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Econ-646 · economics · Economics

External validity of experimental effects

LLM status Open unverified

Statement

Original identifier: B1 (atlas item 61). Classification: Causal transportability research program. Original tractability label: Medium (editorial, not a complexity result).

Definitions and assumptions

Let $S=1$ denote participation in a source experiment and $S=0$ membership in a specified target population. Let $X$ be pretreatment covariates, $A\in\{0,1\}$ randomized treatment, and $c\in\mathcal C$ an institutional context. The potential outcome $Y(a,c)$ is measured on a common, prespecified scale. Assume consistency and no interference within each context; interference requires a different estimand. Randomization identifies source conditional effects wherever treatment has positive probability. Sampling positivity requires every target covariate profile to occur with positive source probability. Distinguish selection into the sample from changing the intervention itself.

Formal research question

For fixed source and target contexts $c_s,c_t$, identify or sharply bound \[ \tau_t=\mathbb E[Y(1,c_t)-Y(0,c_t)\mid S=0]. \] Determine substantively testable conditions under which \[ \tau_t=\int \mathbb E[Y(1,c_s)-Y(0,c_s)\mid X=x,S=1]\,dF_{X\mid S=0}(x). \] This equality follows from conditional equality of treatment effects across populations and contexts, together with overlap; randomization alone does not imply it. When equality is relaxed by a specified discrepancy bound $b(x)$, derive target-effect bounds and confidence regions accounting for source estimation and target sampling.

Interpretation and scope

The central extension is a prospectively validated transport rule for a defined mechanism, outcome, and target population. Paired experiments should hold the treatment protocol constant where possible and independently vary stakes, scrutiny, experience, and institutional rules. Reweighting removes measured compositional differences only under the stated conditional effect restriction. It cannot automatically remove changes in incentives or unmeasured participation motives. A multisite model must distinguish site-level heterogeneity from individual-level sampling error, and predictive coverage should be evaluated on held-out contexts. A finding that the average effect transports need not establish transport of distributional effects or welfare. The relevant failure criterion is an error exceeding a prespecified policy tolerance, rather than a failure to reject equal effects in a small sample.

Source audit and status

All three author-year references are verified. [S1] supplies a field-context taxonomy; [S2] discusses generalization threats. [S3] concerns replication of laboratory studies and is only indirectly relevant: replicability does not establish external validity. None states the displayed transport equation as a published open conjecture. This is an editorial identification and validation problem motivated by these sources; its unresolved scope is conditional on the selected target and model restrictions, not certified for every application in 2026.

References

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  1. Glenn W. Harrison and John A. List (2004). Field Experiments. Journal of Economic Literature 42(4), 1009--1055. https://www.aeaweb.org/articles?id=10.1257/0022051043004577

    Source role: Taxonomy of field settings; methodological foundation. Locator: Abstract; six dimensions of field context.

  2. Steven D. Levitt and John A. List (2007). What Do Laboratory Experiments Measuring Social Preferences Reveal About the Real World?. Journal of Economic Perspectives 21(2), 153--174. https://www.aeaweb.org/articles?id=10.1257/jep.21.2.153

    Source role: Methodological discussion of generalizability. Locator: Abstract; discussion of scrutiny, context, selection, and stakes.

  3. Colin F. Camerer, Anna Dreber, Eskil Forsell, Teck-Hua Ho, Jürgen Huber, Magnus Johannesson, Michael Kirchler, Johan Almenberg, Adam Altmejd, Taizan Chan, Emma Heikensten, Felix Holzmeister, Taisuke Imai, Siri Isaksson, Gideon Nave, Thomas Pfeiffer, Michael Razen, and Hang Wu (2016). Evaluating Replicability of Laboratory Experiments in Economics. Science 351(6280), 1433--1436. https://doi.org/10.1126/science.aaf0918

    Source role: Evidence on laboratory replicability, not field transportability. Locator: Abstract; author repository metadata and supplemental materials.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-646-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-646

External validity of experimental effects Sharp transport bounds under a declared context metric

Target and the missing transport assumption

Let $S=1$ denote a source experiment and $S=0$ a target population. Randomization identifies source conditional effects of a specified binary treatment, but the policy target is \[ \tau_t={\mathbb{E}}[Y(1,c_t)-Y(0,c_t)\mid S=0], \] where $c_t$ includes stakes, institutions, experience, scrutiny, and the treatment version. This attempt derives sharp conditional bounds when context changes obey an explicit Lipschitz restriction. It gives a finite optimization for incorporating source and target sampling error. It does not establish that this restriction holds in economic applications.

Outcomes are normalized to $[0,1]$. Consistency, no interference, and randomization with positive probabilities hold within each source experiment. Covariates $X$ are measured before treatment. Write $u=(x,c)$ for a descriptor in a specified metric space $(\mathcal U,d)$, and $h(u)$ for the conditional treatment effect. Different sites with the same descriptor must have the same effect under this model; omitted site characteristics cannot silently be ignored. Suppose effects are known at finitely many source descriptors $u_1,\ldots,u_m$: \[ h(u_j)=a_j,\qquad -1\le a_j\le1. \] The maintained restriction is \[ |h(u)-h(v)|\le Ld(u,v), \tag{1} \] with a prespecified finite $L\ge0$. The metric, normalization, and $L$ are sensitivity inputs. They are not learned from randomization alone.

Sharp pointwise and average bounds

Assume the anchor values are compatible: $|a_j-a_k|\le Ld(u_j,u_k)$ for every pair. Define \[ \ell(u)=\max\{-1,\max_j[a_j-Ld(u,u_j)]\},\qquad r(u)=\min\{1,\min_j[a_j+Ld(u,u_j)]\}. \tag{2} \] Let $Q_t$ be the target distribution of descriptors, including target context. Its support need not coincide with the source support.
Proposition: Under the stated model the sharp identified interval for the target average effect is \[ \tau_t\in\left[\int\ell(u)\,dQ_t(u),\, \int r(u)\,dQ_t(u)\right]. \tag{3} \] In particular both endpoint functions in (2) are feasible simultaneously across all target descriptors.
Proof: Every feasible $h$ obeys each lower and upper cone in (2), by (1), and also the outcome range. Thus the displayed integral bounds are valid. For each anchor $u_k$, compatibility implies $a_j-Ld(u_k,u_j)\le a_k\le a_j+Ld(u_k,u_j)$; using $j=k$ shows that $\ell(u_k)=r(u_k)=a_k$. Every distance cone is $L$-Lipschitz by the triangle inequality. Finite maxima, finite minima, and clipping at fixed constants preserve that Lipschitz bound. Therefore both $\ell$ and $r$ satisfy (1) and match all anchors. They are ordered because $a_j-Ld(u,u_j)\le a_k+Ld(u,u_k)$ follows from anchor compatibility and the triangle inequality, and both are clipped to the admissible range. They attain the respective endpoints of (3). Their convex combinations preserve all restrictions and attain every intermediate average.

To realize these effects as potential outcomes at unobserved target points, take Bernoulli marginals with means ${\mathbb{E}} Y(1,u)=(1+h(u))/2$ and ${\mathbb{E}} Y(0,u)=(1-h(u))/2$. Retain the actual observed outcome distributions at source anchors; their mean differences already equal $a_j$. There is no cross-context restriction on the separate baseline means in this model. Couple potential outcomes arbitrarily and randomize source assignment independently. This constructs compatible outcome models, so the effect-function attainability is also statistical sharpness under the declared model. ∎

For a numerical check, use the real line, $d(u,v)=|u-v|$, anchors $(u_1,a_1)=(0,0.2)$ and $(u_2,a_2)=(1,0.4)$, and $L=0.4$. At $u=1/2$, (2) gives $[0.2,0.4]$; at $u=2$, it gives $[0,0.8]$. A target with weights $3/4$ and $1/4$ on those points has sharp average bounds $[0.15,0.5]$. The width is not sampling noise: even exact knowledge of both source experiments leaves it. A more distant target can have nearly the unrestricted interval $[-1,1]$.

A direct discrepancy formulation

For a single source context with covariate overlap, write the identified source effect as $a(x)$. If the only restriction is $|h(x,c_t)-a(x)|\le b(x)$, for a declared nonnegative function $b$, then \[ \tau_t\in\left[ {\mathbb{E}}_t\max\{-1,a(X)-b(X)\},\, {\mathbb{E}}_t\min\{1,a(X)+b(X)\}\right]. \tag{4} \] This interval is sharp when no cross-$x$ restrictions are imposed: choose each lower or upper endpoint as the target effect and use the same Bernoulli construction. Setting $b=0$ recovers conditional-effect transportability and reweighting. Covariate overlap supplies the source $a(x)$ where it is needed; it does not justify setting $b$ to zero. A metric restriction links cells and contexts and may tighten (4), but it is an additional assumption, not information supplied by the target covariate distribution.

A finite-sample confidence construction

Suppose there are finitely many source descriptor cells and target cells. For source cell $j$, let $n_{j1},n_{j0}>0$ independent sampled outcomes be available in the two randomized arms. Conditional on arm counts, use \[ e_{ja}=\sqrt{\frac{\log(4m/\alpha_s)}{2n_{ja}}},\qquad A_j=[\widehat a_j-e_{j1}-e_{j0},\, \widehat a_j+e_{j1}+e_{j0}]\cap[-1,1]. \tag{5} \] Hoeffding bounds for the $2m$ arm means and a union bound give simultaneous coverage at least $1-\alpha_s$. Cluster dependence requires replacing this individual-sampling calculation with one valid for the assignment and sampling clusters.

For target descriptors $v_1,\ldots,v_q$ with known weights $w_l$, minimize and maximize $\sum_lw_l t_l$ over variables $a_j\in A_j$ and $t_l\in[-1,1]$, enforcing all pairwise Lipschitz inequalities among source and target descriptors. This is a linear program after replacing absolute values by two inequalities. On the coverage event, the true effect vector is feasible, so the resulting interval covers $\tau_t$. Pairwise compatibility matters: independently selecting a favorable endpoint from each $A_j$ may yield no common Lipschitz function.

If target weights are estimated from $n_t$ independent representative observations, intersect their probability simplex with \[ |w_l-\widehat w_l|\le \sqrt{\frac{\log(2q/\alpha_t)}{2n_t}}\quad(l=1,\ldots,q). \] Optimize the same objective over these weights as well. The problem is then generally bilinear, but its finite compact feasible set has attained extrema. A union bound gives coverage at least $1-\alpha_s-\alpha_t$ under the model. An empty feasible set signals incompatibility between the model and the confidence constraints; it must not be reported as a zero-width successful estimate.

Prospective validation and limits

Before seeing a new site's outcomes, one can construct (3) or its confidence version and specify a policy tolerance $\varepsilon$. After running a target experiment, compare an independent confidence interval for its effect with the predicted interval. Disjoint intervals falsify the joint transport-and-sampling claims at the corresponding error bound. Overlap merely means this test failed to refute them; a wide prediction interval can overlap almost anything. For a claimed predictive error at most $\varepsilon$, a sufficient verification is that the whole confidence interval for observed-minus-predicted effect lies inside $[-\varepsilon,\varepsilon]$. Site selection and repeated validation require prespecified populations and simultaneous error accounting.

Harrison and List's publisher abstract and Levitt and List's publisher abstract explain several ways experimental populations and contexts can differ. They motivate the descriptors here but do not determine $d$, $L$, or $b$. The cone-extension proof is an elementary mathematical construction, not a new empirical result attributed to those papers. The original problem remains open at its intended research-program scope: no evidence here validates a context metric, a transport discrepancy, or prospective coverage across real economic settings. Transporting a mean effect also does not identify individual-effect distributions or welfare.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp average-effect transport bounds under declared discrepancy or context-metric restrictions and constructs confidence regions incorporating source effects and target weights. Prospectively validated transport still requires substantively credible descriptors and discrepancy limits, paired-context evidence and informative held-out coverage within the intended policy tolerance.

References

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