Problem ID: C54-125 Model: GPT 6 Astra Ultra Version: 1
Robust structural counterfactuals
Status. Unresolved in the general structural family. We solve a bounded, explicitly specified weak-IV experiment, including matching orders for confidence-set width. These are deductions for that experiment, not claims of new general IV theory.
Target and restriction
The catalogue asks for honest coverage of all structural counterfactuals within observational total variation $\delta$, while instruments become weak. We retain full identified-set coverage and measure expected diameter. Our restriction is that the observed law belongs to the following one-parameter family; misspecification enlarges the compatible structural family, but the sampling law itself is on the family. Instrument strength $\kappa\in[0,1/2]$ is known.
For $\theta\in[-1,1]$ and $(z,d,y)\in\{-1,1\}^3$, set \[
P_\theta(z,d,y)=\frac18(1+\kappa zd)(1+\kappa\theta zy).
\] Thus $Z$ is symmetric, and $D,Y$ are conditionally independent given $Z$. Define $U=Y-\theta D$ and structural responses $Y(d)=\theta d+U$ for real interventions $d$. The moment restriction is ${\mathbb{E}}[ZU]=0$; full independence of $Z$ and $U$ is not assumed. A policy increasing each unit's treatment by one has average effect $\psi=\theta$, bounded by one. This is a scalar linear-IV model with ${\mathbb{E}}[ZD]=\kappa$ and ${\mathbb{E}}[ZY]=\kappa\theta$. It specifies its latent law by the above construction; endogenous $D$ is permitted.
For actual law $P_\eta$, define \[
I_\delta(\eta)=\{\theta\in[-1,1]:{\operatorname{TV}}(P_\theta,P_\eta)\leq\delta\}.
\] All intervals below can be empty on exceptional samples; their diameter is then zero. Such emptiness is a statistical event, not a claim that the population identified set is empty.
Proposition (Exact observational geometry): For every $\theta,\eta\in[-1,1]$, \[
{\operatorname{TV}}(P_\theta,P_\eta)=\frac\kappa2|\theta-\eta|.
\] Consequently, for $\kappa>0$, \[
I_\delta(\eta)=[-1,1]\cap[\eta-2\delta/\kappa,\eta+2\delta/\kappa].
\] If $\kappa=0$, the set is $[-1,1]$, including when $\delta=0$.
Proof: Subtract the displayed mass functions and sum absolute values. The difference has magnitude $\kappa|\theta-\eta|(1+\kappa zd)/8$. Summing over the eight cells gives $\kappa|\theta-\eta|$; total variation is half that sum. The interval description follows by solving the resulting scalar inequality. At zero strength all mass functions coincide. ∎
Theorem (Finite-sample coverage and width): Observe $n$ independent draws from $P_\eta$. For $\kappa>0$, put $\bar T=n^{-1}\sum_{j=1}^nZ_jY_j$ and $r_n=\sqrt{2\log(2/\alpha)/n}$. The interval \[
C_n=[-1,1]\cap\left[\frac{\bar T-2\delta-r_n}{\kappa},
\frac{\bar T+2\delta+r_n}{\kappa}\right]
\] satisfies, simultaneously for every $\eta\in[-1,1]$, $\delta\geq0$ and $\kappa\in(0,1/2]$, \[
{\mathbb{P}}_\eta\{I_\delta(\eta)\subseteq C_n\}\geq1-\alpha,
\qquad {\operatorname{diam}} C_n\leq\min\{2,(4\delta+2r_n)/\kappa\}.
\] At $\kappa=0$, set $C_n=[-1,1]$.
Proof: The variables $Z_jY_j\in[-1,1]$ have mean $\kappa\eta$. Hoeffding's inequality gives ${\mathbb{P}}(|\bar T-\kappa\eta|>r_n)\leq\alpha$. On its complement, every $\theta\in I_\delta(\eta)$ obeys $|\kappa\theta-\bar T|\leq2\delta+r_n$. Intersection with the parameter range proves coverage; the deterministic width bound follows from the two interval lengths. Hoeffding's bound has no inverse-strength remainder, so coverage is uniform even when $\kappa$ tends to zero. ∎
Theorem (Width cannot improve in order): Fix $0<\alpha\leq1/8$ and $\kappa>0$. Let $C$ be any measurable random set satisfying whole-set coverage at least $1-\alpha$ for every $\eta\in[-1,1]$. Then \[
\sup_\eta{\mathbb{E}}_\eta{\operatorname{diam}} C\ \geq
\max\left\{(1-\alpha)\min(2,4\delta/\kappa),
\frac12\min\left(\frac12,\frac1{4\kappa\sqrt n}\right)\right\}.
\] In particular, for fixed $\alpha$, minimax expected diameter has order $\min\{1,(\delta+n^{-1/2})/\kappa\}$, up to constants independent of $n,\delta,\kappa$.
Proof: At $\eta=0$, covering the whole identified interval forces diameter at least $\min(2,4\delta/\kappa)$ on an event of probability $1-\alpha$. For the sampling term, let $h=\min(1/2,1/(4\kappa\sqrt n))$ and compare $\eta=-h$ and $\eta=h$. The only changing factor in the likelihood is $T=ZY$, whose mean is $\pm a$ with $a=\kappa h$. Its one-observation divergence is \[
{\operatorname{KL}}(P_{-h},P_h)=a\log\frac{1+a}{1-a}\leq4a^2
\] for $0\leq a\leq1/2$: integrate $2/(1-t^2)\leq 8/3<4$ from zero to $a$. Independence and Pinsker's inequality imply ${\operatorname{TV}}(P_{-h}^n,P_h^n)\leq\sqrt{2na^2}\leq1/2$. Each $\eta$ belongs to its own identified set. Under $P_{-h}^n$, the events $-h\in C$ and $h\in C$ therefore hold together with probability at least $1-2\alpha-1/2\geq1/4$. Their intersection forces diameter at least $2h$, giving $h/2$. The maximum of the two lower bounds is comparable to the claimed capped sum, and the preceding theorem gives the upper bound. ∎
What the result does and does not identify
The calculation distinguishes the observational uncertainty $\delta$ from the counterfactual amplification $1/\kappa$. Even infinitely many observations leave diameter $\min(2,4\delta/\kappa)$ at the central law. A validation experiment that bounds $\delta$ improves this component; additional observations from the same experiment improve only the sampling component. The calculation also shows why reporting only coverage would conceal the central weak-identification issue.
The general problem still permits sampling laws outside this curve, unknown first-stage strength, nonlinear equilibrium counterfactuals and latent restrictions stronger than one IV moment. None is covered by this minimax theorem. Under arbitrary off-curve laws, a confidence set for the full eight-cell distribution followed by TV inversion would be valid, but a sharp general excess-width result needs further analysis.
References and provenance
Isaiah Andrews, Nano Barahona, Matthew Gentzkow, Ashesh Rambachan and Jesse M. Shapiro,
Structural Estimation Under Misspecification: Theory and Implications for Practice, January 2025 author draft, Section 4, Definition 6 and Proposition 3.
https://web.stanford.edu/~gentzkow/research/includediv.pdf. The inspected draft maintains strong identification in that robustness result; it is background for the question, not the source of our finite experiment or its bounds. All substantive model-specific arguments are given above.
Completion Estimate
35%