Notes on causal inference with No Overlap – Regression Discontinuity

Here is my notes on regression discontinuity from Prof. Ding’s textbook (2024) and Prof. Imai’s lecture notes.

Motivation

We often cannot run a randomized experiment and have to use/design observational studies to find a setting where credible causal inference is possible.

The key is the knowledge of treatment assignment mechanism.

Regression discontinuity design (RD Design):

RD Design is a simple and widely used quasi-experimental design. The term “quasi experimental” is to emphasize that these approaches are still framed using concepts from randomized experiments but require econometric innovations to compensate for the lack of random treatment assignment.

1. Sharp RD Design: treatment assignment is based on a deterministic rule (i.e. we have full knowledge of how treatment is assigned)

2. Fuzzy RD Design: encouragement to receive treatment is based on a deterministic rule

Setting

• Binary treatment $Z\in \{0,1\}$

• Potential outcomes $\{Y(0),Y(1)\}$

• There is a running variable $X\in \mathbb{R}$ such that $Z=I(X\ge x_0)$, where $x_0$ is a pre-determined threshold. Note that, the treatment assignment is deterministic

• The unconfoundedness assumption holds automatically $$Z\perp \perp \{Y(1),Y(0)\}\mid X$$

• The overlap assumption does not hold

Identification

• RD can identify a local average causal effect at the cutoff point $x_0$

• Estimand:

• Assumption 1 (continuity assumption).

  1. $E\{Y(1)\mid X=x\}$ is continuous from the right at $x_0$
  2. $E\{Y(0)\mid X=x\}$ is continuous from the left at $x_0$

• We have $$\begin{array}{rl}\text{(continuity)}& E\{Y(1)\mid X=x_0\}& =\underset{\epsilon \to 0+}{lim}E\{Y(1)\mid X=x_0+\epsilon \}\text{(def of Z)}& & =\underset{\epsilon \to 0+}{lim}E\{Y(1)\mid Z=1,X=x_0+\epsilon \}& =\underset{\epsilon \to 0+}{lim}E(Y\mid Z=1,X=x_0+\epsilon ),\end{array}$$ Similarly, $$E\{Y(0)\mid X=x_0\}=\underset{\epsilon \to 0+}{lim}E(Y\mid Z=0,X=x_0-\epsilon )$$ So the local average causal effect at $x_0$ can be identified by the difference of the two limits

• Advantage: internal validity

• Disadvantage: external validity

Key Theorem

$\tau \left(x_0\right)$ is nonparametrically identified.

Reference

Ding, P. (2024). A First Course in Causal Inference. CRC Press.

Lecture notes: Regression Discontinuity Design