27.16 Evaluation of an RD

• Evidence for (either formal tests or graphs)

  • Treatment and outcomes change discontinuously at the cutoff, while other variables and pre-treatment outcomes do not.

  • No manipulation of the assignment variable.

• Results are robust to various functional forms of the forcing variable

• Is there any other (unobserved) confound that could cause the discontinuous change at the cutoff (i.e., multiple forcing variables / bundling of institutions)?

• External Validity: How likely the result at the cutoff will generalize?

General Model

$$Y_i = \beta_0 + f(x_i) \beta_1 + [I(x_i \ge c)]\beta_2 + \epsilon_i$$

where $f(x_i)$ is any functional form of $x_i$

Simple case

When $f(x_i) = x_i$ (linear function)

$$Y_i = \beta_0 + x_i \beta_1 + [I(x_i \ge c)]\beta_2 + \epsilon_i$$

RD gives you $\beta_2$ (causal effect) of $X$ on $Y$ at the cutoff point

In practice, everyone does

$$Y_i = \alpha_0 + f(x) \alpha _1 + [I(x_i \ge c)]\alpha_2 + [f(x_i)\times [I(x_i \ge c)]\alpha_3 + u_i$$

where we estimate different slope on different sides of the line

and if you estimate $\alpha_3$ to be no different from 0 then we return to the simple case

Notes:

• Sparse data can make $\alpha_3$ large differential effect

• People are very skeptical when you have complex $f(x_i)$, usual simple function forms (e.g., linear, squared term, etc.) should be good. However, if you still insist, then non-parametric estimation can be your best bet.

Bandwidth of $c$ (window)

• Closer to $c$ can give you lower bias, but also efficiency

• Wider $c$ can increase bias, but higher efficiency.

• Optimal bandwidth is very controversial, but usually we have to do it in the appendix for research article anyway.

• We can either

  • drop observations outside of bandwidth or

  • weight depends on how far and close to $c$