8. Regression Discontinuity Flashcards
What is the regression discontinuity design (RDD)?
Regression Discontinuity Design (RDD) is a quasi-experimental research method used to estimate causal effects in situations where treatment assignment is based on a threshold (or cutoff) in a continuous variable. The key idea behind RDD is that the cutoff creates a natural experiment. Units near the cutoff are presumed to be nearly identical in unobservable characteristics, with the treatment status being the only systematic difference. By comparing outcomes on either side of the cutoff, RDD isolates the causal effect of the treatment.
Components
* Running variable (X): A continuous variable that determines treatment assignment based on a pre-specified cutoff (c0).
* Cutoff point (c0): A threshold that determines whether a unit receives the treatment (D=1) or not (D=0).
* Outcome Variable (Y): The variable of interest that is hypothesized to be affected by the treatment.
* LATE: RDD estimates the causal effect of the treatment for units near the cutoff (c0), known as the Local Average Treatment Effect. Since LATE applies only to individuals close to the cutoff, the results cannot be generalized to the entire population
Two types of RDD:
* Sharp RDD: Treatment assignment changes deterministically at the cutoff.
* Fuzzy RDD: Treatment assignment increases probabilistically at the cutoff but is not guaranteed.
What is the identifying assumption for RDD?
- Continuity assumption (identifying assumption): The expected potential outcomes E[Yi^0|X] and E[Yi^1|X] are continous at the cutoff. Thus, any abrupt change in the outcome (Y) at the cutoff is assumed to be caused by the treatment –> In the absence of treatment, Y would have remained continuous across c_0.
What are key requirements/conditions for a valid RDD?
- Treatment assignment rule: The assignment rule has to be known, precise and free of manipulation. –> The cutoff (c0) must be clearly defined and transparent.
- No manipulation of the running variable (X): Units (individuals or groups) must not be able to manipulate their value of the running variable (X) to cross the cutoff. Manipulation (e.g., inflating test scores) would invalidate the assumption that treatment assignment is quasi-random near c0.
- Local comparability/randomization: Units just above and just below the cutoff should be similar in observable and unobservable characteristics, except for treatment status.
- Sufficient data near the cutoff: A large number of observations near the cutoff ensures robust estimation of the treatment effect. Sparse data near c0 weakens the ability to detect a discontinuity or separate it from noise. Large sample sizes are characteristic features of the RDD.
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Smooth relationship between running X and Y: The relationship between X (running variable) and
Y (outcome) must be appropriately modeled (e.g., linear, quadratic, or nonparametric). Mis-specifying the functional form can lead to biased estimates of the treatment effect. - Continuity of covariates: Pretreatment covariates (e.g., age, income, education) should not exhibit a discontinuity at c0. A jump in covariates suggests systematic differences between groups, violating the condition of comparability.
- Exclusion of competing interventions: No other policies or interventions should coincide with c0 and independently affect Y.
What are characteristics of sharp RDD?
In a sharp RDD, the probability of receiving the treatment jumps abruptly from 0 to 1 at the cutoff (c0). The cutoff perfectly determines treatment (0 or 1) –> treatment is a deterministic and discontinuous function of the running variable (X)
* If you know Xi, you know treatment assignment (Di) with certainty: For any given unit i, if you know the value of Xi, you can perfectly predict whether the unit is treated (Di=1) or not (Di=0).
* No overlap: Because treatment is perfectly determined by the cutoff, there is no overlap or “common support” between treated and untreated units at any given X.
* Extrapolation: Sharp RDD relies on extrapolating trends in the outcome variable (Y) from both sides of the cutoff. Units below c0 (X<c) provide an estimate of the untreated outcome. Units above c0 (X≥c0) provide an estimate of the treated outcome.The difference between these trends at c0 gives the treatment effect.
* Potential outcomes: The observed outcome (Yi) depends on whether a unit is treated or untreated: Yi=Yi^0 +(Yi^1-Yi^0 )∙Di
* Local Average Treatment Effect (LATE): the causal effect of treatment at the cutoff: δSRD=E[Yi^1-Yi^0 |Xi=c_0] This effect is local because it applies only to individuals close to the cutoff (Xi=c0) and may not generalize to individuals farther away.
What are characteristics of fuzzy RDD?
In a fuzzy RDD, the probability of receiving the treatment increases sharply at the cutoff but does not jump from 0 to 1. Treatment assignment is influenced by the cutoff but remains probabilistic rather than deterministic.
* If you know Xi, you only know the likelihood of treatment (Di): For any given unit i, knowing the value of Xi tells you the probability that the unit receives treatment (Di=1), but not with certainty. Units with
Xi≥c0 are more likely to be treated, but some may remain untreated. Similarly, units with Xi<0 are less likely to be treated, but some may still receive treatment.
* Overlap: In contrast to sharp RDD, fuzzy RDD exhibits overlap or “common support” in treatment and control groups: There are treated and untreated units on both sides of the cutoff, due to imperfect compliance with the cutoff-based treatment rule.
* Extrapolation: Fuzzy RDD also relies on extrapolating trends in the outcome variable (Y) from both sides of the cutoff, but it accounts for the probabilistic jump in treatment status. The difference in trends at c0, scaled by the jump in treatment probability, gives the treatment effect.
* Potential outcomes: The observed outcome (Yi) depends on whether a unit is treated or untreated: Yi=Yi^0 +(Yi^1-Yi^0 )∙Di
* Local Average Treatment Effect (LATE): Fuzzy RDD estimates the treatment effect for compliers—those whose treatment status is determined by the cutoff. The LATE is calculated using the Wald estimator. The estimation process is conceptually similar to using instrumental variables (IV) because the cutoff serves as an instrument for treatment.
What are possible challenges to identification when using RDD?
Manipulation of the running variable:
* The assignment rule is known in advance: If individuals are aware of how treatment is assigned, they may alter their behavior to fall on the favorable side of the threshold.
* Agents are interested in adjusting: Individuals or groups have a strong incentive to manipulate their position relative to the cutoff
* Agents have time to adjust: When there is enough time between learning about the assignment rule and the implementation of the treatment, individuals are more likely to adjust their behavior strategically.
The cutoff is endogenous to factors that independently cause potential outcomes to shift: If the cutoff value itself is influenced by factors that also affect the outcome, the design becomes invalid.
There is nonrandom heaping along the running variable: Observations might cluster unnaturally at certain values of the running variable. This nonrandom “heaping” can distort the smooth relationship between the running variable and the outcome.
What is the McCrary’s density test?
The McCrary density test is a diagnostic tool used in RDD to assess whether the running variable has been manipulated around the cutoff. The test evaluates whether the density of the running variable is continuous at the cutoff, as required by the continuity assumption in RDD.
* Hypotheses: H0: The density of the running variable is continuous at the cutoff. H1: The density of the running variable is discontinuous at the cutoff.
* How it works: 1. Partition the running variable into small bins. 2. Calculate the frequency of observations in each bin. 3. Fit local linear regressions on the log-density of the running variable on both sides of the cutoff. 4. Compare densities just above and below the cutoff to detect jumps.
The test requires a large number of observations near the cutoff to distinguish genuine discontinuities from random noise.
What are covariate balance and placebo tests
Covariate balance tests: Assess whether the continuity assumption holds by checking for jumps in pretreatment covariates at the cutoff. Since these covariates are unaffected by treatment, their values should change smoothly across the cutoff. No discontinuity supports the validity of the design, while a discontinuity suggests potential manipulation or violations of assumptions.
Other placebo tests: Test for outcome jumps at arbitrary cutoffs (not the true cutoff). Select a fake cutoff and check for discontinuities in the outcome variable. No discontinuity confirms the validity of the design, while observed jumps suggest potential confounding or model misspecification.There should be no observed discontinuity at arbitrary cutoffs because treatment does not actually change there.
What is the donut hole solution?
The donut hole solution is a technique in RDD to address issues of manipulation or heaping near the cutoff:
* Heaping: Unnatural clustering of observations at specific points on the running variable due to rounding, measurement error, or manipulation. This introduces selection bias, as manipulated cases may systematically differ in unobserved ways.
* The donut hole solution is to remove (“drop”) observations very close to the cutoff, creating a “donut hole” around c0. This focuses on units farther from the cutoff, reducing bias caused by heaping or manipulation.
Disadvantages:
* Reduces sample size, weakening statistical power.
* The treatment effect now reflects units farther from the cutoff, making it less representative of those near
* Smaller sample sizes make it harder to detect significant effects.
Parametic vs. non-parametic RDD (!)
