R/verbyla.R

In skedastic: Handling Heteroskedasticity in the Linear Regression Model

#' Verbyla's Test for Heteroskedasticity in a Linear Regression Model
#'
#' This function implements the residual maximum likelihood test of
#'    \insertCite{Verbyla93;textual}{skedastic} for testing for heteroskedasticity
#'    in a linear regression model.
#'
#' @details Verbyla's Test entails fitting a generalised auxiliary regression
#'    model in which the response variable is the vector of standardised
#'    squared residuals \eqn{e_i^2/\hat{\omega}} from the original OLS model
#'    and the design matrix is some function of \eqn{Z}, an \eqn{n \times q}
#'    matrix consisting of \eqn{q} exogenous variables, appended to a column of
#'    ones. The test statistic is half the residual sum of squares from this
#'    generalised auxiliary regression. Under the null hypothesis of
#'    homoskedasticity, the distribution of the test statistic is
#'    asymptotically chi-squared with \eqn{q} degrees of freedom. The test is
#'    right-tailed.
#'
#' @inheritParams breusch_pagan
#'
#' @return An object of \code{\link[base]{class}} \code{"htest"}. If object is
#'    not assigned, its attributes are displayed in the console as a
#'    \code{\link[tibble]{tibble}} using \code{\link[broom]{tidy}}.
#' @references{\insertAllCited{}}
#' @importFrom Rdpack reprompt
#' @export
#'
#' @examples
#' mtcars_lm <- lm(mpg ~ wt + qsec + am, data = mtcars)
#' verbyla(mtcars_lm)
#' verbyla(mtcars_lm, auxdesign = "fitted.values")
#'

verbyla <- function(mainlm, auxdesign = NA, statonly = FALSE) {

  auxfitvals <- ifelse(all(is.na(auxdesign)) | is.null(auxdesign), FALSE,
                                    auxdesign == "fitted.values")
  processmainlm(m = mainlm, needy = auxfitvals, needyhat = auxfitvals,
                needp = FALSE)

  if (all(is.na(auxdesign)) || is.null(auxdesign)) {
    Z <- X
  } else if (is.character(auxdesign)) {
    if (auxdesign == "fitted.values") {
      Z <- t(t(yhat))
    } else stop("Invalid character value for `auxdesign`")
  } else {
    Z <- auxdesign
    if (nrow(auxdesign) != nrow(X)) stop("No. of observations in `auxdesign`
                                         must match\nno. of observations in
                                         original model.")
  }

  hasintercept <- columnof1s(Z)
  if (hasintercept[[1]]) {
    Z <- Z[, -hasintercept[[2]], drop = FALSE]
  }

  p <- ncol(X)
  q <- ncol(Z)
  n <- nrow(Z)

  Z <- cbind(1, Z)

  M <- fastM(X, n)
  sigma_hatbar <- sum(e ^ 2) / (n - p)
  term1 <- t(t(e ^ 2 / sigma_hatbar - diag(M)))

  mat_to_invert <- t(Z) %*% (M ^ 2) %*% Z
  if (det(mat_to_invert) == 0) {
    Z <- Z[, -1, drop = FALSE]
    mat_to_invert <- t(Z) %*% (M ^ 2) %*% Z
    if (det(mat_to_invert) == 0) stop("Linear dependency detected in auxiliary matrix")
    message("Intercept not included in auxiliary design in order to avoid linear dependency")
  }

  teststat <- as.double(1 / 2 * t(term1) %*% Z %*%
                          solve(mat_to_invert)
                        %*% t(Z) %*% term1)
  if (statonly) return(teststat)

  pval <- stats::pchisq(teststat, df = q, lower.tail = FALSE)
  rval <- structure(list(statistic = teststat, parameter = q, p.value = pval,
                         null.value = "Homoskedasticity",
                         alternative = "greater", method = "Verbyla"),
                         class = "htest")
  broom::tidy(rval)
}