Is the estimand in a Regression Discontinuity Design the ATE, ATT, etc?

Question

For the Regression Discontinuity Design, it exploits a cutoff $c$ which is assumed to assumed to sharply delineate two groups, with individuals just below the cutoff acting as the control group and individuals just above the cutoff serving as the treatment group. Given that this occurs at $c$, it would seem the estimand is

$$ \tau = E[Y(1)-Y(0)|X=c] $$

where $X$ is the running variable. This seems to be a strange estimand and neither the ATE nor the ATT. But the estimand is spoken as if it is an ATE for units around $X=c$. What is the proper classification here?

Answer 1

Quoting the excellent book by Nick Huntington-Klein:

Finally, if we do this all, what kind of treatment effect are we estimating (Chapter 10)? Regression discontinuity is fairly easy to figure out on this front. We’re using variation only from just around the cutoff. So we get the effect of treatment for people who are just around the cutoff. This is a local average treatment effect, getting the weighted average treatment effect for those just on the margin of being given treatment.