R/bootlm.R

In skedastic: Handling Heteroskedasticity in the Linear Regression Model

#' Nonparametric Bootstrapping of Heteroskedastic Linear Regression Models
#'
#' Generates \eqn{B} nonparametric bootstrap replications of a linear
#'    regression model that may have heteroskedasticity.
#'
#' @details Each replication of the pairs bootstrap entails drawing a sample
#'    of size \eqn{n} (the number of observations) with replacement from the
#'    indices \eqn{i=1,2,\ldots,n}. The pair or case \eqn{(y_i, X_i)} is
#'    included as an observation in the bootstrapped data set for each sampled
#'    index. An Ordinary Least Squares fit to the bootstrapped data set is then
#'    computed.
#'
#' Under the wild bootstrap, each replication of the linear regression model
#'    is generated by first independently drawing \eqn{n} random values
#'    \eqn{r_i}, \eqn{i=1,2,\ldots,n}, from a distribution with zero mean and
#'    unit variance. The \eqn{i}th bootstrap response is then computed as
#'    \deqn{X_i'\hat{\beta} + f_i(e_i) r_i}, where \eqn{X_i} is the \eqn{i}th
#'    design observation, \eqn{\hat{\beta}} is the Ordinary Least Squares
#'    estimate of the coefficient vector \eqn{\beta}, \eqn{e_i} is the
#'    \eqn{i}th Ordinary Least Squares residual, and \eqn{f_i(\cdot)} is a
#'    function performing some transformation on the residual. An Ordinary
#'    Least Squares fit is then computed on the original design matrix and the
#'    bootstrap response vector.
#'
#' @param object Either an object of class \code{"lm"} (generated by
#'    \code{\link[stats]{lm}}), an object of class \code{"alvm.fit"}
#'    (generated by \code{\link{alvm.fit}}), or an object of class
#'    \code{"anlvm.fit"} (generated by \code{\link{anlvm.fit}})
#' @param sampmethod A character, either \code{"pairs"} or \code{"wild"},
#'    indicating the method to be used to generate the resampled models:
#'    bootstrapping pairs \insertCite{Efron93}{skedastic} or the wild bootstrap
#'    \insertCite{Davidson08}{skedastic}. Defaults to \code{"pairs"}.
#' @param B An integer representing the number of bootstrapped linear
#'    regression models to generate; defaults to \code{1000L}
#' @param resfunc Either a character naming a function to call to apply a
#'    transformation to the Ordinary Least Squares residuals, or a function
#'    to apply for the same purpose. This argument is ignored if
#'    \code{sampmethod} is \code{"pairs"}. The only two character values
#'    accepted are \code{"identity"}, in which case no transformation is
#'    applied to the residuals, and \code{"hccme"}, in which case the
#'    transformation corresponds to a heteroskedasticity-consistent
#'    covariance matrix estimator calculated from \code{\link{hccme}}. If
#'    \code{resfunc} is a function, it is assumed that its first argument
#'    is the numeric vector of residuals.
#' @param fastfit A logical indicating whether the \code{\link[stats]{lm.fit}}
#'    function should be used to fit the bootstrapped regression models for
#'    greater computational speed; defaults to \code{TRUE}. This affects the
#'    class of the returned value (see below). If \code{FALSE},
#'    \code{\link[stats]{lm}} is used to fit the bootstrapped regression
#'    models.
#' @param ... Other arguments to pass to \code{\link{hccme}}
#'
#' @return A list object of class \code{"bootlm"}, containing \code{B} objects,
#'    each of which is a bootstrapped linear regression model fit by Ordinary
#'    Least Squares. If \code{fastfit} was set to \code{TRUE}, each of these
#'    objects will be a list containing named objects \code{y} (the bootstrap
#'    response vector), \code{X} (the bootstrap design matrix, which is just
#'    the original design matrix under the wild bootstrap), \code{e} (the
#'    residual vector from the Ordinary Least Squares fit to this bootstrap
#'    data set), \code{beta.hat} (the vector of coefficient estimates from the
#'    Ordinary Least Squares fit to this bootstrap data set),
#'    \code{sampmethod}, and \code{ind} (a vector of the indices from the
#'    original data set used in this bootstrap sample; ignored under the
#'    wild bootstrap)
#'    of the kind returned by
#'    \code{\link[stats]{lm.fit}}; otherwise, each will be an object of class
#'    \code{"lm"}.
#' @references{\insertAllCited{}}
#' @importFrom Rdpack reprompt
#' @export
#' @seealso \code{\link[lmboot]{paired.boot}} and
#'    \code{\link[lmboot]{wild.boot}} for the pairs bootstrap and wild
#'    bootstrap, respectively. The latter function does not appear to allow
#'    transformations of the residuals in the wild bootstrap.
#'
#' @examples
#' mtcars_lm <- lm(mpg ~ wt + qsec + am, data = mtcars)
#' mybootlm <- bootlm(mtcars_lm)
#'


bootlm <- function(object, sampmethod = c("pairs", "wild"),
                   B = 1000L, resfunc = c("identity", "hccme"),
                   fastfit = TRUE, ...) {

  sampmethod <- match.arg(sampmethod, c("pairs", "wild"))
  resfunc <- match.arg(resfunc, c("identity", "hccme"))
  if (inherits(object, "lm")) {
    X <- stats::model.matrix(object)
    y <- stats::model.response(stats::model.frame(object))
    e <- stats::resid(object)
    yhat <- stats::fitted(object)
  } else if (inherits(object, "alvm.fit") ||
             inherits(object, "anlvm.fit")) {
    X <- stats::model.matrix(object$ols)
    y <- stats::model.response(stats::model.frame(object$ols))
    e <- stats::resid(object$ols)
    yhat <- stats::fitted(object$ols)
  } else stop("object should be of class lm or alvm.fit or anlvm.fit")
  n <- nrow(X)
  p <- ncol(X)

  if (sampmethod == "wild") {
    myargs <- list(...)
    rvec <- lapply(1:B, function(b) {
      sample(c(-1, 1), n, replace = TRUE)
    })
    if (resfunc == "identity") {
      fi_ei <- e
    } else if (resfunc == "hccme") {
      myargs$object <- list("X" = X, "esq" = e ^ 2)
      myargs$sandwich <- FALSE
      myargs$as_matrix <- FALSE
      fi_ei <- sqrt(do.call(what = "hccme", args = myargs))
    } else {
      fi_ei <- do.call(what = resfunc,
                       args = c(e, myargs))
    }
    yboot <- lapply(1:B, function(b) {
      yhat + fi_ei * rvec[[b]]
    })
    if (!fastfit) {
      olsboot <- lapply(1:B, function(b) {
        stats::lm(yboot[[b]] ~ 0 + X)
      })
    } else {
      olsboot <- lapply(1:B, function(b) {
        thelm <- stats::lm.fit(x = X, y = yboot[[b]])
        list("y" = yboot[[b]], "X" = X, "e" = stats::resid(thelm),
             "beta.hat" = stats::coef(thelm),
             "sampmethod" = sampmethod, "ind" = NULL)
      })
    }
  } else if (sampmethod == "pairs") {
    bootind <- lapply(1:B, function(b) {
      sample(n, n, replace = TRUE)
    })
    yboot <- lapply(1:B, function(b) {
      y[bootind[[b]]]
    })
    Xboot <- lapply(1:B, function(b) {
      X[bootind[[b]], , drop = FALSE]
    })
    if (!fastfit) {
      olsboot <- lapply(1:B, function(b) {
        stats::lm(yboot[[b]] ~ 0 + Xboot[[b]])
      })
    } else {
      olsboot <- lapply(1:B, function(b) {
        thelm <- stats::lm.fit(x = Xboot[[b]], y = yboot[[b]])
        list("y" = yboot[[b]], "X" = Xboot[[b]],
             "e" = stats::resid(thelm), "beta.hat" = stats::coef(thelm),
             "sampmethod" = sampmethod, "ind" = bootind[[b]])
      })
    }
  }

  class(olsboot) <- "bootlm"
  olsboot
}