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In a typical regression discontinuity design, the standard error of the discontinuity is $se(\beta_2)$ in the following model:

$$Y = \beta_0 + \beta_1 X_1 + \beta_2 D + \beta_3 (D X_1) + U$$

However, how is the same standard error calculated when using local regression (i.e. weighted OLS with a linear or gaussian kernel)?

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Functions of your original independent variables just yield new independent variables. Yes, you might know particular interpretations of the transformations, but the math neither knows nor cares.

If you have an approach to calculating the standard error of a regression coefficient, then use it.

I have written about this on the statistics Stack and like the video I linked in my answer.

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