we-solve (beta)

Econ-185 · economics · Economics

Weak and many instruments

LLM status Open unverified

Statement

Original identifier: E3 (atlas item 103). Field: Econometrics and causal inference. Classification: editorial research family with established restricted solutions; current open status not certified.

Definitions and assumptions

Consider independent observations $(Y_i,D_i,Z_i)$ for $i=1,\ldots,n$, after any stated exogenous controls have been partialled out. Here $Y_i,D_i\in\mathbb R$ and $Z_i\in\mathbb R^{k_n}$ with $k_n$ potentially growing. A linear structural model is $Y_i=D_i\beta+u_i$, $D_i=Z_i^\top\pi_n+v_i$, with finite stated moments. Under instrument validity $\mathbb E[Z_i u_i]=0$ and normalized nonsingular $\mathbb E[Z_iZ_i^\top]$, relevance may weaken, for example $\pi_n=c/\sqrt n$ for fixed dimension and bounded $c$. Neither a nonzero sample first-stage estimate nor the number of instruments establishes identification.

Formal research question

Find identification-robust confidence sets and an explicit efficiency frontier when $k_n$ grows, first-stage strength can approach zero, and instrument invalidity is bounded rather than excluded. For a supplied norm $\|\cdot\|_*$ and tolerance $\varepsilon_n\ge0$, define \[ \mathcal B(P,\varepsilon_n)= \{b\in\mathbb R:\|\mathbb E_P[Z(Y-Db)]\|_*\le\varepsilon_n\}. \] For a specified class $\mathcal P_n$ with moment, dependence and dimensional restrictions, construct $C_n$ with $\liminf_n\inf_{P\in\mathcal P_n}\Pr_P\{\mathcal B(P,\varepsilon_n)\subseteq C_n\}\ge1-\alpha$, where $\alpha\in(0,1)$. Determine necessary growth restrictions on $k_n$, appropriate covariance regularization, and whether selection of instruments on the same data can preserve this guarantee. Compare expected diameter only over submodels where finite informative sets are possible, and establish impossibility results where identification disappears.

Interpretation and scope

The norm and tolerance must fix instrument scaling; otherwise an invalidity bound changes when an instrument is multiplied by a constant. With zero relevance, $\beta$ need not be point identified and an honest confidence set may be unbounded. Anderson--Rubin-type tests already solve important weak-instrument cases, while many-instrument asymptotics have their own regimes. Arbitrary invalid instruments cannot identify a causal slope without additional restrictions. Clustered sampling changes the experiment and requires enough independent clusters, not simply a large observation count. A causal interpretation further requires a defensible structural equation.

Source audit and status

[S1] develops local-to-zero asymptotics; [S2] surveys weak identification in IV and GMM; [S3] reviews both theory and practice, including heteroskedastic and dependent settings. Reliable inference with weak instruments is therefore not an entirely unsolved question. The combined growing-dimension, bounded-invalidity and adaptive-selection target is an editorial extension; the cited reviews do not certify its exact 2026 status.

References

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  1. Douglas Staiger and James H. Stock (1997). Instrumental Variables Regression with Weak Instruments. Econometrica 65(3), 557–586. https://www.nber.org/papers/t0151. Locator: Publisher or institutional abstract and bibliographic record. Support and limits: Local-to-zero weak-instrument asymptotics; NBER page verifies the 1997 publication, although the working paper is 1994.

  2. James H. Stock, Jonathan H. Wright and Motohiro Yogo (2002). A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments. Journal of Business & Economic Statistics 20(4), 518–529. https://stock.scholars.harvard.edu/publications/survey-weak-instruments-and-weak-identification-generalized-method-moments. Locator: Publisher or institutional abstract and bibliographic record. Support and limits: Survey of established weak-identification methods; does not establish inference with arbitrary invalid instruments.

  3. Isaiah Andrews, James H. Stock and Liyang Sun (2019). Weak Instruments in Instrumental Variables Regression: Theory and Practice. Annual Review of Economics 11, 727–753. https://doi.org/10.1146/annurev-economics-080218-025643. Locator: Publisher or institutional abstract and bibliographic record. Support and limits: Reviews identification-robust inference, including nonhomoskedastic data; the basic question is already partly solved.

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-185-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-185

Weak and many instruments Whole-set inversion with bounded invalidity and adaptive coordinate selection

Problem ID: C26-181. Status: Unresolved; rigorous restricted results.

Short recap and the declared experiment

The target is an honest confidence set for the entire moment-identified set when relevance can vanish, invalidity is bounded, and the number of instruments grows. We give a finite-sample construction requiring no covariance inverse, establish explicit diameter bounds where relevance is separated from zero, and prove two distinct obstructions to precision. The construction is conservative and does not determine the efficiency frontier for general norms, covariance structures, or adaptive instruments.

Identification-robust IV inference has a substantial existing literature. The reviewed abstract of Andrews, Stock and Sun [review] explicitly covers weak-instrument inference and nonhomoskedastic settings. We do not attribute the combined bounded-invalidity and adaptive-selection theorem below to that review; its proof is supplied here.

Observe $n$ independent identically distributed triples $(Y,D,Z)$ with $Z\in{\mathbb{R}}^k$, where $k=k_n$ is arbitrary. Fix instrument scaling before sampling and assume known constants $A,B>0$ satisfy \[ |Z_jY|\leq A,\quad |Z_jD|\leq B\quad\text{almost surely},\quad j\leq k. \] Assume finite second moments for any structural equation under discussion. Set $a={\mathbb{E}}[ZY]$, $d={\mathbb{E}}[ZD]$, and use the supplied invalidity norm $\|\cdot\|_\infty$. The target is exactly \[ \mathcal B=\{b\in{\mathbb{R}}:\|a-bd\|_\infty\leq\varepsilon\}. \] For example each $b\in\mathcal B$ induces a structural residual $u=Y-Db$ whose moment invalidity is at most $\varepsilon$. This alone does not establish the substantive causal interpretation of that equation.

Uniform inversion without a first-stage division

Let $\widehat a=n^{-1}\sum_iZ_iY_i$ and $\widehat d=n^{-1}\sum_iZ_iD_i$. For $\gamma\in(0,1)$ put \[ t_Y=A\sqrt{\frac{2\log(4k/\gamma)}n},\qquad t_D=B\sqrt{\frac{2\log(4k/\gamma)}n}, \] and define the random closed set \[ C=\{b\in{\mathbb{R}}:\|\widehat a-b\widehat d\|_\infty \leq\varepsilon+t_Y+|b|t_D\}. \tag{1} \] Each side is continuous and piecewise linear in the scalar $b$; hence membership and endpoints are measurable, including unbounded and empty cases.
Theorem (Finite-sample coverage of the whole set): For every law satisfying the stated bounds, $\Pr\{\mathcal B\subseteq C\}\geq1-\gamma$, uniformly in $k$, $d$, and $\varepsilon$. The same assertion holds if (1) is imposed only on any data-selected subset $S\subseteq\{1,\ldots,k\}$, using the same full-dictionary thresholds. If $S$ is empty the resulting set is ${\mathbb{R}}$.
Proof: A mean of independent variables in $[-A,A]$ obeys $\Pr(|\widehat a_j-a_j|>t)\leq2\exp[-nt^2/(2A^2)]$. One proof centers the summands and bounds their moment generating functions by $\exp(\lambda^2A^2/2)$: for a bounded variable the second derivative of the centered log moment generating function is the variance under exponential tilting, at most $(2A)^2/4$; integrate twice from zero. Markov's inequality and optimization over $\lambda$ give the bound. The analogous bound holds for $\widehat d_j-d_j$. A union bound over the $2k$ coordinates gives probability at least $1-\gamma$ to the event \[ \mathcal A=\{\|\widehat a-a\|_\infty\leq t_Y, \|\widehat d-d\|_\infty\leq t_D\}. \] On this one event, for every real $b$ simultaneously, $\|\widehat a-b\widehat d-(a-bd)\|_\infty\leq t_Y+|b|t_D$. Thus every member of $\mathcal B$ belongs to $C$. The conclusion already holds for every coordinate on $\mathcal A$, and therefore for every realization of a data-selected subset. No independence of the selector and the observations is needed. ∎

This protects selection from a fixed dictionary of $k$ instruments. It does not protect a newly optimized, unrestricted function of the same outcomes: such a function was not included in the simultaneous event. Nor may one replace $k$ by the smaller realized $|S|$ without another argument. Selection only discards constraints here, so it cannot improve the set's diameter relative to imposing all coordinates.

Proposition (Observable informativeness certificate): Let $s=\|\widehat d\|_\infty$. If $s>t_D$, every $b\in C$ satisfies \[ |b|\leq L:=\frac{\|\widehat a\|_\infty+\varepsilon+t_Y}{s-t_D}, \qquad {\operatorname{diam}} C\leq\frac{2(\varepsilon+t_Y+Lt_D)}s. \tag{2} \] The diameter of an empty set is defined to be zero.
Proof: Choose a coordinate $j$ with $|\widehat d_j|=s$. For $b\in C$, $|b|s\leq|\widehat a_j|+\varepsilon+t_Y+|b|t_D$, proving the first bound. For $b,c\in C$, subtract their two residuals at coordinate $j$: $s|b-c|\leq2(\varepsilon+t_Y)+(|b|+|c|)t_D$. Insert the first bound and take the supremum over pairs. ∎

Expected diameter on an informative submodel

For an expected-diameter comparison, specify a known $B_0>0$ and restrict the law class to laws with $\mathcal B\subseteq[-B_0,B_0]$ and $\|d\|_\infty\geq\kappa>0$. This is a substantive restriction on the whole identified set, not an unjustified truncation of weak-IV confidence intervals. Let $C^0=C\cap[-B_0,B_0]$.
Corollary (Diameter bound and a sufficient growth condition): If $t_D\leq\kappa/2$, then on $\mathcal A$, \[ {\operatorname{diam}} C^0\leq\min\left\{2B_0, \frac{4(\varepsilon+t_Y+B_0t_D)}\kappa\right\}. \] Consequently \[ {\mathbb{E}}{\operatorname{diam}} C^0\leq \frac{4(\varepsilon+t_Y+B_0t_D)}\kappa+2B_0\gamma. \tag{3} \] With $\gamma=\min(\alpha,n^{-2})$, informative vanishing expected diameter follows if $[\varepsilon+\sqrt{\log(kn)/n}]/\kappa\to0$, for fixed $A,B,B_0$. Coverage remains at least $1-\alpha$ for every $n$.
Proof: On $\mathcal A$, $s\geq\|d\|_\infty-t_D\geq\kappa/2$. For $b,c\in C^0$ the second inequality in the proof of (2) gives $s|b-c|\leq2(\varepsilon+t_Y+B_0t_D)$. Divide by $s$, and also use the deterministic $2B_0$ diameter bound. On the complement of $\mathcal A$ use that deterministic bound and probability at most $\gamma$, proving (3). The proposed sequence of thresholds has the stated order and $t_D/\kappa\to0$. The coverage assertion follows from the preceding theorem and the law-class assumption about $\mathcal B$. ∎

No claim of a necessary restriction on $k$ follows. The construction has logarithmic dictionary dependence because it is simultaneous coordinatewise inversion. Highly redundant instruments need not incur a corresponding intrinsic loss; different covariance and norm classes require separate frontiers. In particular a sample covariance matrix need not be invertible for (1) to be valid, including when $k>n$.

Two lower bounds and the zero-relevance boundary

Proposition (Invalidity cannot be averaged away): For any law with $a=0$ and $\|d\|_\infty=\kappa>0$, $\mathcal B=[-\varepsilon/\kappa,\varepsilon/\kappa]$. Any random set covering the whole $\mathcal B$ with probability at least $1-\alpha$ has ${\mathbb{E}}{\operatorname{diam}} C\geq(1-\alpha)2\varepsilon/\kappa$. If $d=0$, then $\mathcal B={\mathbb{R}}$ when $\|a\|_\infty\leq\varepsilon$, and $\mathcal B=\varnothing$ otherwise. In the former case honest whole-set coverage requires $C={\mathbb{R}}$ with probability at least $1-\alpha$.
Proof: For $a=0$, $\|a-bd\|_\infty=|b|\kappa$, which gives the exact interval. Its two endpoints must both belong to $C$ on the coverage event, so the diameter there is at least their distance. Taking expectations proves the bound. When $d=0$ the defining inequality is independent of $b$, which gives the two stated cases. A subset of ${\mathbb{R}}$ containing all of ${\mathbb{R}}$ equals ${\mathbb{R}}$, proving the final assertion. ∎
Proposition (Sampling obstruction with bounded observations): Fix $0<\kappa\leq1$, $B_0>0$, and $0<\alpha<3/8$. There exists a one-instrument model satisfying $|ZY|\leq1$, $|ZD|\leq1$, normalized ${\mathbb{E}} Z^2=1$, and exact validity $\varepsilon=0$, such that every confidence set honest over this model obeys, at one of its laws, \[ {\mathbb{E}}{\operatorname{diam}} C\geq 2h(1-2\alpha-1/4),\qquad h=\min\{B_0/2,1/(8\kappa\sqrt n)\}. \tag{4} \] Thus order $\min\{B_0,1/(\kappa\sqrt n)\}$ cannot in general be improved.
Proof: Let $Z$ be a fair random sign, $D=\kappa Z$, and $Y=ZV$, where $V$ is an independent random sign with mean $\kappa b$, for $b\in[-h,h]$. These laws have $d=\kappa$, $a=\kappa b$, and identified set $\{b\}$. Writing $u=Y-Db$ gives ${\mathbb{E}} Zu=0$, and all variables are bounded. Compare $b=h$ and $b=-h$. Put $x=\kappa h\leq1/(8\sqrt n)$. The divergence between sign distributions with means $x$ and $-x$ is $x\log((1+x)/(1-x))\leq4x^2$: indeed the logarithm is $\int_{-x}^x(1+t)^{-1}dt\leq2x/(1-x)\leq4x$ for $x\leq1/2$. Thus ${\operatorname{KL}}(P_h^n,P_{-h}^n)\leq4nx^2\leq1/16$, and therefore ${\operatorname{TV}}(P_h^n,P_{-h}^n)\leq\sqrt{1/32}<1/4$. Here Pinsker follows from binary reduction and the curvature bound $\operatorname{kl}(p,q)\geq2(p-q)^2$. Honesty gives $P_h(h\in C)\geq1-\alpha$ and $P_{-h}(-h\in C)\geq1-\alpha$. Transporting the second event to $P_h$ loses at most total variation. The union bound implies $P_h(\{-h,h\}\subseteq C)\geq1-2\alpha-1/4$. On this event the diameter is at least $2h$, proving (4). ∎

An asymptotically honest procedure yields the same conclusions with $\alpha$ replaced by $\alpha+o(1)$ along these sequences. In particular if $n\kappa_n^2$ stays bounded, even the zero-invalidity one-instrument experiment need not permit shrinking confidence sets. The invalidity obstruction and the sampling obstruction are separate: the former remains with perfectly known population moments.

Remaining questions and source limits

The theorem supplies full-set coverage for bounded coordinates, any finite dimension, and data-dependent coordinate deletion. It leaves unresolved optimal covariance regularization under heavier tails, sharp dependence on correlation and the chosen invalidity norm, instrument construction outside a fixed dictionary, and an optimal expected-diameter frontier. Even in the bounded class, (3)'s $\sqrt{\log k}$ factor has not been proved necessary. The cited review was checked at its publisher abstract and bibliographic record; no uninspected theorem of that review is used in these proofs.

References

Completion Estimate

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