# R/rd_est.R In rddapp: Regression Discontinuity Design Application

#### Documented in rd_est

#' Regression Discontinuity Estimation
#'
#' \code{rd_est} estimates both sharp and fuzzy RDDs using parametric and non-parametric
#' (local linear) models.
#' It is based on the \code{RDestimate} function in the "rdd" package.
#' Sharp RDDs (both parametric and non-parametric) are estimated using \code{lm} in the
#' \pkg{stats} package.
#' Fuzzy RDDs (both parametric and non-parametric) are estimated using two-stage least-squares
#' \code{ivreg} in the \pkg{AER} package.
#' For non-parametric models, Imbens-Kalyanaraman optimal bandwidths can be used,
#'
#' @param formula The formula of the RDD; a symbolic description of the model to be fitted. This is supplied in the
#'   format of \code{y ~ x} for a simple sharp RDD or \code{y ~ x | c1 + c2}
#'   for a sharp RDD with two covariates. A fuzzy RDD may be specified as
#'   \code{y ~ x + z} where \code{x} is the running variable, and
#'   \code{z} is the endogenous treatment variable. Covariates are included in the
#'   same manner as in a sharp RDD.
#' @param data An optional data frame containing the variables in the model. If not found in \code{data},
#'   the variables are taken from \code{environment(formula)}.
#' @param subset An optional vector specifying a subset of observations to be used in the fitting process.
#' @param cutpoint A numeric value containing the cutpoint at which assignment to the treatment is determined. The default is 0.
#' @param bw A vector specifying the bandwidths at which to estimate the RD.
#'   Possible values are \code{"IK09"}, \code{"IK12"}, and a user-specified non-negative numeric vector specifying the bandwidths at which to estimate the RD.
#'   The default is \code{"IK12"}. If \code{bw} is \code{"IK12"}, the bandwidth is calculated using the Imbens-Kalyanaraman
#'   2012 method. If \code{bw}  is \code{"IK09"}, the bandwidth is calculated using
#'   the Imbens-Kalyanaraman 2009 method. Then the RD is estimated
#'   with that bandwidth, half that bandwidth, and twice that bandwidth.
#'   If only a single value is passed into the function,
#'   the RD will similarly be estimated at that bandwidth, half that bandwidth,
#'   and twice that bandwidth.
#' @param kernel A string indicating which kernel to use. Options are \code{"triangular"}
#'   (default and recommended), \code{"rectangular"}, \code{"epanechnikov"}, \code{"quartic"},
#'   \code{"triweight"}, \code{"tricube"}, and \code{"cosine"}.
#' @param se.type This specifies the robust standard error calculation method to use,
#'   from the "sandwich" package. Options are,
#'   as in \code{\link{vcovHC}}, \code{"HC3"}, \code{"const"}, \code{"HC"}, \code{"HC0"},
#'   \code{"HC1"}, \code{"HC2"}, \code{"HC4"}, \code{"HC4m"}, \code{"HC5"}.
#'   The default is \code{"HC1"}. This option is overridden by \code{cluster}.
#' @param cluster An optional vector of length n specifying clusters within which the errors are assumed
#'   to be correlated. This will result in reporting cluster robust SEs. This option overrides
#'   anything specified in \code{se.type}. It is suggested that data with a discrete running
#'   variable be clustered by each unique value of the running variable (Lee and Card, 2008).
#' @param verbose A logical value indicating whether to print additional information to
#'   the terminal. The default is \code{FALSE}.
#' @param less Logical. If \code{TRUE}, return the estimates of linear and optimal. If \code{FALSE}
#'   return the estimates of linear, quadratic, cubic, optimal, half and double. The default is \code{FALSE}.
#' @param est.cov Logical. If \code{TRUE}, the estimates of covariates will be included.
#'   If \code{FALSE}, the estimates of covariates will not be included. The default is \code{FALSE}. This option is not
#'   applicable if method is \code{"front"}.
#' @param est.itt Logical. If \code{TRUE}, the estimates of ITT will be returned.
#'   The default is \code{FALSE}.
#' @param t.design A string specifying the treatment option according to design.
#'   Options are \code{"g"} (treatment is assigned if \code{x} is greater than its cutoff),
#'   \code{"geq"} (treatment is assigned if \code{x} is greater than or equal to its cutoff),
#'   \code{"l"} (treatment is assigned if \code{x} is less than its cutoff),
#'   and \code{"leq"} (treatment is assigned if \code{x} is less than or equal to its cutoff).
#'
#' @return \code{rd_est} returns an object of \link{class} "\code{rd}".
#'   The functions \code{summary} and \code{plot} are used to obtain and print a summary and
#'   plot of the estimated regression discontinuity. The object of class \code{rd} is a list
#'   containing the following components:
#' \item{type}{A string denoting either \code{"sharp"} or \code{"fuzzy"} RDD.}
#' \item{est}{Numeric vector of the estimate of the discontinuity in the outcome under
#'   a sharp RDD or the Wald estimator in the fuzzy RDD, for each corresponding bandwidth.}
#' \item{se}{Numeric vector of the standard error for each corresponding bandwidth.}
#' \item{z}{Numeric vector of the z statistic for each corresponding bandwidth.}
#' \item{p}{Numeric vector of the p-value for each corresponding bandwidth.}
#' \item{ci}{The matrix of the 95% confidence interval, \code{c("CI Lower Bound", "CI Upper Bound")}
#'   for each corresponding bandwidth.}
#' \item{d}{Numeric vector of the effect size (Cohen's d) for each estimate.}
#' \item{cov}{The names of covariates.}
#' \item{bw}{Numeric vector of each bandwidth used in estimation.}
#' \item{obs}{Vector of the number of observations within the corresponding bandwidth.}
#' \item{call}{The matched call.}
#' \item{na.action}{The number of observations removed from fitting due to missingness.}
#' \item{impute}{A logical value indicating whether multiple imputation is used or not.}
#' \item{model}{For a sharp design, a list of the \code{lm} objects is returned.
#'   For a fuzzy design, a list of lists is returned, each with two elements:
#'   \code{firststage}, the first stage \code{lm} object, and \code{iv}, the \code{ivreg} object.
#'   A model is returned for each corresponding bandwidth.}
#' \item{frame}{Returns the dataframe used in fitting the model.}
#'
#' @references Lee, D. S., Lemieux, T. (2010).
#'   Regression Discontinuity Designs in Economics.
#'   Journal of Economic Literature, 48(2), 281-355.
#'   \doi{10.1257/jel.48.2.281}.
#' @references Imbens, G., Lemieux, T. (2008).
#'   Regression discontinuity designs: A guide to practice.
#'   Journal of Econometrics, 142(2), 615-635.
#'   \doi{10.1016/j.jeconom.2007.05.001}.
#' @references Lee, D. S., Card, D. (2010).
#'   Regression discontinuity inference with specification error.
#'   Journal of Econometrics, 142(2), 655-674.
#'   \doi{10.1016/j.jeconom.2007.05.003}.
#' @references Angrist, J. D., Pischke, J.-S. (2009).
#'   Mostly harmless econometrics: An empiricist's companion.
#'   Princeton, NJ: Princeton University Press.
#' @references Drew Dimmery (2016). rdd: Regression Discontinuity Estimation. R package
#'    version 0.57. https://CRAN.R-project.org/package=rdd
#' @references Imbens, G., Kalyanaraman, K. (2009).
#'   Optimal bandwidth choice for the regression discontinuity estimator
#'   (Working Paper No. 14726). National Bureau of Economic Research.
#'   \url{https://www.nber.org/papers/w14726}.
#' @references Imbens, G., Kalyanaraman, K. (2012).
#'   Optimal bandwidth choice for the regression discontinuity estimator.
#'   The Review of Economic Studies, 79(3), 933-959.
#'
#' @importFrom AER ivreg
#' @importFrom sandwich estfun sandwich vcovHC
#' @importFrom lmtest coeftest
#' @importFrom Formula as.Formula
#' @importFrom stats model.frame na.pass complete.cases lm coef pnorm qnorm as.formula
#'
#' @include bw_ik12.R
#' @include bw_ik09.R
#' @include wt_kern.R
#' @include treat_assign.R
#'
#' @export
#'
#' @examples
#' set.seed(12345)
#' x <- runif(1000, -1, 1)
#' cov <- rnorm(1000)
#' y <- 3 + 2 * x + 3 * cov + 10 * (x >= 0) + rnorm(1000)
#' rd_est(y ~ x, t.design = "geq")
#' # Efficiency gains can be made by including covariates (review SEs in "summary" output).
#' rd_est(y ~ x | cov, t.design = "geq")

rd_est <- function(formula, data, subset = NULL, cutpoint = NULL, bw = NULL,
kernel = "triangular", se.type = "HC1", cluster = NULL, verbose = FALSE, less = FALSE,
est.cov = FALSE, est.itt = FALSE, t.design = NULL) {

if (is.null(t.design)){
stop("Specify t.design.")
}

call <- match.call()

if (missing(data))
data <- environment(formula)
formula <- as.Formula(formula)
X <- model.frame(formula, rhs = 1, lhs = 0, data = data, na.action = na.pass)[[1]]
Y <- model.frame(formula, rhs = 0, lhs = NULL, data = data, na.action = na.pass)[[1]]

# if only a subset of data is needed for the model
if (!is.null(subset)) {
X <- X[subset]
Y <- Y[subset]
if (!is.null(cluster))
cluster <- cluster[subset]
}

# if data is clustered, a clustered estimator of covariance is needed
if (!is.null(cluster)) {
cluster <- as.character(cluster)
robust.se <- function(model, cluster) {
M <- length(unique(cluster))
N <- length(cluster)
K <- model$rank dfc <- (M / (M - 1)) * ((N - 1) / (N - K)) uj <- apply(estfun(model), 2, function(x) tapply(x, cluster, sum)) rcse.cov <- dfc * sandwich(model, meat. = crossprod(uj) / N) rcse.se <- coeftest(model, rcse.cov) return(rcse.se) } } na.ok <- complete.cases(X) & complete.cases(Y) # if another variable is provided in addition to x, it will be considered as z if (length(all.vars(formula(formula, rhs = 1, lhs = FALSE))) > 1) { type <- "fuzzy" Z <- model.frame(formula, rhs = 1, lhs = 0, data = data, na.action = na.pass)[[2]] if (!is.null(subset)) Z <- Z[subset] na.ok <- na.ok & complete.cases(Z) # if more than one variable is provided in addition to x1 and x2, it is redundant if (length(all.vars(formula(formula, rhs = 1, lhs = FALSE))) > 2) stop("Invalid formula. Read ?rd_est for proper syntax.") } else { type <- "sharp" } covs <- NULL num_covs <- 0 # if variables are provided after the first part of the formula, # they will be considered as covariates if (length(formula)[2] > 1) { covs <- model.frame(formula, rhs = 2, lhs = 0, data = data, na.action = na.pass) if (!is.null(subset)) covs <- subset(covs, subset) na.ok <- na.ok & complete.cases(covs) covs <- subset(covs, na.ok) num_covs <- ncol(covs) } X <- X[na.ok] Y <- Y[na.ok] if (type == "fuzzy") Z <- as.double(Z[na.ok]) if (is.null(cutpoint)) { cutpoint <- 0 if (verbose) cat("No cutpoint provided. Using default cutpoint of zero.\n") } if (type == "sharp") { if (!is.null(covs)) dat.out <- data.frame(X, Y, covs) else dat.out <- data.frame(X, Y) } else { if (!is.null(covs)) dat.out <- data.frame(X, Y, Z, covs) else dat.out <- data.frame(X, Y, Z) } if (is.null(bw) || bw == "IK12") { bw <- try(bw_ik12(X = X, Y = Y, cutpoint = cutpoint, kernel = kernel, verbose = verbose), silent = TRUE) if (inherits(bw, "try-error")) { bws <- c(NA, NA, NA, -1, -1, -1) warning("Fail to calculate the IK12 bandwidth, nonparametric models will be NA.") } else { bws <- c(NA, NA, NA, bw, 0.5 * bw, 2 * bw) } names(bws) <- c("Linear", "Quadratic", "Cubic", "Opt", "Half-Opt", "Double-Opt") } else if (bw == "IK09") { bw <- try(bw_ik09(X = X, Y = Y, cutpoint = cutpoint, kernel = kernel, verbose = verbose), silent = TRUE) if (inherits(bw, "try-error")) { bws <- c(NA, NA, NA, -1, -1, -1) warning("Fail to calculate the IK09 bandwidth, nonparametric models will be NA.") } else { bws <- c(NA, NA, NA, bw, 0.5 * bw, 2 * bw) } names(bws) <- c("Linear", "Quadratic", "Cubic", "Opt", "Half-Opt", "Double-Opt") } else if (length(bw) == 1 && is.numeric(bw)) { bws <- c(NA, NA, NA, bw, 0.5 * bw, 2 * bw) names(bws) <- c("Linear", "Quadratic", "Cubic", "Usr", "Half-Usr", "Double-Usr") } else { stop("Invalid bandwidth. Read ?rd_est for proper syntax.") } # if only linear and local linear models are needed if (less) { bws <- bws[c(1, 4)] } # Setup values to be returned o <- list() class(o) <- "rd" o$type <- type
o$call <- call if (est.cov) { o$est <- vector(length = length(bws) * (1 + num_covs), mode = "numeric")
names(o$est) <- rep(names(bws), each = 1 + num_covs) o$se <- vector(length = length(bws) * (1 + num_covs), mode = "numeric")
if (type == "sharp") {
names(o$se) <- rep(c("Tr", names(covs)), length(bws)) } else { names(o$se) <- rep(c("Z", names(covs)), length(bws))
}
o$ci <- matrix(NA, nrow = length(bws) * (1 + num_covs), ncol = 2) } else { o$est <- vector(length = length(bws), mode = "numeric")
names(o$est) <- names(bws) o$se <- vector(length = length(bws), mode = "numeric")
if (type == "sharp") {
names(o$se) <- rep("Tr", length(bws)) } else { names(o$se) <- rep("Z", length(bws))
}
o$ci <- matrix(NA, nrow = length(bws), ncol = 2) } o$bw <- bws
o$z <- vector(mode = "numeric") o$p <- vector(mode = "numeric")
o$obs <- vector(mode = "numeric") o$cov <- names(covs)
o$model <- list() if (type == "fuzzy") { o$model$firststage <- list() o$model$iv <- list() } o$frame <- dat.out
o$na.action <- which(na.ok == FALSE) o$impute <- FALSE

X <- X - cutpoint
Tr <- treat_assign(X, 0, t.design)
Xl <- (1 - Tr) * X
Xr <- Tr * X

if (type == "fuzzy" && est.itt) {
Z <- Tr
}

degree <- c(1, 2, 3)
for (ibw in 1:length(bws)) {
bw <- bws[ibw]

if (est.cov) {
pos <- (1 + (ibw - 1) * (1 + num_covs)):(ibw * (1 + num_covs))
} else {
pos <- ibw
}

if (!is.na(bw) && bw <= 0) {
o$obs[ibw] <- NA o$est[pos] <- NA
o$se[pos] <- NA o$z[pos] <- NA
o$p[pos] <- NA o$ci[pos, ] <- NA
o$model[[ibw]] <- NA } else { # ibw <- which(bw == bws) # Subset to within the bandwidth, except for when using gaussian weighting # sub <- X >= (-bw) & X <= (+bw) # if (kernel == "gaussian") # sub <- TRUE if (is.na(bw)) { o$obs[ibw] <- length(X)
} else {
w <- wt_kern(X, 0, bw, kernel = kernel)
o$obs[ibw] <- sum(w > 0) } if (type == "sharp") { if (verbose) { cat("Running Sharp RD\n") cat("Running variable:", all.vars(formula(formula, rhs = 1, lhs = FALSE))[1], "\n") cat("Outcome variable:", all.vars(formula(formula, rhs = FALSE, lhs = 1))[1], "\n") if (!is.null(covs)) cat("Covariates:", paste(names(covs), collapse = ", "), "\n") } if (!is.null(covs)) { if (is.na(bw)) { data <- data.frame(Y, Tr, Xl, Xr, covs) form <- as.formula(paste("Y ~ Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE) + ", paste(names(covs), collapse = "+", sep = ""), sep = "")) } else { data <- data.frame(Y, Tr, Xl, Xr, covs, w) form <- as.formula(paste("Y ~ Tr + Xl + Xr + ", paste(names(covs), collapse = "+", sep = ""), sep = "")) } } else { if (is.na(bw)) { data <- data.frame(Y, Tr, Xl, Xr) form <- as.formula(paste("Y ~ Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE)", sep = "")) } else { data <- data.frame(Y, Tr, Xl, Xr, w) form <- as.formula(Y ~ Tr + Xl + Xr) } } if (is.na(bw)) { mod <- try(lm(form, data = data), silent = TRUE) } else { mod <- try(lm(form, weights = w, data = subset(data, w > 0)), silent = TRUE) } if (inherits(mod, "try-error")) { o$est[pos] <- NA
o$se[pos] <- NA o$z[pos] <- NA
o$p[pos] <- NA o$ci[pos, ] <- NA
o$model[[ibw]] <- NA } else { if (verbose == TRUE) { cat("Model:\n") print(summary(mod)) } if (est.cov) { o$est[pos] <- coef(mod)[c("Tr", names(covs))]
} else {
o$est[pos] <- coef(mod)["Tr"] } if (is.null(cluster)) { test_tab <- coeftest(mod, vcovHC(mod, type = se.type)) } else { if (is.na(bw)) { test_tab <- robust.se(mod, cluster[na.ok]) } else { test_tab <- robust.se(mod, cluster[na.ok][w > 0]) } } if (est.cov) { o$se[pos] <- rep(NA, 1 + num_covs)
test_var <- intersect(c("Tr", names(covs)), rownames(test_tab))
o$se[pos][test_var] <- test_tab[test_var, 2] } else { o$se[pos] <- ifelse("Tr" %in% rownames(test_tab), test_tab["Tr", 2], NA)
}

o$z[pos] <- o$est[pos] / o$se[pos] o$p[pos] <- 2 * pnorm(abs(o$z[pos]), lower.tail = FALSE) o$ci[pos, ] <- c(o$est[pos] - qnorm(0.975) * o$se[pos],
o$est[pos] + qnorm(0.975) * o$se[pos])

o$model[[ibw]] <- mod } } else { if (verbose) { cat("Running Fuzzy RD\n") cat("Running variable:", all.vars(formula(formula, rhs = 1, lhs = FALSE))[1], "\n") cat("Outcome variable:", all.vars(formula(formula, rhs = FALSE, lhs = 1))[1], "\n") cat("Treatment variable:", all.vars(formula(formula, rhs = 1, lhs = FALSE))[2], "\n") if (!is.null(covs)) cat("Covariates:", paste(names(covs), collapse = ", "), "\n") } if (!is.null(covs)) { if (is.na(bw)) { data <- data.frame(Y, Tr, Xl, Xr, Z, covs) form <- as.Formula(paste("Y ~ Z + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE) + ", paste(names(covs), collapse = "+"), "|Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE) + ", paste(names(covs), collapse = "+"), sep = "")) form1 <- as.Formula(paste("Z ~ Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE) + ", paste(names(covs), collapse = "+", sep = ""))) } else { data <- data.frame(Y, Tr, Xl, Xr, Z, covs, w) form <- as.Formula(paste("Y ~ Z + Xl + Xr + ", paste(names(covs), collapse = "+"), "|Tr + Xl + Xr + ", paste(names(covs), collapse = "+"), sep = "")) form1 <- as.Formula(paste("Z ~ Tr + Xl + Xr + ", paste(names(covs), collapse = "+", sep = ""))) } } else { if (is.na(bw)) { data <- data.frame(Y, Tr, Xl, Xr, Z) form <- as.Formula(paste("Y ~ Z + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE) | Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE)", sep = "")) form1 <- as.formula(paste("Z ~ Tr + poly(Xl, ", degree[ibw], ", raw = TRUE) + poly(Xr, ", degree[ibw], ", raw = TRUE)", sep = "")) } else { data <- data.frame(Y, Tr, Xl, Xr, Z, w) form <- as.Formula(Y ~ Z + Xl + Xr | Tr + Xl + Xr) form1 <- as.formula(Z ~ Tr + Xl + Xr) } } if (is.na(bw)) { mod1 <- try(lm(form1, data = data), silent = TRUE) mod <- try(ivreg(form, data = data), silent = TRUE) } else { mod1 <- try(lm(form1, weights = w, data = subset(data, w > 0)), silent = TRUE) mod <- try(ivreg(form, weights = w, data = subset(data, w > 0)), silent = TRUE) } if (inherits(mod, "try-error")) { o$est[pos] <- NA
o$se[pos] <- NA o$z[pos] <- NA
o$p[pos] <- NA o$ci[pos, ] <- NA
o$model$firststage[[ibw]] <- NA
o$model$iv[[ibw]] <- NA
} else {
if (verbose == TRUE) {
cat("First stage:\n")
print(summary(mod1))
cat("IV-RD:\n")
print(summary(mod))
}

if (est.cov) {
o$est[pos] <- coef(mod)[c("Z", names(covs))] } else { o$est[pos] <- coef(mod)["Z"]
}

if (is.null(cluster)) {
test_tab <- coeftest(mod, vcovHC(mod, type = se.type))
} else {
if (is.na(bw)) {
test_tab <- robust.se(mod, cluster[na.ok])
} else {
test_tab <- robust.se(mod, cluster[na.ok][w > 0])
}
}

if (est.cov) {
o$se[pos] <- rep(NA, 1 + num_covs) test_var <- intersect(c("Z", names(covs)), rownames(test_tab)) o$se[pos][test_var] <- test_tab[test_var, 2]
} else {
o$se[pos] <- ifelse("Z" %in% rownames(test_tab), test_tab["Z", 2], NA) } o$z[pos] <- o$est[pos] / o$se[pos]
o$p[pos] <- 2 * pnorm(abs(o$z[pos]), lower.tail = FALSE)
o$ci[pos, ] <- c(o$est[pos] - qnorm(0.975) * o$se[pos], o$est[pos] + qnorm(0.975) * o$se[pos]) o$model$firststage[[ibw]] <- mod1 o$model$iv[[ibw]] <- mod } } if (est.cov && num_covs > 0){ d = o$d <- o$est / sd(Y) d[seq(2, length(d), 2)] = NA o$d <- d
}else{
o$d <- o$est / sd(Y)
}

}
}

names(o$se) <- names(o$est)

return(o)
}

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rddapp documentation built on April 6, 2023, 1:15 a.m.