Calculate a single replication of the multiplier bootstrap for an OLS fitted dataset

comp_boot_mul_ind calculates a single replication of the multiplier bootstrap for an OLS fitted dataset based on an lm fitted object in R. This is given by the following expression: $$\frac{1}{n}\sum_{i=1}^{n} e_{i}\widehat{J}^{-1}X_{i}(Y_{i}-X_{i}^{T} \widehat{\beta})$$.

comp_boot_mul_ind(J_inv_X_res, e)

Arguments

J_inv_X_res	A \(d \times \d\) matrix given by the expression \(\sum_{i=1}^{n}\widehat{J}^{-1}X_{i}(Y_{i}-X_{i}^{T} \widehat{\beta})\).
e	multiplier bootstrap weights. This is an \(n \times 1\) vector of mean zero, variance 1 random variables.
n	Number of observations (rows) of the underlying dataset. In the given notation we assume our dataset has \(n\) observations and \(d\) features (including an intercept)

Value

A tibble of the bootstrap standard errors.

Examples

if (FALSE) {
# Run an linear model (OLS) fit
set.seed(162632)
n <- 1e2
X <- stats::rnorm(n, 0, 1)
y <- 2 + X * 1 + stats::rnorm(n, 0, 1)
lm_fit <- stats::lm(y ~ X)

# Calculate the necessary required inputs for the bootstrap
J_inv <- summary.lm(lm_fit)$cov.unscaled
X <- qr.X(lm_fit$qr)
res <- residuals(lm_fit)
n <- length(res)
J_inv_X_res <-
  1:nrow(X) %>%
  purrr::map(~ t(J_inv %*% X[.x, ] * res[.x])) %>%
  do.call(rbind, .)

# Generate a single random vector of length n, containing
# mean 0, variance 1 random variables
e <- rnorm(n, 0, 1)

# Run a single replication of the multiplier bootstrap
mult_boot_single_rep <-
  comp_boot_mul_ind(
    n = n,
    J_inv_X_res = J_inv_X_res,
    e = e
  )
# Display the output
print(mult_boot)
}