# Bandwidth Selection for Distance-Based RD Designs `rdbw2d.dist` implements bandwidth selector for distance-based local polynomial boundary regression discontinuity (RD) point estimators with robust bias-corrected pointwise confidence intervals and uniform confidence bands, developed in Cattaneo, Titiunik and Yu (2025a) with a companion software article Cattaneo, Titiunik and Yu (2025b). For robust bias-correction, see Calonico, Cattaneo, Titiunik (2014). ```r rdbw2d.dist( Y, D, b = NULL, p = 1, kink = c("off", "on"), kernel = c("tri", "triangular", "epa", "epanechnikov", "uni", "uniform", "gau", "gaussian"), bwselect = c("mserd", "imserd", "msetwo", "imsetwo"), vce = c("hc1", "hc0", "hc2", "hc3"), bwcheck = 20 + p + 1, masspoints = c("check", "adjust", "off"), C = NULL, scaleregul = 1, cqt = 0.5 ) ``` ## Arguments - `Y`: Dependent variable; a numeric vector of length $N$, where $N$ is the sample size. - `D`: Distance-based scores $\mathbf{D}_i=(\mathbf{D}_{i}(\mathbf{b}_1),\cdots,\mathbf{D}_{i}(\mathbf{b}_J))$; dimension is $N \times J$ where $N$ = sample size and $J$ = number of cutoffs; non-negative values means data point in treatment group and negative values means data point in control group. - `b`: Optional evaluation points; a matrix or data frame specifying boundary points $\mathbf{b}_j = (b_{1j}, b_{2j})$, dimension $J \times 2$. - `p`: Polynomial order for point estimation. Default is `p = 1`. - `kink`: Logical; whether to apply kink adjustment. Options: `"on"` or `"off"` (default). - `kernel`: Kernel function to use. Options are `"unif"`, `"uniform"` (uniform), `"triag"`, `"triangular"` (triangular, default), and `"epan"`, `"epanechnikov"` (Epanechnikov). - `bwselect`: Bandwidth selection strategy. Options: * `"mserd"`. One common MSE-optimal bandwidth selector for the boundary RD treatment effect estimator for each evaluation point (default). * `"imserd"`. IMSE-optimal bandwidth selector for the boundary RD treatment effect estimator based on all evaluation points. * `"msetwo"`. Two different MSE-optimal bandwidth selectors (control and treatment) for the boundary RD treatment effect estimator for each evaluation point. * `"imsetwo"`. Two IMSE-optimal bandwidth selectors (control and treatment) for the boundary RD treatment effect estimator based on all evaluation points. * `"user provided"`. User-provided bandwidths. If `h` is not `NULL`, then `bwselect` is overwritten to `"user provided"`. - `vce`: Variance-covariance estimator for standard errors. Options: - **`"hc0"`**: Heteroskedasticity-robust variance estimator without small sample adjustment (White robust). - **`"hc1"`**: Heteroskedasticity-robust variance estimator with degrees-of-freedom correction (default). - **`"hc2"`**: Heteroskedasticity-robust variance estimator using leverage adjustments. - **`"hc3"`**: More conservative heteroskedasticity-robust variance estimator (similar to jackknife correction). - `bwcheck`: If a positive integer is provided, the preliminary bandwidth used in the calculations is enlarged so that at least `bwcheck` observations are used. The program stops with “not enough observations” if sample size $N$ < `bwcheck`. Default is `50 + p + 1`. - `masspoints`: Strategy for handling mass points in the running variable. Options: - **`"check"`**: Check for repeated values and adjust inference if needed (default). - **`"adjust"`**: Adjust bandwidths to guarantee a sufficient number of unique support points. - **`"off"`**: Ignore mass points completely. - `C`: Cluster ID variable used for cluster-robust variance estimation with degrees-of-freedom weights.Default is `C = NULL`. - `scaleregul`: Scaling factor for the regularization term in bandwidth selection. Default is `1`. - `cqt`: Constant controlling subsample fraction for initial bias estimation. Default is `0.5`. ## Returns An object of class `"rdbw2d.dist"`, containing: - **`bws`**: Data frame of optimal bandwidths for each evaluation point: - **`b1`**: First coordinate of the evaluation point $b1$. - **`b2`**: Second coordinate of the evaluation point $b2$. - **`h0`**: Bandwidth for observations with distance $D_{i}(\mathbf{b}) < 0$. - **`h1`**: Bandwidth for observations with distance $D_{i}(\mathbf{b}) \geq 0$. - **`Nh0`**: Effective sample size for $D_{i}(\mathbf{b}) < 0$. - **`Nh1`**: Effective sample size for $D_{i}(\mathbf{b}) \geq 0$. - **`mseconsts`**: Data frame of intermediate bias and variance constants used for MSE/IMSE calculations. - **`opt`**: List of options used in the function call. ## Examples ```r set.seed(123) n <- 5000 # Generate running variables x1 and x2 x1 <- rnorm(n) x2 <- rnorm(n) # Define treatment assignment: treated if x1 >= 0 d <- as.numeric(x1 >= 0) # Generate outcome variable y with some treatment effect y <- 3 + 2 * x1 + 1.5 * x2 + 1.5 * d + rnorm(n, sd = 0.5) # Define evaluation points (e.g., at the origin and another point) eval <- data.frame(x.1 = c(0, 0), x.2 = c(0, 1)) # Compute Euclidean distances to evaluation points dist.a <- sqrt((x1 - eval$x.1[1])^2 + (x2 - eval$x.2[1])^2) dist.b <- sqrt((x1 - eval$x.1[2])^2 + (x2 - eval$x.2[2])^2) # Combine distances into a matrix D <- as.data.frame(cbind(dist.a, dist.b)) # Assign positive distances for treatment group, negative for control d_expanded <- matrix(rep(2 * d - 1, times = ncol(D)), nrow = nrow(D), ncol = ncol(D)) D <- D * d_expanded # Run the rd2d.dist function bws <- rdbw2d.dist(y, D, b = eval) # View the estimation results print(bws) summary(bws) ``` ## References * [Cattaneo, M. D., Titiunik, R., Yu, R. R. (2025a).](https://rdpackages.github.io/references/Cattaneo-Titiunik-Yu_2025_BoundaryRD.pdf) Estimation and Inference in Boundary Discontinuity Designs * [Cattaneo, M. D., Titiunik, R., Yu, R. R. (2025b).](https://rdpackages.github.io/references/Cattaneo-Titiunik-Yu_2025_rd2d.pdf) rd2d: Causal Inference in Boundary Discontinuity Designs * [Calonico, S., Cattaneo, M. D., Titiunik, R. (2014)](https://rdpackages.github.io/references/Calonico-Cattaneo-Titiunik_2014_ECMA.pdf) Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs ## See Also `rd2d.dist`, `rd2d`, `summary.rdbw2d.dist`, `print.rdbw2d.dist` ## Author(s) Matias D. Cattaneo, Princeton University. cattaneo@princeton.edu Rocío Titiunik, Princeton University. titiunik@princeton.edu Ruiqi Rae Yu, Princeton University. rae.yu@princeton.edu

rdbw2d.dist function