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Consider a regression discontinuity setting in which $x_2=1$ if and only if $x_1 \ge D$ and $x_2 = 0$ otherwise. The baseline regression discontinuity regression for some outcome, $y$, is:

$$y_i =\beta_0 +\beta_1 (x_{1i} -D)+\beta_2 x_{2i} +\beta_3 (x_{1i}-D)x_{2i} +u_i $$

This could be estimated with OLS. Alternatively, more complicated kernel estimation could be utilized. The key is that data are restricted to observations for which $x_{1i}$ is in a bandwidth around $D$

My understanding is that Calonico, Cattaneo, and Titiunik's (CCT) bandwidth minimizes mean squared error of the estimated coefficient for $x_2$ (that is, $\beta_2$) when using a bias-correction approach. My understanding is that their method is the most recent innovation in terms of regression discontinuity bandwidths (after the Imbens and Kalyanaraman 2012 bandwidth). If there is a more recent development, I would be delighted to learn of it.

It appears to me that the CCT bandwidth is symmetic, i.e., we'd restrict to data for which $x_1 \in [D-h, D+h]$ rather than asymmetric, as in $x_i \in [D-h_1, D+h_2]$ for which $h_1\ne h_2$.

Are there any papers that develop theory in which an asymmetric bandwidth is optimal in some sense? Or are there any major implementations of regression discontinuity with an asymmetric bandwidth?

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Imben and Kalyanaraman (2012) discuss this possibility in section 3.1. They say that there may be theretical benefits, but in practice it would be impossible to implement well, and do not consider this.

However Calonico, Cattaneo, and Titiunik (2014) do allow for this (see Remark 9), ostensibly with no problems. It's rarely implemented. One paper that uses this method is, Carlson, Correia, and Luck (2022).

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