Stata – Parametric vs. Non-Parametric Results in Regression Discontinuity Design

regression-discontinuitystata

I am using the parametric approach and non-parametric (local linear regression) approaches of regression discontinuity design (RDD) to compute the treatment effect using Stata.

To get the user-written rd and the 102nd Congress data, I do this:

net get rd
use votex

The local linear approach:

rd lne d,bw(0.20) mbw(100) ker(rec)
Two variables specified; treatment is 
assumed to jump from zero to one at Z=0. 

 Assignment variable Z is d
 Treatment variable X_T unspecified
 Outcome variable y is lne

Estimating for bandwidth .2
------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwald |  -.1046939   .1147029    -0.91   0.361    -.3295075    .1201197
-----------------------------------------------------------------------------

As far as I understand this is equivalent to following :

gen win_d=win*d
reg lne d win win_d if d>=-0.2 & d<=0.2

      Source |       SS       df       MS              Number of obs =     267
-------------+------------------------------           F(  3,   263) =    0.43
       Model |  .271662326     3  .090554109           Prob > F      =  0.7339
    Residual |  55.7885045   263  .212123591           R-squared     =  0.0048
-------------+------------------------------           Adj R-squared = -0.0065
       Total |  56.0601668   266  .210752507           Root MSE      =  .46057

------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           d |   .8450601   .7855123     1.08   0.283    -.7016333    2.391753
         win |  -.1046939   .1257913    -0.83   0.406    -.3523801    .1429923
       win_d |  -.8707605   1.048807    -0.83   0.407    -2.935887    1.194366
       _cons |   21.44195   .0925378   231.71   0.000     21.25974    21.62415
------------------------------------------------------------------------------

However, when we use the parametric approach (let's say with the polynomial of order one), we use all the observations. But, I am trying to see how parametric approach can be compared with non-parametric approach with the same number of observation as in non-parametric approach. So, I do as follows:

reg lne d win if d>=-0.2 & d<=0.2

      Source |       SS       df       MS              Number of obs =     267
-------------+------------------------------           F(  2,   264) =    0.30
       Model |  .125446108     2  .062723054           Prob > F      =  0.7440
    Residual |  55.9347207   264  .211873942           R-squared     =  0.0022
-------------+------------------------------           Adj R-squared = -0.0053
       Total |  56.0601668   266  .210752507           Root MSE      =   .4603

------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           d |   .3566172   .5201877     0.69   0.494    -.6676274    1.380862
         win |  -.0964314   .1253232    -0.77   0.442    -.3431916    .1503288
       _cons |   21.39136   .0696112   307.30   0.000      21.2543    21.52843
------------------------------------------------------------------------------

My concern is why the non-parametric approach result (-.1046939) is not the same as parametric approach (-.0964314), although we are using the same observation for both.

Best Answer

This is happening because you are restricting the effect of Democratic vote share to be the same on both sides of the cutoff in your third specification, which is a slightly different model. As the magnitude and significance of the interaction term in (2) tells you, the slopes are actually somewhat different:

enter image description here

Graph code:

tw (lfit lne d if inrange(d,-.2,0)) (lfit lne d if inrange(d,0,.2)), legend(off) ylab(#15, angle(0)) ytitle("lne") xtitle("d")

You may want something like my third specification (though it it not clear what you have in mind with the comparison):

. use votex, clear
(102nd Congress)

. /* RD/local linear regression model */
. rd lne d, mbw(100) bw(0.2) ker(rec)
Two variables specified; treatment is 
assumed to jump from zero to one at Z=0. 

 Assignment variable Z is d
 Treatment variable X_T unspecified
 Outcome variable y is lne

Estimating for bandwidth .2
------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwald |  -.1046939   .1147029    -0.91   0.361    -.3295075    .1201197
------------------------------------------------------------------------------

. 
. /* OLS Version With Interactions */
. reg lne c.d##i.win if d > -.2 & d < .2 // note that you can specify interaction on the fly

      Source |       SS       df       MS              Number of obs =     267
-------------+------------------------------           F(  3,   263) =    0.43
       Model |  .271662281     3  .090554094           Prob > F      =  0.7339
    Residual |  55.7885045   263  .212123591           R-squared     =  0.0048
-------------+------------------------------           Adj R-squared = -0.0065
       Total |  56.0601668   266  .210752507           Root MSE      =  .46057

------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           d |     .84506   .7855123     1.08   0.283    -.7016333    2.391753
       1.win |  -.1046939   .1257913    -0.83   0.406    -.3523801    .1429923
             |
     win#c.d |
          1  |  -.8707604   1.048807    -0.83   0.407    -2.935887    1.194366
             |
       _cons |   21.44195   .0925378   231.71   0.000     21.25974    21.62415
------------------------------------------------------------------------------

. 
. /* OLS Model Without Interaction */
. reg lne d if d >= -.2 & d < 0 // fit a line to the left

      Source |       SS       df       MS              Number of obs =     109
-------------+------------------------------           F(  1,   107) =    1.58
       Model |  .245503732     1  .245503732           Prob > F      =  0.2116
    Residual |  16.6357215   107  .155474033           R-squared     =  0.0145
-------------+------------------------------           Adj R-squared =  0.0053
       Total |  16.8812252   108  .156307641           Root MSE      =   .3943

------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           d |     .84506   .6724925     1.26   0.212    -.4880779    2.178198
       _cons |   21.44195   .0792234   270.65   0.000     21.28489      21.599
------------------------------------------------------------------------------

. reg lne d if d >= 0 & d < .2 // fit a line to the right

      Source |       SS       df       MS              Number of obs =     158
-------------+------------------------------           F(  1,   156) =    0.00
       Model |  .000290102     1  .000290102           Prob > F      =  0.9729
    Residual |   39.152783   156  .250979378           R-squared     =  0.0000
-------------+------------------------------           Adj R-squared = -0.0064
       Total |  39.1530731   157  .249382631           Root MSE      =  .50098

------------------------------------------------------------------------------
         lne |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           d |  -.0257003    .755932    -0.03   0.973    -1.518883    1.467483
       _cons |   21.33725   .0926828   230.22   0.000     21.15418    21.52033
------------------------------------------------------------------------------

. 
. di  "RD is "21.33725 - 21.44195 // TE is the diff in the intercepts 
RD is -.1047

Citations

Card, David & Alan Krueger. 1994. Minimum wages and employment: A case study of the fast-food industry in New Jersey and Pennsylvania. The American Economic Review 84(4):772-793. PDF Here.
Ridgeway, Greg. 2006. Assessing the effect of race bias in post-traffic stop outcomes using propensity scores. Journal of Quantitative Criminology 22(1): 1-29. PDF Here

Solved – Graphs in regression discontinuity design in “Stata” or “R”

Is this much different from doing two local polynomials of degree 2, one for below the threshold and one for above with smooth at $K_i$ points? Here's an example with Stata:

use votex // the election-spending data that comes with rd

tw 
(scatter lne d, mcolor(gs10) msize(tiny)) 
(lpolyci lne d if d<0, bw(0.05) deg(2) n(100) fcolor(none)) 
(lpolyci lne d if d>=0, bw(0.05) deg(2) n(100) fcolor(none)), xline(0)  legend(off)

Alternatively, you can just save the lpoly smoothed values and standard errors as variables instead of using twoway. Below $x$ is the bin, $s$ is the smoothed mean, $se$ is the standard error, and $ul$ and $ll$ are the upper and lower limits of the 95% Confidence Interval for the smoothed outcome.

lpoly lne d if d<0, bw(0.05) deg(2) n(100) gen(x0 s0) ci se(se0)
lpoly lne d if d>=0, bw(0.05) deg(2) n(100) gen(x1 s1) ci se(se1)

/* Get the 95% CIs */
forvalues v=0/1 {
    gen ul`v' = s`v' + 1.95*se`v' 
    gen ll`v' = s`v' - 1.95*se`v' 
};

tw 
(line ul0 ll0 s0 x0, lcolor(blue blue blue) lpattern(dash dash solid)) 
(line ul1 ll1 s1 x1, lcolor(red red red) lpattern(dash dash solid)), legend(off)

As you can see, the lines in the first plot are the same as in the second.

Best Answer

Related Solutions

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Citations

Solved – Graphs in regression discontinuity design in “Stata” or “R”

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