rdhte: Empirical Illustration

In rdhte: Heterogeneous Treatment Effects in Regression Discontinuity Designs

knitr::opts_chunk$set(
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Overview

rdhte estimates heterogeneous treatment effects in sharp Regression Discontinuity (RD) designs along one or more pretreatment covariates. The package exposes three main commands:

• rdhte() -- point estimation and robust bias-corrected inference of group-conditional RD effects.
• rdbwhte() -- automatic per-cell bandwidth selection.
• rdhte_lincom() -- linear-combination tests over the estimated CATEs.

This vignette illustrates each feature on a real-data extract from Granzier, Pons, and Tricaud (2023, American Economic Journal: Applied), bundled with the package as rdhte_dataset.

The data and the design

Granzier, Pons, and Tricaud study coordination and bandwagon effects in French two-round elections, where candidates must clear a qualifying-vote threshold in the first round to advance to the runoff. The institutional rule creates a sharp regression-discontinuity design on every candidate's first-round margin against that threshold: candidates just above the cutoff advance, those just below do not, and small differences in first-round support produce a discrete change in runoff participation. The authors use the design to ask whether being just-qualified causally affects subsequent voter and candidate behavior, and document substantive heterogeneity across ideology and candidate-strength dimensions.

The bundled extract has 39,534 candidate-race observations with the following variables:

• y -- 0/1 indicator for advancing to the runoff (the outcome).
• x -- first-round margin against the qualifying threshold (the running variable; cutoff at zero).
• cluster_var -- district identifier, used for cluster-robust inference.
• w_left -- 0/1 indicator for left-of-center parties.
• w_ideology -- unordered party-ideology bucket (4 categories).
• w_strength -- continuous proxy for ex-ante candidate strength (average prior national-level vote share).
• w_strong -- 0/1 indicator for above-median strength.
• w_strength_qrt -- ordered quartile of w_strength (4 levels).

These covariates support every covariate-incorporation pattern rdhte exposes: binary cells, multi-level (unordered and ordered) factor cells, factor-by-factor interactions, continuous slopes, and binary x continuous interactions.

library(rdhte)
library(rdrobust)
library(broom)
data(rdhte_dataset)
attach(rdhte_dataset)

Single binary variable

When covs.hte is binary (or a factor with two levels), rdhte reports one CATE per cell.

summary(rd_left <- rdhte(y = y, x = x, covs.hte = factor(w_left),
                          cluster = cluster_var))

How large is the advantage for left-of-center candidates? Test the difference with rdhte_lincom:

rdhte_lincom(model = rd_left,
             linfct = c("`factor(w_left)1` - `factor(w_left)0` = 0"))

Forcing a common bandwidth across cells (bw.joint = TRUE) can distort the cell-specific inference -- here it makes the right-of-center effect (incorrectly) statistically significant:

summary(rdhte(y = y, x = x, covs.hte = w_left, cluster = cluster_var,
              bw.joint = TRUE))

A 0/1 numeric variable is auto-promoted to a factor; coefficient names then carry the variable name as a prefix:

summary(rd_left2 <- rdhte(y = y, x = x, covs.hte = w_left,
                           cluster = cluster_var))
rdhte_lincom(model = rd_left2,
             linfct = c("`w_left1` - `w_left0` = 0"))

Single categorical variable -- unordered

A multi-level factor produces one CATE per level. The reference category is the factor's first level.

summary(rd_ideology <- rdhte(y = y, x = x,
                              covs.hte = factor(w_ideology),
                              cluster = cluster_var))

Joint test that the three non-reference ideology cells are all indistinguishable from zero:

rdhte_lincom(model = rd_ideology,
             linfct = c("`factor(w_ideology)2` = 0",
                        "`factor(w_ideology)3` = 0",
                        "`factor(w_ideology)4` = 0"))

Single categorical variable -- ordered

For an ordered categorical variable, wrapping in factor() keeps the per-level CATE interpretation:

summary(rdhte(y = y, x = x, covs.hte = factor(w_strength_qrt),
              cluster = cluster_var))

The per-quartile CATE increases monotonically: candidates with higher average strength have larger treatment effects.

A bare numeric variable (no factor()) is treated as continuous -- useful when the ordering is meaningful but you want the linear-in-W parametrization:

summary(rdhte(y = y, x = x, covs.hte = w_strength_qrt,
              cluster = cluster_var))

Two binary variables -- interaction

Interactions of two factors produce one CATE per joint cell. The ordering of cells follows R's interaction() convention.

summary(rdhte(y = y, x = x,
              covs.hte = factor(w_left):factor(w_strong),
              cluster = cluster_var))

The same model can be expressed by pre-building the joint cell variable:

interactions <- 1*(w_left==0)*(w_strong==1) +
                2*(w_left==0)*(w_strong==2) +
                3*(w_left==1)*(w_strong==1) +
                4*(w_left==1)*(w_strong==2)
summary(rdhte(y = y, x = x, covs.hte = factor(interactions),
              cluster = cluster_var))

Single continuous variable

When covs.hte is continuous, rdhte switches to a linear-in-W parametrization of the CATE function:

[ \tau(w) = \beta_T + \beta_{T:W}\,w. ]

summary(rd_continuous <- rdhte(y = y, x = x, covs.hte = w_strength,
                                kernel = "uni",
                                cluster = cluster_var))

To aid interpretation, the slope coefficient is precisely a partial-linear regression slope. With the uniform kernel and a fixed bandwidth, the rdhte point estimates match plain lm() on the same in-bandwidth subset:

trt <- (x > 0)
new.data <- data.frame(y, x, w_strength, trt)
using.lm <- coef(lm(y ~ trt * x * w_strength,
                    data = new.data[abs(new.data$x) < rd_continuous$h[1], ]))
rd_continuous$Estimate[1] - using.lm["trtTRUE"]
rd_continuous$Estimate[2] - using.lm["trtTRUE:w_strength"]

(Inference, however, requires the robust bias correction in rdhte and cannot be obtained from the lm() fit alone.)

Interaction effect: binary x continuous

A fully interacted specification gives a separate intercept and slope for each level of the categorical covariate.

summary(rd_interaction <- rdhte(y = y, x = x,
                                 covs.hte = "w_left*w_strength",
                                 cluster = cluster_var))

Wrapping the binary in factor() is also allowed but shifts the baseline category and therefore the reported intercepts:

summary(rdhte(y = y, x = x, covs.hte = "factor(w_left)*w_strength",
              cluster = cluster_var))

Each cell-specific coefficient may look insignificant individually while the joint test still rejects:

rdhte_lincom(model = rd_interaction,
             linfct = c("`T` = 0",
                        "`T:w_left` = 0",
                        "`T:w_strength` = 0",
                        "`T:w_left:w_strength` = 0"))

With the bandwidth fixed, the fully interacted model matches results from category-specific estimation. Use rdhte_lincom to read off the per-cell intercept and slope:

summary(rd_interaction <- rdhte(y = y, x = x,
                                 covs.hte = "w_left*w_strength",
                                 h = 0.1, cluster = cluster_var))
summary(rdhte(y = y[w_left == 0], x = x[w_left == 0],
              covs.hte = w_strength[w_left == 0], h = 0.1,
              cluster = cluster_var[w_left == 0]))
summary(rdhte(y = y[w_left == 1], x = x[w_left == 1],
              covs.hte = w_strength[w_left == 1], h = 0.1,
              cluster = cluster_var[w_left == 1]))
rdhte_lincom(model = rd_interaction,
             linfct = c("`T` + `T:w_left` = 0",
                        "`T:w_strength` + `T:w_left:w_strength` = 0"))

Standalone bandwidth selection (rdbwhte)

rdbwhte runs the same data-driven bandwidth selectors as rdhte but returns the bandwidths without estimating the CATEs. Useful when you want to inspect or fix h and then explore alternative variance estimators or compare specifications at a common bandwidth.

bw <- rdbwhte(y = y, x = x, covs.hte = factor(w_ideology),
              cluster = cluster_var)
bw$h

bw.joint = TRUE forces a single shared bandwidth across cells:

bw_joint <- rdbwhte(y = y, x = x, covs.hte = factor(w_ideology),
                    cluster = cluster_var, bw.joint = TRUE)
bw_joint$h

Efficiency-improving covariates (covs.eff)

covs.eff adds covariates to the local-polynomial regression to shrink standard errors without changing the identification of the CATE. They enter additively (and as covs.eff x W interactions in the heterogeneity-aware paths) but never with the treatment indicator or the running-variable polynomial. Useful when you have strong pretreatment predictors of the outcome.

m_no_eff <- rdhte(y = y, x = x, covs.hte = factor(w_ideology),
                   cluster = cluster_var)
m_eff    <- rdhte(y = y, x = x, covs.hte = factor(w_ideology),
                   covs.eff = w_strength, cluster = cluster_var)
data.frame(cell       = m_no_eff$W.lev,
           se_no_eff  = round(m_no_eff$se.rb, 4),
           se_eff     = round(m_eff$se.rb,    4),
           pct_change = round(100 * (m_eff$se.rb - m_no_eff$se.rb)
                              / m_no_eff$se.rb, 1))

Plotting

plot() is a post-estimation method for categorical covs.hte: one point per cell at the conventional point estimate, with the robust bias-corrected CI as an error bar.

plot(rd_ideology)

sort = TRUE reorders cells along the x-axis by their point estimate -- often the more useful default for ranking-style narratives:

plot(rd_ideology, sort = TRUE)

The returned object is a ggplot, so additional layers compose naturally. For example, a custom title with horizontal orientation:

p <- plot(rd_ideology, sort = TRUE,
          title = "Heterogeneity by ideology bucket",
          ylab  = "Sharp RD ITT")
if (requireNamespace("ggplot2", quietly = TRUE)) {
  p + ggplot2::theme_minimal(base_size = 11) +
      ggplot2::coord_flip()
}

Building publication-ready tables

rdhte exposes the per-cell results through tidy() and a one-row summary through glance() (both broom-compatible). Combined with knitr::kable() / xtable::xtable() the same numbers can be turned into Markdown, HTML, or LaTeX tables for papers.

(A) per-cell table from a single call

tidy() returns one row per cell with the conventional point estimate, robust BC standard error, z / p-value, BC confidence interval, and per-side h and Nh. Renaming or formatting columns before kable() gives a clean publication panel:

tab_A <- tidy(rd_ideology)
tab_A

knitr::kable(
  tab_A[, c("term", "estimate", "std.error", "conf.low", "conf.high",
            "n.eff.left", "n.eff.right", "h.left", "h.right")],
  digits = c(NA, 3, 3, 3, 3, 0, 0, 3, 3),
  col.names = c("Cell", "Estimate", "SE", "CI lo", "CI hi",
                "Nh-", "Nh+", "h-", "h+"),
  caption = "Per-cell RD effects by ideology bucket"
)

glance() complements tidy() with a one-row description of the fit (N, polynomial orders, kernel, VCE, bandwidth selector):

glance(rd_ideology)

(B) cell x specification comparison

Fix the covs.hte argument and vary vce across columns to see how the choice of variance estimator moves the per-cell standard errors.

specs <- list(HC0 = list(vce = "hc0"),
              HC2 = list(vce = "hc2"),
              HC3 = list(vce = "hc3"),
              CR1 = list(vce = "cr1", cluster = cluster_var))

fit_one <- function(args) {
  args <- c(list(y = y, x = x, covs.hte = factor(w_ideology)), args)
  do.call(rdhte, args)
}

cells  <- sort(unique(w_ideology))
mat_pt <- sapply(specs, function(s) tidy(fit_one(s))$estimate)
mat_se <- sapply(specs, function(s) tidy(fit_one(s))$std.error)
rownames(mat_pt) <- rownames(mat_se) <- as.character(cells)

mat_pt
mat_se

A common publication layout interleaves the point estimate and its SE in parentheses below it:

n_cells <- nrow(mat_pt)
out     <- data.frame(matrix(NA_character_, nrow = 2 * n_cells,
                             ncol = ncol(mat_pt)))
for (i in seq_len(n_cells)) {
  out[2 * i - 1, ] <- sprintf("%.3f", mat_pt[i, ])
  out[2 * i,     ] <- sprintf("(%.3f)", mat_se[i, ])
}
out <- cbind(Cell = rep(rownames(mat_pt), each = 2), out)
out$Cell[seq(2, nrow(out), by = 2)] <- ""
colnames(out)[-1] <- colnames(mat_pt)
knitr::kable(out, caption = "Per-cell estimates by variance option (SE in parentheses)")

(C) LaTeX export

xtable or kableExtra can render the same data frame as LaTeX. The snippet below produces a paper-ready table; uncomment the writeLines() call to write to a file:

tex <- xtable::xtable(
  tab_A[, c("term", "estimate", "std.error", "conf.low", "conf.high")],
  digits  = c(0, 0, 3, 3, 3, 3),
  caption = "Per-cell RD effects by ideology bucket",
  label   = "tab:rdhte-ideology"
)
print(tex, include.rownames = FALSE, booktabs = TRUE,
      caption.placement = "top")
# writeLines(capture.output(print(tex, include.rownames = FALSE,
#                                booktabs = TRUE,
#                                caption.placement = "top")),
#            con = "rdhte_table_A.tex")

Replicating rdrobust

Average effects from rdhte() and rdrobust() will not match under default settings, because the two packages use different defaults for rho and vce:

summary(rdhte(y = y, x = x))
summary(rdrobust(y = y, x = x))

To replicate exactly with rdrobust, set rho = 1, choose a matching vce, and enforce the same bandwidth:

summary(rdhte(y = y, x = x, h = 0.1, vce = "hc3"))
summary(rdrobust(y = y, x = x, h = 0.1, rho = 1, vce = "hc3"))

The same trick reproduces the per-cell rdrobust estimates from a single rdhte call with per-cell bandwidths:

summary(rdhte(y = y, x = x, covs.hte = w_left,
              h = c(0.078, 0.116), vce = "hc3"))
summary(rdrobust(y = y[w_left == 1], x = x[w_left == 1],
                 h = 0.116, rho = 1, vce = "hc3"))

detach(rdhte_dataset)

See also

• rdhte(), rdbwhte(), rdhte_lincom(), plot.rdhte().
• The companion Stata package ships an rdhte_plot command that mirrors the R plot() method.

Reference

Calonico, S., Cattaneo, M.D., Farrell, M.H., Palomba, F., and Titiunik, R. (2025). Treatment Effect Heterogeneity in Regression Discontinuity Designs. arXiv:2503.13696.

Granzier, R., Pons, V., and Tricaud, C. (2023). Coordination and Bandwagon Effects: How Past Rankings Shape the Behavior of Voters and Candidates. American Economic Journal: Applied Economics 15(4): 177-217.

rdhte documentation built on May 27, 2026, 1:06 a.m.