Regression Discontinuity Design

Background

Think of a continuous criterion for treatment.

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Sharp Regression Discontinuity

It occurs when treatment is a deterministic and discontinuous function of covariate $x_i$ .

For example, $D_i$ (whether or not being treated) = $1$ if $x_i\ge x_0$ . $0$ otherwise.

An example would be the National Merit Scholarship in US awarded to students on a GPA-like basis ( $x_i$ ). The core assumption here is that, those around the threshold $x_0$ should have similar outcomes, except that those above $x_0$ received the scholarship.

The lack of this setting is that we could never observe two different treatment for the same covariate $x_i$ . Thus we could only need to extrapolate for $x_i$ , do causal inference around (near) the threshold $x_0$ . Because by taking the limit we could find the approximation.

Identification Assumption: Thus we need the covariate to be continuous!

Consider the function that we regress:

$$Y_i=\alpha +\beta x_i+\rho D_i+{\eta }_i$$

The only thing to be cared is that the relationship btw $x_i$ and $D_i$ . Here $D_i$ is not only correlated with $x_i$ , it is a deterministic function of $x_i$ .

除了用线性模型去拟合以外，我们还可以用非线性的模型去拟合，但是这个问题的缺陷就在于，如果你采取了一种没有那么灵活的策略去拟合，那就会带来模型的 misspecification 可能在阈值处产生一个假的”跳跃”，被你误认为是 treatment effect。

Parametric RDD

最常见的错误就是只写了一个普通的 OLS 回归：

$$Y_i=\alpha +\gamma D_i+X^{\prime }\beta +{ϵ}_i$$

这缺少了 running variable 多项式，导致它只是一个普通的 Dummy Variable regression.

正确的 specification 应该是：

$$Y_i=\alpha +\rho D_i+\sum _{k=1}^p{\gamma }_k{\tilde{R}}_i^k+\sum _{k=1}^p{\delta }_k{D}_i\cdot {\tilde{R}}_i^k+{ϵ}_i$$

Also see Nonparametric RDD

• Manipulation in RDD

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Reference

这条笔记主要会使用 EC423 Topic 2 作为参考资料。