Interpreting results from Regression Kink Design with -rdrobust-

[Christian Rhind]

I am attempting to estimate a Regression Kink Design in Stata using -rdrobust- and -rdplot- commands and I am having a bit of trouble interpreting my results.
My assignment variable is income and I have a binary outcome y. Individuals above some income threshold c receive treatment.

 gen income_c = (income - c)/10000

When I run rdplot

 rdplot y income_c, p(1)

I get the following result



It appears that the proportion of individuals for whom the outcome is equal to one increases up to the cutoff, then becomes flat. This is consistent with the result I obtain if I run a simple OLS regression interacting the centered income variable with the threshold.

gen kink = income if income > 0
replace kink = 0 if income<=0
regress y income kink

The coefficient on 'kink' is negative:

      Source |       SS       df       MS              Number of obs =    5860
-------------+------------------------------           F(  2,  5857) =   91.57
       Model |   24.664251     2  12.3321255           Prob > F      =  0.0000
    Residual |  788.780118  5857  .134673061           R-squared     =  0.0303
-------------+------------------------------           Adj R-squared =  0.0300
       Total |  813.444369  5859  .138836724           Root MSE      =  .36698

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    income_c |   .0259426   .0020376    12.73   0.000     .0219481    .0299371
        kink |   -.031287   .0042029    -7.44   0.000    -.0395264   -.0230477
       _cons |    .251795   .0090023    27.97   0.000     .2341471    .2694429
------------------------------------------------------------------------------

But for some reason (I don't know why) -rdrobust- returns a positive point estimate

 rdrobust y income_c, p(1) deriv(1)

         Cutoff c = 0 | Left of c  Right of c               Number of obs =       5860
----------------------+----------------------               BW type       =      mserd
        Number of obs |      4508        1352               Kernel        = Triangular
   Eff. Number of obs |      1677         767               VCE method    =         NN
 Order loc. poly. (p) |         2           2
       Order bias (q) |         3           3
    BW loc. poly. (h) |     3.086       3.086
          BW bias (b) |     4.525       4.525
            rho (h/b) |     0.682       0.682

Outcome: y. Running variable: income_c.
--------------------------------------------------------------------------------------
               Method |    Coef.   Std. Err.     z      P>|z|     [95% Conf. Interval]
----------------------+---------------------------------------------------------------
         Conventional |   .06264      .091    0.6883    0.491      -.115727    .241003
               Robust |      -         -      0.4178    0.676      -.205286    .316505
--------------------------------------------------------------------------------------

At this stage I am not concerned that the coefficient is insignificant, but I would expect it to at least have the correct sign. Does anyone have any idea what might be going on?

Thanks for any help.

[Vicente Dinis Valentim]

I understand this is old and you may not need an answer anymore. But my answer would be that the OLS regression takes the full scope of the data, and for that reason it estimates a positive slope left of the cutoff. The rdrobust only takes observations in the vicinity of the threshold. It can be that within that bandwidth, the positive slope becomes flatter and the predicted value at the threhold is lower than the one at the right. You can try and have a plot only within the optimal bandwidth given by rdrobust and fit a regression line at both sides of the cutoff, to see if it seems like this is happening.