A Test for the Practical Relevance
of Endogeneity in Semiparametric
Distribution Regression

Worum geht es — und warum ist das relevant?

Statistische Signifikanz ≠ Praktische Relevanz

Beide Diagramme zeigen dieselbe gemessene Verzerrung d̂ = 0,026. Der Unterschied liegt allein in der Entscheidungsschwelle. (Monte-Carlo-Illustration: n = 1.000, ρ = 0,5.)

Abstract

We develop a test for the practical relevance of endogeneity in semiparametric distribution regression (DR). Classical tests ask whether the endogenous regressor is exactly exogenous — a question that is uninformative in large samples because even negligible deviations are detected with probability approaching one. We instead test whether correcting for endogeneity changes the estimated conditional distribution by more than an economically meaningful threshold Δ > 0.

The object of interest is the L²-distance d(x,y₂) = ‖F𝑁(·|x,y₂) − F𝑉(·|x,y₂)‖₂ between the ordinary DR estimator and the control-function IV-DR estimator of Wied (2024). The resulting test statistic is asymptotically pivotal: its limiting distribution under the boundary d = Δ is that of a ratio of Brownian-motion functionals, whose quantiles can be obtained by simulation once and for all. The test is consistent and detects local alternatives at rate n⁻½.

A central challenge in applied microeconometrics is deciding whether to use instrumental variables (IV) estimation. IV is consistent under endogeneity but less precise than OLS. The conventional approach relies on Hausman-type tests of exact exogeneity: if the null is rejected, switch to IV. This creates a well-known trap.

This perspective belongs to the equivalence-testing and minimum-detectable-effect literatures (Wellek 2010; Lakens 2017), but has received little attention in distributional models. This paper fills that gap for semiparametric distribution regression.

Distribution Regression

Distribution regression (DR), introduced by Foresi & Peracchi (1995) and developed by Chernozhukov, Fernández-Val & Melly (2013), models the entire conditional distribution function of an outcome Y given covariates X:

where Λ is a known link function (e.g. probit or logit) and β(y) varies with the threshold y. Each y-binary regression 1{Yₐ ≤ y} = Λ(Xₐ'β(y)) + Uₐ(y) is estimated separately, yielding a flexible nonparametric description of the full conditional distribution. Unlike quantile regression, DR handles mass points and non-smooth distributions naturally, and avoids bandwidth choices.

Structural Model with Endogeneity

When the regressor Y₂ is endogenous — correlated with the latent error U(y) — the standard DR estimator is inconsistent for the structural conditional distribution. The paper adopts a triangular simultaneous-equations structure:

Endogeneity enters through the error decomposition U(y) = α₁(y)V + α₂(y)ε(y), where α₁(y) ≠ 0 induces correlation between the structural error and the first-stage residual V. Using instruments Z and the control-function approach of Rivers & Vuong (1988), Wied (2024) proposes a three-step IV-DR estimator that recovers the structural distribution.

Three-Step IV-DR Estimation

The Endogeneity Distance Functional

The practical endogeneity distance at covariate profile (x,y₂) is the L²-norm of the pointwise difference between the naive and IV-corrected conditional distribution functions over the outcome interval I:

Hypotheses

Rejecting H₀ means that correcting for endogeneity changes the estimated conditional distribution by more than Δ over I. Failing to reject means endogeneity is practically irrelevant at that covariate profile, regardless of statistical significance.

Threshold Calibration

The threshold Δ must be chosen before observing the data on substantive grounds. Two natural approaches:

Test Statistic and Decision Rule

The test statistic is the full-sample integrated squared distance:

The self-normalizing term V̂_n(x,y₂) is constructed from sequential (partial-sample) versions of both estimators, adapting the self-normalization device of Dette, Möllenhoff & Wied (2025) to the one-sample correlated setting. The quantile q_1−α is that of a Brownian-motion ratio W = B(1) / (∫_ε¹(B(t)/t − B(1))² dt)^1/2 — simulatable to arbitrary precision with q_0.95 ≈ 1.755 for ε = 0.1.

The theoretical contribution requires combining three distinct bodies of asymptotic theory, because the naive and IV-DR estimators are estimated from the same data — creating a non-trivial correlation structure absent in the independent two-sample case.

Simulations use a triangular model with one exogenous regressor X, one instrument Z, one endogenous regressor Y₂, and a continuous outcome Y. The endogeneity strength is controlled by ρ ∈ [0,1) — when ρ = 0, Y₂ is exogenous; when ρ > 0, Y₂ is endogenous. Distribution regression is implemented using a probit link, evaluated at the covariate profile (x, y₂) = (0, 0). Sample size n = 1,000; nominal level α = 0.05.

Interior Null: Endogeneity is Present but Not Practically Relevant (Δ = 0.05)

Table 1 shows that even at ρ = 0.5 — substantial correlation between structural error and first-stage residual — the estimated distance d̂ = 0.026 remains below the relevance threshold Δ = 0.05, and the test correctly does not reject.

Power as the Relevance Threshold Varies (ρ = 0.5)

Table 2 shows power as Δ varies. Rejection probability decreases as Δ increases — the test is sensitive to the threshold rather than merely to the presence of nonzero endogeneity. The transition occurs near Δ ≈ 0.027, consistent with the estimated distance d̂ ≈ 0.027.

Boundary Behavior (ρ = 0.5, Δ = 0.027)

When Δ is set close to the true distance d(x,y₂) ≈ 0.027, rejection rates converge to the nominal level α = 0.05 for both n = 1,000 and n = 2,000, confirming the theoretical prediction of Theorem 5.7.

The test is applied to Card's (1995) returns-to-schooling data (n = 3,010), where education is treated as potentially endogenous and proximity to a four-year college (nearc4) serves as the excluded instrument. Controls include experience, experience squared, Black, SMSA, and South indicators.

Benchmark: Mean-Based Analysis Finds No Evidence of Endogeneity

The standard linear Wu–Hausman test fails to reject exogeneity (p-value = 0.215), despite the OLS estimate (0.0740) and the IV estimate (0.1323) differing substantially in magnitude. This motivates asking a distributional question: even if mean endogeneity is not statistically detectable, does the IV correction shift the distribution of log wages by a practically relevant amount?

Profile-Specific Relevant-Endogeneity Test

The test is applied at three covariate profiles representing low (educ=10, exper=4), middle (educ=13, exper=8), and high (educ=17, exper=15) education/experience workers. The findings are strikingly heterogeneous.

Estimated Conditional Distribution Functions

Figures 1 and 2 plot the estimated conditional CDFs of log wages for naive DR (solid) and IV/control-function DR (dashed) at the three covariate profiles. The curves are closest for the middle profile; the low and high profiles show visibly larger discrepancies — especially the high profile, where the IV correction shifts the distribution substantially.

Figure 1

Naive and IV-DR Estimates — Baseline Specification

Baseline controls (experience, experience², Black, SMSA, South). Solid: naive DR; dashed: IV-DR. The high-education profile (right panel) shows the largest gap.

Figure 2

Naive and IV-DR Estimates — Full Specification

Full controls (baseline + smsa66 + region indicators). Qualitative pattern persists: middle profile closest, low and high profiles most discrepant. Formal rejection for the high profile disappears with the richer control set.