Placebo Discontinuity Design

Technical Summary: Placebo Discontinuity Design

1. The Problem
Standard Regression Discontinuity Design (RDD) relies on the identifying assumption that expected potential outcomes are continuous at the cutoff. This assumption ensures that units just above and below the threshold are comparable in the absence of treatment. However, this continuity can be violated when the running variable is subject to strategic manipulation or sorting by agents who are aware of the cutoff (e.g., school administrators manipulating enrollment to trigger class-size reductions, or doctors manipulating birth-weight reports to trigger medical interventions).

When such manipulation occurs, the conditional distribution of unobserved confounders given the running variable becomes discontinuous at the cutoff. Consequently, standard RDD estimators become biased because the discontinuity in the outcome reflects both the treatment effect and the discontinuity in unobserved confounding. While existing literature uses placebo variables (outcomes or covariates that should not be affected by the treatment) as diagnostic tools to detect such violations, current methods offer no path to recover the treatment effect once a violation is detected.

2. Methodology and Identification Strategy
The authors propose a new identification strategy, termed "Placebo Discontinuity Design" (PDD), which leverages a placebo treatment ($Z$) and a placebo outcome ($W$) to correct for unobserved confounding and recover the causal effect even when the standard continuity assumption fails.

• Structural Model: The paper models the outcome $Y$ as a function of treatment $A$, running variable $D$, unobserved confounder $U$, and idiosyncratic noise $\eta_y$. Crucially, the authors allow the density of $U$ given $D$ to be discontinuous at the cutoff $d^*$.

• Key Assumptions:

  • Conditional Continuity: While potential outcomes may be discontinuous in $D$ marginally, they are assumed to be continuous conditional on the unobserved confounder $U$. That is, $E[Y(a, D, U, \eta_y) | D=d, U=u]$ is continuous in $d$ for fixed $u$.
  • Placebo Variables:
    • Placebo Treatment ($Z$): A variable that affects the running variable $D$ (and thus the treatment $A$) but is independent of the unobserved confounder $U$ and the outcome $Y$ conditional on $D$ and $U$.
    • Placebo Outcome ($W$): A variable that is a proxy for the unobserved confounder $U$ but is not directly caused by the treatment $A$, the running variable $D$, or the placebo treatment $Z$.
  • Confounding Bridge: The authors assume the existence of a "confounding bridge" function $h$ such that the conditional expectation of the outcome given $D$ and $U$ can be expressed as an integral of $h$ with respect to the distribution of the placebo outcome $W$ given $D$ and $U$.
  • Completeness: A rank condition ensuring the solution to the integral equation is unique, allowing for point identification.

• Identification Result: Under these assumptions, the treatment effect $\tau_0$ is identified as the difference between the right-limit and the adjusted left-limit of the outcome, where the adjustment is derived from the placebo outcome's discontinuity. Specifically, the estimator decomposes into a standard RDD term minus an adjustment term based on the placebo outcome's discontinuity weighted by a local instrumental variable coefficient.

3. Estimation
The paper proposes a Local Instrumental Variable (LIV) estimator to operationalize the identification strategy.

• Approach: The authors approximate the confounding bridge function using a partially linear specification. They estimate the parameters using a local linear regression framework extended to include instrumental variables.
• Decomposition: The resulting estimator, $\hat{\tau}_{pdd}$, is numerically equivalent to:
  $$\hat{\tau}_{pdd} = \hat{\tau}^y_{rdd} - (\hat{\tau}^w_{rdd})^\top \hat{\gamma}_-$$
  where $\hat{\tau}^y_{rdd}$ is the standard RDD estimate for the target outcome, $\hat{\tau}^w_{rdd}$ is the RDD estimate for the placebo outcome, and $\hat{\gamma}_-$ is a weight vector derived from a local IV regression of the residualized target outcome on the residualized placebo outcome (instrumented by $Z$) below the cutoff.
• Interpretation: If the placebo outcome shows no discontinuity ($\hat{\tau}^w_{rdd} \approx 0$), the adjustment term vanishes, and the estimator reduces to the standard RDD. If a discontinuity exists, the estimator uses it to correct the bias in the target outcome.

4. Theoretical Results

• Consistency: The authors establish that the LIV estimator is consistent for the treatment effect under the assumption that the confounding bridge is partially linear (or well-approximated by it).
• Asymptotic Normality and Bias Correction: Following the framework of Calonico et al. (2014), the paper derives the asymptotic bias of the naive estimator when using Mean Squared Error (MSE) optimal bandwidths. The authors propose a robust bias-corrected inference procedure. They show that after bias correction, the estimator is asymptotically normal at the rate $n^{-2/5}$, allowing for the construction of valid confidence intervals and hypothesis tests.

5. Key Contributions and Significance

• From Diagnosis to Correction: The primary contribution is shifting the role of placebo variables in RDD from mere diagnostic tools (to detect invalidity) to active correction mechanisms (to recover validity).
• Point Identification under Manipulation: Unlike previous approaches that only provide bounds on the treatment effect when the running variable is manipulated (e.g., Gerard et al., 2020), this paper provides conditions for point identification of the treatment effect in the presence of strategic behavior.
• Nonparametric Recovery: The method recovers the RDD treatment effect nonparametrically, avoiding the need for strong parametric assumptions about the production function or the distribution of unobservables required by other methods (e.g., Garicano et al., 2016).
• Practical Implementation: The proposed estimator decomposes into familiar RDD components and a weighted adjustment, making it computationally tractable and interpretable for empirical researchers. It avoids the "curse of dimensionality" often associated with nonparametric estimation involving multiple covariates by utilizing a partially linear approximation.

6. Claims and Limitations
The authors modestly claim to provide a new identification result and a novel variation of the local linear RDD estimator. They acknowledge that the structural assumptions (specifically the existence of the confounding bridge and the completeness condition) are generally untestable, similar to standard IV assumptions. The method relies on the availability of valid placebo variables that satisfy specific exclusion and independence restrictions. The paper does not claim to solve all forms of RDD invalidity but specifically targets settings where strategic manipulation leads to discontinuities in unobserved confounders, a common issue in social science applications involving government benefits or social programs. Future work is suggested to explore bias-aware inference and the behavior of the estimator with weak proxies.