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Econ-407 · economics · Economics

Policy effects beyond IV compliers

LLM status Partially solved unverified

Statement

Original subject group: Econometrics, causal inference and measurement

Audit classification: Background; proposed extension. The verified chapter surveys solutions and recommends target-specific modeling; it does not identify this model class as a single open theorem.

Definitions and precise research target

Consider a population with covariates $X$, binary treatment $D$, bounded potential outcomes $Y(0),Y(1)\in[0,1]$, and a possibly multivalued instrument $Z$. Latent vector $V$ describes treatment choices and effect heterogeneity. Assume consistency, instrument exclusion, and conditional independence of $Z$ from potential outcomes and potential treatment choices given $X$; impose monotonicity only where justified. For each model $M$ satisfying these restrictions, let $D_M^q(V,X)$ denote treatment under a specified new assignment policy $q$, and $D_M^b(V,X)$ treatment under the baseline policy $b$. The policy effect is \[ \psi(M)=E_M[Y(D_M^q(V,X))-Y(D_M^b(V,X))]. \] Define $\mathcal M(P,\mathcal R)$ as models reproducing observed data law $P$ and obeying additional restrictions $\mathcal R$, and define the identified set $\{\psi(M):M\in\mathcal M(P,\mathcal R)\}$. Which credible structural restrictions, new instruments, randomized policy pilots or outcome bounds narrow this set enough to evaluate $q$? Clearly specify how each model maps the proposed policy into treatment choices, rather than equating a complier effect with the policy effect. Existing IV and partial-identification methods already solve important cases; this proposed problem asks which added information is credible in an application with multidimensional heterogeneity.

Definitions, assumptions and mathematical qualifications. A model supplies $Y_M(d,V,X)$, an instrument-to-choice law $D_M(z,V,X)$, the joint law of $(V,X)$, and an explicit policy map $D_M^q$. Baseline policy $b$ reproduces observed treatment assignment. Full instrument exogeneity is $Z\perp\{Y_M(0),Y_M(1),D_M(z):z\in\mathcal Z\}\mid X$; exclusion removes $Z$ from outcomes apart from $D$. Since outcomes are bounded, $\psi(M)\in[-1,1]$. A nonempty observationally compatible class yields an identified set, which need not be an interval unless convexity or closure is established. Policy consistency and stable treatment versions must hold before extrapolating from compliers.

Variants and assumption changes

  1. Sourced threshold benchmark. Use $D=\mathbf1\{U\leq p(Z,X)\}$, $U\mid X\sim U[0,1]$, with conditional instrument independence; MTE integrals identify policy responses over supported propensity ranges under the stated model.
  2. Editorial multidimensional policy. Permit vector $V$ and policies affecting costs or eligibility outside observed support. Use pilots, added instruments and explicit bounds on selection or outcomes to identify a decision-relevant range for $\psi$.

References and source audit

Checked source. 10 December 2024 author chapter, Sections 2.6 (p. 14), 4.3 (pp. 57--76), 4.7 (pp. 88--93), and 6 (p. 99). The chapter surveys policy-relevant IV, MTE and bounding solutions. Full primary text retrieved. https://www.rfberlin.com/wp-content/uploads/2024/12/24030.pdf.
  1. Magne Mogstad; Alexander Torgovitsky. 2024. Instrumental Variables with Unobserved Heterogeneity in Treatment Effects. 10 December 2024 author chapter, Sections 2.6 (p. 14), 4.3 (pp. 57--76), 4.7 (pp. 88--93), and 6 (p. 99). https://www.rfberlin.com/wp-content/uploads/2024/12/24030.pdf. DOI: https://doi.org/10.3386/w32927.

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-407-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-407

Policy effects beyond IV compliers

Question and specified policy

An IV identifies outcomes for treatment choices changed by that instrument under particular assumptions. The target here is a different policy: universal access with compulsory treatment, $D^q=1$, compared with the observed baseline encouragement rule. Specifying this map is essential; voluntary access would require a separate behavioral response. We derive a sharp interval, explain which pilot closes it, and extend the construction to a finite multidimensional latent-type program. The wider application remains unresolved.

Binary instrument benchmark

Suppress covariates, let $Z$ be randomized independently of $(Y_0,Y_1,D_0,D_1)$ with $p=\Pr(Z=1)\in(0,1)$, impose exclusion and $D_1\ge D_0$, and take $0\le Y_d\le1$. Let strata $A,C,N$ denote always, complier and never takers, with probabilities $\pi_A,\pi_C,\pi_N$. The observed law identifies \[ \pi_A={\mathbb{E}}[D\mid Z=0],\quad \pi_N=1-{\mathbb{E}}[D\mid Z=1],\quad \pi_C={\mathbb{E}}[D\mid Z=1]-{\mathbb{E}}[D\mid Z=0]. \] Define $\mu_{dS}={\mathbb{E}}[Y_d\mid S]$ whenever $\pi_S>0$. Positive strata have \[ \mu_{1A}={\mathbb{E}}[Y\mid Z=0,D=1],\quad \mu_{0N}={\mathbb{E}}[Y\mid Z=1,D=0], \] \[ \pi_C\mu_{1C}={\mathbb{E}}[YD\mid Z=1]-{\mathbb{E}}[YD\mid Z=0],\qquad \pi_C\mu_{0C}={\mathbb{E}}[Y(1-D)\mid Z=0]-{\mathbb{E}}[Y(1-D)\mid Z=1]. \] For zero-mass strata use the products in these displays directly; no division by an unidentified zero probability is necessary. These identities follow by conditioning on strata and using instrument independence, not by assuming constant effects. They can also be written for outcome distribution measures, which imposes the usual observed-law compatibility restrictions.
Theorem (Sharp universal-treatment effect): Assume the observed law is compatible with this model. Put $B=\pi_C(1-p)(\mu_{1C}-\mu_{0C})$. For baseline $D^b=D_Z$ and universal treatment $D^q=1$, the sharp identified set is \[ \mathcal I_\psi=[B-\pi_N\mu_{0N},\ B+\pi_N(1-\mu_{0N})]. \tag{1} \]
Proof: Always takers do not change treatment. Compliers change only when baseline $Z=0$, an event independent of their outcomes of probability $1-p$. Never takers always change. Hence $\psi=B+\pi_N(\mu_{1N}-\mu_{0N})$. Since $Y_1\in[0,1]$, this yields (1). Never takers' $Y_1$ is unobserved under either instrument value. Start with any compatible model and replace this potential outcome by a constant $t\in[0,1]$, leaving every observed random variable and all choice strata unchanged. Randomization, exclusion, monotonicity and the full observed outcome law are preserved. As $t$ varies the entire interval is attained, proving sharpness. ∎
The result bounds the policy gain, not the population ATE: $Y_0$ for always takers is irrelevant to this policy and should not needlessly widen its bounds. For example, $p=1/2$, $\pi_C=0.4$, $\pi_N=0.3$, $\tau_C=0.2$, and $\mu_{0N}=0.6$ yield $[-0.14,0.16]$. The complier effect is positive, but a universal-treatment recommendation is not identified. An implementation cost $k$ subtracts $k$ from both endpoints under equal monetary weights.

Which extra information narrows the interval?

A credible restriction $\mu_{1N}\in[\ell,u]$ changes the width from $\pi_N$ to $\pi_N(u-\ell)$. Monotone treatment response for never takers, $Y_1\ge Y_0$, makes their mean contribution nonnegative and changes the lower endpoint to $B$, provided the restriction is compatible. This is an assumption about outcomes; it does not follow from monotone treatment choices.

A new randomized compulsory-treatment pilot on a population representative of never takers identifies $\mu_{1N}$ and collapses (1). Establishing that sampling frame is difficult: a subject untreated when $Z=0$ could be a complier, while one untreated at $Z=1$ is a never taker under the maintained model. A prospective pilot may first observe the latter event, then independently randomize treatment within that selected stratum, with stable treatment versions and no selection based on unobserved follow-up outcomes. A pilot that merely repeats the original encouragement supplies no new outcome support for never takers.

With covariates $X$, all formulas can be conditional, using the known baseline $p(X)$, conditional exogeneity and overlap. Integrate the lower and upper endpoints with respect to the fixed target distribution of $X$. Sharpness requires measurable choices of the missing outcomes. Weak first stages need not destabilize the policy formula if identified products are used directly; they do affect estimation of separately reported stratum means.

A finite program for multidimensional policy response

To make multidimensional heterogeneity explicit, take finite $X,Z$ and binary potential outcomes. A latent atom $v$ records $(y_0,y_1)$, the entire vector $(d_z:z\in\mathcal Z)$, and policy responses $d^q,d^b$ if those are not already determined by the assignment rule. Enumerate only atoms allowed by a declared structural restriction $\mathcal R$. For each $x$ introduce nonnegative masses $\lambda_{v\mid x}$ summing to one. The observable constraints are \[ \sum_v\lambda_{v\mid x}\, \mathbf1\{d_z(v)=d,\ y_d(v)=y\} =\Pr(D=d,Y=y\mid Z=z,X=x). \tag{2} \] Instrument exogeneity is encoded by using the same $\lambda_{v\mid x}$ for all $z$. Under a stochastic baseline, average its atom-level outcome over the specified baseline randomization. Define the resulting known atom contrast $c_{vx}$ for the intervention and minimize and maximize \[ \sum_x\Pr(X=x)\sum_v c_{vx}\lambda_{v\mid x} \tag{3} \] subject to (2) and any extra linear restrictions. Feasibility itself is testable at the population law.
Proposition: For this finite atom model, the feasible values of (3) form exactly the closed interval between the two linear-program optima, unless the program is infeasible.
Proof: Every admissible model induces feasible masses. Conversely feasible masses define a distribution of the finite latent atom conditional on $X$; generate $Z$ independently given $X$ and use the atom's potential variables. This constructs a model with precisely the required observable probabilities and policy effect. The feasible set is compact and convex, so the linear target's image is a closed interval with attained endpoints. ∎
This is an exact finite-class result, not a claim of polynomial complexity in the number of instruments: the type space can grow exponentially. For continuous outcomes, a discretization requires outward rounding and an explicit error bound before it can stand in for the infinite-dimensional problem. If $d^q$ is unconstrained and absent from observations, merely appending that coordinate to the program does not manufacture informative extrapolation.

Remaining work and source relationship

Sampling uncertainty can be handled by enlarging (2) to a simultaneous confidence region for all cell probabilities; any true compatible masses are then retained, so optimizing (3) gives an outer confidence region for the whole identified set. A useful procedure must also report width and positivity, since tiny conditional cells can make coverage uninformative. The missing empirical ingredients remain which policy responses are credible, how pilots represent the target population, and how treatment versions and equilibrium spillovers are controlled.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp universal-treatment bounds under a binary monotone instrument and an exact finite latent-type linear program for policy effects. Credible multidimensional policy responses, continuous-outcome error control, representative pilots, and equilibrium spillovers still require substantive identification.

References

  • [source] M. Mogstad and A. Torgovitsky (2024), Instrumental Variables with Unobserved Heterogeneity in Treatment Effects, 10 December author chapter. Sections 2.6, 4.3 and 4.7 provide context for policy-specific IV and bounding methods. https://www.rfberlin.com/wp-content/uploads/2024/12/24030.pdf. The principal-stratum derivation and finite program above are standard-method specializations with proofs, not claims of a new general IV solution.

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