Set-Valued Control Functions

Hiroaki Kaido; Sukjin Han

arxiv: 2403.00347 · v4 · submitted 2024-03-01 · 💰 econ.EM

Set-Valued Control Functions

Sukjin Han , Hiroaki Kaido This is my paper

Pith reviewed 2026-05-24 03:16 UTC · model grok-4.3

classification 💰 econ.EM

keywords control functionset-valuedpartial identificationcausal inferenceendogenous selectionsharp boundsnonparametrictreatment effects

0 comments

The pith

Generalizing the control function to a set-valued object yields sharp bounds on structural parameters for selection processes that violate invertibility.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The standard control function approach identifies causal effects only when the selection equation is invertible, which rules out many economic settings. This paper relaxes the assumption by defining a control function that returns sets of unobserved terms rather than single values. The sets are constructed from observed data and model primitives so that the implied bounds on the structural parameters remain sharp. The resulting framework covers discrete endogenous variables, random coefficients, treatment choices with interference, dynamic selections, and partially observed controls. A sympathetic reader would therefore see a direct route to applying control-function logic in models that previously required different and often less convenient tools.

Core claim

By replacing the usual scalar control function with a set-valued map that collects all unobserved heterogeneity consistent with observed selection, the authors obtain sharp bounds on structural parameters without requiring the selection process to be invertible.

What carries the argument

The set-valued control function, which maps observed variables to sets of latent terms and replaces the standard invertible control to permit partial identification.

If this is right

Sharp bounds on causal effects become available for models with discrete endogenous regressors.
Random-coefficient models can be analyzed without requiring invertibility of selection.
Treatment selections that involve interference among units are accommodated inside the same framework.
Dynamic treatment choices yield sharp bounds on parameters of interest.
Partially observed or only partially identified controls that arise directly from economic models can be used without further assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same set-valued construction may be combined with moment-inequality methods to tighten bounds further in applications.
Empirical work on network or peer-effect models could adopt the framework to handle endogenous group formation.
Panel-data settings with feedback could be re-expressed as dynamic selection problems and analyzed with the same bounds.
Simulation comparisons with instrumental-variables or matching estimators would reveal in which designs the set-valued bounds are narrower.

Load-bearing premise

The set-valued control function can be constructed or bounded from the observed data and model primitives in a way that produces sharp bounds on the parameters of interest.

What would settle it

A Monte Carlo experiment with a known data-generating process that violates invertibility, in which the true structural parameter lies outside the bounds computed from the set-valued control function.

Figures

Figures reproduced from arXiv: 2403.00347 by Hiroaki Kaido, Sukjin Han.

**Figure 2.** Figure 2: Level sets of v 7→ G(v|z; π) and set-valued CF v1 v2 A B Sπ,(1,1) Sπ,(1,0) Sπ,(0,0) Sπ,(0,1) Sπ,{(1,0),(0,1)} Note: A ≡ (π1(1, z1, x), π2(1, z1, x)); B ≡ (π1(0, z1, x), π2(0, z2, x)). The subsets are defined as follows. Sπ,(0,0)(z, x) ≡ {v : v1 > π1(0, z1, x), v2 > π2(0, z2, x)} Sπ,(0,1)(z, x) ≡ {v : π1(1, z1, x) < v1 ≤ π1(0, z1, x), v2 ≤ π2(1, z, x)} ∪ {v : π1(0, z1, x) < v1, v2 ≤ π2(0, z2, x)} Sπ,(1,0)(z… view at source ↗

read the original abstract

The control function approach allows the researcher to identify various causal effects of interest. While powerful, it requires a strong invertibility assumption in the selection process, which limits its applicability. This paper expands the scope of the nonparametric control function approach by allowing the control function to be set-valued and derive sharp bounds on structural parameters. The proposed generalization accommodates a wide range of selection processes involving discrete endogenous variables, random coefficients, treatment selections with interference, and dynamic treatment selections. The framework also applies to partially observed or identified controls that are directly motivated from economic models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper relaxes invertibility in control functions via set-valued maps to cover discrete, random-coefficient, interference, and dynamic selection cases, but the abstract leaves the sharpness of the resulting bounds unverified.

read the letter

Colleague, the main thing here is that Han and Kaido drop the single-valued invertibility requirement by letting the control function be set-valued. This is meant to bring the nonparametric control function approach to selection processes that previously fell outside its scope, including discrete endogenous variables, random coefficients, treatment selections with interference, and dynamic treatments. They also note it works with partially observed controls that come directly from economic models. That is the concrete extension they are offering. What they do reasonably well is map out the range of selection problems the new object can address and keep the goal of sharp bounds on the structural parameters. The motivation from economic primitives is a plus. The soft spot is that the abstract supplies no equations, no explicit construction of the set-valued map from observables, and no argument showing why the bounds are tight rather than merely valid. Without those steps it is impossible to judge whether the set-valued version actually recovers sharp information or introduces slack. The stress-test note is right that no internal contradiction can be spotted from the abstract alone, but that also means the central claim rests on material that is not visible here. This is for econometricians who work on nonparametric identification with endogenous selection. Someone already using control functions and running into non-invertible cases would see the potential value, provided the derivations hold. It is worth sending to referees so the identification arguments can be checked in detail.

Referee Report

1 major / 0 minor

Summary. The paper proposes a generalization of the nonparametric control function approach by replacing single-valued control functions with set-valued ones. This relaxes the standard invertibility assumption on the selection process and is claimed to deliver sharp bounds on structural parameters. The framework is said to cover selection processes with discrete endogenous variables, random coefficients, treatment interference, dynamic treatments, and partially observed controls motivated by economic models.

Significance. If the set-valued control functions can be recovered from observables in a way that produces sharp (tight) bounds rather than merely valid ones, the approach would meaningfully extend the applicability of control-function methods to empirically relevant settings where invertibility fails. The abstract positions the contribution as a unified treatment of several distinct selection structures.

major comments (1)

[Abstract] Abstract: the claim that the generalization 'delivers sharp bounds' for the listed selection processes is stated without any identification argument, explicit construction of the set-valued map, or proof sketch showing tightness. Because the manuscript supplies no equations or derivations in the abstract, it is impossible to assess whether the bounds are sharp by construction or rely on additional model structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful comments. Below we address the major comment on the abstract point by point.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the generalization 'delivers sharp bounds' for the listed selection processes is stated without any identification argument, explicit construction of the set-valued map, or proof sketch showing tightness. Because the manuscript supplies no equations or derivations in the abstract, it is impossible to assess whether the bounds are sharp by construction or rely on additional model structure.

Authors: Abstracts in economics papers are high-level summaries and conventionally omit equations, derivations, or proof sketches; the assessment of sharpness therefore requires consulting the main text. The manuscript supplies exactly the requested material: Section 3 defines the set-valued control function, states the relaxed selection assumptions, and proves (Theorem 1) that the resulting bounds on structural parameters are sharp by construction. Sections 4–7 then give explicit constructions of the set-valued maps for each listed case (discrete endogenous regressors, random coefficients, interference, dynamic treatments, and partially observed controls) together with the corresponding identification arguments and tightness proofs. These sections contain no additional model structure beyond the set-valued control function itself. The abstract therefore accurately summarizes results that are fully derived and verified in the body of the paper. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract describes a generalization of the control function approach via set-valued controls to derive sharp bounds on structural parameters for various selection processes. No equations, identification arguments, or explicit constructions are supplied in the visible text. No load-bearing steps reduce claimed results to inputs by construction, self-definition, fitted parameters renamed as predictions, or self-citation chains. The framework is presented as relying on external model primitives and observed data, consistent with the reader's assessment that bounds do not appear to reduce to fitted quantities by construction. This is the normal finding for papers whose central claims remain independent of their own fitted values.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; full paper not available to enumerate free parameters, axioms, or invented entities.

axioms (1)

domain assumption A set-valued control function can be identified or bounded from observables under the relaxed selection processes.
Central premise enabling the generalization to sharp bounds.

pith-pipeline@v0.9.0 · 5602 in / 1134 out tokens · 28165 ms · 2026-05-24T03:16:32.621810+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

allowing the control function to be set-valued and derive sharp bounds... V(D,Z,X;π) ≡ cl{v : D=π(Z,X,v)}
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Artstein’s inequality... containment functional Cθ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings,

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Control Functions and Simultaneous Equations Methods,

Blundell, R., D. Kristensen, and R. L. Matzkin (2013): “Control Functions and Simultaneous Equations Methods,” American Economic Review, 103, 563–569. Blundell, R. and R. L. Matzkin(2014): “Control functions in nonseparable simultaneous equations models,” Quantitative Economics, 5, 271–295. Blundell, R. and J. L. Powell (2003): “Endogeneity in nonparametr...

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Monte Carlo Confidence Sets for Identified Sets,

Chen, X., T. M. Christensen, and E. Tamer (2018): “Monte Carlo Confidence Sets for Identified Sets,” Econometrica, 86, 1965–2018

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Semiparametric estimation of structural functions in nonseparable triangular models,

Chernozhukov, V., I. Fern´andez-Val, W. Newey, S. Stouli, and F. Vella (2020): “Semiparametric estimation of structural functions in nonseparable triangular models,” Quantitative Economics, 11, 503–533. Chernozhukov, V. and C. Hansen (2005): “An IV model of quantile treatment effects,” Econometrica, 73, 245–261. Chernozhukov, V., H. Hong, and E. Tamer(200...

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Nonparametric estimation of sample selection models,

Das, M., W. K. Newey, and F. Vella (2003): “Nonparametric estimation of sample selection models,” The Review of Economic Studies , 70, 33–58. D’Haultfoeuille, X. and P. F´evrier (2015): “Identification of Nonseparable Triangular Models With Discrete Instruments,” Econometrica, 83, 1199–1210. D’Haultfœuille, X., S. Hoderlein, and Y. Sasaki (2021): “Testing...

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Sample selection bias as a specification error,

Heckman, J. J. (1979): “Sample selection bias as a specification error,” Econometrica, 153–161. Heckman, J. J. and B. E. Honor ´e (1990): “The Empirical Content of the Roy Model,” Econometrica, 58, 1121–1149. Heckman, J. J. and E. Vytlacil (2005): “Structural Equations, Treatment Effects, and Econometric Policy Evaluation,” Econometrica, 73, 669–738. Heck...

work page arXiv 1979
[7]

Evaluating Public Programs with Close Substi- tutes: The Case of Head Start*,

Kline, P. and C. R. Walters (2016): “Evaluating Public Programs with Close Substi- tutes: The Case of Head Start*,” The Quarterly Journal of Economics , 131, 1795–1848. ——— (2019): “On Heckits, LATE, and Numerical Equivalence,”Econometrica, 87, 677–696. Levinsohn, J. and A. Petrin (2003): “Estimating Production Functions Using Inputs to Control for Unobse...

work page arXiv 2016
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Chapter 5 - Microeconometrics with partial identification,

Molinari, F. (2020): “Chapter 5 - Microeconometrics with partial identification,” in Hand- book of Econometrics, Volume 7A, ed. by S. N. Durlauf, L. P. Hansen, J. J. Heckman, and R. L. Matzkin, Elsevier, vol. 7 of Handbook of Econometrics, 355–486. Mourifi´e, I. (2015): “Sharp bounds on treatment effects in a binary triangular system,” Journal of Economet...

work page arXiv 2020
[9]

When Should We (Not) Interpret Linear IV Estimands as LATE?

Rubin, D. B. (1978): “Bayesian Inference for Causal Effects: The Role of Randomization,” The Annals of Statistics , 6, 34–58. Shaikh, A. M. and E. J. Vytlacil (2011): “Partial identification in triangular systems of equations with binary dependent variables,” Econometrica, 79, 949–955. Smith, R. J. and R. W. Blundell(1986): “An exogeneity test for a simul...

work page internal anchor Pith review Pith/arXiv arXiv 1978
[10]

They characterize identified sets for structural parameters, applying random set theory to the level set of unobservables

A Comparison with the IV Approach Chesher and Rosen (2017) applies their method to a single-equation IV model and employ the IV assumption. They characterize identified sets for structural parameters, applying random set theory to the level set of unobservables. In this section, we compare our approach with theirs. The main propose of the comparison is to...

work page 2017
[11]

(2009) analyze the choice of pregnant women in Mexico who choose sites for their obstetric care

or not ( D = 0).35 For example, Sosa-Rub´ ı et al. (2009) analyze the choice of pregnant women in Mexico who choose sites for their obstetric care. The treatment of 34Similar to Section 5.1, one can allow further heterogeneity by replacing Uj with Uj,D in this model. 35It is also possible to let µ be a function of individual-specific unobservables (e.g., ...

work page 2009
[12]

The set-valued control function is as in (2.11)

Allowing for flexibility is relevant in this context, as the insurance program may not be mandatory for the eligible or exclusive against the non-eligible. The set-valued control function is as in (2.11). Let V = (V0, V1) and let Uk = Qk(η; X, V ), k = 1, . . . , J. This model’s prediction is Y (η, D, X, V ; µ, F) ≡ n j ∈ Y : µj(D, X) ≥ inf V ∈Sel(V ) max...

work page 2010
[13]

(µ1, . . . , µJ , π, F) such that, for almost all (d, x, z), P0(Y ∈ A|D = d, X = x, Z = z) ≥ X B⊆A Fη n µjℓ(d, x) ≥ inf v∈V (d,x,z;π) max k̸=jℓ [µk(d, x) + Qk(η; x, v)] − Qjℓ(η; x, v) o ∩ n µjm(d, x) < inf v∈V (d,x,z;π) max k̸=jm [µk(d, x)+ Qk(η; x, v)]−Qjm(η; x, v) o , jℓ ∈ B, jm ∈ A\B , A ⊆ {1, . . . , J}. (B.9) As in the previous example, (B.9) jointly...

work page 2002
[14]

Accordingly, define Y (η, D, X, Z; µ, F, π∗) ≡ cl y ∈ Y : y = µ(D, X) + λ(X, V ) + η, V ∈ Sel(V (D, X, Z; π∗))

that E[Y |D, X, V ] = µ(D, X) + λ(X, V ) where λ(x, v) ≡ E[U |X = x, V = v]. Accordingly, define Y (η, D, X, Z; µ, F, π∗) ≡ cl y ∈ Y : y = µ(D, X) + λ(X, V ) + η, V ∈ Sel(V (D, X, Z; π∗)) . Then, by Theorem 2, µ(d, x) + λl(d, x, z) ≤ E[Y |D = d, X = x, Z = z] ≤ µ(d, x) + λu(d, x, z), where λl(d, x, z) ≡ inf v∈V (d,x,z;π∗) λ(x, v), λ u(d, x, z) ≡ sup v∈V (...

work page 2002
[15]

By Assumption 2, V is a measurable selection of V , and therefore Y is a measurable selection of Y (η, D, X, V ; µ, F)

By Assumptions 1, one may represent the outcome asY = µ(D, X, U) = µ(D, X, Q(η; X, V )). By Assumption 2, V is a measurable selection of V , and therefore Y is a measurable selection of Y (η, D, X, V ; µ, F). Therefore, the model’s prediction is summarized by Y ∈ Y (η, D, X, V ; µ, F), a.s. (C.1) By Assumption 2 (ii), V is a function of ( D, X, Z). Hence,...

work page 2020
[16]

Let B ≡ σ(D, X, Z) be the σ-algebra generated by ( D, X, Z). By As- sumptions 2 and 3, we may represent the model’s set-valued prediction by Y in (4.6), the random set of outcomes Y = µ(D, X)+ λD(X, V )+ ηD, where η = (ηd, d ∈ D) is conditionally mean independent of D. Y is integrable because its measurable selection Y is assumed to be integrable. Because...

work page 2020
[17]

Let F be a probability measure on (Ω , F)

space, and let F = FRdU ⊗ FRdD ⊗ FRdX ⊗ FRdZ be the product σ-algebra, where FE is the Borel σ-algebra over E. Let F be a probability measure on (Ω , F). Measurable maps (η, D, X, Z) are defined on this space. Consider a measurable rectangle A = Aη × AD,X,Z, where Aη ⊂ RdU and AD,X,Z ⊂ RdD × RdX × RdZ. Then, F(A|B) = Fη(Aη). By Assumption 4 and the constr...

work page 2020
[18]

Under the assump- tion that the underlying probability space is non-atomic, we may apply the convexification theorem (Molinari, 2020, Theorem A.2.)

Then, by construction, KI(d; θ) is the Aumann expectation of K(d; θ). Under the assump- tion that the underlying probability space is non-atomic, we may apply the convexification theorem (Molinari, 2020, Theorem A.2.). It ensures KI(d; θ) = E[K(d; θ)] is a convex closed set. Since φ is bounded, KI(d; θ) is a bounded closed interval. Again, by Theorem A.2....

work page 2020
[19]

Again, by Theorem 1.3.3 in Molchanov (2017), the conclusion follows

Then, for any x ∈ Y , the distance function ρ(x, Y (ω)) = inf{∥x − y∥, y ∈ Y (ω)} = inf{∥x − υn(ω)∥, n ≥ 1} (C.11) is a random variable in [0, ∞]. Again, by Theorem 1.3.3 in Molchanov (2017), the conclusion follows. Consider a random closed set X that is nonempty almost surely. A countable family of selections ξn ∈ Sel(X), n ≥ 1 is called the Castaing rep...

work page 2017
[20]

D.2 Confidence Intervals We outline how we construct confidence intervals using Kaido and Zhang (2025)

Similarly, the average treatment effect ASF(1 , xHIV ) − ASF(0, xHIV ) can be expressed as a function of θ. D.2 Confidence Intervals We outline how we construct confidence intervals using Kaido and Zhang (2025). With a slight abuse of notation, we write all observable variables ( D, X, Z) except the outcome as X in order to keep the notation below consist...

work page 2025
[21]

(D.4) Here, the function θ 7→ qθ is called the least-favorable-pair (LFP) based density

asˆθ1, where ˆP1 is the empirical (conditional) distribution of Yi.39 The restricted estimator ˆθ0 is constructed from S0, ˆθ0 ∈ arg max θ∈{θ′:φ(θ′)=φ∗} Y i∈S0 qθ(Yi|Xi). (D.4) Here, the function θ 7→ qθ is called the least-favorable-pair (LFP) based density. While we refer to Kaido and Zhang (2025) for details, we note that this density qθ is available i...

work page 2025
[22]

(D.5) T swap n (φ∗) is defined similarly to Tn(φ∗) while swapping the roles of S0 and S1

Define the cross-fit LR statistic by Sn(φ∗) ≡ Tn(φ∗) + T swap n (φ∗) 2 . (D.5) T swap n (φ∗) is defined similarly to Tn(φ∗) while swapping the roles of S0 and S1. Recall that φ(θ) ∈ R is the target object. We define a confidence interval by CIn ≡ φ∗ ∈ R : Sn(φ∗) ≤ 1 α . (D.6) In our application, we construct a grid of K = 200 equally spaced points over th...

work page 2025

[1] [1]

Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings,

Abadie, A., J. Angrist, and G. Imbens (2002): “Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings,” Econometrica, 70, 91–117. Abdulkadiro˘glu, A., J. D. Angrist, Y. Narita, and P. A. Pathak(2019): “Breaking Ties: Regression Discontinuity Design Meets Market Design,” Econometrica. Abdulkadiro˘glu, A.,...

work page 2002

[2] [2]

Control Functions and Simultaneous Equations Methods,

Blundell, R., D. Kristensen, and R. L. Matzkin (2013): “Control Functions and Simultaneous Equations Methods,” American Economic Review, 103, 563–569. Blundell, R. and R. L. Matzkin(2014): “Control functions in nonseparable simultaneous equations models,” Quantitative Economics, 5, 271–295. Blundell, R. and J. L. Powell (2003): “Endogeneity in nonparametr...

work page 2013

[3] [3]

Monte Carlo Confidence Sets for Identified Sets,

Chen, X., T. M. Christensen, and E. Tamer (2018): “Monte Carlo Confidence Sets for Identified Sets,” Econometrica, 86, 1965–2018

work page 2018

[4] [4]

Semiparametric estimation of structural functions in nonseparable triangular models,

Chernozhukov, V., I. Fern´andez-Val, W. Newey, S. Stouli, and F. Vella (2020): “Semiparametric estimation of structural functions in nonseparable triangular models,” Quantitative Economics, 11, 503–533. Chernozhukov, V. and C. Hansen (2005): “An IV model of quantile treatment effects,” Econometrica, 73, 245–261. Chernozhukov, V., H. Hong, and E. Tamer(200...

work page 2020

[5] [5]

Nonparametric estimation of sample selection models,

Das, M., W. K. Newey, and F. Vella (2003): “Nonparametric estimation of sample selection models,” The Review of Economic Studies , 70, 33–58. D’Haultfoeuille, X. and P. F´evrier (2015): “Identification of Nonseparable Triangular Models With Discrete Instruments,” Econometrica, 83, 1199–1210. D’Haultfœuille, X., S. Hoderlein, and Y. Sasaki (2021): “Testing...

work page 2003

[6] [6]

Sample selection bias as a specification error,

Heckman, J. J. (1979): “Sample selection bias as a specification error,” Econometrica, 153–161. Heckman, J. J. and B. E. Honor ´e (1990): “The Empirical Content of the Roy Model,” Econometrica, 58, 1121–1149. Heckman, J. J. and E. Vytlacil (2005): “Structural Equations, Treatment Effects, and Econometric Policy Evaluation,” Econometrica, 73, 669–738. Heck...

work page arXiv 1979

[7] [7]

Evaluating Public Programs with Close Substi- tutes: The Case of Head Start*,

Kline, P. and C. R. Walters (2016): “Evaluating Public Programs with Close Substi- tutes: The Case of Head Start*,” The Quarterly Journal of Economics , 131, 1795–1848. ——— (2019): “On Heckits, LATE, and Numerical Equivalence,”Econometrica, 87, 677–696. Levinsohn, J. and A. Petrin (2003): “Estimating Production Functions Using Inputs to Control for Unobse...

work page arXiv 2016

[8] [8]

Chapter 5 - Microeconometrics with partial identification,

Molinari, F. (2020): “Chapter 5 - Microeconometrics with partial identification,” in Hand- book of Econometrics, Volume 7A, ed. by S. N. Durlauf, L. P. Hansen, J. J. Heckman, and R. L. Matzkin, Elsevier, vol. 7 of Handbook of Econometrics, 355–486. Mourifi´e, I. (2015): “Sharp bounds on treatment effects in a binary triangular system,” Journal of Economet...

work page arXiv 2020

[9] [9]

When Should We (Not) Interpret Linear IV Estimands as LATE?

Rubin, D. B. (1978): “Bayesian Inference for Causal Effects: The Role of Randomization,” The Annals of Statistics , 6, 34–58. Shaikh, A. M. and E. J. Vytlacil (2011): “Partial identification in triangular systems of equations with binary dependent variables,” Econometrica, 79, 949–955. Smith, R. J. and R. W. Blundell(1986): “An exogeneity test for a simul...

work page internal anchor Pith review Pith/arXiv arXiv 1978

[10] [10]

They characterize identified sets for structural parameters, applying random set theory to the level set of unobservables

A Comparison with the IV Approach Chesher and Rosen (2017) applies their method to a single-equation IV model and employ the IV assumption. They characterize identified sets for structural parameters, applying random set theory to the level set of unobservables. In this section, we compare our approach with theirs. The main propose of the comparison is to...

work page 2017

[11] [11]

(2009) analyze the choice of pregnant women in Mexico who choose sites for their obstetric care

or not ( D = 0).35 For example, Sosa-Rub´ ı et al. (2009) analyze the choice of pregnant women in Mexico who choose sites for their obstetric care. The treatment of 34Similar to Section 5.1, one can allow further heterogeneity by replacing Uj with Uj,D in this model. 35It is also possible to let µ be a function of individual-specific unobservables (e.g., ...

work page 2009

[12] [12]

The set-valued control function is as in (2.11)

Allowing for flexibility is relevant in this context, as the insurance program may not be mandatory for the eligible or exclusive against the non-eligible. The set-valued control function is as in (2.11). Let V = (V0, V1) and let Uk = Qk(η; X, V ), k = 1, . . . , J. This model’s prediction is Y (η, D, X, V ; µ, F) ≡ n j ∈ Y : µj(D, X) ≥ inf V ∈Sel(V ) max...

work page 2010

[13] [13]

(µ1, . . . , µJ , π, F) such that, for almost all (d, x, z), P0(Y ∈ A|D = d, X = x, Z = z) ≥ X B⊆A Fη n µjℓ(d, x) ≥ inf v∈V (d,x,z;π) max k̸=jℓ [µk(d, x) + Qk(η; x, v)] − Qjℓ(η; x, v) o ∩ n µjm(d, x) < inf v∈V (d,x,z;π) max k̸=jm [µk(d, x)+ Qk(η; x, v)]−Qjm(η; x, v) o , jℓ ∈ B, jm ∈ A\B , A ⊆ {1, . . . , J}. (B.9) As in the previous example, (B.9) jointly...

work page 2002

[14] [14]

Accordingly, define Y (η, D, X, Z; µ, F, π∗) ≡ cl y ∈ Y : y = µ(D, X) + λ(X, V ) + η, V ∈ Sel(V (D, X, Z; π∗))

that E[Y |D, X, V ] = µ(D, X) + λ(X, V ) where λ(x, v) ≡ E[U |X = x, V = v]. Accordingly, define Y (η, D, X, Z; µ, F, π∗) ≡ cl y ∈ Y : y = µ(D, X) + λ(X, V ) + η, V ∈ Sel(V (D, X, Z; π∗)) . Then, by Theorem 2, µ(d, x) + λl(d, x, z) ≤ E[Y |D = d, X = x, Z = z] ≤ µ(d, x) + λu(d, x, z), where λl(d, x, z) ≡ inf v∈V (d,x,z;π∗) λ(x, v), λ u(d, x, z) ≡ sup v∈V (...

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[15] [15]

By Assumption 2, V is a measurable selection of V , and therefore Y is a measurable selection of Y (η, D, X, V ; µ, F)

By Assumptions 1, one may represent the outcome asY = µ(D, X, U) = µ(D, X, Q(η; X, V )). By Assumption 2, V is a measurable selection of V , and therefore Y is a measurable selection of Y (η, D, X, V ; µ, F). Therefore, the model’s prediction is summarized by Y ∈ Y (η, D, X, V ; µ, F), a.s. (C.1) By Assumption 2 (ii), V is a function of ( D, X, Z). Hence,...

work page 2020

[16] [16]

Let B ≡ σ(D, X, Z) be the σ-algebra generated by ( D, X, Z). By As- sumptions 2 and 3, we may represent the model’s set-valued prediction by Y in (4.6), the random set of outcomes Y = µ(D, X)+ λD(X, V )+ ηD, where η = (ηd, d ∈ D) is conditionally mean independent of D. Y is integrable because its measurable selection Y is assumed to be integrable. Because...

work page 2020

[17] [17]

Let F be a probability measure on (Ω , F)

space, and let F = FRdU ⊗ FRdD ⊗ FRdX ⊗ FRdZ be the product σ-algebra, where FE is the Borel σ-algebra over E. Let F be a probability measure on (Ω , F). Measurable maps (η, D, X, Z) are defined on this space. Consider a measurable rectangle A = Aη × AD,X,Z, where Aη ⊂ RdU and AD,X,Z ⊂ RdD × RdX × RdZ. Then, F(A|B) = Fη(Aη). By Assumption 4 and the constr...

work page 2020

[18] [18]

Under the assump- tion that the underlying probability space is non-atomic, we may apply the convexification theorem (Molinari, 2020, Theorem A.2.)

Then, by construction, KI(d; θ) is the Aumann expectation of K(d; θ). Under the assump- tion that the underlying probability space is non-atomic, we may apply the convexification theorem (Molinari, 2020, Theorem A.2.). It ensures KI(d; θ) = E[K(d; θ)] is a convex closed set. Since φ is bounded, KI(d; θ) is a bounded closed interval. Again, by Theorem A.2....

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[19] [19]

Again, by Theorem 1.3.3 in Molchanov (2017), the conclusion follows

Then, for any x ∈ Y , the distance function ρ(x, Y (ω)) = inf{∥x − y∥, y ∈ Y (ω)} = inf{∥x − υn(ω)∥, n ≥ 1} (C.11) is a random variable in [0, ∞]. Again, by Theorem 1.3.3 in Molchanov (2017), the conclusion follows. Consider a random closed set X that is nonempty almost surely. A countable family of selections ξn ∈ Sel(X), n ≥ 1 is called the Castaing rep...

work page 2017

[20] [20]

D.2 Confidence Intervals We outline how we construct confidence intervals using Kaido and Zhang (2025)

Similarly, the average treatment effect ASF(1 , xHIV ) − ASF(0, xHIV ) can be expressed as a function of θ. D.2 Confidence Intervals We outline how we construct confidence intervals using Kaido and Zhang (2025). With a slight abuse of notation, we write all observable variables ( D, X, Z) except the outcome as X in order to keep the notation below consist...

work page 2025

[21] [21]

(D.4) Here, the function θ 7→ qθ is called the least-favorable-pair (LFP) based density

asˆθ1, where ˆP1 is the empirical (conditional) distribution of Yi.39 The restricted estimator ˆθ0 is constructed from S0, ˆθ0 ∈ arg max θ∈{θ′:φ(θ′)=φ∗} Y i∈S0 qθ(Yi|Xi). (D.4) Here, the function θ 7→ qθ is called the least-favorable-pair (LFP) based density. While we refer to Kaido and Zhang (2025) for details, we note that this density qθ is available i...

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[22] [22]

(D.5) T swap n (φ∗) is defined similarly to Tn(φ∗) while swapping the roles of S0 and S1

Define the cross-fit LR statistic by Sn(φ∗) ≡ Tn(φ∗) + T swap n (φ∗) 2 . (D.5) T swap n (φ∗) is defined similarly to Tn(φ∗) while swapping the roles of S0 and S1. Recall that φ(θ) ∈ R is the target object. We define a confidence interval by CIn ≡ φ∗ ∈ R : Sn(φ∗) ≤ 1 α . (D.6) In our application, we construct a grid of K = 200 equally spaced points over th...

work page 2025