Regression Discontinuity Designs (RDD)

Regression Discontinuity Designs (RDD) offer an effective quasi-experimental approach to estimate causal effects in situations where treatment assignment isn't random but depends deterministically or probabilistically on whether an observed continuous variable (the "running" or "forcing" variable) crosses a specific threshold. This design is particularly valuable when we suspect unobserved confounders might bias naive comparisons, as RDD uses the assignment rule itself to isolate the treatment effect locally around the cutoff.

Imagine a scenario where students are assigned to a remedial program ($T=1$) if their score ($R$) on an entrance exam falls below a certain cutoff ($c$), and receive the standard curriculum ($T=0$) otherwise. Comparing the average outcomes ($Y$) of all students in the remedial program versus all those in the standard program would likely be misleading due to pre-existing differences (confounding). Students with lower scores might differ in motivation, prior knowledge, or socioeconomic background, all of which could affect their final outcomes independently of the remedial program.

RDD exploits the discontinuity in treatment assignment at the cutoff $c$. The core idea is that individuals just below the cutoff ($R = c - \epsilon$) and just above the cutoff ($R = c + \epsilon$) are likely very similar in all relevant characteristics, both observed and unobserved, except for their treatment status. Any abrupt change, or "jump," in the average outcome observed precisely at the cutoff can then be attributed to the treatment itself.

Sharp vs. Fuzzy RDD

There are two primary types of RDD:

1. Sharp RDD: Treatment assignment is a deterministic function of the running variable $R$ relative to the cutoff $c$. For example, everyone with $R \ge c$ receives the treatment, and everyone with $R < c$ does not. The probability of receiving treatment jumps from 0 to 1 exactly at the cutoff.
2. Fuzzy RDD: Crossing the cutoff $c$ changes the probability of receiving treatment, but doesn't determine it perfectly. For instance, individuals with $R \ge c$ might be encouraged or become eligible for a program, but not all eligible individuals enroll. Similarly, some below the cutoff might still access the treatment through other means. In Fuzzy RDD, the probability of treatment $P(T=1|R=r)$ exhibits a discontinuity at $r=c$, but the jump is not necessarily from 0 to 1.

Core Assumptions for Identification

The validity of RDD depends on several assumptions:

1. Discontinuity in Treatment Assignment: The probability of treatment must change discontinuously at the cutoff $c$. This is definitional for RDD. In Sharp RDD, this jump is from 0 to 1. In Fuzzy RDD, the jump must be non-zero.

2. Continuity of Conditional Expectations: This is the most significant identifying assumption. It states that the average potential outcomes, $E[Y(0)|R=r]$ and $E[Y(1)|R=r]$, must be continuous functions of the running variable $R$ around the cutoff $c$. In simpler terms, if the treatment had no effect, the plot of the average outcome against the running variable would show no jump at the cutoff. This assumption allows us to attribute any observed discontinuity in the actual outcome $Y$ at $c$ to the causal effect of the treatment $T$.

3. No Precise Manipulation of the Running Variable: Individuals should not be able to perfectly control their score $R$ to strategically place themselves just above or below the cutoff. If manipulation occurs, individuals just below and just above the cutoff might differ systematically, violating the principle that they are comparable except for treatment status. A common diagnostic is the McCrary density test, which checks for a discontinuity in the density of the running variable $R$ at the cutoff.

   A smooth density plot around the cutoff suggests no manipulation. A significant jump or drop exactly at the cutoff would raise concerns.

4. Exclusion Restriction (Fuzzy RDD only): The cutoff $c$ should only affect the outcome $Y$ through its effect on the treatment $T$. This is analogous to the exclusion restriction in Instrumental Variables.

Estimation Strategies

The goal is to estimate the magnitude of the jump in the outcome $Y$ at the cutoff $c$. Since we typically don't have data exactly at the cutoff, estimation involves comparing outcomes for units just below and just above it.

Sharp RDD Estimation: The causal effect in Sharp RDD is the difference in the expected outcome just above and just below the cutoff:

$$\tau_{SRD} = E[Y | R=c, T=1] - E[Y | R=c, T=0] = \lim_{r \downarrow c} E[Y|R=r] - \lim_{r \uparrow c} E[Y|R=r]$$

The standard approach is local polynomial regression:

1. Choose a bandwidth $h$ around the cutoff $c$. Use only data where $c-h \le R \le c+h$.
2. Fit separate polynomial regressions of the outcome $Y$ on the centered running variable $(R-c)$ for units below the cutoff ($R<c$) and units above the cutoff ($R \ge c$). Typically, local linear (polynomial degree 1) or local quadratic (degree 2) regressions are used.
3. The estimated treatment effect $\hat{\tau}_{SRD}$ is the difference in the predicted values (intercepts) from these two regressions evaluated exactly at the cutoff ($R=c$).

Fuzzy RDD Estimation: In Fuzzy RDD, the jump in the outcome needs to be scaled by the jump in the probability of treatment. It resembles an Instrumental Variable (IV) approach where the discontinuity assignment (being above vs. below the cutoff) serves as the instrument for the actual treatment received. The effect is estimated as the ratio of the jump in the outcome to the jump in the treatment probability:

$$\tau_{FRD} = \frac{\lim_{r \downarrow c} E[Y|R=r] - \lim_{r \uparrow c} E[Y|R=r]}{\lim_{r \downarrow c} E[T|R=r] - \lim_{r \uparrow c} E[T|R=r]}$$

This is often estimated using a local IV framework (like two-stage least squares, 2SLS) within a bandwidth $h$ around the cutoff. The first stage models the treatment $T$ as a function of the cutoff indicator and polynomial terms of $(R-c)$. The second stage models the outcome $Y$ as a function of the predicted treatment from the first stage and polynomial terms of $(R-c)$.

Bandwidth Selection: Choosing the bandwidth $h$ is important. A smaller bandwidth reduces bias (by using observations closer to the cutoff, strengthening the continuity assumption) but increases variance (fewer data points). A larger bandwidth increases statistical power but risks introducing bias if the relationship between $R$ and $Y(0)$ or $Y(1)$ is highly non-linear. Data-driven methods like Imbens-Kalyanaraman (IK) or Calonico-Cattaneo-Titiunik (CCT) are commonly used.

Visual Diagnostics

Plotting the data is essential for assessing RDD validity.

1. Outcome vs. Running Variable: Plot the average outcome $Y$ within bins of the running variable $R$. Look for a clear jump at the cutoff $c$. Superimposing the fitted local polynomial lines helps visualize the estimated discontinuity.

   Visualization of Sharp RDD. The jump in the fitted lines at the cutoff represents the estimated treatment effect. Data points show the underlying relationship.

2. Treatment vs. Running Variable (Fuzzy RDD): Plot the proportion of units receiving treatment within bins of $R$. Look for a jump at $c$ to confirm the first stage in Fuzzy RDD.

3. Covariates vs. Running Variable: Plot pre-treatment covariates against $R$. These should not show a discontinuity at the cutoff. If they do, it casts doubt on the assumption that units just above and below the cutoff are comparable.

Interpretation and Limitations

RDD estimates a Local Average Treatment Effect (LATE), specifically the average causal effect for the subpopulation whose running variable values are near the cutoff $c$. This effect might not generalize to individuals far from the cutoff, who could have different characteristics and potentially different treatment effect magnitudes.

RDD requires sufficient data density around the cutoff for reliable estimation. It's also sensitive to the choice of bandwidth and polynomial degree. Violations, such as manipulation of the running variable or the presence of other treatments or policies changing at the same cutoff, can invalidate the results.

RDD in Machine Learning

While often seen in econometrics and policy evaluation, RDD concepts are relevant in ML:

• Validation Benchmark: In settings with a known assignment threshold, RDD can provide a credible estimate of the local causal effect. This can serve as a benchmark for evaluating the performance of more complex causal ML models (e.g., CATE estimators like Causal Forests or Double ML) for the specific subpopulation around the threshold.
• Identifying Natural Experiments: Thresholds are common in ML systems (e.g., classification thresholds for recommending an action, risk scores triggering an intervention, content promotion cutoffs). Recognizing these as potential RDD settings allows for causal evaluation of the system's components.
• Advanced Estimation: While local polynomials are standard, techniques from ML, such as local random forests or kernel methods, can sometimes be adapted for RDD estimation, potentially offering flexibility in capturing complex relationships near the cutoff, though careful implementation is needed to maintain theoretical guarantees.

RDD provides a rigorous framework for causal inference when assignment follows a threshold rule, offering a way to address unobserved confounding locally without needing to measure all confounders directly. Careful implementation and thorough diagnostic checks are essential for obtaining credible estimates.