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Learning-Enabled Estimation: Tight Characterizations under Sample Selection Biases

Vikram Kher, Jane H. Lee, Anay Mehrotra, Manolis Zampetakis

arXiv:2609.38608v1cs.LGstat.ML

TL;DR

Regression from selected samples is difficult when outcomes are observed only after an outcome-dependent filter, and the conditions for identifying the regression function have lacked a complete characterization. The paper characterizes when identification is possible and establishes finite-sample estimation guarantees under stronger conditions; it also shows estimation can fail even when identification remains possible.

  • Problem

    Outcome-dependent selection can bias regression on observed samples, leaving the conditions for learning the regression function from selected data to be characterized.

    Source: §1 · p. 4, §1.2.1 · p. 6

  • Method

    The authors model regression, selection, covariate, and noise distributions through a joint concept class, then characterize identification and derive finite-sample estimation guarantees under a stronger condition.

    Source: §1.1 · p. 5, §4 · p. 14

  • Results

    The paper characterizes when the joint concept class enables identification and establishes finite-sample estimation under a strengthened condition; its proof reduces target discrepancy to total variation error.

    Source: §4 · p. 14, §4.2.1 · p. 19

  • Takeaways & Limitations

    The conditions apply to classic and new selection settings beyond those covered by existing sufficient conditions.

    Source: §5 · p. 24

    When expected selection probability is zero, finite samples contain no uncensored outcomes, so estimation can be impossible even when identification remains possible.

    Source: §4.2.2 · p. 20

Abstract

from arXiv · show

When can we learn from biased samples? We study regression when outcomes are observed only after passing through selection filters that depend on both covariates and outcomes themselves, a ubiquitous challenge spanning clinical trials with patient dropout, labor markets with self-selection, and auctions with strategic entry. Ignoring such selection yields systematically biased conclusions with real-world consequences. This challenge has a long history in econometrics and statistics, starting with Heckman's seminal two-stage model and followed by numerous generalizations. While these works provide various sufficient conditions for identification, a complete characterization of when such regression is possible has remained elusive. In this work, we provide a characterization for when regression is possible in the presence of sample selection bias. Our results establish the minimal assumptions required on the functional forms of selection processes under which regression remains possible, which are particularly relevant in modern settings where selection mechanisms are increasingly complex and opaque. As a corollary of our characterization, we show that there are settings where the regression function can be identified even when the selection filter itself cannot. This observation already goes beyond the ``estimate selection filter, then debias regression'' paradigm that is followed by virtually all existing approaches. Under natural strengthenings of our identification conditions, we also establish finite-sample estimation guarantees with explicit convergence rates and provide oracle-efficient algorithms. This yields the first general-purpose estimation method for this broad class of selection problems. Finally, we explore the implications of our results for several well-studied econometric settings with complex selection mechanisms such as auctions with entry costs and labor markets.

1 Introduction

The introduction frames regression under sample-selection bias, where outcomes are observed only after filters that may depend on covariates and outcomes. It presents a general framework and necessary-and-sufficient identification condition, relating it to established selection models and extending the discussion to censoring, truncation, and identification across fields.

1 Introduction

  • Classical selection models require application-specific assumptions, while learning the filter typically depends on clean auxiliary data that may be unavailable.Misspecification can severely bias estimates, and first-stage estimation errors can also disrupt second-stage methods.Source: §1 · p. 3

1.1 Our Contributions

  • The paper gives a necessary-and-sufficient condition for identifying the regression function from censored data and finite-sample estimation guarantees under stronger conditions.The estimator’s computational bottleneck reduces to a single density-estimation oracle call.Source: §1.1 · p. 5
  • Applications show regression can be identifiable when selection is not, extend Das, Newey, and Vella’s conditions, and characterize extended Roy and auction models.The paper also reports identification for outcome-dependent truncation mechanisms in physical-science pipelines, despite their violating the prior framework’s requirements.Source: §1.1 · pp. 5–6

1.2.1 Heckman’s Sample-Selection Model

  • In Heckman’s model, correlated outcome and selection errors make observed samples nonrandom, so ordinary least squares is biased.Heckman’s two-step method estimates the selected-sample error mean and uses it as an additional regressor.Source: §1.2.1 · pp. 6–7

1.2.2 A Generalization of Heckman’s Model

  • Lee’s model allows known, continuous, non-Gaussian error marginals; a monotone probability-integral transformation preserves selection and enables Heckman’s two-step approach.The paper states that its identification condition also holds for Lee’s model.Source: §1.2.2 · pp. 7–8

1.2.3 Ahn–Powell’s Sample-Selection Model

  • Ahn and Powell identify linear coefficients under an index restriction: selected-error bias is a smooth function of propensity, enabling pairwise differencing.A non-singularity condition ensures sufficient variation in covariates after conditioning on selection probability.Source: §1.2.3 · pp. 8–9

1.2.4 Censoring on Outcomes: Tobit Model and Variants

  • Tobit and related models represent outcome-based censoring through selection functions such as thresholds or probit/logit probabilities.Deterministic truncation can restrict observation to a specified survival set.Source: §1.2.4 · p. 9

1.3 Related Work

  • The work provides the previously missing necessary-and-sufficient condition for identifying regression under sample selection, extending beyond sufficient conditions in earlier work.Heckman introduced the canonical model and two-step correction under strong parametric assumptions; later work relaxed assumptions in different directions, including Lee, Ahn and Powell, and Das, Newey, and Vella.Source: §1.3 · p. 10
  • The characterization covers outcome-only censoring and determines when regression is identifiable under unknown or joint (x, y)-dependent censoring, while joint-dependence makes truncation identification non-trivial.In truncation, when D = 0, neither X nor Y is observed; the paper suggests combining its characterization with recent algorithmic techniques for efficient estimators.Source: §1.3 · p. 10
  • The paper contributes a necessary-and-sufficient identification condition for regression under outcome-dependent selection, connecting identification questions across econometrics, computer science, and statistics.Related examples include auction models, self-selection models, language identification from positive data, and mixture and latent-variable models.Source: §1.3 · p. 10

2 Preliminaries

The preliminaries establish notation for distributions, random variables, concept classes, and vectors. They also define the paper’s conventions for densities, support, and distances between distributions.

2 Preliminaries

  • Distributions are written with calligraphic letters; the section defines density and mass notation, support, almost-sure conventions, Gaussian notation, and total variation distance.The underlying measure is Lebesgue measure on R unless otherwise specified.Source: §2 · p. 11
  • Capital letters denote random variables, ξ denotes noise, 1{E} denotes an event indicator, and X⊥Y denotes independence; vector norms and inner products use standard ℓ2, ℓ∞, and sum notation.For vectors v and w, the inner product is ∑i viwi.Source: §2 · p. 11
  • Concept classes are sets of functions or distributions that represent model families, and a ⋆ marks the ground-truth function or distribution.Examples include Gaussian mixtures, generalized linear models, diffusion-model distributions, and neural-network regression functions.Source: §2 · p. 11

3 Model of Regression with Sample-Selection Bias

The model treats regression outcomes as observed only when a potentially unknown selection function retains a sample, with function and distribution components drawn from specified classes. It formalizes the conditions for identifying the regression function and estimating it from censored samples.

3 Model of Regression with Sample-Selection Bias

  • The regression target is f⋆ in a function class, with noise from Q⋆ and covariates from D⋆; outcomes are revealed only with probability s⋆(x, y), otherwise only x is observed.The selection function and the feature and noise distributions are unknown members of their respective concept classes.Source: §3 · pp. 11–12
  • The framework permits broad classes of selection functions, including unknown parametric forms and classes such as neural networks or decision trees.The formulation also captures the Heckman and Lee selection models.Source: §3 · p. 12
  • The model’s distributions of features and noise belong to specified classes, which can include families such as Gaussian mixtures, heavy-tailed distributions, or nonnegative distributions.The framework assumes homoskedastic noise; extending it to heteroskedastic settings is possible but yields more complex characterizations.Source: §3 · p. 12

Formal Model and Main Questions

  • The formal sampling process draws covariates and noise, forms the outcome, applies the selection probability, and reveals the outcome only for retained samples.Admissible parameter tuples may impose joint restrictions across components, and realizability means the censored data distribution arises from such a tuple.Source: Formal Model and Main Questions · p. 13
  • The central questions are which assumptions on the joint class permit identification of f⋆ from the censored distribution and estimation with vanishing expected prediction error.Estimation implies identification, and examples show estimation can be strictly harder.Source: Formal Model and Main Questions · p. 14

4 Our Results

The results characterize when the regression function is identifiable from censored data and establish finite-sample estimation guarantees under a quantitative strengthening of the identification condition. They also develop an estimation analysis, including a total-variation identity, and discuss computational implementation and limitations.

4.1 Characterization of the Identifiability of f ⋆

  • Under a noise-density assumption, Θ identifies f⋆ up to D⋆-almost-everywhere equivalence if and only if Condition 1 holds.Condition 1 requires that admissible tuples with different regressors and the same covariate distribution induce observed-outcome kernels that differ on a set of positive measure.Source: §4.1 · p. 15
  • The characterization can identify f⋆ even when the selection filter s⋆ is not identifiable, and it yields characterizations for selective-entry auctions and an extended Roy labor-market model.The paper presents this as the first characterization of when regression is possible under sample selection bias.Source: §4.1 · pp. 15–16

4.2 Estimation of f ⋆from Finite Samples

  • Condition 2 strengthens identification by requiring quantitative separation: ε-separated regressors must differ in covariate distributions or observed-outcome kernels by a rate-controlled amount.Unlike Condition 1, it imposes a quantitative lower bound on the discrepancy between the induced observations.Source: §4.2 · p. 18

4.2.1 Technical Overview: Challenges in Estimating f ⋆and Our Approach

  • 5 dTV(Cbθ, C⋆) bounds dTV(bD, D⋆) + dD⋆(π⋆, bπ), enabling estimation without directly learning the unknown outcome kernel.The method selects a candidate censored distribution from a cover using Yatracos’s finite-sample procedure, then converts its total-variation error into the target discrepancy bound.Source: §4.2.1 · p. 19

4.2.2 Our Estimation Result

  • Condition 2, together with regularity assumptions and E[s⋆(X, Y)] ≥ η, yields a finite-sample estimator of f⋆ with a guarantee holding with probability at least 1 − δ.The guarantee’s sample complexity depends on the separation rate R(ε) and covering numbers; the expected observation-rate lower bound is necessary for finite-sample estimation.Source: §4.2.2 · pp. 19–20
  • Finite-sample estimation can fail when the expected observation probability is zero, even though identification may remain possible from censored densities on measure-zero sets.This distinguishes finite-sample learnability from population identification.Source: §4.2.2 · p. 20
  • For the ordered pair (θ⋆, bθ), the contrapositive of Condition 2 gives E_{X~D⋆}|bf(X)−f⋆(X)|<ε; substituting (6) into the choice of n yields the stated sample-complexity bound.Source: §4.2.2 · p. 22

Proof of Lemma 4.7

  • Changing only the covariate marginal while preserving the conditional law leaves total variation exactly equal to the distance between the covariate marginals.This identity supports the comparison between the selected censored law and an auxiliary law using the true covariate distribution.Source: §4.2.2 · pp. 22–23

4.2.3 Oracle Efficiency and Implementation Considerations

  • The estimator is sample-efficient but may be computationally slow for large finite covers; its computational bottleneck is a single density-estimation oracle call.Unlike the method of Kontonis, Tzamos, and Zampetakis, it does not require an auxiliary optimization for each candidate.Source: §4.2.3 · pp. 23–24
  • In parametric settings, maximum likelihood estimation on censored data can implement density estimation and jointly learn the selection correction and outcome regression in the Heckman model.This differs from Heckman’s two-step procedure, which estimates the probit selection equation first and then adds the inverse Mills ratio to the regression.Source: §4.2.3 · p. 24

5 Implications of Our Results

The section applies the framework to classic and newer sample-selection settings, including extended Roy models and entry-cost auctions, to show where Conditions 1 and 2 enable identification and estimation beyond earlier sufficient conditions, and where additional structure is needed. It also gives instrumental-variable conditions that imply the estimation condition with an explicit rate function.

5.1 f ⋆Can Be Identified Even When s⋆Is Not Identifiable

  • Condition 1 can identify f⋆ even when s⋆ is unidentifiable, so recovering the selection function first is not always necessary.The separation uses simple component classes and one-sided noise supported on R≥0; non-identifiability persists under model enlargement and despite selectors bounded below by η.Source: §5.1 · p. 25

5.2.1 Heckman’s Sample-Selection Model

  • For Heckman’s model, Condition 1 holds when the covariate distribution has support with nonempty interior, establishing identification for its specified concept classes.Source: §5.2.1 · pp. 27, 29

5.2.2 Ahn–Powell’s Sample-Selection Model

  • For the Ahn–Powell class, Condition 1 holds when the instrument-based matrix is well-defined and nonsingular.The class has a linear outcome function and a smooth correction function of the propensity score.Source: Suppose that the matrix · pp. 29–30, §5.2.2 · p. 29

5.2.3 Censoring on Outcomes: Tobin Model and Variants

  • With full-dimensional covariate support, outcome-only censoring satisfies Condition 1; the section also considers censoring by unknown survival sets depending jointly on covariates and outcomes.Source: §5.2.3 · p. 31
  • Condition 1 need not hold for arbitrary measurable survival sets, so additional structure on the survival sets is necessary.Recent work develops algorithms and identifiability results under complexity assumptions such as low Gaussian surface area.Source: §5.2.3 · p. 31

5.3 Revisiting the Identification Condition of Das, Newey, and Vella [DNV03]

  • The Das–Newey–Vella model embeds in the framework under its stated assumptions, and its joint concept class satisfies Condition 1.The embedding result assumes independent noise with a density; the paper also makes explicit support conditions it says the earlier proof implicitly requires.Source: §5.3 · pp. 32–33
  • Outcome-dependent survival-set truncation can remain identifiable under Condition 1 even when the Das–Newey–Vella scalar-index restriction fails.The example allows selection rules that record nontrivial outcome ranges at each feature value; the propensity score need not summarize their selection effects.Source: §5.3 · p. 33

5.4 Bounded-Friction Participation Costs

  • Bounded-friction participation selectors satisfy Condition 1 despite allowing joint covariate-outcome frictions, equal propensities with different biases, and locally nonmonotone participation.The stated selector uses a positive bounded responsiveness parameter and bounded friction; these examples violate scalar-index and monotone-entry restrictions while remaining identifiable.Source: §5.4 · pp. 34, 36

5.5 Characterizations of Related Models: Extended Roy and Auctions with Entry Costs

  • Condition 1 yields necessary and sufficient identifiability conditions for extended Roy and auction-entry models when their noise components admit densities.Previously, only sufficient conditions were known for these models.Source: §5.5 · p. 36

5.5.1 Extended Roy Model

  • Each regime of the extended Roy model reduces to a sample-selection regression problem, so each potential-outcome regression can be learned from its chosen-outcome subsample.The paper explicitly gives the reduction for regime 1 and states that an analogous result holds for regime 0.Source: §5.5.1 · p. 37
  • In the extended Roy model, Condition 1 applied to admissible regime-specific parameter tuples characterizes identification of f⋆1 or f⋆0 without generally requiring Lee and Park’s monotonicity conditions.The observed triples yield a censored outcome by setting Y(1) to ∅ for nonparticipants, matching Definition 4’s structure.Source: §5.5.1 · p. 38

5.5.2 Auctions with Entry Costs

  • Entry-cost auctions create selection because only entrants’ bids are observed, and Condition 1 characterizes value-regression identification after entry is modeled as selection.The characterization weakly generalizes Gentry and Li’s sufficient conditions based on exclusion restrictions and monotonicity.Source: §5.5.2 · p. 38

5.6 Instrumental Variables and the Estimation Condition

  • Index, overlap, moment, rank, and anti-concentration conditions suffice to satisfy Condition 2 with an explicit rate function for linear regressors.This provides a quantitative version of instrumental-variable intuition behind semi-parametric selection estimators such as Ahn and Powell.Source: §5.6 · pp. 38, 40, 42

A Characterization of Identifiability of s⋆

This section characterizes when the selection rule itself can be recovered from censored data. Under density assumptions, selector identifiability holds exactly when Condition 3 is satisfied.

A Characterization of Identifiability of s⋆

  • The selection rule is a primary object in applications such as bioinformatics and astronomy, where detection or censoring mechanisms are also of interest.These settings motivate studying selector identifiability in addition to recovering the regression signal.Source: A Characterization of Identifiability… · p. 49
  • Condition 3 requires that distinct selectors induce different observed kernels for every allowed regressor, noise distribution, and covariate distribution.The kernels are (x, y) ↦ s(x, y)Q(y − f(x)); they must differ on a set of positive (D ⊗ ν)-measure.Source: A Characterization of Identifiability… · p. 49
  • With densities for all distributions in D and Q, every selector is identifiable from censored data if and only if Condition 3 holds.The proof uses equality of censored laws to compare the observable conditional densities and show that unequal selectors would violate Condition 3.Source: A Characterization of Identifiability… · pp. 49–51

B Details Omitted from Proof of Theorem 4.3 (Cover Size)

This section constructs a finite cover of the censored distributions induced by the component classes. The cover supports finite-sample estimation by reducing candidate selection to a finite family of laws.

B Details Omitted from Proof of Theorem 4.3 (Cover Size)

  • A finite cover of the component classes yields a finite candidate family of censored laws that can be searched using standard finite-class distribution-selection routines.The section identifies this cover as the main combinatorial input to the finite-sample estimation result.Source: B Details Omitted from Proof of… · pp. 51, 55
  • The construction assumes finite covers for F and S in L1 norms, for D and Q in total variation, smoothness conditions, and translation-Lipschitz noise densities.It also assumes the distributions Pf,D,Q are σ²-smooth with respect to µ.Source: B Details Omitted from Proof of… · p. 52
  • The censored-law family admits a finite total-variation ε-cover whose size is bounded by the product of the component-class covering numbers.The proof combines covers for F, S, D, and Q, then controls both the non-null density and the null outcome.Source: B Details Omitted from Proof of… · pp. 51, 52, 55

C Omitted Details from Section 5.1

The omitted details construct an example in which the regression function remains identifiable although the selection rule is not. The construction uses two noise distributions and a selector family with distinct constant and step-shaped rules.

C Omitted Details from Section 5.1

  • The example uses a uniform noise distribution on [0, 1] and a piecewise-constant alternative with density η/c on [0, 1/2].Here c is set to (1 + η)/2; the alternative assigns density 1/c on the remaining interval.Source: C Omitted Details from Section 5.1 · p. 55
  • The construction shows non-identifiability of the selector alongside identifiability and estimation of the regression function under the stated class conditions.The selection and noise classes may contain the constructed families, while regression identifiability and estimation hold when each noise distribution has (0, 1) in its support and F admits a cover.Source: C Omitted Details from Section 5.1 · p. 55

C.1 Proof of Theorem 5.1

Theorem 5.1 gives classes where the regression function is identifiable despite non-identifiability of the selection rule. The proof establishes these properties through separate conditions on the component classes.

C.1 Proof of Theorem 5.1

  • There are component classes satisfying Condition 1 for which the selection function is not identifiable from realizable censored distributions.The theorem’s proof constructs these classes using a nonempty regressor class, selectors bounded between η and 1, and noise distributions supported on nonnegative values and positive on (0, 1).Source: §C.1 · p. 56
  • Condition 1 holds because distinct regressors produce observed kernels that differ on a set of positive measure.The proof selects covariates where one regressor is larger, then uses outcomes between the regressor values to obtain positive kernel mass for one tuple and zero for the other.Source: §C.1 · p. 56
  • Condition 3 fails because distinct selectors and noise distributions can yield identical observed kernels while the regressor remains fixed.The construction uses s(x, y)Q(t) = s′(x, y)Q′(t) across t outside [0, 1], [0, 1/2], and (1/2, 1], so the selectors differ despite matching kernels.Source: §C.1 · pp. 56–57
  • Theorem A.1 then implies that some selector in Sη is not identifiable from censored data.This follows from the failure of Condition 3 for the constructed classes.Source: §C.1 · pp. 56–57

C.2 Finite-Sample Version of Theorem 5.1

The construction gives a class satisfying the estimation condition with a linear rate even though its selection function is not identifiable. The proof establishes the regularity and overlap conditions needed for Theorem 4.3 and verifies the class’s covering numbers.

C.2 Finite-Sample Version of Theorem 5.1

  • The class satisfies Condition 2 with a linear rate, yet its selection function is not identifiable from realizable censored distributions.The regularity and overlap hypotheses needed to invoke Theorem 4.3 also hold.Source: §C.2 · pp. 57, 60
  • The classes (F, D, Q) satisfy the regularity conditions in Assumption 1 with respect to the product measure μX × μY.Here μX is D⋆ and μY is uniform on [−BR, BR + 1].Source: §C.2 · p. 59
  • The regressor class F admits a ζ-cover in the supremum norm of size at most (1 + 2BR/ζ)^d.The finite-cover lemma obtains this bound from a net of the Euclidean ball of parameters.Source: §C.2 · pp. 58, 61
  • The selection class S and noise class Q each have two elements, while the distribution class D is a singleton.Their covering numbers are therefore 2 and 1, respectively, at every scale.Source: §C.2 · pp. 57, 61

D.1 Proof of Lemma 5.6

The proof constructs an observationally equivalent model by defining its selection function as the probability of selection conditional on the original outcome and covariate. Under the stated restrictions and density assumption, the constructed tuple belongs to ΘDNV.

D.1 Proof of Lemma 5.6

  • Setting s⋆(x, y) to the original instance’s conditional probability of D = 1 given X = x and Y⋆ = y defines the selector for the constructed model.The constructed model uses parameters (f⋆, s⋆, DX, Q⋆).Source: §D.1 · p. 61
  • Because ξ is independent of X, the constructed model matches the original joint distribution of (X, Y⋆) and the conditional distribution of D given (X, Y).The two models therefore have the same joint distribution and the same observations.Source: §D.1 · p. 61
  • The restrictions in Definition 8 and Assumption 2, together with the assumed density of Q⋆, place the constructed tuple in ΘDNV.Source: §D.1 · p. 61

D.2 Proof of Theorem 5.7

The proof shows that agreement of observed kernels forces the regressors to agree. It first equates propensity scores and selected conditional means, then uses the rank condition and the common zero anchor to eliminate any constant difference.

D.2 Proof of Theorem 5.7

  • Agreement of observed kernels implies equality of propensity scores almost everywhere, which extends throughout int(X) by continuity and supp(D) = X.The proof denotes the common propensity score by p.Source: §D.2 · p. 61
  • The selected conditional means then yield g1(w) + λ1(p(w, z)) = g2(w) + λ2(p(w, z)) throughout int(X).This follows from equality of the means almost everywhere and the index restriction and continuity.Source: §D.2 · p. 61
  • The rank condition makes the regressor difference constant, and the common zero anchor forces that constant to zero, so f1 = f2 on X.Differentiating first with respect to z and then w yields the constant-difference conclusion.Source: §D.2 · p. 61

D.3 Proof of Lemma 5.9

The proof gives a counterexample to the index restriction: both covariate values have the same propensity, but their selected conditional means differ in sign.

D.3 Proof of Lemma 5.9

  • The construction takes f⋆ ≡ 0, ξ distributed as N(0, 1), and a selector that selects ξ above 1 for x > 0 and below −1 otherwise.Source: §D.3 · p. 62
  • By symmetry, both x values in {0, 1} have propensity p(x) = Pr(ξ > 1).Source: §D.3 · p. 62
  • Yet the selected conditional mean is at least 1 for X = 1 and at most −1 for X = 0, so it cannot be a function of p(X).Thus the index restriction fails in this construction.Source: §D.3 · p. 62

D.4 Proof of Lemma 5.10

The proof establishes that the observed kernels differ on a set of positive measure whenever the outcome functions differ on a set of positive D⋆-measure.

D.4 Proof of Lemma 5.10

  • A covariate value with positive D⋆-mass has different outcome values under f and f′.Because X = {0, 1}, such a value exists when f and f′ differ on a set of positive D⋆-measure.Source: §D.4 · p. 62
  • If the selectors differ on a positive-ν set, the observed kernels differ there because the Gaussian density is positive everywhere.Thus ν(Ax) > 0.Source: §D.4 · p. 62
  • If the selectors agree ν-almost everywhere, their shared selector equals one on a nontrivial open interval Ix.This follows from the definition of ST.Source: §D.4 · p. 62
  • The distinct Gaussian outcome densities agree at only one point, so the kernels differ almost everywhere on Ix and ν(Ax) > 0.The crossing point is (f(x) + f′(x))/2.Source: §D.4 · p. 62

E OLS-Based Method with Sub-Gaussian Noise with (Approximately) Known s⋆

The section develops an OLS-based estimator for linear outcome functions under pointwise overlap, with finite-sample guarantees under sub-Gaussian noise and bounded covariates. It also examines selection-function misspecification and gives asymptotically valid confidence regions under weaker noise and covariate-moment conditions.

OLS-Based Method

Finite-Sample Results

  • Under bounded covariates, the selection-weighted noise remains sub-Gaussian, with projected variance proxy L²σ²/α².The transformed noise is also sub-Gaussian conditional on D = 1, with variance proxy σ²/α².Source: E OLS-Based Method with Sub-Gaussian… · pp. 64–65
  • Theorem E.3 gives a finite-sample error bound for the OLS estimator with probability at least 1 − δ when n ≥ n0.The bound assumes sub-Gaussian noise and bounded covariates; its threshold depends on L, α, λmin(M), and δ.Source: E OLS-Based Method with Sub-Gaussian… · p. 65

Misspecified Selection Function

  • The misspecified-selection analysis bounds population target bias, while the simpler bound based on Assumption 6 is weaker because it depends on 1/α.The population-bias result does not itself bound finite-sample error; estimating the selection function on the same sample also requires accounting for first-stage dependence.Source: E OLS-Based Method with Sub-Gaussian… · p. 69

Asymptotic Distribution

  • The asymptotic confidence-region result assumes fixed dimension, known selection function with overlap, finite noise variance, and E∥X∥⁴ < ∞.Under these conditions, the paper gives an asymptotically valid (1 − δ) confidence region for β⋆.Source: E OLS-Based Method with Sub-Gaussian… · pp. 69–71
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