Regression Discontinuity Design

• Regression discontinuity design (RDD) is an approach to estimate the causal effect of treatment

Observational Alternatives to Experiments

• Selection on observables involves differing treatment and control groups based on observable characteristics
• Selection on unobservables involves differences based on unobservable characteristics
• Exogenous variables induce variation in treatment through instrumental variables
• Regression discontinuity designs have a known selection mechanism
• Treatment and controls are observed before and after treatment in difference-in-differences

RDD Basics

• RDD was introduced by Thistlethwaithe and Cambell in 1960
• Used to study merit awards impact where awards are given if test score exceeds a cutoff
• RDD reappeared and formalized in economics in late 90s
• Has proven to be a powerful causal tool in empirical economics and other disciplines
• Disciplines include: political science, education, epidemiology, criminology
• RDD has strong internal validity but is very data intensive
• There needs to be a lot of observations near the cutoff

Causality and RDD

• RDD isolates the causal effect of treatment where individuals become treated after crossing an arbitrary cutoff
• RDD has three fundamental components: running variable, cutoff, and treatment
• Individuals become treated after crossing a cutoff in the running variable
• Sharp RDD involves treatment received with probability zero below the cutoff and one above it
• Fuzzy RDD involves the probability of receiving treatment increasing discontinuously at the threshold
• RDD assumes potential outcomes evolve smoothly across the cutoff
• If there is no manipulation of the running variable, observations just below the threshold are similar to those just above, forming a control group

Running Variables and Cutoffs

• Examples of running variables and cutoffs: entry to high school/university depending on test scores/GPA
• Eligibility to vote or buy alcohol after a certain age
• Access to services based on residential location and catchment areas
• Candidate vote share determining election status (treatment)
• Speeding fines in Finland based on exceeding the speed limit by more than 20 km

Sharp RDD

• Test scores determine admission to a course or school as an example
• The running variable is the test score
• A threshold score is required for passing the exam, like 50/100, which is the cutoff
• Treatment involves attending the course or school

Causal Inference Problem in RDD

• The fundamental problem of causal inference shows because it is only possible to observe the outcome under control for units below the cutoff
• Additionally, can only observe the outcome under treatment for units above the cutoff
• As an example: test scores determine entry to education
• Two people with the same score and a score above the cutoff, the other rejected so there is no common support in scores between the accepted and rejected group. Potential outcomes, like ability and motivation, are different between those who score high and low

Local Causal Effect

• If units cannot perfectly "sort" around the cutoff, the discontinuous change in treatment probability can still be used to understand the local causal effect
• Units with scores barely below the cutoff can be used as a control group
• Units with scores barely above the cutoff can be the treatment

Key Point and Assumption of RDD

• Units with similar score values on opposite sides of the cutoff are comparable in all aspects except treatment status
• In a small neighborhood around the cutoff conditions mimic a randomized experiment with local randomization
• There is a continuity of average potential outcomes near the cutoff with continuity-based framework

Minimum Legal Drinking Age Example

• At the age of 21 is the legal drinking age in the U.S
• T is legal access to alcohol and Y is the likelihood of dying (and specific cause)

Alcohol and Deaths

• The treatment status is a deterministic function of age so age determines status
• Treatment status is a discontinuous function of age where after the cutoff is reached, remains unchanged.

Testing RDD Assumptions

• RDD assumes units cannot precisely manipulate their running variable value
• This can be tested checking for sorting, looking at if observations just above and below the threshold are similar regarding observables
• Placebo tests also confirm assumption, using noneffective cutoffs or other ages in drinking example.

Sorting or Manipulation of the Running Variables

• RDD assumes units do not have ability to manipulate the value
• If treatment is beneficial, units would want to receive the treatment and sort on the right side of the cutoff
• With no manipulation, treated observations above the cutoff the control observations below it
• Test: Use histogram of the running variable to see number of observations near cutoff similar
• A formal statistical density test can also be used (McCrary test)

Test of Observable Variables

• One of the most important RDD falsification tests examines if treated units are similar to control units near the cutoff in terms of observable characteristics
• If units lack the ability to alter the running variable, there should be no systematic differences
• All predetermined variables should be analyzed using RDD as the outcome of interest

Placebo Tests

• Placebo test 1 replaces the true cutoff value with a fake cutoff in the running variable
• A value at which the treatment status does not really change, performing estimation and inference using this “fake” cutoff. A significant treatment effect should occur only at the true value and not elsewhere.
• Placebo test 2 runs placebos at the true cutoff but replaces the outcome Y with other outcomes that should not be affected by the treatment

Technical Issues

• RDD is to comparing means for those just above to those just below the cut-off
• Often, there isn't enough data to estimate the treatment effect simply by comparing means at the cutoff
• This requires the choice of bandwith, which is a balance between accurate estimation and having enough data points

How to Address Limitations - Robustness

• In an a RDD paper, must show robustness to different modelling, data choices, and demonstrate results are similar with different bandwidths around cutoff

Fuzzy RD (Regression Discontinuity)

• When passing the cutoff creates a jump in treatment probabilities or treatment intensity
• Rather than switching the treatment on/off completely, here the resulting RD design is fuzzy
• Different schools have different cut-offs. Suppose there are 3 schools, and school 1 has the highest cut-off for intake. Examples include but are not inclusive to exam school or charter school.
• Scoring > cut-off for school 1: this increases your probability of attending an exam school, but not to p=1

Sarvimäki, Uusitalo & Jäntti (2021)

• After WW II approx 11% of the Finnish population was moved from ceded Soviet union.
• Re-homed to remaining parts of Finland.
• Displaced farmers approximately make up 50% of those displaced. They were given land and assistance to establish new farms in areas with soil and climate as origin regions.
• Former neighbours got resettled close to each other to preserve social networks.
• Since 1948 the displaced would cease to be issued subsidies, and instead were at liberty to sell land and move from the area.

Meyersson (2014)

• Uses a regression discontinuity design to compare municipalities where the Islamic party barely won or [barely] lost elections
• The despite negative raw correlations, the RD results reveal that, over six years, Islamic rule increased female secular high school education

Speeding Tickets

• In Finland, speeding tickets become income-dependent if the driver's speed exceeds the speeding limit by more than 20 km/h
• This Leads to a substantial jump in the size of the fine.
• No bunching below the threshold, but smooth speed distributions are around the speed threshold.
• Discontinuity may assist determining, estimating the the effect of punishment with fine size relative likelihood to re-offend.
• To compare similar individuals who drove 19 and 21 km too fast.

RDD Recap

• If a rule determines treatment due to a hard to predict cut-off, one can use the rule to estimate causal effect without RCT.
• The necessary criteria using RDD: the running variable, treatment, and cutoff and treatment assignments.
• Discontinuously change as function running the variable and the cutoff.
• Units just below and above the cut-off are very similar and comparable.
• Tests for Validity and Design such as Density tests, Test for balance covariate or and Test of placebo.
• The challenges requires a lot of observations that are in range of cut off. Or and cannot extrapolate results for to units that are far from the cut- off using Local causal effects!

This lesson covers the features of Regression Discontinuity (RD) designs, including common support, estimation with imperfect sorting, and challenges in estimating treatment effects. It also discusses how units near the cutoff are used and the role of pre-treatment characteristics.