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Econ-345 · economics · Economics

Measurement error in digital and administrative data

LLM status Open unverified

Statement

Original identifier: E8 (atlas item 108). Field: Econometrics and causal inference. Classification: editorial research family with established restricted solutions; current open status not certified.

Definitions and assumptions

Let latent research variables $L=(Y,D,X)$ follow a population law, with $D\in\{0,1\}$, bounded outcome $Y\in[a,b]$, and covariates $X$. Operational data report $W=(\widetilde Y,\widetilde D,\widetilde X)$ according to a measurement kernel $K_s(dw\mid L)$ indexed by system version $s$. A validation indicator $V$ reveals $L$ when $V=1$, in addition to $W,s$. Specify validation sampling probabilities $v(W,s)$, and assume validation is independent of $L$ conditional on $(W,s)$ only if defensible. Under latent-variable consistency, unconfoundedness and overlap, the causal target is $\theta=\mathbb E[m_1(X)-m_0(X)]$, $m_d(x)=\mathbb E[Y\mid D=d,X=x]$.

Formal research question

Characterize the identified set for $\theta$ and estimators uniformly robust to nonclassical measurement and system changes. In a representative validation design with $v(W,s)\ge\eta>0$, the joint latent-observed law is recoverable in principle; quantify the information needed when the validation fraction tends to zero. With no validation for a new version $s$, instead impose a declared kernel distance $d(K_s,K_{s_0})\le\delta$ from a validated version $s_0$, where $d$ and $\delta$ are supplied. Find sharp causal-effect bounds over all latent laws and kernels matching the observed law and these restrictions, and confidence sets covering the bounds uniformly as sample size, validation size and drift vary.

Interpretation and scope

An algorithmic score is not inherently a noisy measurement of the desired construct: the construct and intervention must first be defined. Record linkage can create mismatched outcomes, selective missingness and dependent observations, so a linkage error rate alone may not characterize bias. Multiple proxies identify latent variables only with additional conditional-independence or structural restrictions; more observations cannot identify an unrestricted measurement kernel. Classical errors-in-variables attenuation holds in particular linear models and should not be assumed for misclassified treatment or outcomes. Correcting measurement does not remove unobserved causal confounding. Provenance identifies the versioning structure and validation design on which credible comparisons depend.

Source audit and status

[S1] and [S2] are verified measurement-error references, covering classical and nonclassical concerns. The third input item, modern record-linkage literature, is a field label and cannot be audited as a publication; it is retained as unresolved [S3]. The formal kernel-and-validation question is an editorial specialization. The references motivate it but do not supply a universal modern-digital-data theorem or a certified 2026 open-status claim.

References

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  1. Jerry Hausman (2001). Mismeasured Variables in Econometric Analysis: Problems from the Right and Problems from the Left. Journal of Economic Perspectives 15(4), 57–67. https://doi.org/10.1257/jep.15.4.57. Locator: Publisher or institutional abstract and bibliographic record. Support and limits: Measurement-error overview; classical attenuation needs specific assumptions.

  2. John Bound, Charles Brown and Nancy Mathiowetz (2001). Measurement Error in Survey Data. Handbook of Econometrics 5, chapter 59, 3705–3843. https://www.sciencedirect.com/science/article/pii/S1573441201050127. Locator: Publisher or institutional abstract and bibliographic record. Support and limits: Classical and nonclassical measurement, validation and proxies; not platform-specific digital measurement.

  3. Unresolved original citation: modern record-linkage literature. Not an identifiable publication: no authors, title, year or persistent identifier supplied.

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-345-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-345

Measurement error in digital and administrative data: a research attempt

Target and outcome

Status: unresolved. Version changes, erroneous linkage and selective validation can make a large administrative dataset less informative than its record count suggests. This attempt solves a finite-state bounding problem with a validated reporting kernel, bounded subsequent drift and a known randomized assignment rule. It also gives a finite-sample outer confidence construction and a validation-rate obstruction. Unrestricted observational treatment assignment and arbitrary record linkage remain outside this result.

Hausman's review [hausman], whose primary abstract was checked, provides background on why different measurement-error structures require different identifying assumptions. No empirical error rates are taken from it. All kernel restrictions below are declared rather than inferred from the label "administrative data."

A finite reporting model

Let the latent record be $l=(y,d,x)$, with finitely many values, $y\in[0,1]$, $d\in\{0,1\}$. Let $W$ be a finite reported record. A baseline validation exercise identifies a stochastic kernel $K_0(w\mid l)$ for every relevant latent state. In the new version, the unknown kernel satisfies the row-wise restriction \[ \frac12\sum_w|K(w\mid l)-K_0(w\mid l)|\leq\delta, \qquad 0\leq\delta\leq1. \] This bounds arbitrary nonclassical misclassification, including dependence on outcome and treatment, within the finite representation. It does not model unmatched people absent from the sampling frame unless an explicit missing-record category and population denominator are included.

Assume treatment in the latent population was randomized conditional on $X$ with a known probability $e_x\in(0,1)$. Assume consistency and conditional independence of treatment and the two potential outcomes. The observed new-version distribution is $q_w={\mathbb{P}}(W=w)$. The desired average treatment effect is \[ \tau=\sum_x{\mathbb{P}}(X=x)\{E[Y\mid D=1,x]-{\mathbb{E}}[Y\mid D=0,x]\}. \] This identification statement requires the randomization assumptions; arbitrary association of $D$ and $Y$ would not suffice.

An exact linear-program representation

Introduce variables $z_{lw}={\mathbb{P}}(L=l,W=w)$, $\pi_l=\sum_wz_{lw}$ and nonnegative auxiliaries $u_{lw}$. Impose \begin{align*} z_{lw}&\geq0,&\sum_lz_{lw}&=q_w,&\pi_l&=\sum_wz_{lw},\\ u_{lw}&\geq z_{lw}-K_0(w\mid l)\pi_l,& u_{lw}&\geq -z_{lw}+K_0(w\mid l)\pi_l,\\ \sum_wu_{lw}&\leq2\delta\pi_l,&&&\\ \sum_y\pi_{(y,1,x)}&=e_x\sum_{y,d}\pi_{(y,d,x)}&&&\text{for every }x. \end{align*} Minimize and maximize the linear functional \[ T(\pi)=\sum_{x,y}y\left\{\frac{\pi_{(y,1,x)}}{e_x} -\frac{\pi_{(y,0,x)}}{1-e_x}\right\}. \] An empty feasible set means that the data and restrictions are incompatible, not that the effect equals zero.
Proposition (Sharpness and attainment): If the feasible set is nonempty, the minimum and maximum of these programs are attained and their interval is exactly the identified set for $\tau$ in the specified finite-state model.
Proof: Any admissible model produces $z=\pi K$. Multiplying each total-variation row bound by $\pi_l$ gives the displayed inequalities, and treatment balance follows from the known assignment probability. Substitution in the treatment-effect formula gives $T(\pi)$.

Conversely, take feasible $(z,\pi,u)$. If $\pi_l>0$, define $K(w\mid l)=z_{lw}/\pi_l$. The inequalities imply the required row bound. If $\pi_l=0$, set that unobserved row to $K_0$; it affects neither the observed law nor the target. Generate $X$ with mass $\sum_{y,d}\pi_{(y,d,x)}$. In a positive-mass stratum choose $Y(0)$ and $Y(1)$ with the conditional distributions encoded by the corresponding $\pi$ rows, coupling them independently if desired. Generate $D$ independently with probability $e_x$, set $Y=Y(D)$, and then generate $W$ using $K$. This constructs a valid causal model with exactly the requested joint distribution and observed law.

The feasible set of $z,\pi$ is closed and bounded; auxiliaries can also be bounded by their finite row totals. Thus extrema exist. Convex combinations of feasible points remain feasible and the objective is linear. The construction therefore realizes all intermediate target values as well as the endpoints. ∎

Known propensities are essential to linearity. If they depend on unknown latent frequencies under observational assignment, the displayed denominators cannot simply be inserted as constants. Likewise, estimating $K_0$ from cells absent in validation is an unsupported extrapolation, not a consequence of the program.

Finite validation and outer confidence sets

Suppose an independently selected validation sample contains paired true and reported records from the baseline version. Conditional on each latent cell's count, the reported counts are multinomial. If simultaneous confidence statements establish \[ \operatorname{TV}(K_0(\cdot\mid l),\widehat K_0(\cdot\mid l)) \leq\eta_l\quad\text{for all }l, \] then the triangle inequality puts the new kernel within radius $\min(1,\delta+\eta_l)$ of $\widehat K_0$ in every row. A zero-count cell must receive an uninformative radius, rather than an estimated zero error rate. Finite-state coordinate Hoeffding bounds and a union bound provide one conservative way of constructing the $\eta_l$.

Construct simultaneous multinomial confidence intervals for the observed $q_w$ as well. Replace exact column sums by membership in those intervals and the probability simplex, and replace the row centers and radii as just described. The optimization remains linear. On the joint confidence event the true model is feasible, so its effect belongs to the projected interval. Allocating failure probabilities across the validation and main-sample statements yields uniform coverage of at least $1-\alpha$ under their sampling assumptions. This is a conservative outer confidence interval; it is not asserted to be the sharp confidence procedure or the exact population identified set.

There is a basic limit to what shrinking validation can accomplish. With constant independent validation probability $v_n$, $N_V\sim\operatorname{Binomial}(n,v_n)$. In an unrestricted submodel where all ordinary reports equal the same symbol, only validation distinguishes different latent outcome laws. If $nv_n$ stays bounded along a subsequence, then ${\mathbb{P}}(N_V=0)$ stays bounded away from zero (unless the trivial finite-sample design is changed). On that event two separated latent-effect models have identical observed records. A two-point argument therefore rules out uniformly consistent estimation over that submodel. The condition $nv_n\to\infty$ is necessary there; positive information per ordinary report can change the conclusion in more restricted models.

A scalar check and the remaining gap

For a simpler model, suppose treatment is recorded without error and outcome misclassification in each arm is at most $\delta$ in probability. Then true and reported outcome means differ by at most $\delta$. If reported means are $q_1,q_0$, the sharp effect bounds under otherwise unrestricted binary misclassification are \[ [\max(0,q_1-\delta)-\min(1,q_0+\delta),\quad \min(1,q_1+\delta)-\max(0,q_0-\delta)]. \] They are attained by moving outcome mass in opposite directions in the two arms. At $q_1=0.6,q_0=0.4,\delta=0.05$, the interval is $[0.10,0.30]$. This scalar model is a separate simplified error restriction, not an automatic reduction for every multidimensional kernel.

The main result provides an implementable benchmark for known randomization and explicitly bounded version drift. The full catalogue target still needs defensible validation sampling, changing linkage populations, sparse latent cells, dependence across records and causal assumptions for observational exposures. The magnitude of $\delta$ must be supported externally or reported as a sensitivity parameter. More records do not identify it by themselves.

Completion Estimate

58%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt gives an attained sharp linear programming identified set for finite reporting states with known randomized assignment and bounded kernel drift, plus uniformly valid outer intervals. The broader problem still needs observational assignment, defensible selective validation, sparse cell information rates, and changing or dependent record linkage populations.

References

  • [hausman] J. Hausman (2001), Mismeasured Variables in Econometric Analysis: Problems from the Right and Problems from the Left, Journal of Economic Perspectives 15(4), 57--67. Primary abstract: https://pubs.aeaweb.org/doi/10.1257/jep.15.4.57.

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