Regression Discontinuity

Motivating Regression Discontinuity-Design

• In an experiment, we can determine causal effects accurately when our experiment is randomized

  • Specifically, a randomized experiment means treatment assignment is random
  • Roughly, this happens when there aren't any confounding variables (or colliders)

• Stratification (or matching) and regression can be used to randomize an experiment with measured confounders

  • Thus, bias is removed when an experiment is stratified
  • Stratification isn't very effective when a covariate $X$ is continuous
  • Since, it's almost impossible to stratify to each unique value of $X_{i}$
  • Thus, regression can be more appealing

• Regression discontinuity-design (or RDD) can also be used to eliminate selection bias

  • Specifically, we'll use RDD if our experiment is without measured confounders
• In other words, stratification can eliminate bias with measured confounders, whereas RDD can eliminate bias without measured confounders

Motivating the Use of RDD in Natural Experiments

• Typically, RDD is used in natural experiments

  • A natural experiment refers to an experiment without a controlled treatment assignment
  • Rather, treatment is assigned by some random, external factor
• Whereas, a controlled experiment refers to an experiment where all factors are held constant except for one

• As a result, RDD uses a running variable $X$

  • Don't mistake this for an ordinary covariate
• Don't confuse this $X$ with observed or unobserved covariates

• Here, $X$ refers to a running variable, which represents a continuous variable that assigns units to a treatment $D$

  • We usually use a continuous $X$ instead of binary treatment variable $D$ in RDD, since RDDs are usually used in natural experiments
  • Remember, we don't usually have a treatment variable $D$ give to us in natural experiments

Assuming Continuity for RDD

• Continuity assumes an absence of selection bias

  • Thus, we can remove any selection bias by verifying that the continuity assumption is satisfied
  • To do this, we can use RDD

• Suppose we model the following potential outcomes:

  • Our potential outcomes $Y^{0}$ against some covariate $X$
  • Out potential outcomes $Y^{1}$ against some covariate $X$
• The continuity assumption says that if we place any cutoff $c_{0}$ on the range of $X$ and assign $Y$ to be $Y^{0}$ when $X > c_{0}$ and $Y^{1}$ when $X < c_{0}$, then any discontinuity implies some causality

Defining Requirements for RDD

• The requirement for RDD to estimate a causal effect are the continuity assumptions
• Meaning, the expected potential outcomes change smoothly as a function of the running variable through the cutoff

• This means that the only thing that causes the outcome to change abruptly at $c_{0}$ is the treatment (and not confounders)

  • Implying, we can only correctly intrepret causal effects about the treatment if there is a discontinuity between the control and treatment, but not at the cutoff for a confounder

• But, this can be violated in practice if any of the following is true:

  • The assignment rule is known in advance
  • Agents are interested in adjusting
  • Agents have time to adjust
  • The cutoff is endogenous to factors that independently cause potential outcomes to shift

Introducing Two Types of RDD

• There are generally accepted two kinds of RDD studies
• A sharp RDD is a design where the probability of treatment goes from $0$ to $1$ at the cutoff
• A fuzzy RDD is a design where the probability of treatment discontinuously increases at the cutoff
• In both designs, there is always a running variable $X$ where the likelihood of receiving treatment flips when reaching some cutoff $c_{0}$

Estimating Causal Effects using a Sharp RDD

• In sharp RDD, treatment is a deterministic function of the running variable $X$

  • Don't confuse this $X$ with observed or unobserved covariates
  • Here, $X$ refers to a running variable, which represents a continuous variable that assigns units to a treatment $D$

• An example of this might be Medicare enrollment

  • Which, happens sharply at age $65$
• The sharp RDD estimation is interpreted as an average causal effect of the treatment as the running variable $X$ approaches the cutoff $c_{0}$
• This average causal effect is the local average treatment effect (LATE)

Estimating Causal Effects using a Fuzzy RDD

• A fuzzy RDD represents a discontinuous jump in the probability of treatment when $X > c_{0}$

• The identifying assumptions are the same under fuzzy designs as they are under sharp designs

  • Meaning, they satisfy the continuity assumptions

References

• Lecture Slides about Causality and RDD
• Lecture Slides about Causality and Stratification
• Causal Inference Textbook
• Python Causality Handbook
• Comprehensive Causal Inference Textbook
• RDD Animation on Twitter