Re: st: Polynomial Fitting and RD Design

[Nick Cox <njcoxstata@gmail.com>]

Even if you can get this to work as intended, look at the sizes of
those coefficients! The resultant curve may look about right, but this
is a dubious thing to do numerically and statistically. I can't
comment on the underlying scientific rationale for quartics here,
although I will guess wildly that there isn't one.

Nick

On Thu, Sep 1, 2011 at 3:59 AM, Austin Nichols <austinnichols@gmail.com> wrote:
> Patrick Button <pbutton@uci.edu>:
> Try redefining your x so that the discontinuity is at zero.
>
> On Wed, Aug 31, 2011 at 9:54 PM, Patrick Button <pbutton@uci.edu> wrote:
>> Hello Stata users,
>>
>> I've been getting some unexpected Stata output when fitting polynomials
>> using a pretty simple OLS regression.
>>
>> I am replicating a regression discontinuity design paper (Lee, Moretti and
>> Butler 2004). The paper is here:
>> http://emlab.berkeley.edu/~moretti/final.pdf Code and data are here:
>> http://emlab.berkeley.edu/~moretti/data3.html (I am using enricoall2.dta).
>>
>> I need to run a regression that fits a 4th degree polynomial separately
>> for points of the running variable, x, below 0.5 and above 0.5. The
>> regression includes a dummy variable for if x >= 0.5 or not as well. If
>> there is a discontinuity at 0.5, then this is picked up in the coefficient
>> on that dummy variable.
>>
>> In this case the running variable is the vote share that the Democratic
>> candidate got in U.S. House of Representatives elections, including just
>> the Democratic and Republican votes. So x < 0.5 means a Republican won,
>> and >= 0.5 means a Democrat won.
>>
>> I would like to pool the data instead of running a separate regression for
>> each side. This is one of the recommended methods in the RD literature.
>> For some reason this method does not appear in the authors' code so I need
>> to do it myself.
>>
>> I'm running and setting up the regression as follows:
>>
>> ***
>> gen x = demvoteshare
>>
>> gen D = 1 if x >=0.5
>> replace D = 0 if x < 0.5
>>
>> *Left Side Polynomial
>> gen xa = (1-D)*x
>> gen x2a = (1-D)*x^2
>> gen x3a = (1-D)*x^3
>> gen x4a = (1-D)*x^4
>>
>> *Right Side Polynomial
>> gen xb = D*x
>> gen x2b = D*x^2
>> gen x3b = D*x^3
>> gen x4b = D*x^4
>>
>> regress realincome D xa x2a x3a x4a xb x2b x3b x4b
>>
>> ***
>>
>> Based on what the authors of the paper got, graphical analysis, and logic,
>> there should be no jump in realincome at 0.5. There is no reason why
>> income should be suddenly much different for districts that democrats just
>> barely won or just barely lost. If it is, this invalidates the regression
>> discontinuity design. So the coefficient on D should be statistically
>> insignificant. However, I get the following results:
>>
>> ------------------------------------------------------------------------------
>>  realincome |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
>> Interval]
>> -------------+----------------------------------------------------------------
>>           D |   497414.5   94802.12     5.25   0.000       311589
>> 683240.1
>>          xa |   34396.25   27783.67     1.24   0.216    -20063.66
>> 88856.17
>>         x2a |  -22571.61   234577.9    -0.10   0.923    -482377.5
>> 437234.3
>>         x3a |  -429659.3   655505.3    -0.66   0.512     -1714542
>> 855223.6
>>         x4a |   667813.9   598416.4     1.12   0.264    -505166.7
>> 1840795
>>          xb |   -2805647   534665.3    -5.25   0.000     -3853667
>> -1757628
>>         x2b |    5828381    1112850     5.24   0.000      3647038
>> 8009724
>>         x3b |   -5281210    1012800    -5.21   0.000     -7266441
>> -3295979
>>         x4b |    1754682   339914.5     5.16   0.000      1088402
>> 2420963
>>       _cons |   31536.64   501.1422    62.93   0.000     30554.33
>> 32518.95
>> ------------------------------------------------------------------------------
>>
>> I have no idea why D is statistically significant, and why only the
>> polynomial on the right side is statistically significant. This is not
>> just a problem with this regression. I get messed up results for every
>> regression I run that has a 4th degree polynomial on each side of 0.5.
>>
>> However, I do not get weird results like this when I use just one 4th
>> degree polynomial (one for the entire thing) with the D dummy.
>>
>> Does anyone know what I am doing wrong? I have no idea but I have a
>> feeling that i'm missing something obvious.
>>
>> Thank you very much for your time and consideration.
>>
>> --

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