## 24.10 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 $$c$$