Regression discontinuity: Meaning, Criticisms & Real-World Uses

What Is Regression Discontinuity?

Regression discontinuity (RD) is a quasi-experimental research design used in econometrics and other social sciences to estimate the causal effect of an intervention or "treatment" when its assignment is determined by whether an observed continuous variable crosses a specific cutoff point. This methodology allows researchers to infer causality in situations where a true randomized controlled trial is not feasible, by comparing outcomes for individuals or entities just above and just below the threshold. The core idea behind regression discontinuity is that units just on either side of the cutoff are very similar in all unobserved characteristics, making the treatment assignment "as good as random" at that precise point.

History and Origin

The regression discontinuity design was first introduced by Donald L. Thistlethwaite and Donald T. Campbell in 1960. Their seminal work aimed to evaluate the impact of merit awards on students' future academic outcomes by leveraging the fact that these awards were granted based on a specific test score cutoff²⁶. Students who scored just above the threshold received the award, while those just below did not. The researchers then compared the outcomes of these two groups, arguing that individuals near the cutoff were sufficiently similar that any difference in outcomes could be attributed to the award.

Despite its invention in the early 1960s, the regression discontinuity method did not gain widespread attention in economics until the late 1990s and early 2000s²⁴, ²⁵. Its resurgence was partly due to theoretical advancements and a growing interest in robust causal inference methods beyond traditional experimental designs. Economists and statisticians, including Guido Imbens and Thomas Lemieux, further formalized and popularized the design, publishing influential guides on its practical implementation²³.

Key Takeaways

Regression discontinuity is a powerful quasi-experimental design for estimating causal effects.
It is applied when treatment assignment is based on a strict cutoff of a continuous assignment variable.
The method compares outcomes for observations just above and just below the threshold, where treatment assignment is considered "as good as random."
Regression discontinuity designs offer a high degree of internal validity for the treatment effect at the cutoff point.
Its limitations include potentially lower statistical power compared to randomized trials and challenges with generalizing findings beyond the cutoff.

Formula and Calculation

A sharp regression discontinuity design (SRDD) typically models the relationship between the outcome variable ( Y ) and the assignment variable ( X ). The treatment ( D ) is a binary variable (1 if treated, 0 if not). The cutoff point is denoted by ( c ).

The basic model can be expressed as:

Y_i = \alpha + \tau D_i + f(X_i) + \epsilon_i

Where:

( Y_i ): The outcome variable for individual ( i ).
( D_i ): A dummy variable indicating treatment status ( ( D_i = 1 ) if ( X_i \ge c ), and ( D_i = 0 ) if ( X_i < c )).
( \alpha ): The intercept.
( \tau ): The treatment effect at the cutoff, which is the parameter of interest. This represents the jump in the outcome variable at the threshold.
( f(X_i) ): A function of the assignment variable ( X_i ), often modeled using a polynomial regression or local linear regression. This function captures the smooth relationship between ( X ) and ( Y ) in the absence of the treatment.
( \epsilon_i ): The error term.

The estimation involves fitting separate regression analysis lines on either side of the cutoff, or a single regression with a functional form that accounts for the discontinuity. The difference in the predicted outcome at the cutoff from these two fitted lines provides the estimate of ( \tau ).

For a fuzzy regression discontinuity design (FRDD), the probability of receiving treatment changes discontinuously at the cutoff, but not sharply from 0 to 1. In this case, the analysis often employs an instrumental variable approach.

Interpreting Regression Discontinuity

Interpreting a regression discontinuity analysis primarily involves examining the "jump" or discontinuity in the outcome variable at the predetermined cutoff point. If a statistically significant jump is observed, it suggests a causal effect of the intervention for individuals or entities located precisely at that threshold.

The key insight is that observations just below the cutoff serve as a valid control group for those just above, because any other factors influencing the outcome are assumed to evolve smoothly around the cutoff²². Therefore, any abrupt change in the outcome can be attributed to the treatment itself. Visualizing the relationship between the assignment variable and the outcome, with data points plotted around the cutoff, is a standard and effective way to present and interpret regression discontinuity findings. A clear graphical representation helps confirm whether the jump in the outcome variable at the cutoff is unusually large compared to the trends away from the cutoff²¹.

Hypothetical Example

Consider a hypothetical scholarship program designed to boost academic performance. The scholarship is awarded automatically to students who score 80% or higher on a standardized exam, which serves as the assignment variable.

Define Variables:
- Outcome variable (( Y )): Subsequent GPA (e.g., in the following semester).
- Assignment variable (( X )): Standardized exam score.
- Cutoff (( c )): 80%.
- Treatment (( D )): Receiving the scholarship.
Data Collection: Researchers collect data on exam scores and subsequent GPAs for a large group of students.
Plot Data: A scatter plot shows GPA on the y-axis against exam score on the x-axis. A vertical line is drawn at 80% to mark the cutoff.
Observe Discontinuity: While there's a general trend of higher exam scores correlating with higher GPAs, the regression discontinuity analysis looks specifically at students scoring, for example, between 75% and 85%. If students scoring 80% and above (who received the scholarship) show a notably higher average GPA compared to students scoring 79% (who did not receive the scholarship), that abrupt difference is the estimated treatment effect.
Interpretation: If the average GPA for students with scores just above 80% is significantly higher than for students just below 80%, this suggests that the scholarship causally improved academic performance for students at the margin of eligibility. This provides a direct insight into the program's impact, controlling for the inherent differences across the full range of scores.

Practical Applications

Regression discontinuity designs are widely used in various fields for program evaluation and policy analysis, especially when controlled experiments are not feasible.

Education: Evaluating the impact of merit scholarships based on test scores, the effect of class size reductions at specific enrollment thresholds, or the impact of grade retention policies²⁰.
Public Policy and Health: Assessing the impact of social programs where eligibility is determined by an income threshold (e.g., poverty lines), or the effect of health interventions triggered by a specific diagnostic marker like blood pressure¹⁸, ¹⁹. For instance, a study might evaluate the impact of subsidies to private schools in Pakistan where schools had to achieve a certain average on an achievement test¹⁷.
Finance and Economics: Analyzing the effects of regulatory changes that apply to companies above a certain size threshold, the impact of credit scores on loan approvals, or the effect of monetary policy changes tied to specific economic indicators. Researchers might use this design to study the effects of policies and political institutions that depend on whether a municipality is above or below arbitrary population thresholds, affecting factors like electoral systems or mayors' salaries¹⁶.
Political Science: Examining the impact of winning an election by a narrow margin on policy outcomes.

These applications leverage the sharp cutoff rule to isolate the causal impact of the intervention, providing robust insights for decision-makers.

Limitations and Criticisms

Despite its strengths, regression discontinuity has several limitations and criticisms that researchers must consider:

Local Average Treatment Effect: The primary output of a regression discontinuity analysis is the treatment effect at the specific cutoff point, known as the local average treatment effect (LATE)¹⁴, ¹⁵. This means the findings cannot necessarily be generalized to individuals far from the threshold, limiting the external validity of the results¹³.
Statistical Power: Regression discontinuity designs often require larger sample sizes compared to randomized controlled trials to achieve similar statistical power, especially when focusing on observations very close to the cutoff¹¹, ¹². The narrower the "bandwidth" (the range of data around the cutoff used for estimation), the fewer observations are available, which can reduce precision⁹, ¹⁰.
Manipulation of the Assignment Variable: If individuals or entities can precisely manipulate their score or value on the assignment variable to fall just above or below the cutoff, the "as good as random" assumption is violated⁷, ⁸. For example, if students know an exact test score cutoff for a scholarship, some might exert undue effort or even engage in unethical behavior to barely cross the line, potentially making them systematically different from those just below⁶.
Smoothness Assumption: A crucial assumption is that all other factors influencing the outcome evolve smoothly across the cutoff⁵. If other unobserved factors also change discontinuously at the same point, it becomes difficult to isolate the causal effect of the treatment of interest⁴.
Functional Form Sensitivity: The estimated treatment effect can be sensitive to the choice of functional form (e.g., linear, polynomial) used to model the relationship between the assignment variable and the outcome, especially when data is scarce around the cutoff², ³.
Lack of Overlap and Confounding Variables: If there's a significant lack of overlap in other relevant pre-treatment variables between the groups just above and below the cutoff, or if the analysis fails to adjust for other important predictors, the results can be biased¹.

Understanding these potential drawbacks is crucial for a balanced data analysis and interpretation of regression discontinuity findings.

Regression Discontinuity vs. Randomized Controlled Trial

Regression discontinuity (RD) and randomized controlled trials (RCTs) are both powerful methods for causal inference, but they differ fundamentally in their mechanism for assigning treatment.

Feature	Regression Discontinuity	Randomized Controlled Trial
Treatment Assignment	Determined by a strict, pre-defined cutoff on a continuous "assignment" variable. Individuals above/below the cutoff receive/do not receive treatment.	Random assignment of individuals or groups to treatment or control groups.
Randomness	"As good as random" only at the cutoff point. Individuals just on either side are assumed to be similar.	Randomness applies across the entire study sample, theoretically balancing all observed and unobserved characteristics.
Applicability	Ideal for situations with naturally occurring or administratively imposed thresholds (e.g., age limits, test score cutoffs, poverty lines).	Requires the ability to randomly assign treatment, which may not be ethical or feasible in many real-world policy contexts.
Generalizability	Estimates the treatment effect locally at the cutoff, limiting direct generalization to the broader population.	Estimates an average treatment effect for the studied population, often with broader generalizability, assuming adequate sample size and study design.
Internal Validity	High, assuming the key assumptions (e.g., no manipulation of assignment variable, smoothness of other factors) hold.	Generally considered the "gold standard" for internal validity due to random assignment.

While RCTs are often preferred when possible due to their strong causal identification, regression discontinuity provides a robust quasi-experimental design alternative when true randomization is not an option.

FAQs

What is the main assumption of regression discontinuity?

The main assumption is that individuals or units just above and just below the cutoff point are statistically similar in all unobserved characteristics that could affect the outcome. This makes treatment assignment at the cutoff essentially random, allowing for robust causal inference.

What is the difference between sharp and fuzzy regression discontinuity?

In a sharp regression discontinuity (SRD) design, the cutoff perfectly determines treatment assignment; everyone above the threshold receives treatment, and everyone below does not. In a fuzzy regression discontinuity (FRD) design, the cutoff influences the probability of receiving treatment, but doesn't perfectly determine it. For instance, individuals above a certain score might be more likely to receive a scholarship, but not guaranteed. Fuzzy designs often require an instrumental variable approach for statistical analysis.

Can regression discontinuity estimate effects for everyone?

No, regression discontinuity estimates the treatment effect only for individuals or units who are at the very margin of eligibility – that is, those whose assignment variable values are precisely at the cutoff point. This is known as the local average treatment effect. It does not directly provide an estimate for those far from the cutoff or for the entire population.

How is regression discontinuity used in finance?

In finance, regression discontinuity can be used to evaluate the impact of regulations that apply based on firm size (e.g., asset thresholds), the effect of credit scoring