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Econ-223 · economics · Economics

Generalization when observed outcomes depend on unobserved outcomes

LLM status Open unverified

Statement

CI-58 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

For known symmetric $B_N$ with $\|B_N\|_{\mathrm{op}}\leq1$, observe all randomized treatments but only outcomes indexed by an independently drawn, recorded sample $S_N$ of size $n$. Assume $n\to\infty$ and $n/N\to0$. Potential outcomes satisfy \[ Y(w)=M_\delta\{\alpha\mathbf1+\beta w+\gamma B_Nw+U\}, \qquad M_\delta=(I-\delta B_N)^{-1}, \] where $|\delta|\leq1-\eta$ and $U\sim N(0,\sigma^2 I)$ independently of assignment and sampling. Unknown parameters occupy a fixed compact set with $\sigma$ bounded away from zero. For supplied policy probability vectors $q^0,q^1$, the population contrast is \[ \psi_N=\frac1N\mathbf1^\top M_\delta(\beta I+\gamma B_N)(q^1-q^0). \] The observation law uses the mean restricted to $S_N$ and covariance $\sigma^2(M_\delta M_\delta^\top)_{S_N,S_N}$, integrating rather than setting unsampled outcomes to zero.

Formal question

Characterize graph, sampling, and treatment-design sequences that identify $\psi_N$ from this marginal observation law, even when some structural parameters are not individually identified. On identified classes, derive matching squared-error minimax bounds and confidence intervals with uniform coverage $1-\alpha$ and expected length tending to zero. Relate attainable precision to information about the target in the sampled marginal likelihood; establish impossibility results when that information stays bounded.

Scope and status

Stability makes potential outcomes well defined, but does not itself ensure identification from a small subsample. In particular, graph symmetry or missing connections can conceal propagation parameters. The Gaussian model imposes independent innovations and an invariant structural mechanism; arbitrary hidden outcome dependence is not thereby identifiable.

References for the source of the problem

Subhankar Bhadra and Michael Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808v1, 9 August 2025, Section 8.2; revised version arXiv:2508.06808v2, 9 August 2026, Section 7.6, External Validity. https://arxiv.org/html/2508.06808v1#S8.SS2 https://arxiv.org/html/2508.06808v2#S7.SS6

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-223-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-223

Generalization when observed outcomes depend on unobserved outcomes: an exact sampled-network information criterion

Scope

In the specified Gaussian network system, write \[ M_\delta=(I-\delta B)^{-1},\qquad Y=M_\delta(\alpha\mathbf1+\beta W+\gamma BW+\varepsilon), \quad\varepsilon\sim N(0,\sigma^2I). \] The treatment vector and network are observed but outcomes are observed only on a sampled subset $S$. The policy contrast is $\Delta=c'(\alpha,\beta,\gamma)'$, where \[ c=\left(0,\ N^{-1}\mathbf1'M_\delta(q'-q),\ N^{-1}\mathbf1'M_\delta B(q'-q)\right)'. \] Section 7.6 of [network] identifies external validity with unsampled-outcome contagion as a challenge. The results below solve the conditional Gaussian information problem when $\delta$ and $\sigma$ are supplied, and isolate a sampling obstruction that persists with arbitrarily many observations.
Proposition (Conditional target identification and exact inference): Assume $B$ is symmetric with operator norm at most one, $|\delta|<1$, $\sigma>0$, and sampling is independent of the structural errors and treatment. Condition on observed $(W,S)$. Set \[ X_S=[M_\delta\mathbf1,M_\delta W,M_\delta BW]_S, \quad\Sigma_S=\sigma^2(M_\delta^2)_{SS},\quad I_S=X_S'\Sigma_S^{-1}X_S. \] Over unrestricted $(\alpha,\beta,\gamma)\in\mathbb R^3$, the target is identified by this conditional experiment if and only if $c\in\operatorname{range}(I_S)$. If it is identified, with $I_S^+$ the Moore--Penrose inverse, \[ \widehat\Delta=c'I_S^+X_S'\Sigma_S^{-1}Y_S, \qquad\widehat\Delta-\Delta\sim N(0,v_S),\quad v_S=c'I_S^+c. \] Thus the usual Gaussian interval has exact conditional coverage; $v_S=0$ means an exactly known zero target.
Proof: The inverse $M_\delta$ exists and is positive definite, so $\Sigma_S$ is positive definite for nonempty $S$. Two parameter vectors induce the same conditional law exactly when their difference lies in $\ker X_S=\ker I_S$. The target agrees on all such vectors exactly when $c$ is orthogonal to this kernel, equivalently in the range of the symmetric matrix $I_S$. Under this condition $c'I_S^+I_S=c'$, proving unbiasedness. The estimator is Gaussian and its variance is $c'I_S^+I_S I_S^+c=c'I_S^+c$. If the range condition fails, a null vector $h$ with $c'h\ne0$ gives indistinguishable parameters with different targets. ∎
For compact parameter sets the range condition remains necessary at any relative interior point permitting small perturbations in the offending direction; boundary restrictions can of course identify quantities not identified in an unrestricted linear model.
Proposition (The conditional variance is a local lower bound): Suppose $v_S>0$, and a parameter set contains both $\vartheta_0$ and $\vartheta_0+t I_S^+c/\sqrt{v_S}$, with fixed sufficiently small $t>0$. Then every estimator has maximum conditional mean squared error at these two points at least $c_t v_S$, where $c_t>0$ depends only on $t$.
Proof: The two Gaussian means have squared Mahalanobis distance $t^2c'I_S^+I_S I_S^+c/v_S=t^2$. Their relative entropy is $t^2/2$, while their target separation is $t\sqrt{v_S}$. Total variation is at most $t/2$ by Pinsker's inequality. Thresholding any estimator at the target midpoint turns an error below half the separation into a correct test, so its maximum squared risk is at least $t^2v_S(1-t/2)/8$. ∎
Example (A vanishing sampling fraction can be harmless): The formal matrix class permits $B=I$. Then \[ Y_i=\frac\alpha{1-\delta}+\frac{\beta+\gamma}{1-\delta}W_i +\frac{\varepsilon_i}{1-\delta},\qquad \Delta=\frac{\beta+\gamma}{1-\delta}\,\overline{(q'-q)}. \] Although $\beta$ and $\gamma$ cannot be separated, the target depends only on their identifiable sum. With iid treatment probabilities bounded away from zero and one and an independent sample of size $n$, the two sampled arm counts are proportional to $n$ with exponentially high probability. The difference of arm means then estimates the slope with variance $O(1/n)$ uniformly over a fixed stable compact parameter set. If an arm count is zero, use a bounded fallback estimate; its exponentially small probability does not change the risk order. Therefore $n/N\to0$ alone does not imply an external-validity failure. This example uses the matrix class literally; models forbidding diagonal entries would require a different network construction.
Proposition (Positive inclusion probabilities are insufficient): For even $N$, take $B=\operatorname{diag}(0_{N/2},I_{N/2})$ and a policy difference zero on the first block $A$ and one on the second block $D$. There are stable bounded parameters and independent sampling schemes with positive inclusion probability for every vertex, $n\to\infty$, and $n/N\to0$, for which every estimator has nonvanishing worst-case risk for $\Delta$.
Proof: Fix $\alpha=\beta=\delta=0$, $\sigma=1$, and compare $\gamma=g$ and $\gamma=-g$, with $g>0$ fixed. Outcomes in $A$ are independent standard Gaussian errors under both laws. The target equals $\gamma/2$, so the target separation is $g$. Sample a uniform $n$-subset of $A$ with probability $1-n^{-2}$, and a uniform $n$-subset of $D$ with probability $n^{-2}$, independently of all other variables. Every vertex has positive inclusion probability. The observed experiments agree on the first sampling event, and their total variation is at most $n^{-2}$. The two-point risk bound is at least $g^2(1-n^{-2})/8$. Likewise any interval honest at both laws has expected length at least $g(1-2\alpha-n^{-2})$ under either fixed reference law. Choose any even $N$ growing faster than $n$ with $n\le N/2$. ∎

Remaining gap

The information criterion makes the role of unsampled outcomes explicit through the full resolvent and covariance. It does not solve the nonlinear identification of unknown $\delta$, establish uniform information bounds for general random networks, or characterize which sampling designs yield efficient transport. The examples show why neither sample size nor positivity of inclusion alone is an adequate condition.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt gives an exact conditional target-identification criterion, Gaussian inference and local risk bounds when propagation and noise parameters are supplied. Unknown propagation, general graph and sampling sequences, and uniform shrinking intervals over the full marginal-likelihood model remain uncharacterized.

References

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