Manipulation of the running variable - RD design

[Kathrin Me]

Hi all,

I am performing an estimation using a sharp RDD. A graphical inspection of bunching at the cutoff gives the following picture:

I created the graph by using:

twoway__histogram_gen logwpop if logwpop>=-.15, start(-.15) width(0.01) generate(h x, replace)
twoway (bar h x if x<0, barwidth(0.01) bstyle(histogram)) (bar h x if x>=0, barwidth(0.01) bstyle(histogram)) (lpoly h x if x<0, degree(1) kernel(epanechnikov) lwidth(1) color(blue)) (lpoly h x if x>=0, degree(1) kernel(epanechnikov) lwidth(1) color(red)), ytitle("Density") xtitle("Log population") legend(off)

As my running variable (log population) is discrete I cannot use the McCrary density test without further adaptions.

Yet, I would like to provide a formal test and not just a graphical inspection. I would like to perform a regression analysis using the bin averages and a third-order polynomial specification in the log of population.

I tried the following:

gen Dx=x>0
gen x_2=x^2
gen x_3=x^3
gen x_Dx1=x*Dx
gen x_Dx2=x_2*Dx
gen x_Dx3=x_3*Dx
reg h Dx x x_2 x_3 x_Dx1 x_Dx2 x_Dx3, robust

I get a coefficient of .9317 (significant at the 1-percent level). However, I am not sure whether I can do it as simple as this as my Stata output shows me that in the regression only 30 observations are used (as I have 30 bins).

So my question is whether I can do it like this and if not, how can I then perform a regression analysis using the bin averages and a third-order polynomial specification in the log of population.

Thank you in advance already
Best, Kathrin

[Jackson Monroe]

How is the log of population not functionally continuous? Use a McCrary test. Nothing in life is continuous continuous.

[Kathrin Me]

Yet the McCrary Density Test would not be valid as the chosen bin sizes would be too small and the the McCrary test would falsely reject (it is written so in a critique paper of the study I want to replicate). Maybe I could fix this? But I don't know how using

DCdensity logwpop, breakpoint(0) generate (Xj Yj r0 fhat se_fhat)

Thanks again!

[Jackson Monroe]

Perhaps a permutation test (authors call it Randomization Inference)? https://www.degruyter.com/document/d...2013-0010/html

Doesn't have parametric restrictions and is underutilized in my opinion. I know Stata has some ability to implement these procedures.

[Kathrin Me]

Thanks for the advice! I will definitely look into the paper.

Inbetween I checked how the problem is handled in the critique paper:
The authors of the critique-paper run a McCrary test making the following adaptions: 1) It is required that the bin width of the historam takes an integer value (or alternatively set the treshold to -0.5) and 2) using absolute deviations (i.e. the number of inhabitants from the threshold) than relative deviation (i.e. the percentage distance from thresholds)

For 1) I simply write:

DCdensity logwpop, breakpoint(-0.5) generate (Xj Yj r0 fhat se_fhat)

but I wouldn't know how to implement 2)?