Regression Discontinuity with Measurement Error
An RD comparison relies on knowing which side of the assignment cutoff an observation occupies. Measurement error may preserve that classification even while changing which latent values occur near the observed cutoff. (Dong and Kolesár 2023) separates these questions: whether the observed jump identifies an effect, and which population’s effect it identifies.
1. What the observed discontinuity identifies
Let \(X^*=X+e\) be the latent running variable, with cutoff zero. Suppose assignment agrees almost surely with \(Z=1\{X\ge0\}\). This is a substantive restriction on the measurement process; ordinary additive noise need not satisfy it.
Theorem 1. In a sharp design, let \(T=Z\). If \(g_t(x)=\mathbb{E}[Y(t)\mid X=x]\) is continuous at zero for \(t=0,1\), the observed outcome jump identifies
If, additionally, conditioning on the measurement error does not alter the conditional treatment effect at a given \(X^*\), then
The first conclusion follows by taking limits of the treated and untreated conditional means. The second is an iterated-expectations calculation: among observations with \(X=0\), the latent running variable equals \(e\).
2. The estimand can move
The observed cutoff selects a distribution of latent values. It does not generally select \(X^*=0\). For example, if \(\tau^*(x)=a+bx\), then
Even perfect classification leaves a difference from the latent-cutoff effect \(a\) when the conditional error mean is nonzero. Constant treatment effects make this distinction disappear; unrestricted heterogeneity does not.
3. Fuzzy assignment adds a second weighting
Now let \(T(z)\) be potential treatment status under assignment \(z\), and let \(C=\{T(1)>T(0)\}\) denote compliers. Impose exclusion, monotonicity, a positive treatment jump, and continuity at zero of the relevant potential-treatment and potential-outcome means.
Theorem 2. The ratio of the outcome jump to the treatment jump identifies
Under the corresponding nondifferential-error restrictions for compliance and complier effects,
The outcome jump averages the treatment effect multiplied by the compliance indicator. The treatment jump averages that indicator alone. Their ratio therefore normalizes the complier weights. A small denominator makes the ratio unstable, as in weak-instrument inference.
Once the weighted effect is identified, inference still requires smooth observed conditional means and control of local-polynomial bias. The measurement restriction preserves the assignment comparison; it does not supply those estimation conditions.
4. Deriving the observed jump
In the sharp design, observed outcomes satisfy
Hence the right conditional mean is \(g_1(x)\) and the left conditional mean is \(g_0(x)\). Continuity gives
At a continuously distributed cutoff, these are limits of the specified continuous conditional-mean versions. Classification permits using treated potential outcomes on the right and untreated potential outcomes on the left, which is the essential step in the jump argument.
To see why this condition cannot be omitted, suppose instead that treatment is \(\mathbf1\{X^*\geq0\}\) and \(X=X^*+u\) contains ordinary continuous noise. Even with a constant treatment effect \(\tau\), the observed regression can take the form
If assignment probability is smooth at the observed cutoff, the regression has no jump identifying \(\tau\). More data estimate the smooth regression more accurately without restoring the lost classification. The theorem instead assumes assignment agrees with the observed side.
5. From the observed cutoff to a latent-effect average
Let \(\Delta Y=Y(1)-Y(0)\). Iterated expectations always imply
Since \(X^*=X+e\), the inner conditioning fixes \(X^*=u\). To replace that inner expectation by \(\tau^*(u)\), one needs precisely the restriction that conditioning additionally on the measurement error does not change the mean effect at that latent value. Without it, two people at the same \(X^*\) but with different measurement errors could have different effects, and the latent-effect function alone would not describe the selected population.
For a quadratic latent effect, \(\tau^*(x)=a+bx+cx^2\), the corresponding calculation is
Thus even conditionally mean-zero error does not generally recover the latent-cutoff effect. Curvature matters. More generally, if \(\tau^*\) is Lipschitz with constant \(K\), then
The bound states what measurement-quality information would imply under the weighting representation. It does not identify the unknown error law from the observed RD data.
6. The fuzzy numerator and denominator, term by term
Under exclusion, the potential observed outcome under assignment \(z\) is \(Y(T(z))\). For binary treatment,
Monotonicity makes \(T(1)-T(0)=\mathbf1_C\). Continuity of the appropriate conditional means therefore gives an outcome jump and treatment jump equal to
Provided \(J_T>0\), their ratio is the mean effect among compliers at the observed cutoff. Always-takers and never-takers contribute zero to both differences; defiers would contribute with the opposite sign and invalidate this nonnegative-weight interpretation.
For the latent representation, write \(c(u)=\mathbb P(C\mid X^*=u)\) and \(\tau_C^*(u)=\mathbb{E}[\Delta Y\mid C,X^*=u]\). Under the additional error restrictions,
Dividing gives the theorem’s weights. Latent values with more compliers receive more weight even if the measurement-error distribution itself is symmetric. For instance, two equally likely latent values with compliance probabilities \(1/4\) and \(3/4\) receive normalized weights \(1/4\) and \(3/4\), not \(1/2\) each.
The compliance reweighting has a useful covariance decomposition. Take all expectations in the next display with respect to \(F_{e\mid X}(\cdot\mid0)\), and suppose \(\mathbb{E}c>0\). Then
The first term averages latent complier effects over the cutoff’s error distribution. The covariance term accounts for greater weight at latent values with more compliers. Centered, symmetric error need not make that covariance zero.
If \(\tau_C^*\) is Lipschitz with constant \(K\), its distance from the latent-cutoff complier effect is bounded by
The relevant error magnitude is itself weighted by compliance. Knowing the unweighted mean absolute measurement error can be insufficient for a sharp interpretation bound when compliance varies strongly with \(e\).
Correct classification preserves the observed-cutoff discontinuity. Measurement error determines which latent values that cutoff represents, and compliance reweights those values in a fuzzy design. These changes in interpretation remain even with perfectly estimated jumps.
Finally, for estimated jumps \(\widehat J_Y,\widehat J_T\), a first-order ratio expansion is
Both jumps use the same observations, so their covariance enters the variance of the expanded numerator. A small \(J_T\) magnifies error and smoothing bias in either jump. Establishing the weighted estimand is therefore distinct from obtaining an accurate confidence interval.