# RDD, wages, and day of birth

What I am doing: I use the regression discontinuity design (henceforth RDD) to study differences in wages received by people born at the end of the year and by those people born at the beginning of the year. I have a panel dataset with observations on 5 years in a row and use Stata. My running variable is the amount of days between the date of birth and the end of the calendar year. The cut-off date is January 1st.

The problem: I pool observations across years, but in doing so now the running variable presents no value on its left.

My proposed solution to this problem: I move the time window of interest from Jan-Dec to July-June. This implies the creation of a new running variable where 0 is assigned to people born on July 1st,..., 30 is assigned to people born on July 31st,...,and 364 is assigned to people born on June 30th. Now the cut-off value is 183, which still corresponds to January 1st. Finally, I also center this new variable as recommended by the literature on RDD, so that July 1st is now -183, January 1st is now 0, and December 31st is +183. Now I have observations both on the left and the right of the cut-off value of the running variable; this new variable also allows me to run the RDD with different bandwidths.

My questions:

1. What do you think about this new re-scaled variables? Could this solution be considered data manipulation in bad sense? (like, am I inventing data and get results that do not reflect reality?)
2. I am using the sharp RDD, is this appropriate?
3. If sharp RDD is appropriate, should I use also the fuzzy RDD as robustness check? (As robustness checks I already use different tools, as proposed here)

This question is present also in "Cross Validated."

(3) Sharp RD is just a special case of fuzzy RD. Say $Y$ is your outcome of interest (in this case, wages), $X$ is your running variable (day of the year), $D$ is a dummy variable for treatment assignment where $D=1$ if $X>c$ and $D=0$ if $X\leq c$, and $c$ is the cutoff (December 31). Then the treatment effect for the fuzzy design is (from Lee and Lemieux 2010):
$$\tau_{fuzzy} = \dfrac{\lim_{\varepsilon\downarrow0} \mathbb{E} \left[ Y | X = c + \varepsilon \right] - \lim_{\varepsilon\uparrow0} \mathbb{E} \left[ Y | X = c + \varepsilon \right]}{\lim_{\varepsilon\downarrow0} \mathbb{E} \left[ D | X = c + \varepsilon \right] - \lim_{\varepsilon\uparrow0} \mathbb{E} \left[ D | X = c + \varepsilon \right]}$$
Because $X>c$ means by definition $D=1$, the denominator reduces to $1$, leaving us with the sharp treatment effect.