Bunching and Endogeneity: Testing, Identification, and Correction
At a boundary mass point, nearby observed treatment values can represent very different people. The atom may pool an entire range of latent choices, whereas positive treatment approaching the boundary selects a narrow range. The test in (Caetano 2015) compares their outcome means to detect a consequence of selection under maintained structural continuity.
1. The testable implication
Let \(Y=Y(T)\), with a mass at \(T=0\) and positive treatment values arbitrarily close to zero. Conditional on covariates \(X=x\), define
Proposition 1. Under the paper’s support, continuity, and integrability conditions, \(\Delta(x)\) is identified. Conditional exogeneity of treatment, together with continuity of the structural mean response at zero, implies \(\Delta(x)=0\). A nonzero discontinuity therefore rejects this joint implication.
The test rejects a joint implication of exogeneity and structural continuity. Interpreting the jump as evidence of endogeneity requires maintaining continuity, since a discontinuous causal effect could also create it.
2. Decompose the boundary comparison
For small \(t>0\),
Add and subtract the untreated potential-outcome mean among those with positive treatment. The first part is
and the second part is
Continuity and domination make the causal difference vanish as \(t\downarrow0\). The second term remains: it compares untreated potential outcomes for people approaching the boundary with those pooled at the atom. This is the selection contrast.
For smoking and birthweight, for example, the maintained claim would be that an arbitrarily small amount of smoking has an arbitrarily small causal effect. The test then asks whether nonsmokers and people smoking arbitrarily little nevertheless have different expected birthweights after conditioning on the chosen controls.
3. Why the atom is informative
A censored latent choice \(T=\max(T^*,0)\) maps an entire range of latent values to the atom, while \(T=t>0\) reveals \(T^*=t\). The two groups can therefore have different unobservables even when observed treatment values are arbitrarily close.
A regression containing \(1\{T=0\}\) and a sufficiently flexible continuous function of \(T\) estimates this boundary contrast under the required specification and support conditions. An inflexible polynomial can create an apparent jump by extrapolation error, so the continuous-side fit is part of the inferential problem.
4. What nonrejection means
A zero contrast checks only this implication at the boundary. Selection may occur elsewhere or cancel in the conditional mean at zero. Nonrejection therefore does not establish exogeneity throughout the treatment support.
Correcting selection requires more information than detecting its boundary implication. The jump does not identify a full causal response curve. The boundary-density identification argument obtains a local correction under additional transformation and deconvolution assumptions.
5. Seeing selection in a censored normal model
Take \(T^*=a+V\), \(T=\max(T^*,0)\), and suppose \(V\sim \mathcal{N}(0,\sigma^2)\). Let the conditional outcome mean be \(\beta T+\gamma+\delta V\). At a positive treatment value \(t\), the latent residual is observed: \(V=t-a\). Thus
At the atom, \(V\leq-a\). Integration of the normal density gives
For example, when \(a=0\), the continuous-side limit is \(\gamma\), while the atom mean is \(\gamma-\delta\sigma\sqrt{2/\pi}\). Their difference is \(\delta\sigma\sqrt{2/\pi}\). The treatment effect \(\beta t\) vanishes as \(t\downarrow0\), but the selection difference remains. The mass point pools a whole half of the latent residual distribution; the continuous-side limit selects residuals close to zero.
The normal example illustrates the mechanism rather than an assumption required for the test. Censoring creates an observable boundary difference even when selection along the positive support would be difficult to distinguish from a causal slope.
6. From a test to an identified correction
The stronger model in (Caetano, Caetano, and Nielsen 2023) takes
Define \(\psi(X)=\mathbb{E}[T^*\mid T^*\leq0,X]\). For \(T>0\), \(T^*=T\); for \(T=0\), only its truncated conditional mean is available. Consequently
and iterated expectations yield
Proposition 2. If the latent-tail restrictions identify \(\psi\) and the displayed regressors have full rank, the conditional-mean model identifies \(\beta\). The paper supplies tail-symmetry and location-scale conditions for recovering the latent negative tail, together with conditions for estimation and inference.
In this parameterization, estimate the coefficient \(b_T\) on \(T\) and the coefficient \(b_\psi\) on \(\psi(X)\mathbf1\{T=0\}\). Then \(\beta=b_T-b_\psi\). Alternatively, regress on \(T\), \(X\), and \(H=T+\psi(X)\mathbf1\{T=0\}\); the coefficient on \(T\) is directly \(\beta\) and the coefficient on \(H\) is \(\delta\).
Without the atom, \(H=T\), so the regression cannot separate the two coefficients. The atom creates additional variation, and the tail assumption fixes its scale. The observed jump alone identifies neither the full control function nor that scale.
At a fixed covariate value, the model gives a direct relation between the boundary test and the correction. The positive-side slope is \(b_T=\beta+\delta\), while the continuous-side limit minus the atom mean is
If \(\psi(x)<0\) is identified, these two features recover
In the centered normal example, \(\psi=-\sigma\sqrt{2/\pi}\), so the correction subtracts \(\Delta/(\sigma\sqrt{2/\pi})\) from the observed positive-side slope. The discontinuity supplies the size of a selection difference, while the latent-tail mean supplies the scale needed to turn that difference into a selection slope.
Without the tail restriction, the observed \(\Delta\) identifies only the product \(-\delta\psi\). Different pairs \((\delta,\psi)\) can produce the same boundary jump, and different values of \(\beta=b_T-\delta\) then fit the same positive-side slope. This is the precise missing equation that separates the endogeneity test from the identification result.
Under the maintained constant-\(\delta\) model, covariate variation also imposes \(\Delta(x)/\psi(x)=-\delta\) wherever the ratio is defined. A correction estimated at one covariate value must agree with the others. That implication can diagnose the stronger specification, although satisfying it does not establish every latent-tail assumption. When \(|\psi(x)|\) is small, the division magnifies error in both the jump and the recovered tail mean.
7. Recovering the missing negative tail
For an integrable latent variable,
To derive this, write the negative part as \((-T^*)_+=\int_0^\infty\mathbf1\{T^*<-s\}\,ds\) and interchange the nonnegative integral and expectation. Therefore
The atom probability gives the denominator. The numerator also depends on how far the latent choices extend below zero, information removed by censoring.
In the normal location model with \(a=x'\pi\), the calculation above gives
This is negative, as a conditional mean of a nonpositive latent variable must be. A specified family can identify its parameters from the uncensored portion and the censoring probabilities under suitable support and rank conditions. Symmetry restrictions provide another route. Neither extrapolation follows from the existence of a mass point itself.
8. Why the first stage enters the standard error
Use \(W=(T,X',H)'\) with coefficient vector \(\vartheta=(\beta,(\gamma-\delta\pi)',\delta)'\), and let \(Q=\mathbb{E}[WW']\). Replacing \(\psi\) by \(\widehat\psi\) changes the last regressor by \(\mathbf1\{T=0\}(\widehat\psi-\psi)\). At the true coefficients, the resulting regression error is
Under the rate and empirical-process conditions controlling the other product terms, the coefficient expansion contains
The first-stage and outcome terms can be correlated. Treating the generated regressor as known discards the second term from the expansion. A bootstrap must re-estimate both stages, with conditions that justify their joint approximation.
The correction therefore needs a conditional-mean model, identification of the unobserved tail, and a rank condition. The boundary test checks a weaker observable implication. Neither rejection nor nonrejection supplies those additional assumptions.
A sufficient continuity condition for the boundary comparison
The vanishing causal term in Section 2 concerns changing conditional populations, so pointwise continuity of each response curve alone should not be substituted for its integrability condition. One transparent sufficient condition is
for a nonnegative envelope \(H\). Then the absolute causal contribution is bounded by \(t\mathbb E[H\mid T=t,X=x]\to0\). Conditional exogeneity gives another direct route: it replaces this conditional expectation by the common potential-outcome mean, to which dominated continuity can be applied. Either argument states explicitly why approaching the observed boundary removes the causal difference while allowing the selection contrast to remain.