What equation corresponds to rdrobust in Stata?

[Izzy Benasso]

I'm using rdrobust in Stata for a sharp Regression Discontinuity design and need to state the equation that corresponds to my results table, but I'm unsure of the correct form. I estimate the effect of crossing an age-80 cutoff on the probability of invasive treatment:
rdrobust received_invasive days_from_80, c(0) masspoints(check)
I'm currently using the standard local-linear RD specification (see image) as the equation to go alongside my stata results output.

Two questions:

Is this the right equation to present alongside rdrobust output (photo above)? My understanding is that it represents the local-linear fit within the MSE-optimal bandwidth, with β₁ as the discontinuity. One thing I want to get right: β₁ from this equation reproduces the conventional estimate, whereas the CI and p-value rdrobust reports are the robust bias-corrected ones. Is it standard to present this equation as the estimand even though the reported inference comes from the bias-corrected procedure?

I've recovered the intercepts and slopes on each side from the stored results and the intercept difference reproduces the RD effect exactly. But rdrobust doesn't report inference for these slopes. Am I right that they shouldn't appear in the results table?

[Nick Cox]

Cross-posted at https://stats.stackexchange.com/ques...obust-in-stata

[Andrew Musau]

I'm using rdrobust in Stata for a sharp Regression Discontinuity design and need to state the equation that corresponds to my results table

rdrobust is from the Stata Journal (FAQ Advice #12). It estimates the same underlying local linear RD model regardless of whether the inference is conventional or robust bias-corrected. The bias correction affects the estimated standard errors, confidence intervals, and \(p\)-values and not the underlying regression specification. This is no different from estimating a linear regression model using OLS with conventional versus heteroskedasticity-robust standard errors: the regression equation is identical; only the inference changes. Your equation is therefore appropriate. However, since this is an RD design, you'll see that most papers present the model using the standard local-polynomial notation rather than as an interaction regression. The local linear RD model is


\[ Y_i = \alpha + \tau D_i + \beta_1(X_i-c) + \beta_2D_i(X_i-c) + \varepsilon_i, \] where \[ D_i=\mathbf{1}(X_i\ge c), \]


and \(X_i\) is the running variable (age), \(c=80\) is the cutoff, and \(\tau\) is the treatment effect (the jump at the cutoff). This specification is exactly equivalent to your interaction model after defining

\[ d_i=X_i-c, \qquad \text{over80}_i=D_i=\mathbf{1}(X_i\ge c). \]

Note that the reported point estimate is the local linear RD estimate of \(\tau\) and corresponds to the uncorrected local-linear estimator. The reported confidence intervals and \(p\)-values are the robust bias-corrected inference recommended by Calonico, Cattaneo, and Titiunik. Finally, the slope coefficients on either side of the cutoff are nuisance parameters used to estimate the discontinuity and construct the bias correction. They are generally not reported, and rdrobust does not provide inference for them because they are not the object of interest.