RDD, wages, and day of birth

Question

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."

Answer 1

(1) Since the RD design identifies an effect only at the cutoff (in your case, at precisely 12am on the last day of the year), I don't think it's an issue of data manipulation for you to be removing the tails. Your estimates really shouldn't be sensitive to observations far away from the cutoff, but this depends somewhat on how granular your data is and how many observations you have (a lot of birth data is quarterly, which may make this strategy difficult).

(2) Sharp vs. fuzzy design refers to the assignment of the treatment. If your treatment of interest is being born December 31st vs. being born January 1st, then this is a sharp design because the running variable deterministically assigns treatment. If there is noncompliance with treatment, (i.e you were born December 30 but you somehow end up getting "treated" as though you were born in January the following year) this is when you would use the fuzzy design.

(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.