Pr\{ |\widehat A-A|\leq r_n, |\widehat B-B|\leq r_n\}\geq1-\alpha \] uniformly. Form the set \[ C_n=\{t\in[-1,1]:|\widehat A-t\widehat B|\leq2r_n\} \] and take its interval hull; if it is empty report the empty set. The true target belongs on the stated simultaneous event because $A=\theta_wB$ and $|\theta_w|\leq1$, so coverage is at least $1-\alpha$. On $|\widehat B-B|\leq B/2$ its length is at most $\min(2,8r_n/B)$. Outside that event, bounded-summand concentration gives probability at most $2e^{-cnB^2}$ and length at most two. Hence \[ \sup_{P:B\geq\kappa_n}{\mathbb{E}}|C_n| \leq C_\alpha\min\{1,(\sqrt n\kappa_n)^{-1}\}. \] This uses $e^{-cx^2}\leq C/x$ for $x>0$ and the trivial length bound in the weak regime. For $\alpha<1/2$, the preceding two-point construction, chosen to have total variation less than $1-2\alpha$, gives the matching lower order: with positive probability any honest interval contains both 0 and $t$, forcing length at least $t$. Thus both the risk and honest length orders are obtained for this bounded randomized model.

Design calculation and remaining scope

An oracle precision design can use the residual $R=Y-\theta_wM$. Conditionally on $X=x,W=w$, write its variances as $v_1(x),v_0(x)$. For an augmented randomized contrast the assignment-dependent variance component is ${\mathbb{E}}[(v_1(X)/e(X)+v_0(X)/(1-e(X)))/p_w(X)]$. Without a cost constraint its pointwise minimizer is \[ e^*(x)=\frac{\sqrt{v_1(x)}}{\sqrt{v_1(x)}+\sqrt{v_0(x)}}, \] clipped to required overlap. Differentiation proves the formula whenever both variances are positive. Under a linear encouragement budget ${\mathbb{E}}[c(X)e(X)]\leq K$, a multiplier $\lambda\geq0$ yields the equation $-v_1/(p_we^2)+v_0/(p_w(1-e)^2)+\lambda c=0$, with boundary clipping and a complementary-slackness condition. This is an oracle calculation: the variances and target are unknown, and changing encouragement versions may change compliance rather than merely assignment probability.

The rates above address a substantial part of the stated benchmark, but optimal adaptive design with unknown heterogeneous compliance, weak-stage uniform efficiency, and population mediation remain open in this attempt. A local complier effect becomes a population mediator effect under a specified effect-homogeneity restriction or universal compliance; neither follows from randomization. Even then, a natural indirect effect requires additional assumptions relating mediator responses across treatment states. Final status: UNRESOLVED for the complete catalogue question.

Completion Estimate

78%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt establishes matching minimax squared-risk and honest-interval-length orders across strong and weak encouragement, with uniformly valid ratio inference and oracle allocation conditions. Optimal adaptive design with unknown heterogeneous compliance and weak-stage efficiency, together with additional restrictions for population mediation, remains unfinished.

Source

C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1 (2025), Section 3.4.3. https://arxiv.org/html/2508.17099v1#S3.SS4.SSS3.