Treatment Effects at an Overtime Notch
The slope of an observed outcome regression combines a causal response with changing composition of the treated population. (Caetano, Caetano, and Nielsen 2022) uses the latent choices pooled at a treatment boundary to identify the selection component under additional assumptions. In its overtime application, the identifying calculation is a transformation of densities at the boundary.
1. Separate response and selection
For positive treatment values, let \(m(t)=\mathbb{E}[Y\mid T=t]\) and define the average marginal response \(\beta(t)=\mathbb{E}[Y'(t)\mid T=t]\). Under differentiability and interchange-of-expectation conditions, write
At the boundary, use the limits from positive treatment: \(\beta(0)=\lim_{t\downarrow0}\beta(t)\) and \(u(0)=m(0+)\). These need not equal the marginal response or observed mean averaged over the entire atom at zero. Once \(\beta\) is defined, \(B(0)=0\) fixes the level of the decomposition. The observed regression \(m\) still reveals only the sum of the causal and selection derivatives.
2. The boundary result
Suppose \(T=\max(T^*,0)\) and extend the selection function to the censored latent range by \(u(t^*)=\mathbb{E}[Y(0)\mid T^*=t^*]\) for \(t^*\leq0\). Assume this extension is monotone and joins the positive-side selection function with the smoothness required below. This is an additional restriction on latent selection, not a consequence of decomposing the observed regression alone. Let \(V=u(T^*)\) among those with \(T=0\).
Proposition 1. Under the paper’s smoothness, monotonicity, nondegeneracy, and deconvolution-identification conditions, the boundary response is identified by
where, in the nonconstant-selection case,
The last equality uses continuity of the latent density at zero. The numerator is identified from the observed atom and the density approaching it from above.
3. Why a density ratio gives the slope
For a monotone differentiable transformation \(v=u(t)\), the change-of-variables formula is
Solving the change-of-variables formula gives the magnitude of the selection slope. The monotonicity and boundary-mean restrictions determine its sign. Both densities in the ratio must condition on the atom; using an unconditional density in only one place would introduce a probability-mass error.
4. Where deconvolution enters
At \(T=0\), observed outcomes mix the latent selection component \(V\) with outcome noise. The continuous-side limit identifies the noise law under the stipulated stability and independence conditions. If \(Y(0)=V+E\) with the required independence, characteristic functions satisfy
The quotient recovers the selection characteristic function only when the deconvolution problem is identified. Fourier inversion needs its own regularity conditions, and zeros of \(\varphi_E\) cannot be divided out without justification.
The paper supplies conditions for the required density value and thus identifies the local response \(\beta(0)\). It does not thereby identify the response at every treatment value. The endogeneity test needs only the boundary mean contrast and avoids this density-recovery problem.
5. Deriving the selection component in a smooth model
Suppose potential outcomes can be represented by a smooth function \(y(t,v)\), with conditional latent-type density \(f(v\mid T=t)\). Differentiating the observed mean gives
The first integral averages causal derivatives among individuals observed at \(t\). The second holds their outcome function fixed and differentiates their type weights. Differentiability and domination are needed to pass the derivative through the integral.
When the conditional density is positive, the selection term can also be written as the expectation of \(y(t,V)\) times the conditional score \(\partial_t\log f(V\mid T=t)\). That score has mean zero because the conditional density integrates to one. Selection therefore depends on how outcomes co-move with the changes in type weights as treatment varies.
6. The boundary density transformation in detail
On the latent interval mapped to the atom, suppose \(u\) is strictly increasing with nonzero derivative at zero. For \(v=u(t)\),
Differentiate with respect to \(t\) to obtain
If \(u\) is decreasing, the event reverses direction and the derivative changes sign, giving the absolute derivative in the general change-of-variables formula. Evaluating at the one-sided boundary and solving yields the proposition’s ratio. The density at the boundary means the appropriate interior limit of the conditional density.
The conditional latent density is
Continuity of \(f_{T^*}\) through zero and the fact that positive \(T\) equals \(T^*\) imply \(f_{T^*}(0)=f_T(0+)\). The atom probability is therefore essential in translating the observed positive-side density into the conditional density used in the ratio.
7. A case where every quantity can be checked
Take \(T^*\) uniform on \([-a,b]\), with \(a,b>0\), and \(u(t)=u_0+\lambda t\), \(\lambda\ne0\). Then
Within the atom, \(V\) is uniform over an interval of length \(|\lambda|a\), so its interior boundary density is \(1/(|\lambda|a)\). The ratio equals \(|\lambda|\), exactly the magnitude of the selection slope.
Also \(\mathbb{E}[V\mid T=0]=u_0-\lambda a/2\). If the outcome noise has a common mean on the two sides, the continuous-side mean minus the atom mean is \(\lambda a/2\), revealing the sign of \(\lambda\). If \(m'(0+)=b_0+\lambda\), subtracting the identified selection slope recovers the causal boundary response \(b_0\).
If \(u\) is constant on the censored region, \(V\) is a point mass and this density-ratio formula is not applicable. That case must be handled separately: the selection slope is zero under the corresponding smooth continuation assumptions. Plugging an undefined density into the nonconstant formula would not constitute identification.
8. Deconvolution identifies a distribution before estimating a slope
Under the additive independent-noise representation within the atom,
If the quotient is integrable, Fourier inversion gives
The inversion uses two pieces of information: the outcome distribution at the atom and the noise law recovered through the continuous-side stability assumptions. One observed convolution cannot generally separate two unknown factors. Other inversion conditions are possible, but they must establish that separation.
Even when \(\varphi_E\) has no zeros, estimation can be ill-conditioned. For Gaussian noise with variance \(\sigma^2\), division multiplies high-frequency error by \(e^{\sigma^2s^2/2}\). A regularized estimator truncates or smooths high frequencies, creating a bias-variance tradeoff. Identification of \(f_V\) does not imply that its boundary value can be estimated at a parametric rate.
A frequency cutoff makes the instability quantitative. Temporarily suppose the noise characteristic function is known and Gaussian, and estimate \(\varphi_{Y\mid T=0}\) from \(n_0\) independent observations at the atom. At each frequency, the empirical characteristic function has expected absolute error at most \(n_0^{-1/2}\). For the Fourier integral truncated at \(|s|\leq T\), Fubini and the triangle inequality give
The omitted Fourier tail is a separate approximation error. Taking \(T=c\sqrt{\log n_0}\) makes the displayed stochastic bound of order \(\sqrt{\log n_0}\,n_0^{(\sigma^2c^2-1)/2}\), which vanishes if \(\sigma^2c^2<1\). How quickly the truncation bias vanishes depends on the regularity of the selection density. Estimating the noise distribution adds another error beyond this illustrative known-noise calculation.
The formula identifies a derivative by relating density near the latent boundary to density after transformation. Observation noise smooths that distribution. Undoing the smoothing amplifies high-frequency estimation error, explaining why exact identification need not lead to precise estimation.
If the densities are positive and differentiable near the evaluation point, the local error in the magnitude \(s=|u'(0)|=f_T(0+)/[\pi_0f_V(u_0)]\) also obeys the differential identity
The differential \(df_V(u_0)\) measures density-estimation error with the evaluation point held fixed. The final term accounts for estimating the point itself. Both contribute to the recovered slope’s uncertainty.
The recovered magnitude divides \(f_T(0+)\) by \(\Pr(T=0)f_V(u(0))\), evaluated at the estimated point \(u(0)\). Small denominator density magnifies errors in every component. Implementing the boundary identification result therefore requires derivative, density, and deconvolution estimation, in addition to the structural assumptions.
Connecting the latent and observed derivatives at zero
Let \(H(t,s)=\mathbb E[Y(t)\mid T^*=s]\) be continuously differentiable near \((0,0)\), with differentiation under the conditional expectation justified. For \(t>0\), \(m(t)=H(t,t)\) and \(\beta(t)=\partial_1H(t,t)\). The chain rule gives
For \(s\leq0\), the latent extension is \(u(s)=H(0,s)\), so its left derivative at zero is \(\partial_2H(0,0)\). Continuity therefore matches this derivative to \(u'(0+)\) from the observed positive side. In contrast, the observed atom mean is
which averages the whole censored range and generally differs from \(H(0,0)=m(0+)\). The density transformation recovers the slope at the edge of that range. It does not replace the boundary causal response by an average over all censored individuals.