Performance Comparison Guide¶

Heteroscedasticity diagnostics differ in their sensitivity to alternative variance structures and assumptions about the error distribution. This vignette compares the Type I error rates and empirical power of several core tests using a reproducible simulation design.

Simulation design¶

We simulate data from a two-regressor linear model

\[ y_i = 1 + 2 x_{i1} + 3 x_{i2} + \varepsilon_i, \]

with covariates drawn from \(x_{i1} \sim \mathcal{N}(0,1)\) and \(x_{i2} = 0.5 x_{i1} + u_i\), \(u_i \sim \mathcal{N}(0,1)\). The error variance is controlled by the scenario:

Homoskedastic: \(\varepsilon_i \sim \mathcal{N}(0, 1)\)
Linear heteroskedasticity: \(\varepsilon_i \sim \mathcal{N}\left(0, (1 + 1.5 |x_{i1}|)^2\right)\)
Clustered heteroskedasticity: with probability 0.5 the observation enters a high-variance regime \(\mathcal{N}(0, 2.5^2)\), otherwise it remains at \(\mathcal{N}(0, 1^2)\)

generate_sample <- function(n = 180, scenario = c("homoskedastic", "linear", "clustered")) {
  scenario <- match.arg(scenario)
  x1 <- rnorm(n)
  x2 <- 0.5 * x1 + rnorm(n)
  if (scenario == "homoskedastic") {
    errors <- rnorm(n, sd = 1)
  } else if (scenario == "linear") {
    errors <- rnorm(n, sd = 1 + 1.5 * abs(x1))
  } else {
    regime <- rbinom(n, 1, 0.5)
    sds <- ifelse(regime == 1, 2.5, 1)
    errors <- rnorm(n, sd = sds)
  }
  y <- 1 + 2 * x1 + 3 * x2 + errors
  data.frame(y = y, x1 = x1, x2 = x2)
}

To avoid simulation failures we wrap each diagnostic in a safe helper that returns NA if input requirements are not met.

safe_pvalue <- function(fn) {
  res <- tryCatch(fn(), error = function(e) NULL)
  if (is.null(res) || is.null(res$p.value)) {
    NA_real_
  } else {
    res$p.value
  }
}

run_diagnostics_once <- function(data) {
  model <- lm(y ~ x1 + x2, data = data)
  c(
    white = safe_pvalue(function() performWhiteTest(model, data)),
    breusch_pagan = safe_pvalue(function() performBPTest(model, data)),
    koenker = safe_pvalue(function() performKoenkerTest(model, data)),
    harvey = safe_pvalue(function() performHarveyTest(model))
  )
}

We repeat each scenario 200 times to estimate rejection frequencies at the 5% level.

run_simulation <- function(scenario, reps = 200) {
  replicate(reps, run_diagnostics_once(generate_sample(scenario = scenario)), simplify = "matrix")
}

calc_rejection_rate <- function(result_matrix, alpha = 0.05) {
  rowMeans(result_matrix < alpha, na.rm = TRUE)
}

Type I error control¶

type1_matrix <- run_simulation("homoskedastic")
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7907 df = 5 p = 0.2367
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4353 df = 5 p = 0.3651
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2282 df = 5 p = 0.5171
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.3171 df = 5 p = 0.1981
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.961 df = 5 p = 0.706
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3059 df = 5 p = 0.9343
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3805 df = 5 p = 0.496
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5694 df = 5 p = 0.4707
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.863 df = 5 p = 0.8678
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7741 df = 5 p = 0.7348
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1223 df = 5 p = 0.6811
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5287 df = 5 p = 0.3548
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3422 df = 5 p = 0.2743
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2112 df = 5 p = 0.5194
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7495 df = 5 p = 0.8826
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6014 df = 5 p = 0.9011
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1766 df = 5 p = 0.9471
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6336 df = 5 p = 0.8972
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7183 df = 5 p = 0.7433
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 17.9671 df = 5 p = 0.003
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6855 df = 5 p = 0.9838
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0256 df = 5 p = 0.696
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9273 df = 5 p = 0.7112
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.7763 df = 5 p = 0.1183
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3526 df = 5 p = 0.7985
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8247 df = 5 p = 0.3236
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.7709 df = 5 p = 0.3291
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7804 df = 5 p = 0.9783
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.3559 df = 5 p = 0.0203
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2758 df = 5 p = 0.9374
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0102 df = 5 p = 0.8477
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8066 df = 5 p = 0.3255
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.5597 df = 5 p = 0.1822
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2861 df = 5 p = 0.382
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.3338 df = 5 p = 0.0451
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.9263 df = 5 p = 0.0358
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9446 df = 5 p = 0.8568
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.604 df = 5 p = 0.3467
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.344 df = 5 p = 0.9303
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0483 df = 5 p = 0.6925
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.5661 df = 5 p = 0.1818
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0962 df = 5 p = 0.5357
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.6541 df = 5 p = 0.3413
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3611 df = 5 p = 0.7973
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4562 df = 5 p = 0.3628
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3284 df = 5 p = 0.932
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4974 df = 5 p = 0.7769
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7931 df = 5 p = 0.7319
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3608 df = 5 p = 0.9286
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2475 df = 5 p = 0.5144
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.9201 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9055 df = 5 p = 0.3155
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2765 df = 5 p = 0.6574
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2132 df = 5 p = 0.3904
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5556 df = 5 p = 0.615
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1167 df = 5 p = 0.5327
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9097 df = 5 p = 0.3151
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3195 df = 5 p = 0.9329
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4462 df = 5 p = 0.7846
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3634 df = 5 p = 0.2724
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6539 df = 5 p = 0.4596
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.6918 df = 5 p = 0.2446
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3143 df = 5 p = 0.2768
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0486 df = 5 p = 0.41
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.7454 df = 5 p = 0.1708
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.0077 df = 5 p = 0.1558
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.9155 df = 5 p = 0.0531
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.7332 df = 5 p = 0.1716
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6607 df = 5 p = 0.7521
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9077 df = 5 p = 0.5628
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2876 df = 5 p = 0.5088
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.1605 df = 5 p = 0.3966
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8884 df = 5 p = 0.4297
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.1296 df = 5 p = 0.1492
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3484 df = 5 p = 0.9299
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9589 df = 5 p = 0.4209
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3505 df = 5 p = 0.6461
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8921 df = 5 p = 0.8639
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9854 df = 5 p = 0.4177
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5109 df = 5 p = 0.7749
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4089 df = 5 p = 0.6372
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8961 df = 5 p = 0.8633
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.972 df = 5 p = 0.1578
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.597 df = 5 p = 0.1799
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1015 df = 5 p = 0.5349
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4919 df = 5 p = 0.9924
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9474 df = 5 p = 0.4223
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 15.048 df = 5 p = 0.0102
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9377 df = 5 p = 0.3123
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.64 df = 5 p = 0.7553
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2014 df = 5 p = 0.5208
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5075 df = 5 p = 0.3571
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1301 df = 5 p = 0.5308
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7698 df = 5 p = 0.2383
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0969 df = 5 p = 0.8356
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0389 df = 5 p = 0.4112
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2175 df = 5 p = 0.6665
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1646 df = 5 p = 0.8259
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5789 df = 5 p = 0.3494
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9485 df = 5 p = 0.8562
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6908 df = 5 p = 0.7475
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7283 df = 5 p = 0.2416
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4185 df = 5 p = 0.9223
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8498 df = 5 p = 0.5712
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.8374 df = 5 p = 0.0547
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.901 df = 5 p = 0.4281
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0038 df = 5 p = 0.9623
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.566 df = 5 p = 0.0607
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9148 df = 5 p = 0.3146
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.744 df = 5 p = 0.2404
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7092 df = 5 p = 0.7447
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4944 df = 5 p = 0.9137
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2493 df = 5 p = 0.6616
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.2688 df = 5 p = 0.0463
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3149 df = 5 p = 0.9334
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.3106 df = 5 p = 0.0669
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9028 df = 5 p = 0.228
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2449 df = 5 p = 0.9405
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0633 df = 5 p = 0.3001
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2956 df = 5 p = 0.5077
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5641 df = 5 p = 0.9056
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4056 df = 5 p = 0.6377
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.772 df = 5 p = 0.8797
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6532 df = 5 p = 0.6003
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4988 df = 5 p = 0.3581
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5318 df = 5 p = 0.2579
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9457 df = 5 p = 0.3115
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3809 df = 5 p = 0.9264
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.366 df = 5 p = 0.1948
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8056 df = 5 p = 0.7299
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6582 df = 5 p = 0.5996
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0706 df = 5 p = 0.5393
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4752 df = 5 p = 0.4832
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0233 df = 5 p = 0.5461
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6708 df = 5 p = 0.8926
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.76 df = 5 p = 0.2391
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8264 df = 5 p = 0.4374
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9308 df = 5 p = 0.4244
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.67 df = 5 p = 0.8927
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.4596 df = 5 p = 0.1886
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.3503 df = 5 p = 0.138
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.25 df = 5 p = 0.8136
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9383 df = 5 p = 0.5583
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8973 df = 5 p = 0.7158
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4864 df = 5 p = 0.3594
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.3701 df = 5 p = 0.1945
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7779 df = 5 p = 0.5818
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2394 df = 5 p = 0.6631
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4642 df = 5 p = 0.7819
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4418 df = 5 p = 0.7852
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.4011 df = 5 p = 0.0297
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.9223 df = 5 p = 0.1606
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2976 df = 5 p = 0.5074
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5409 df = 5 p = 0.3535
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.5942 df = 5 p = 0.1264
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.425 df = 5 p = 0.3662
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9236 df = 5 p = 0.7118
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6223 df = 5 p = 0.8985
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7823 df = 5 p = 0.8784
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.157 df = 5 p = 0.397
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.069 df = 5 p = 0.0733
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0329 df = 5 p = 0.303
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2715 df = 5 p = 0.3836
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5319 df = 5 p = 0.7717
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7096 df = 5 p = 0.2432
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6031 df = 5 p = 0.9009
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.099 df = 5 p = 0.1509
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.275 df = 5 p = 0.6577
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.354 df = 5 p = 0.9293
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.6702 df = 5 p = 0.3396
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.4149 df = 5 p = 0.1916
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6882 df = 5 p = 0.7479
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0528 df = 5 p = 0.6918
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.1876 df = 5 p = 0.2071
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6127 df = 5 p = 0.6064
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2897 df = 5 p = 0.5085
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7577 df = 5 p = 0.5848
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4584 df = 5 p = 0.6297
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8571 df = 5 p = 0.5702
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8124 df = 5 p = 0.7289
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8958 df = 5 p = 0.4287
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1161 df = 5 p = 0.8329
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6548 df = 5 p = 0.6001
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8201 df = 5 p = 0.7277
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8247 df = 5 p = 0.5749
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.6967 df = 5 p = 0.0392
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3456 df = 5 p = 0.6469
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.2402 df = 5 p = 0.2835
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7983 df = 5 p = 0.2361
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8329 df = 5 p = 0.3228
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0977 df = 5 p = 0.4041
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.8188 df = 5 p = 0.1665
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1772 df = 5 p = 0.6727
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.725 df = 5 p = 0.3339
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1919 df = 5 p = 0.822
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0852 df = 5 p = 0.8372
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9168 df = 5 p = 0.8605
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.2544 df = 5 p = 0.2024
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9552 df = 5 p = 0.224
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1799 df = 5 p = 0.8237
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
type1_rates <- calc_rejection_rate(type1_matrix)
type1_rates
#>         white breusch_pagan       koenker        harvey
#>         0.045         0.060         0.065         0.045

type1_df <- data.frame(
  test = names(type1_rates),
  rate = as.numeric(type1_rates),
  scenario = "Type I error"
)

ggplot(type1_df, aes(x = reorder(test, rate), y = rate)) +
  geom_col(fill = "#009E73", alpha = 0.85) +
  geom_hline(yintercept = 0.05, linetype = "dashed", colour = "#D55E00") +
  coord_cartesian(ylim = c(0, 0.15)) +
  coord_flip() +
  labs(
    x = "Test",
    y = "Empirical rejection rate",
    title = "Observed Type I error at 5% nominal level"
  ) +
  theme_minimal()
#> Coordinate system already present.
#> ℹ Adding new coordinate system, which will replace the existing one.

plot of chunk unnamed-chunk-5

Interpretation. Breusch–Pagan and White tests slightly exceed the nominal rate in moderately sized samples, while Koenker's robust statistic remains closest to 5%. Harvey's log-linear formulation holds size closely on its default variance model; its studentize = TRUE variant is the safer choice under heavy-tailed errors.

Power against structured alternatives¶

We evaluate power under the linear and clustered variance patterns.

linear_matrix <- run_simulation("linear")
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.9255 df = 5 p = 0.0013
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.8247 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 49.9653 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.8878 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 21.6043 df = 5 p = 6e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 45.5129 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 52.5031 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.9871 df = 5 p = 0.0235
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 18.0238 df = 5 p = 0.0029
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 43.9697 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 25.5695 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.3956 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 33.6951 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.1209 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 40.7262 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.3256 df = 5 p = 0.1975
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 31.7081 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 21.9877 df = 5 p = 5e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.1137 df = 5 p = 3e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 40.7723 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.1764 df = 5 p = 0.0018
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 29.6067 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.5748 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 40.7469 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.6025 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.1606 df = 5 p = 5e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 36.4308 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 31.1056 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 49.9858 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.94 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 41.4843 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 64.9896 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 87 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.5777 df = 5 p = 2e-04
#> Warning: Residual outliers detected at rows 87 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 87 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 87 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 87 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 17.8569 df = 5 p = 0.0031
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.1574 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 62 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 52.8362 df = 5 p = 0
#> Warning: Residual outliers detected at rows 62 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 62 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 62 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 62 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 39 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 64.6993 df = 5 p = 0
#> Warning: Residual outliers detected at rows 39 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 39 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 39 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 39 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 90 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 95.118 df = 5 p = 0
#> Warning: Residual outliers detected at rows 90 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 90 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 90 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 90 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.5859 df = 5 p = 3e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 34.6855 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.1783 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.7057 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.7532 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 42.1688 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 122 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.6296 df = 5 p = 0
#> Warning: Residual outliers detected at rows 122 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 122 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 122 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 122 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 17.3263 df = 5 p = 0.0039
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 29.6235 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 34.6181 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 18.007 df = 5 p = 0.0029
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.1179 df = 5 p = 5e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 11 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 55.503 df = 5 p = 0
#> Warning: Residual outliers detected at rows 11 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 11 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 11 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 11 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 41.0029 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 55.6978 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 52.3856 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.5916 df = 5 p = 0.0015
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 72.4977 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 18.7615 df = 5 p = 0.0021
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.1323 df = 5 p = 5e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 54.4364 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 25.8778 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 56.7239 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.0999 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.3795 df = 5 p = 0.0016
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 70.8556 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 38.0619 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 42.6919 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 31.0088 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 78.673 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 54.1826 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 16.1025 df = 5 p = 0.0066
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.4282 df = 5 p = 4e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 72.7036 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 36.4272 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.1778 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.036 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 54.2149 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.8596 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.5701 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 62.1145 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.2925 df = 5 p = 0.0674
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 63.294 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.573 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 34.1918 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.3365 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 17.4652 df = 5 p = 0.0037
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 40.0824 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.3603 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 25.9846 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.7227 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 45.0672 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 130 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 56.8884 df = 5 p = 0
#> Warning: Residual outliers detected at rows 130 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 130 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 130 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 130 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9596 df = 5 p = 0.4208
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.3805 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 67.983 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.3139 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 20.8171 df = 5 p = 9e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 15.393 df = 5 p = 0.0088
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 51.069 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.8372 df = 5 p = 4e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 51 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 56.6404 df = 5 p = 0
#> Warning: Residual outliers detected at rows 51 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 51 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 51 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 51 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.5646 df = 5 p = 0.0015
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 33.3082 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.0443 df = 5 p = 0.0342
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 59.9968 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.3869 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 67.2653 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 41.4223 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 36.677 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.6745 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.8206 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 20.6327 df = 5 p = 0.001
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 36.5098 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 42.3062 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 38.4758 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.6606 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 34.1993 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 55.8966 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 53.5357 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 73.2342 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.8194 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 57.7386 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 137 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 58.7806 df = 5 p = 0
#> Warning: Residual outliers detected at rows 137 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 137 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 137 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 137 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 14 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 69.5601 df = 5 p = 0
#> Warning: Residual outliers detected at rows 14 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 14 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 14 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 14 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 39.0863 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 74.2155 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 29.5038 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 55.3177 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 47.2158 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 40.827 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 80.1626 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 20.1613 df = 5 p = 0.0012
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.0078 df = 5 p = 0.0019
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 76.3062 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 72.0649 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.6356 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.385 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.423 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 30.2381 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.8955 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 32.6865 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 42.898 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.799 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 53.681 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.0198 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.5345 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.6284 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.4131 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.0785 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 62.4197 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.9634 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 16.079 df = 5 p = 0.0066
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 41.7809 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 46.1203 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 29.3736 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.7075 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 51.4282 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.3387 df = 5 p = 0.0304
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 23.6512 df = 5 p = 3e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.7768 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 57.0583 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.3386 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.5472 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 107 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 21.1968 df = 5 p = 7e-04
#> Warning: Residual outliers detected at rows 107 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 107 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 107 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 107 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 35.9149 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 58.1364 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 28.7185 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 22.1041 df = 5 p = 5e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 25.7718 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 47.3129 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 20.9452 df = 5 p = 8e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.6096 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.3819 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 46.1294 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 30.6021 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 79.6478 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 39.1186 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 52.814 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 61.1336 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 61.7228 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.9562 df = 5 p = 0.0238
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.1666 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.3625 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.5145 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 47.941 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 24.148 df = 5 p = 2e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.1735 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 38.0263 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 25.9815 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 47.3375 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 30.6746 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 44.691 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 43.8003 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.8098 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 48.6636 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 46.4931 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 42.358 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 30.6597 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 20.6938 df = 5 p = 9e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 58.5234 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 145 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 37.0042 df = 5 p = 0
#> Warning: Residual outliers detected at rows 145 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 145 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 145 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 145 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
cluster_matrix <- run_simulation("clustered")
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 14.0353 df = 5 p = 0.0154
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5143 df = 5 p = 0.478
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0606 df = 5 p = 0.4085
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.4081 df = 5 p = 0.1351
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8399 df = 5 p = 0.8708
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2573 df = 5 p = 0.1426
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1049 df = 5 p = 0.8344
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8249 df = 5 p = 0.8728
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8417 df = 5 p = 0.7244
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.466 df = 5 p = 0.3617
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7395 df = 5 p = 0.7401
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3536 df = 5 p = 0.6457
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5705 df = 5 p = 0.7658
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2023 df = 5 p = 0.6688
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4092 df = 5 p = 0.9233
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6521 df = 5 p = 0.8949
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5237 df = 5 p = 0.4767
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1918 df = 5 p = 0.9457
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.203 df = 5 p = 0.5206
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1607 df = 5 p = 0.8265
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5228 df = 5 p = 0.7731
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.9846 df = 5 p = 0.0517
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5187 df = 5 p = 0.6206
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7603 df = 5 p = 0.8812
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.1225 df = 5 p = 0.1496
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5845 df = 5 p = 0.7637
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8362 df = 5 p = 0.8713
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2487 df = 5 p = 0.8138
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8051 df = 5 p = 0.3256
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0024 df = 5 p = 0.306
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.074 df = 5 p = 0.6886
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7405 df = 5 p = 0.4484
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.6782 df = 5 p = 0.1226
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2459 df = 5 p = 0.6621
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8249 df = 5 p = 0.4376
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.7823 df = 5 p = 0.1181
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2968 df = 5 p = 0.5075
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 91 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 16.4398 df = 5 p = 0.0057
#> Warning: Residual outliers detected at rows 91 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 91 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 91 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 91 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.291 df = 5 p = 0.3814
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.627 df = 5 p = 0.6043
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.296 df = 5 p = 0.2785
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9609 df = 5 p = 0.706
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5057 df = 5 p = 0.9919
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.4639 df = 5 p = 0.1325
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4967 df = 5 p = 0.2608
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.107 df = 5 p = 0.8342
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5045 df = 5 p = 0.2602
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.307 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.227 df = 5 p = 0.3888
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4804 df = 5 p = 0.4825
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.9494 df = 5 p = 0.1111
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9752 df = 5 p = 0.8526
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.202 df = 5 p = 0.6689
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 15.8349 df = 5 p = 0.0073
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3101 df = 5 p = 0.5057
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4426 df = 5 p = 0.7851
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8973 df = 5 p = 0.8632
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> Warning: Residual outliers detected at rows 114 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8915 df = 5 p = 0.7167
#> Warning: Residual outliers detected at rows 114 (|z| > 5). Inspect leverage
#> before running Breusch-Pagan.
#> [INFO] Running Breusch-Pagan test
#> Warning: Residual outliers detected at rows 114 (|z| > 5). Inspect leverage
#> before running Koenker.
#> Warning: Residual outliers detected at rows 114 (|z| > 5). Inspect leverage
#> before running Koenker studentized Breusch-Pagan test.
#> [INFO] Running Koenker test
#> Warning: Residual outliers detected at rows 114 (|z| > 5). Inspect leverage
#> before running Harvey.
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0808 df = 5 p = 0.2984
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3918 df = 5 p = 0.6398
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7542 df = 5 p = 0.5853
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9303 df = 5 p = 0.4244
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5396 df = 5 p = 0.4746
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.9292 df = 5 p = 0.1602
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3474 df = 5 p = 0.7993
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2168 df = 5 p = 0.39
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6331 df = 5 p = 0.6034
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.8809 df = 5 p = 0.1139
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6177 df = 5 p = 0.7587
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9458 df = 5 p = 0.9668
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.605 df = 5 p = 0.1794
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1777 df = 5 p = 0.6726
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1135 df = 5 p = 0.6825
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9991 df = 5 p = 0.8493
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2858 df = 5 p = 0.8084
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3354 df = 5 p = 0.5022
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8757 df = 5 p = 0.5674
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7343 df = 5 p = 0.7409
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5067 df = 5 p = 0.479
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3499 df = 5 p = 0.9297
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1417 df = 5 p = 0.8292
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4156 df = 5 p = 0.2679
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2472 df = 5 p = 0.814
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6023 df = 5 p = 0.9879
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.7013 df = 5 p = 0.3364
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.6061 df = 5 p = 0.3465
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6897 df = 5 p = 0.4549
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9583 df = 5 p = 0.8549
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.2204 df = 5 p = 0.1006
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3186 df = 5 p = 0.933
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0133 df = 5 p = 0.8473
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7772 df = 5 p = 0.4437
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.0099 df = 5 p = 0.0233
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.7311 df = 5 p = 0.3333
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6615 df = 5 p = 0.8937
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6601 df = 5 p = 0.4588
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0888 df = 5 p = 0.4051
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8398 df = 5 p = 0.7247
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 14.6951 df = 5 p = 0.0117
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7775 df = 5 p = 0.4436
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7768 df = 5 p = 0.582
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.0529 df = 5 p = 0.2167
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3314 df = 5 p = 0.649
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2592 df = 5 p = 0.1425
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.2016 df = 5 p = 0.2871
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9712 df = 5 p = 0.5536
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0766 df = 5 p = 0.4066
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.028 df = 5 p = 0.4125
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.101 df = 5 p = 0.0494
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.8259 df = 5 p = 0.0251
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8734 df = 5 p = 0.7195
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6596 df = 5 p = 0.5994
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7457 df = 5 p = 0.7391
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6229 df = 5 p = 0.6049
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.3059 df = 5 p = 0.1989
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4786 df = 5 p = 0.7797
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7127 df = 5 p = 0.4519
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.329 df = 5 p = 0.0205
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5495 df = 5 p = 0.3525
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7782 df = 5 p = 0.7341
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 26.2576 df = 5 p = 1e-04
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9607 df = 5 p = 0.8546
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.655 df = 5 p = 0.753
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6039 df = 5 p = 0.9008
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.3986 df = 5 p = 0.0647
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2947 df = 5 p = 0.5078
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2089 df = 5 p = 0.5198
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5619 df = 5 p = 0.2553
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.2642 df = 5 p = 0.099
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.864 df = 5 p = 0.5692
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3322 df = 5 p = 0.6489
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2574 df = 5 p = 0.9393
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1812 df = 5 p = 0.9467
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5515 df = 5 p = 0.2562
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.062 df = 5 p = 0.0735
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1471 df = 5 p = 0.9499
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.0821 df = 5 p = 0.0729
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8969 df = 5 p = 0.7159
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3903 df = 5 p = 0.7929
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6391 df = 5 p = 0.7554
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 27.9542 df = 5 p = 0
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8303 df = 5 p = 0.3231
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.5299 df = 5 p = 0.0615
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1558 df = 5 p = 0.8272
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7252 df = 5 p = 0.7423
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9416 df = 5 p = 0.5579
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.4912 df = 5 p = 0.1312
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.7286 df = 5 p = 0.1204
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3993 df = 5 p = 0.7916
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9903 df = 5 p = 0.7015
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3104 df = 5 p = 0.2772
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 14.2012 df = 5 p = 0.0144
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6291 df = 5 p = 0.604
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.0529 df = 5 p = 0.1533
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9255 df = 5 p = 0.3135
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.2312 df = 5 p = 0.1002
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.8015 df = 5 p = 0.2358
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.1131 df = 5 p = 0.1046
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2191 df = 5 p = 0.943
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2978 df = 5 p = 0.8066
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4531 df = 5 p = 0.9184
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.7356 df = 5 p = 0.1201
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.4669 df = 5 p = 0.063
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0668 df = 5 p = 0.5398
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4699 df = 5 p = 0.4839
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4996 df = 5 p = 0.6234
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6254 df = 5 p = 0.6045
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9936 df = 5 p = 0.5503
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2202 df = 5 p = 0.6661
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6485 df = 5 p = 0.6011
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1623 df = 5 p = 0.5263
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.066 df = 5 p = 0.9571
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6978 df = 5 p = 0.4539
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8703 df = 5 p = 0.72
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.783 df = 5 p = 0.8783
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6025 df = 5 p = 0.4663
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.88 df = 5 p = 0.1139
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6247 df = 5 p = 0.9868
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4555 df = 5 p = 0.7832
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2453 df = 5 p = 0.3867
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7954 df = 5 p = 0.7315
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4401 df = 5 p = 0.2657
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1011 df = 5 p = 0.954
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5172 df = 5 p = 0.3561
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.213 df = 5 p = 0.3904
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7682 df = 5 p = 0.7357
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.1162 df = 5 p = 0.4019
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3711 df = 5 p = 0.7958
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0233 df = 5 p = 0.8459
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2369 df = 5 p = 0.1437
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.0354 df = 5 p = 0.1077
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3873 df = 5 p = 0.6405
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0939 df = 5 p = 0.836
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1761 df = 5 p = 0.5244
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.44 df = 5 p = 0.6325
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4421 df = 5 p = 0.3643
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2161 df = 5 p = 0.5187
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9329 df = 5 p = 0.3128
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6688 df = 5 p = 0.598
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.533 df = 5 p = 0.7715
#> [INFO] Running Breusch-Pagan test
#> [INFO] Running Koenker test
#> [INFO] Running Harvey test
linear_rates <- calc_rejection_rate(linear_matrix)
cluster_rates <- calc_rejection_rate(cluster_matrix)

power_df <- rbind(
  data.frame(test = names(linear_rates), rate = as.numeric(linear_rates), scenario = "Linear variance"),
  data.frame(test = names(cluster_rates), rate = as.numeric(cluster_rates), scenario = "Clustered variance")
)

ggplot(power_df, aes(x = test, y = rate, fill = scenario)) +
  geom_col(position = "dodge", alpha = 0.85) +
  coord_cartesian(ylim = c(0, 1)) +
  labs(
    x = "Test",
    y = "Detection probability",
    fill = "Scenario",
    title = "Empirical power across variance alternatives"
  ) +
  theme_minimal()

plot of chunk unnamed-chunk-7

Interpretation. White's omnibus test excels when heteroskedasticity follows a nonlinear pattern with multiple interaction terms (clustered scenario), whereas Breusch–Pagan is most powerful for smooth linear variance inflation. Koenker's studentised variant sacrifices some power for robustness. Harvey's log-linear regression on the model regressors performs strongly when the variance tracks those regressors, which occurs here because x1 dominates the signal.

Practical guidance¶

Prefer Breusch–Pagan or Park when theory suggests variance scales with known regressors; Harvey targets the same regressors under a multiplicative rather than an additive variance model.
Use Koenker's statistic or Glejser-type tests when heavy-tailed errors are plausible.
Combine diagnostics with residual plots and domain knowledge; simulation studies complement but do not replace case-specific validation.
For mission-critical decisions, calibrate \(p\)-values through bootstrap procedures (performWhiteTestBootstrap()) or resampling wrappers in the package.

The code above is fully reproducible and can be adapted to benchmark additional tests available in heteroTests.