
  set.seed(1)
  DGP_OLS <- function() {
    X <- runif(100,2,10)
    Y <- 2*X + rnorm(100, sd = sqrt(X))
    return(
      c("beta_1_hat" = sum(X*Y)/sum(X^2))
    )
  }
  estimates <- replicate(1000, DGP_OLS())
  est_var_OLS <- var(estimates)


  set.seed(1)
  # define the function `DGP_WLS()`
  
  
  # estimate the variance, assign the value to `var_est_WLS`
  
  
  # compare the estimated variances


  set.seed(1)
  # define the function `DGP_WLS()`
  DGP_WLS <- function() {
  X <- runif(100,2,10)
  Y <- 2*X + rnorm(100, sd = sqrt(X))
  w <- 1/sqrt(X)
  return(
    c("beta_1_hat" = sum(w^2*X*Y)/sum((w*X)^2))
  )
  }
  # estimate the variance, assign the value to `var_est_WLS`
  est_var_WLS <- var(replicate(1000, DGP_WLS()))

  # compare the estimated variances
  est_var_WLS < est_var_OLS


  test_predefined_objects(c("DGP_OLS","est_var_OLS"))
  test_function_definition("DGP_WLS",
                           function_test =
                           test_expression_result("DGP_OLS()")
  
  )
  test_object("est_var_WLS")
  test_or({
    test_student_typed("est_var_WLS < est_var_OLS")
  },{
    test_student_typed("est_var_WLS > est_var_OLS")
  },{
    test_student_typed("est_var_OLS > est_var_WLS")
  },{
    test_student_typed("est_var_OLS < est_var_WLS")
  })
  success_msg("Nice! This indicates that the WLS estimator indeed has lower variance than OLS under heteroskedasticity.")