Comparing fuzzy RD estimates to an "OLS" analogue In IV, I've sometimes seen the estimates compared to OLS to give a sense of how the LATE may compare to ATE.  
What would the analogue be in fuzzy RD?  What is the benchmark estimate?  
I'd imagine the analogue would be the structural equation without instrumenting for the endogenous variable?  Would you include the RD polynomial in that?  Would you limit it to a narrow bandwidth?  
Does the following make sense?
Suppose the RD is for a test-score cutoff for a acceptance to an academic program, and the fuzzy RD looks at the impact of being apart of that program on future earnings.
The OLS baseline would be regress wages on a dummy for participation in the academic program and the level of the test score?  The instrument is not 
The IV baseline would be regress wages on participation in the program and the level of the test score, and instrument participation with a first stage that regresses participation on a dummy for whether the test score was above the threshold plus the level of the test score?  
Then, the RD nails that by doing 2 things: First, it replaces the level of the test score with a range of flexible polynomials in the level of test score: piecewise linear, quadratic etc ... Second, it limits the sample to narrower and narrower bandwidths around the discontinuity.  Presenting estimates of numerous permutations between bandwidth and polynomial order?
Is that a sensible framework for comparing ATE, LATE, and Weighted LATE (ie fuzzy RD)?
 A: First of all, you seem to mix up a few of the treatment effect concepts. IV results are not compared to OLS to see how the LATE compares to the ATE. That's because OLS doesn't estimate the ATE which is the reason why you instrumented in the first place.
If the techniques you mention provide consistent estimates, then


*

*OLS estimates the ATE, $E[Y_{i1} - Y_{i0}]$

*IV estimates the LATE for a subpopulation affected by the instrument (the compliers), $E[Y_{i1} - Y_{i0}\mid Z_i]$

*RD estimates the ATE at the cutoff where you compare treated and untreated units just before and after this cutoff, $E[Y_{i1}-Y_{i0}\mid X_i = \tilde{x}]$


but again: had OLS given you consistent estimates to begin with, there would not have been the need to use IV or RD. When you compare your estimates, you can compare fuzzy RD to OLS as you would have done with your IV results. In fact, fuzzy RD is IV.
You have a running/forcing variable $X_i$ and a cutoff value $X_i = \tilde{x}$ but now the treatment $D_i$ is not a deterministic function of this cutoff anymore (as in the sharp RD case) but the probability of getting the treatment is affected by being above the threshold. Then you have a dummy $T_i$ which is one for those above the cutoff and zero otherwise, and you run the two regressions:
$$
\begin{align}
D_i &= \gamma_0 + \gamma_1 X_i + \gamma_2 X_i^2 + ... + \gamma_p X_i^p + \rho T_i + e_i \newline
Y_i &= \beta_0 + \beta_1 X_i + \beta_2 X_i^2 + ... + \beta_p X_i^p + \delta D_i + u_i
\end{align}
$$
with the appropriate number of polynomials.
Fuzzy RD estimates the LATE and the compliers are those individuals whose treatment status changes as we increase their $X_i$ from just before the threshold to just after the threshold. Having clarified the concept of fuzzy RD this should help you to know what you can compare it to.
The main selling point with RD though has to be made by the graphs and less so by the numbers. For some practical guidelines and further reading see Lee and Lemieux (2010) "Regression Discontinuity Designs in Economics".
