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Econ-557 · economics · Economics

Long-run effects from proxies

LLM status Open unverified

Statement

Original subject group: Econometrics, causal inference and measurement

Audit classification: Published research agenda. The2026 paper explicitly lists these broad challenges in Section6. The catalogue interval formula is editorial and already tractable under supplied tolerances; the earlier target-Q formulation omitted proxy-law transport from the new experiment to target users.

Definitions and precise research target

Consider a new randomized platform experiment with treatment $A\in\{0,1\}$, baseline covariates $X$, short-run proxy $Z$, and long-run potential outcome $Y_H(a)\in[0,1]$ at horizon $H$. Historical experiments observe $A,X,Z,Y_H$; the new experiment initially observes only $A,X,Z$. The target population has a specified covariate distribution $Q$, possibly differing from historical participants, and the target is $\tau_H=E_Q[Y_H(1)-Y_H(0)]$. Let $m_a(x,z)$ be the historical conditional mean of $Y_H$, and let $\mathcal R_\delta$ restrict deviations between new and historical conditional means by $\delta_a(x,z)\geq0$, including deviations from complete mediation and treatment-version changes. Assume treatment randomization, consistent measurement and overlap where historical extrapolation is used. Derive sharp bounds for $\tau_H$ by integrating admissible means over each new treatment arm's proxy distribution and the target covariate distribution. Which long-run validation subsamples, persistent-treatment experiments or substantively credible restrictions justify smaller deviation bounds? Treat missing support separately with outcome bounds. The source explicitly identifies evolving treatments, incomplete proxy mediation and population mismatch as open challenges; this specified identified-set formulation adapts that agenda rather than assuming short-run surrogates universally identify persistent effects.

Definitions, assumptions and mathematical qualifications. Treatment $a$ denotes a fully specified persistent intervention through horizon $H$, including updates if any. Let $K_a(dz\mid x)$ be the new-trial proxy law under randomized $A=a$. To integrate it under target covariate law $Q$, assume the target proxy law given $X=x$ equals $K_a$; otherwise supply bounds or target proxy data. Define historical predictive means $m_a(x,z)=E[Y_H\mid A=a,X=x,Z=z,\text{historical}]$, not a causal effect of setting the proxy. Target conditional means obey $l_a=\max(0,m_a-\delta_a)\leq\mu_a\leq u_a=\min(1,m_a+\delta_a)$. Off historical support set $l_a=0,u_a=1$. With unrestricted conditional means within these bounds, the sharp effect interval is $[\int(\int l_1dK_1-\int u_0dK_0)dQ,\int(\int u_1dK_1-\int l_0dK_0)dQ]$. Computing this interval is elementary once kernels and tolerances are supplied; credible calibration and transport are the research challenges. Conditional kernel transport additionally requires target covariate absolute continuity with respect to the new experimental covariate law, and absolute continuity of target arm-specific $X,Z$ laws with respect to historical arm-specific laws wherever $m_a$ is used. Null-set failures use the full $[0,1]$ bounds rather than an arbitrary version of a conditional mean.

Variants and assumption changes

  1. Editorial, stable fully mediated benchmark. Assume a fixed treatment version, no direct long-run channel after conditioning on the proxy, stable conditional means and target proxy transport. Set $\delta_a=0$ on common support.
  2. Editorial, evolving/persistent treatment. Index outcomes by treatment paths $Y_H(a_1,\ldots,a_H)$, allow direct long-run channels and population drift, and use representative long-run holdouts to calibrate deviations and proxy-law transport separately.

References and source audit

Checked source. arXiv:2608.08043v1 (8 August 2026), Section 6, propositions 6.3--6.6. Explicit research challenges concern evolving treatments, persistent effects, transport and decision accuracy. Full primary HTML and complete author list checked. https://arxiv.org/html/2608.08043v1.
  1. Leif Sigerson; Tom Cunningham; Winston Chou; Sana Pandey; Jonathan Stray; Lo-Hua Yuan; Eytan Bakshy; Timothy Chan; Molly Davies; Maria Dimakopoulou; Simon Ejdemyr; Kenneth Hung; Nathan Kallus; Madhav Kumar; Thu Le; A. Demetri Pananos; Lee Richardson; Brennan Schaffner; Rose Tan; Martin Tingley; Nadia Tomova; Panagiotis Toulis; Wenjing Zheng; Zander Arnao; Dean Eckles. 2026. Evaluating for the Long Term: Learnings from Industry. arXiv:2608.08043v1 (8 August 2026), Section 6, propositions 6.3--6.6. https://arxiv.org/html/2608.08043v1. DOI: https://doi.org/10.48550/arXiv.2608.08043.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-557-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-557

Long-run effects from proxies: representative validation and the value of residual sampling

Target and scope

The estimand is the effect of two fully specified persistent treatment paths, \[ \tau_H=E_Q[Y_H(1)-Y_H(0)],\qquad 0\le Y_H(a)\le1. \] Historical predictive means and new short-run proxy laws do not alone identify this effect. The supplied pointwise sensitivity interval is already elementary. We instead analyze how randomized long-run validation can replace a tolerance by information, and how to allocate that validation. The industry research agenda in [longrun], Section 6, explicitly distinguishes evolving treatments, persistent channels and population mismatch. No universal surrogate-validity assertion is inferred from it.

A validation experiment

Let new-trial observations be iid. Write $e_a(x)=P(A=a\mid X=x)$, with both arms bounded below by $\epsilon>0$. Assignment is independent of the potential proxy and long-run outcomes given $X$, and consistency holds for the specified treatment path. Let $P_X$ be the trial covariate law and suppose the supplied ratio $r=dQ/dP_X$ is bounded. Assume both the proxy law and the conditional long-run response law in this trial transport to the target given $X$ and the realized arm-specific proxy. This last assumption is substantive: validating trial participants need not validate a different institution.

At the short-run assessment, randomize long-run outcome collection with indicator $R$ and supplied probability $\pi_a(X,Z)>0$. Require $R\perp Y_H\mid X,A,Z$. All participants remain under their original treatment path regardless of whether $R=1$; sampling an outcome is not permission to end treatment. Let $m_a(X,Z)\in[0,1]$ be a supplied or independently trained historical predictor, with arbitrary bounded values where historical support is absent.

Theorem (Validation removes predictive drift bias): Put $s_1=1$, $s_0=-1$ and \[ T=\sum_{a=0}^1\frac{s_a r(X)1\{A=a\}}{e_a(X)} \left[m_a(X,Z)+\frac R{\pi_a(X,Z)}\{Y_H-m_a(X,Z)\}\right]. \] Then $ET=\tau_H$, without assuming that the historical predictor equals the new long-run conditional mean. Conditional on $(X,A=a,Z=z)$, write $\mu_a=E[Y_H\mid X,A=a,Z=z]$, $\sigma_a^2=\operatorname{Var}(Y_H\mid X,A=a,Z=z)$ and $v_a=E[(Y_H-m_a)^2\mid X,A=a,Z=z]$. The conditional variance of the bracket is \[ \sigma_a^2+\left(\frac1{\pi_a}-1\right)v_a. \] Consequently the variance of $T$ equals a term independent of $\pi$ plus \[ E\left[\frac{r(X)^2}{e_A(X)^2}\frac{v_A(X,Z)}{\pi_A(X,Z)}\right]. \]
Proof: Conditional independent collection gives $E[R(Y_H-m_a)/\pi_a\mid X,A,Z]=\mu_a-m_a$. The bracket therefore has mean $\mu_a$. Averaging randomized treatment yields the arm-specific proxy integral; change of measure by $r$ and the stated transport restrictions give $ET=\tau_H$. For the variance, the random part is $R(Y_H-m_a)/\pi_a$, with second moment $v_a/\pi_a$ and mean $\mu_a-m_a$. Its variance is $v_a/\pi_a-(\mu_a-m_a)^2$, which equals the claimed expression because $v_a=\sigma_a^2+(\mu_a-m_a)^2$. Finally use total variance and multiply the conditional variance by the squared arm weight. All conditional means are independent of the collection probability. ∎
Thus a poor surrogate remains usable for unbiased correction, but costs precision through its squared residual, including its calibration error. Optimizing solely on conditional outcome variance is justified only when $m_a=\mu_a$.
Proposition (Cost-aware validation allocation): Suppose collection costs $c(X,A,Z)$ are positive and bounded away from zero, and impose a floor $0<\ell<1$. Fix expected budget $b$ strictly between $\ell Ec$ and $Ec$. If $r^2v_A/e_A^2>0$ almost surely, the variance-minimizing probability subject to $\ell\le\pi\le1$ and $E[c\pi]\le b$ is \[ \pi^*(X,A,Z)=\min\left\{1,\max\left\{\ell, \frac{r(X)\sqrt{v_A(X,Z)}}{e_A(X)\sqrt{\lambda c(X,A,Z)}}\right\}\right\}, \] where $\lambda>0$ is chosen so that $E[c\pi^*]=b$.
Proof: Let $a=r^2v_A/e_A^2$. The objective depending on allocation is $E[a/\pi]$. For fixed $\lambda>0$, minimize $a/u+\lambda cu$ pointwise on $[\ell,1]$. Differentiation and strict convexity give the truncated square-root formula. Its expected cost is continuous in $\lambda$ by dominated convergence, with limits $Ec$ at zero and $\ell Ec$ at infinity. The intermediate value theorem supplies a multiplier with the required cost. For any competing feasible $\widetilde\pi$, pointwise optimality yields $E[a/\pi^*]+\lambda b\le E[a/\widetilde\pi]+\lambda E[c\widetilde\pi]\le E[a/\widetilde\pi]+\lambda b$. Cancel the budget term. If the residual is zero on some cells, allocation there need not be unique and slack budget can occur; the positive-residual hypothesis avoids that qualification. ∎
If the ratio, propensity inverse and validation inverse are bounded, then $T$ is bounded and its sample mean admits a finite-sample Hoeffding interval. If the historical predictor is learned independently, every statement holds conditionally on that training sample. Learning the allocation from a separate pilot likewise preserves conditional validity when its realized probabilities are recorded; oracle variance optimality then requires controlling pilot error and is not automatic.
Proposition (What unvalidated support and treatment changes leave unknown): Let a collection design have zero validation probability on a measurable arm-specific region of target probability $w_a$. Without a restriction on the long-run means there, that region contributes width $w_a$ to the effect's identified interval. More generally, supplying mean ranges $[l_a,u_a]$ there contributes width $E_Q\int(u_a-l_a)\,dK_a$ over that region. These widths are sharp jointly across the two arms when no further cross-cell restrictions are imposed.
Proof: Hold all observed laws and validated responses fixed. On each unvalidated cell choose a Bernoulli long-run outcome with any mean in its admissible interval, using an independent uniform disturbance. This changes no observed outcome or proxy law. For the upper effect choose upper treated and lower control endpoints, reversing them for the lower effect. Independent construction of potential outcomes across arms realizes both endpoints, and mixing realizes intermediate values. The width is the sum of the two arm contributions. ∎
The same obstruction applies to a persistent treatment path never followed to $H$: observations under a discontinued or subsequently modified path impose no restriction on its unobserved endpoint without a version-transport assumption. A probability sample of those alternative paths is not representative validation for the original estimand.

Remaining gap

The design gives an explicit route to learning long-run effects despite surrogate misspecification, and shows where scarce validation is most valuable. It supplies neither evidence that a historical tolerance transports to a new platform nor an optimal sequential pilot-and-validation policy. Target population sampling, learned density ratios, evolving treatment versions and joint deployment decisions remain unresolved. The estimator uses familiar inverse-probability correction; its derivation is a targeted benchmark, not a novelty claim.

Completion Estimate

65%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves unbiased long-run validation correction despite predictive drift, derives cost-optimal validation probabilities, and sharply bounds unvalidated support. Full proxy-based transport still requires credible target response and proxy-law assumptions, calibrated treatment-version changes, learned sampling and density inputs, and validated sequential collection and deployment decisions.

References

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