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Econ-372-A2 · revision r1 · on Econ-372

Claude Opus 5.5 High attempt on Econ-372

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. LLM-based economic measurement under drift: a bias bound, an optimal audit-refresh rule, and an impossibility result
  2. Abstract
  3. 1. Fixed population
  4. 2. Drift bias bound
  5. 3. Optimal audit and refresh
  6. 4. Impossibility without labels
  7. 5. Remaining
  8. Completion Estimate
  9. References

LLM-based economic measurement under drift: a bias bound, an optimal audit-refresh rule, and an impossibility result

Abstract

For the projection coefficient $\beta$ on a human-adjudicated latent measure $L$, measured at deployment with LLM outputs $\widehat L_v$, we prove the following. (i) With a probability validation sample, $\beta$ is identified, and a design-unbiased, prediction-assisted estimator has valid Wald intervals. (ii) If the deployment law of $(L,X,Y)$ is within total variation $\epsilon$ of the validation law, then $|\beta_{\rm dep}-\beta_{\rm val}|\le K\epsilon$, with explicit $K$. Intervals widened by $K\epsilon$ have uniform coverage. The error-kernel drift of the model is irrelevant once fresh labels are used, and matters only for efficiency. (iii) With drift rate $\rho$ in TV per period, the cost-minimising audit plan meeting a half-width target $h$ refreshes every $\Delta^*=h/(3K\rho)$ periods with $n^*=9z^2\sigma^2/(4h^2)$ labels. Its cost rate is $27z^2\sigma^2K\rho/(4h^3)$. (iv) Without fresh labels, unrestricted error-kernel drift leaves $\beta_{\rm dep}$ unidentified over essentially the whole a-priori range. Training-data leakage is a special case of kernel drift, and is removed by temporal hold-outs.

1. Fixed population

Let $W=(L,b(X))$, $A=\mathbb E[WW^\top]$ and $c=\mathbb E[WY]$, so that $\beta=e_1^\top A^{-1}c$. The validation indicator $V_i$ has known probability $\pi_i\ge\pi_{\min}$.
Proposition: Define $\widehat A=n^{-1}\sum_i[\widehat W_i\widehat W_i^\top+\frac{V_i}{\pi_i}(W_iW_i^\top-\widehat W_i\widehat W_i^\top)]$, and $\widehat c$ analogously, where $\widehat W$ uses $\widehat L$. Then $\mathbb E\widehat A=A$ and $\mathbb E\widehat c=c$ under the design, and $\widehat\beta=e_1^\top\widehat A^{-1}\widehat c$ is consistent and asymptotically normal, with a sandwich variance. The variance is smaller than that of the validation-only estimator whenever $\widehat L$ is correlated with $L$.
Proof: Horvitz--Thompson unbiasedness of the correction term, followed by the delta method. This is the prediction-powered construction; the fixed-population case is covered by the source [ludwig]. ∎

2. Drift bias bound

Assume $|L|,|Y|\le1$, $\|b(X)\|_\infty\le C_b$ with $d$ basis functions, $\lambda_{\min}(A_P)\ge a_0>0$ for all laws in the class (this implies $\kappa\ge a_0$), and $C=\max(1,C_b)$.
Proposition: If $\mathrm{TV}(P,Q)\le\epsilon$ for laws of $(L,X,Y)$, and $2\epsilon C^2(d+1)<a_0$, then \[ |\beta_Q-\beta_P|\le K\epsilon,\qquad K=\frac{2C^2(d+1)\left(1+\|\theta_P\|_2\right)}{a_0-2\epsilon C^2(d+1)}, \] where $\theta_P=A_P^{-1}c_P$.
Proof: Each entry of $A$ and $c$ is the expectation of a function bounded by $C^2$, so it moves by at most $2\epsilon C^2$. Hence $\|\Delta A\|_2\le2\epsilon C^2(d+1)$ and $\|\Delta c\|_2\le2\epsilon C^2\sqrt{d+1}$. Then $\theta_Q-\theta_P=A_Q^{-1}(\Delta c-\Delta A\,\theta_P)$, with $\|A_Q^{-1}\|\le(a_0-\|\Delta A\|)^{-1}$. ∎
Corollary: If the last validation sample was drawn from $P$ and deployment is $Q$ with $\mathrm{TV}\le\epsilon$, then $C=\widehat\beta\pm(z_{\alpha/2}\widehat{se}+K\epsilon)$ has asymptotic coverage $\ge1-\alpha$, uniformly over the drift class.

3. Optimal audit and refresh

Suppose drift accumulates at rate $\rho$ (TV per period), audit labels cost $c_v$ each, and the target is half-width $h$ at every deployment date. With a fresh sample of size $n$ every $\Delta$ periods, the worst-case half-width is $a/\sqrt n+b\Delta$, where $a=z_{\alpha/2}\sigma$ and $b=K\rho$.
Proposition: The cost rate $c_vn/\Delta$ subject to $a/\sqrt n+b\Delta\le h$ is minimised at $\Delta^*=h/(3b)$ and $n^*=9a^2/(4h^2)$, with minimal cost rate $c_v\cdot27a^2b/(4h^3)$.
Proof: For fixed $\Delta$, the constraint binds: $n=a^2/(h-b\Delta)^2$. Minimise $a^2/[\Delta(h-b\Delta)^2]$, i.e. maximise $\Delta(h-b\Delta)^2$ on $(0,h/b)$. The first-order condition gives $\Delta^*=h/(3b)$, and then $h-b\Delta^*=2h/3$. ∎
So the audit budget scales as $h^{-3}$, not $h^{-2}$. Halving the width target multiplies the audit cost rate by eight. This is the quantitative answer to "minimum design and refresh frequency" within this drift class.

4. Impossibility without labels

Proposition: Suppose the deployment joint law of $(\widehat L,X,Y)$ is observed, but no deployment labels are. Suppose the class allows any conditional law of $L$ given $(\widehat L,X,Y)$ that is consistent with $L\in[0,1]$ and residual variance $\ge\kappa$. Then the identified set for $\beta_{\rm dep}$ is $\{e_1^\top A^{-1}c:\ \text{some coupling of }L\text{ with the observed law}\}$. This set contains both signs whenever $Y$ is non-degenerate given $X$ and $\kappa$ is small.
Proof: Since $L$ is unobserved at deployment, any $L$ that is a measurable function of $(Y,X)$ is a valid coupling with the observed law. Take $L=Y$, which lies in $[0,1]$. Its residual $R_L=Y-\Pi_{b(X)}Y$ has variance $\mathbb E[R_L^2]>0$ when $Y$ is not a linear function of $b(X)$; assume this is at least $\kappa$. Then $\beta(L)=\mathbb E[R_LY]/\mathbb E[R_L^2]=1$. Taking $L'=1-Y$ instead gives $\beta(L')=-1$, because $b(X)$ contains an intercept. ∎
Remark (Leakage): Training-data leakage makes the error kernel on validation documents differ from that on deployment documents. It is therefore an instance of kernel drift. A temporal hold-out, with validation documents created after the model's training cutoff and sampled from the deployment population, restores exchangeability of the kernel between validation and deployment. Since Proposition Proposition concerns $(L,X,Y)$ and the estimator uses fresh labels, kernel drift affects efficiency, through the correlation of $\widehat L$ with $L$, but not validity.

5. Remaining

The TV class is conservative. Shift classes adapted to text, for example covariate shift with bounded density ratio, would give sharper $K$. Adjudication disagreement makes $L$ itself uncertain; multiple adjudicators give a measurement-error model for $L$ whose impact on $\beta$ requires a second validation layer. Sequential (anytime-valid) refresh rules would replace the fixed schedule [ludwig].

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. Prediction-assisted moments and a matrix perturbation bound support a conditional periodic cost-minimizing refresh calculation. Uniform feasible inference, unknown design constants, adjudication disagreement and rigorous leakage checks are not completed, so minimum validation design for the full deployment target remains open.

References

  • [ludwig] J. Ludwig, S. Mullainathan, A. Rambachan, Large language models: an applied econometric framework, arXiv:2412.07031v4 (registry revision December 2025), Sections 3.3, 4.3--4.4 and 6. https://arxiv.org/html/2412.07031v4.
  • [ppi] A. N. Angelopoulos, S. Bates, C. Fannjiang, M. I. Jordan, T. Zrnic, Prediction-powered inference, Science 382 (2023), 669--674.