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Econ-585 · economics · Economics

Calibrating sensitivity assumptions

LLM status Open unverified

Statement

Original subject group: Econometrics, causal inference and measurement

Audit classification: Published research agenda. The substantive-calibration obstacle is verified in indexed primary text, and the paper already supplies sensitivity algebra. Linear-projection residual variance must not be confused with conditional variance.

Definitions and precise research target

Consider a study population with outcome $Y$, treatment $D$, measured covariates $X$, and an unmeasured scalar confounder $U$. Assume a constant-effect linear causal model $Y=\beta D+g(X)+\gamma U+\varepsilon$, with $g$ linear in the recorded covariate basis and $E[\varepsilon\mid D,X,U]=0$; otherwise the coefficient need not equal a population treatment effect. Let $\beta_{\rm obs}$ be the coefficient from a specified linear projection omitting $U$. Define $r_D$ as the partial squared correlation of $D$ and $U$ after projecting on $X$, and $r_Y$ as that of $Y$ and $U$ after projecting on $D,X$. Assume the partial correlations are well-defined and both displayed residual standard deviations are positive. If $r_D<1$, the omitted-variable bias satisfies \[ |\beta_{\rm obs}-\beta|=\frac{\sigma_{Y\mid D,X}}{\sigma_{D\mid X}}\sqrt{\frac{r_Yr_D}{1-r_D}}, \] where the two $\sigma$'s are population residual standard deviations for the stated projections. Known sensitivity algebra therefore supplies bounds once credible bounds on $r_D,r_Y$ are given. The research task is to calibrate those bounds using external validation samples, measured confounders, negative controls or substantive restrictions, without assuming measured and unmeasured confounders are interchangeable. State validity conditions for each calibration source, propagate estimation uncertainty, and evaluate coverage and informativeness across a declared class of confounding mechanisms. The agenda concerns credible calibration, not rediscovery of the bias identity.

Definitions, assumptions and mathematical qualifications. Assume finite second moments and a full-rank covariate basis including an intercept. For random variables $V$ and regressor vector $W$, $\sigma_{V\mid W}^2$ denotes the population variance of the residual from linear projection on $W$, not general conditional variance. Define $r_D=\operatorname{Corr}(D-\Pi_XD,U-\Pi_XU)^2$ and $r_Y=\operatorname{Corr}(Y-\Pi_{D,X}Y,U-\Pi_{D,X}U)^2$, with positive residual variances. The bias identity is algebraic for the stated constant-effect structural model. A calibration source must bound the same population parameters; external validation requires a transport assumption, a measured-confounder benchmark requires an explicit comparative-strength assumption, and negative controls require specified exclusion relations.

Variants and assumption changes

  1. Sourced methodology benchmark. Given deterministic upper bounds $r_D\leq d<1$, $r_Y\leq y$, the displayed identity bounds bias by $(\sigma_{Y\mid D,X}/\sigma_{D\mid X})\sqrt{yd/(1-d)}$.
  2. Editorial calibration design. Obtain uncertain bounds from a probability validation sample observing a candidate confounder and state a transfer restriction to unobserved confounders; construct joint coverage for calibration and causal-effect intervals.

References and source audit

Checked source. JRSS B 82(1) (2020), pp. 39--67, introduction, third obstacle; DOI 10.1111/rssb.12348. The indexed introduction identifies the difficulty of substantive calibration. Publisher metadata and summary retrieved; exact introductory paragraph verified in indexed primary text. Direct Oxford full text unavailable. https://academic.oup.com/jrsssb/article/82/1/39/7056023.
  1. Carlos Cinelli; Chad Hazlett. 2020. Making Sense of Sensitivity: Extending Omitted Variable Bias. JRSS B 82(1) (2020), pp. 39--67, introduction, third obstacle; DOI 10.1111/rssb.12348. https://academic.oup.com/jrsssb/article/82/1/39/7056023. DOI: https://doi.org/10.1111/rssb.12348.

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-585-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-585

Calibrating sensitivity assumptions: what validation can and cannot calibrate

Target and scope

For the specified constant-effect linear causal model, the known omitted-variable identity is \[ |\beta_{\rm obs}-\beta|=s\sqrt{\frac{r_Yr_D}{1-r_D}},\qquad s=\frac{\sigma_{Y\mid D,X}}{\sigma_{D\mid X}}. \] The variances are linear-projection residual variances. The problem is credible calibration of $r_D,r_Y$, not derivation of this identity. Cinelli and Hazlett [sensitivity] provide the sensitivity framework and discuss formal benchmarking against observed covariates. Here a Gaussian construction shows why validation of a candidate covariate alone cannot calibrate an unrestricted omitted confounder. A separate finite-sample construction then propagates an explicitly declared comparative-strength assumption.
Proposition (Unlimited bias with the same observed experiment): Fix $b\in\mathbb R$ and any $0<r<1$. There is a family of centered Gaussian structural models with the identical observable law $D\sim N(0,1)$ and $Y=bD+E$, $E\sim N(0,1)$ independent of $D$, but with causal coefficients unbounded in magnitude. Additional validated variables independent of $(D,E)$ can have the same joint observed law throughout this family.
Proof: Let $V,D,E$ be independent standard Gaussians. For $-1<c<1$, set \[ U=cD+\sqrt{1-c^2}\{\sqrt r E+\sqrt{1-r}V\},\quad \gamma=\frac{\sqrt r}{\sqrt{1-c^2}},\quad \beta=b-\frac{c\sqrt r}{\sqrt{1-c^2}}. \] Then $Y=\beta D+\gamma U+\varepsilon$, where $\varepsilon=(1-r)E-\sqrt{r(1-r)}V$. Its covariance with $D$ is zero, and its covariance with $U$ is $\sqrt{1-c^2}\{(1-r)\sqrt r-r\sqrt{1-r}\sqrt{1-r}\}=0$. Joint Gaussianity therefore gives $E[\varepsilon\mid D,U]=0$, as required by the structural model. The observable equation remains exactly $Y=bD+E$. The two partial squared correlations are $r_D=c^2$ and $r_Y=r$, so the causal coefficient diverges as $c$ approaches either endpoint. Add any fixed independent validated vector to both the observed covariates and the validation records; its law and its associations with $D,Y$ remain unchanged. It places no restriction on this omitted $U$. ∎
Thus even a perfectly measured benchmark with zero observed associations does not bound an omitted confounder without a relationship between their strengths. Negative controls likewise need exclusion and relevance restrictions; their label alone supplies neither.

A declared validation bridge

Now let a probability validation sample observe a candidate benchmark $V$ along with $(Y,D,X)$. The following assumptions are substantive inputs. Validation observations are iid from the study population; if they come from another population, an explicit moment-transport restriction replaces equality. Define $r_D^V,r_Y^V$ by the same population projections used for $U$. Assume \[ r_D\le\min\{d_*,\kappa_D r_D^V\},\qquad r_Y\le\min\{1,\kappa_Y r_Y^V\},\qquad d_*<1, \] with prespecified $\kappa_D,\kappa_Y\ge0$. The benchmark is not asserted to be the actual omitted confounder; these inequalities are the scientific calibration bridge.
Proposition (A simultaneous covariance confidence region): Let $Z=(Y,D,X',V)'\in\mathbb R^p$ have each coordinate bounded by $M\ge1$, where the nonconstant covariate basis is recorded separately from its intercept. For an iid validation sample of size $n$, let $\widehat\mu$ and $\widehat H$ be empirical first and raw second moments. Define \[ t_n=M^2\sqrt{\frac{2\log\{2(p+p^2)/\alpha\}}n}. \] With probability at least $1-\alpha$, all entries of the population moments satisfy $|\mu_j-\widehat\mu_j|\le t_n$ and $|H_{jk}-\widehat H_{jk}|\le t_n$ simultaneously. Intersect this box with realizable moment pairs and any maintained positive residual-variance restrictions, and map it by $C=H-\mu\mu'$ to a covariance region $\mathcal C_n$. The true covariance belongs to this region on the same event.
Proof: Each first coordinate lies in an interval of length $2M$ and each raw product in an interval of length at most $2M^2$. Hoeffding's two-sided bound is therefore at most $2\exp\{-nt_n^2/(2M^4)\}$ for each entry. There are at most $p+p^2$ entries. A union bound proves simultaneous coverage. Intersecting with true maintained restrictions and applying the deterministic covariance map preserves membership. ∎
For a covariance $C$, all required projection variances and covariances are Schur complements. For example, after removing the recorded $X$ basis, \[ C_{DV\cdot X}=C_{DV}-C_{DX}C_{XX}^{-1}C_{XV},\qquad r_D^V(C)=\frac{C_{DV\cdot X}^2}{C_{DD\cdot X}C_{VV\cdot X}}. \] For $r_Y^V$ residualize on $(D,X)$ instead. Constants in $X$ are handled by centering, rather than inverting an intercept's zero variance.
Theorem (An honest calibrated causal interval): Suppose the preceding bridge holds, and every covariance admitted to $\mathcal C_n$ has the inverses and positive residual variances needed for these definitions. Let $b(C)$ and $s(C)$ be the omitted regression coefficient and residual standard-deviation ratio. Put \[ d(C)=\min\{d_*,\kappa_Dr_D^V(C)\},\quad y(C)=\min\{1,\kappa_Yr_Y^V(C)\},\quad B(C)=s(C)\sqrt{\frac{y(C)d(C)}{1-d(C)}}. \] Then the interval \[ \left[\inf_{C\in\mathcal C_n}\{b(C)-B(C)\},\ \sup_{C\in\mathcal C_n}\{b(C)+B(C)\}\right] \] has coverage at least $1-\alpha$ for $\beta$. An empty region is reported separately. Suprema and infima need not be replaced by attained extrema unless the region is compact and the maps are continuous there.
Proof: On the covariance-coverage event the true $C$ is feasible. The bridge and monotonicity of $yd/(1-d)$ in both arguments bound the true omitted-variable bias by $B(C)$. Thus $\beta$ belongs to the interval associated with the true covariance, which is contained in the displayed outer interval. This argument requires no independence between estimated calibration parameters and the estimated omitted coefficient: they are covered jointly by one moment region. ∎
Uniform lower bounds on relevant covariance eigenvalues and residual variances make this procedure informative as moments concentrate. Indeed $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ bounds inverse perturbations by $\lambda^{-2}\|A-B\|$ when eigenvalues exceed $\lambda$. Products, Schur complements and bounded-away-from-zero ratios are then continuous uniformly. The sensitivity width generally converges to a positive limit; more validation removes sampling error, not the maintained allowance for omitted confounding. Near-zero residual variances or $d_*=1$ can destroy this control.

Remaining gap

The note supplies a reproducible uncertainty calculation and a precise impossibility without a calibration bridge. It does not justify the strength multipliers, identify every hidden confounder, or verify external moment transport. Its interval is a coverage guarantee, not an assertion that every covariance-box endpoint is a sharp causal model. Establishing credible bridges using institutional knowledge or specially designed negative controls remains unresolved.

Completion Estimate

58%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves unlimited causal bias despite identical observed and validation data, then constructs joint finite-sample sensitivity intervals under an explicit comparative-strength bridge. Credible justification of that bridge, negative-control exclusion and relevance, and informative coverage across broader confounding mechanisms and transported validation populations remain central missing steps.

References

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