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Bootstrap White test

Source: R/additional_tests.R
performWhiteTestBootstrap.Rd

Approximates the finite-sample distribution of White's LM statistic by resampling the fitted residuals, refitting the model, and recalculating the test statistic across many bootstrap replications.

Usage

performWhiteTestBootstrap(model, data, B = 1000, parallel = FALSE)

Arguments

model

A fitted stats::lm object representing the mean specification to be diagnosed.

data

A base::data.frame (or object coercible to one) containing the variables referenced by model. It must include the observations used to fit model and will be checked for missing values.

B

Integer scalar giving the number of bootstrap replications. Larger values yield smoother p-value estimates at the cost of additional runtime.

parallel

Logical scalar; if TRUE the bootstrap loop is executed with parallel::mclapply() when the parallel package and forked processing are available.

Value

A htest object reporting both the asymptotic White statistic and its bootstrap p-value. The returned object also stores the simulated test statistics in boot_statistics for further inspection.

Details

For a fitted model \(\hat{y} = X\hat{\beta}\), the algorithm proceeds as follows:

  1. Compute White's LM statistic \(n R^2\) from the auxiliary regression on squares and cross-products of the regressors.

  2. Generate \(B\) bootstrap samples by resampling the centred residuals with replacement, forming \(y^{*(b)} = \hat{y} + \hat{e}^{*(b)}\).

  3. Refit the model to each bootstrap sample and recompute White's statistic \(T^{*(b)}\) using the same auxiliary specification.

  4. Estimate the bootstrap p-value as \(\hat{p} = B^{-1} \sum_{b = 1}^B I\{T^{*(b)} \ge T_{\text{obs}}\}\).

The bootstrap distribution offers improved size control for moderate sample sizes or high-dimensional designs where the chi-squared approximation may be inaccurate. When parallel = TRUE the resampling step exploits available CPU cores to reduce computation time.

References

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC.

Davidson, R., & MacKinnon, J. G. (2006). The power of bootstrap and asymptotic tests. Journal of Econometrics, 133(2), 421–441. doi:10.1016/j.jeconom.2005.02.002

See also

performWhiteTest() for the classical statistic and performWhiteTestRobust() for enriched diagnostics.

Examples

# The bootstrap needs at least 50 observations, so mtcars (32) is too small.
set.seed(42)
n <- 120
sim <- data.frame(x = runif(n, 1, 5))
sim$y <- 1 + 2 * sim$x + rnorm(n, sd = 0.3 + 0.6 * sim$x)
mod <- lm(y ~ x, data = sim)
performWhiteTestBootstrap(mod, sim, B = 199)
#> [INFO] Running Bootstrap White test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.8014 df = 2 p = 0.001
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3446 df = 2 p = 0.8417
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2837 df = 2 p = 0.1936
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3391 df = 2 p = 0.5119
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3425 df = 2 p = 0.114
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2326 df = 2 p = 0.8902
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8126 df = 2 p = 0.1486
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0022 df = 2 p = 0.9989
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0196 df = 2 p = 0.9903
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2459 df = 2 p = 0.8843
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3111 df = 2 p = 0.8559
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8817 df = 2 p = 0.1436
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0204 df = 2 p = 0.3642
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2161 df = 2 p = 0.5444
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0329 df = 2 p = 0.9837
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0131 df = 2 p = 0.9935
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3472 df = 2 p = 0.8406
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2298 df = 2 p = 0.8915
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9291 df = 2 p = 0.3812
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2165 df = 2 p = 0.5443
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.8608 df = 2 p = 0.0072
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.6006 df = 2 p = 0.0369
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9695 df = 2 p = 0.3735
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0509 df = 2 p = 0.9749
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3438 df = 2 p = 0.8421
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2669 df = 2 p = 0.8751
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0216 df = 2 p = 0.9893
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5935 df = 2 p = 0.2734
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3695 df = 2 p = 0.1125
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2488 df = 2 p = 0.3248
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3767 df = 2 p = 0.3047
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3948 df = 2 p = 0.8209
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.717 df = 2 p = 0.257
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9433 df = 2 p = 0.624
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3313 df = 2 p = 0.5139
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1961 df = 2 p = 0.9066
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1901 df = 2 p = 0.3345
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.976 df = 2 p = 0.6139
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7063 df = 2 p = 0.035
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7958 df = 2 p = 0.6717
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3729 df = 2 p = 0.8299
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0386 df = 2 p = 0.9809
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8084 df = 2 p = 0.6675
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.069 df = 2 p = 0.586
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3402 df = 2 p = 0.5117
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.866 df = 2 p = 0.6486
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3051 df = 2 p = 0.5207
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.1977 df = 2 p = 0.0274
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.552 df = 2 p = 0.7588
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1139 df = 2 p = 0.3475
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3545 df = 2 p = 0.508
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6617 df = 2 p = 0.2642
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0408 df = 2 p = 0.9798
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.394 df = 2 p = 0.0674
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8958 df = 2 p = 0.639
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9377 df = 2 p = 0.1396
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0087 df = 2 p = 0.2222
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8406 df = 2 p = 0.0539
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8321 df = 2 p = 0.4001
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1602 df = 2 p = 0.3396
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9351 df = 2 p = 0.6265
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5704 df = 2 p = 0.2766
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4317 df = 2 p = 0.8059
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6341 df = 2 p = 0.7283
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0935 df = 2 p = 0.3511
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3635 df = 2 p = 0.3067
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0528 df = 2 p = 0.2173
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6037 df = 2 p = 0.1001
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0781 df = 2 p = 0.0789
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3043 df = 2 p = 0.316
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.247 df = 2 p = 0.5361
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3873 df = 2 p = 0.8239
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0273 df = 2 p = 0.9864
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9375 df = 2 p = 0.1396
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0141 df = 2 p = 0.993
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4861 df = 2 p = 0.175
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7028 df = 2 p = 0.7037
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0702 df = 2 p = 0.9655
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7504 df = 2 p = 0.6872
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8487 df = 2 p = 0.6542
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9788 df = 2 p = 0.083
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2445 df = 2 p = 0.3255
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8242 df = 2 p = 0.6623
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3464 df = 2 p = 0.841
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1308 df = 2 p = 0.209
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1867 df = 2 p = 0.9109
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.336 df = 2 p = 0.8453
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6805 df = 2 p = 0.1588
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0811 df = 2 p = 0.3533
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4613 df = 2 p = 0.1772
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4808 df = 2 p = 0.7863
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6652 df = 2 p = 0.7171
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1094 df = 2 p = 0.2113
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.714 df = 2 p = 0.6998
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2647 df = 2 p = 0.876
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9194 df = 2 p = 0.383
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8729 df = 2 p = 0.6463
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4724 df = 2 p = 0.7896
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8234 df = 2 p = 0.4018
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4619 df = 2 p = 0.4814
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2021 df = 2 p = 0.3325
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9248 df = 2 p = 0.0517
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2903 df = 2 p = 0.3182
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3617 df = 2 p = 0.5062
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 19.323 df = 2 p = 1e-04
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3711 df = 2 p = 0.5038
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.6322 df = 2 p = 0.0363
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4734 df = 2 p = 0.7892
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2135 df = 2 p = 0.8987
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4698 df = 2 p = 0.2909
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1132 df = 2 p = 0.5732
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.729 df = 2 p = 0.6945
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1631 df = 2 p = 0.559
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.869 df = 2 p = 0.3928
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0499 df = 2 p = 0.9754
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4626 df = 2 p = 0.4813
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.333 df = 2 p = 0.5135
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6118 df = 2 p = 0.4467
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4333 df = 2 p = 0.8052
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2834 df = 2 p = 0.1175
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1807 df = 2 p = 0.9136
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5441 df = 2 p = 0.7618
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5106 df = 2 p = 0.4699
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6232 df = 2 p = 0.7323
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5378 df = 2 p = 0.4635
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1565 df = 2 p = 0.9247
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6747 df = 2 p = 0.7137
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.295 df = 2 p = 0.8629
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.165 df = 2 p = 0.2055
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6772 df = 2 p = 0.4323
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7635 df = 2 p = 0.4141
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1968 df = 2 p = 0.5497
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3865 df = 2 p = 0.3032
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6458 df = 2 p = 0.4392
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3062 df = 2 p = 0.5204
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4605 df = 2 p = 0.7944
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2312 df = 2 p = 0.8908
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4729 df = 2 p = 0.2904
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2189 df = 2 p = 0.0164
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.514 df = 2 p = 0.7734
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5715 df = 2 p = 0.2764
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7288 df = 2 p = 0.4213
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3422 df = 2 p = 0.5111
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.739 df = 2 p = 0.1542
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2764 df = 2 p = 0.3204
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0887 df = 2 p = 0.2135
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9965 df = 2 p = 0.1356
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0816 df = 2 p = 0.96
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2934 df = 2 p = 0.8635
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0018 df = 2 p = 0.9991
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1926 df = 2 p = 0.5509
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9461 df = 2 p = 0.031
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6326 df = 2 p = 0.4421
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0059 df = 2 p = 0.1349
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.516 df = 2 p = 0.1046
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7496 df = 2 p = 0.4169
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5417 df = 2 p = 0.4626
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2715 df = 2 p = 0.8731
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.576 df = 2 p = 0.1015
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0999 df = 2 p = 0.9513
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9765 df = 2 p = 0.6137
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.804 df = 2 p = 0.669
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5574 df = 2 p = 0.7568
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1413 df = 2 p = 0.5651
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2204 df = 2 p = 0.8957
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8408 df = 2 p = 0.3984
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2889 df = 2 p = 0.5249
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8255 df = 2 p = 0.6618
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6314 df = 2 p = 0.7293
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6262 df = 2 p = 0.4435
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0082 df = 2 p = 0.3664
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.702 df = 2 p = 0.0953
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2486 df = 2 p = 0.8831
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0176 df = 2 p = 0.6012
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1708 df = 2 p = 0.9181
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0795 df = 2 p = 0.3535
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2251 df = 2 p = 0.542
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3173 df = 2 p = 0.5175
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7174 df = 2 p = 0.4237
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5102 df = 2 p = 0.285
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1498 df = 2 p = 0.207
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0216 df = 2 p = 0.6
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7324 df = 2 p = 0.2551
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1358 df = 2 p = 0.9344
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0209 df = 2 p = 0.9896
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7581 df = 2 p = 0.6845
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.552 df = 2 p = 0.4602
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3622 df = 2 p = 0.5061
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7445 df = 2 p = 0.6892
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3095 df = 2 p = 0.8566
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4532 df = 2 p = 0.7972
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2992 df = 2 p = 0.8611
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3343 df = 2 p = 0.3113
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5872 df = 2 p = 0.1009
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3228 df = 2 p = 0.851
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1882 df = 2 p = 0.9102
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1922 df = 2 p = 0.9084
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.26 df = 2 p = 0.1959
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5071 df = 2 p = 0.0386
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7897 df = 2 p = 0.4087
#> 
#> 	Bootstrap White test
#> 
#> data:  mod
#> X-squared = 13.801, B = 199, p-value = 0.01
#> 

# \donttest{
# More replications give a smoother p-value estimate
performWhiteTestBootstrap(mod, sim, B = 499)
#> [INFO] Running Bootstrap White test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.8014 df = 2 p = 0.001
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1122 df = 2 p = 0.9455
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5489 df = 2 p = 0.2796
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5436 df = 2 p = 0.4622
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1299 df = 2 p = 0.3447
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5301 df = 2 p = 0.7672
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6918 df = 2 p = 0.2603
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3419 df = 2 p = 0.042
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4966 df = 2 p = 0.7801
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4611 df = 2 p = 0.4816
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7545 df = 2 p = 0.153
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7888 df = 2 p = 0.6741
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.316 df = 2 p = 0.8539
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5299 df = 2 p = 0.1712
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5749 df = 2 p = 0.1674
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9193 df = 2 p = 0.383
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2744 df = 2 p = 0.8718
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3293 df = 2 p = 0.5144
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1286 df = 2 p = 0.9377
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0036 df = 2 p = 0.2227
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0597 df = 2 p = 0.5887
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6483 df = 2 p = 0.266
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7658 df = 2 p = 0.2509
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1754 df = 2 p = 0.2044
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4842 df = 2 p = 0.4761
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5186 df = 2 p = 0.1044
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9914 df = 2 p = 0.6091
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3304 df = 2 p = 0.0422
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1013 df = 2 p = 0.3497
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9277 df = 2 p = 0.0851
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3861 df = 2 p = 0.5
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9322 df = 2 p = 0.0849
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8695 df = 2 p = 0.6474
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5559 df = 2 p = 0.7573
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2857 df = 2 p = 0.3189
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1402 df = 2 p = 0.9323
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2051 df = 2 p = 0.9025
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1762 df = 2 p = 0.9157
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4039 df = 2 p = 0.8171
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3123 df = 2 p = 0.5188
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7577 df = 2 p = 0.4153
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0092 df = 2 p = 0.9954
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1268 df = 2 p = 0.3453
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2756 df = 2 p = 0.1944
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7727 df = 2 p = 0.1516
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3754 df = 2 p = 0.8289
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8326 df = 2 p = 0.6595
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9094 df = 2 p = 0.3849
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7963 df = 2 p = 0.4073
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7574 df = 2 p = 0.2519
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4611 df = 2 p = 0.1075
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8801 df = 2 p = 0.3906
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2207 df = 2 p = 0.0735
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3097 df = 2 p = 0.8566
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.025 df = 2 p = 0.599
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0817 df = 2 p = 0.96
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8473 df = 2 p = 0.3971
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3493 df = 2 p = 0.8397
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5682 df = 2 p = 0.4565
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1463 df = 2 p = 0.3419
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2211 df = 2 p = 0.1212
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9396 df = 2 p = 0.6251
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2316 df = 2 p = 0.3277
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6097 df = 2 p = 0.4472
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1518 df = 2 p = 0.9269
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.655 df = 2 p = 0.2651
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9958 df = 2 p = 0.3687
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1508 df = 2 p = 0.5625
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7373 df = 2 p = 0.6917
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3516 df = 2 p = 0.3086
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.4921 df = 2 p = 0.0087
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7459 df = 2 p = 0.2534
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0338 df = 2 p = 0.3617
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.036 df = 2 p = 0.5957
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.743 df = 2 p = 0.4183
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4329 df = 2 p = 0.1797
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8368 df = 2 p = 0.1468
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4659 df = 2 p = 0.7922
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1577 df = 2 p = 0.2062
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9445 df = 2 p = 0.2294
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1321 df = 2 p = 0.9361
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3459 df = 2 p = 0.8412
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4131 df = 2 p = 0.2992
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9473 df = 2 p = 0.6227
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6932 df = 2 p = 0.7071
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3518 df = 2 p = 0.5087
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8515 df = 2 p = 0.2403
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1465 df = 2 p = 0.9294
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1116 df = 2 p = 0.5736
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3233 df = 2 p = 0.313
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8404 df = 2 p = 0.3984
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5441 df = 2 p = 0.7618
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.461 df = 2 p = 0.1075
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6065 df = 2 p = 0.7384
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8705 df = 2 p = 0.6471
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0747 df = 2 p = 0.3544
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5234 df = 2 p = 0.7698
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3092 df = 2 p = 0.8568
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8506 df = 2 p = 0.6536
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0587 df = 2 p = 0.3572
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.1335 df = 2 p = 0.0014
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9368 df = 2 p = 0.1397
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3932 df = 2 p = 0.4983
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4861 df = 2 p = 0.175
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.167 df = 2 p = 0.2053
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.183 df = 2 p = 0.9126
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4898 df = 2 p = 0.1059
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9738 df = 2 p = 0.6145
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9183 df = 2 p = 0.6318
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3666 df = 2 p = 0.8325
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5206 df = 2 p = 0.2836
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0764 df = 2 p = 0.2148
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0159 df = 2 p = 0.6017
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1605 df = 2 p = 0.9229
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5087 df = 2 p = 0.7754
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0939 df = 2 p = 0.9541
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5672 df = 2 p = 0.7531
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2044 df = 2 p = 0.9029
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7385 df = 2 p = 0.4193
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0982 df = 2 p = 0.2124
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.272 df = 2 p = 0.1948
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3445 df = 2 p = 0.8417
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.574 df = 2 p = 0.7505
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8716 df = 2 p = 0.6468
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1461 df = 2 p = 0.9295
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2255 df = 2 p = 0.5419
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.526 df = 2 p = 0.0232
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.245 df = 2 p = 0.8847
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2379 df = 2 p = 0.5385
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4491 df = 2 p = 0.7989
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4409 df = 2 p = 0.2951
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8975 df = 2 p = 0.6384
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5382 df = 2 p = 0.7641
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0546 df = 2 p = 0.5902
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9884 df = 2 p = 0.2244
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7717 df = 2 p = 0.6799
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8234 df = 2 p = 0.4018
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9611 df = 2 p = 0.6184
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3163 df = 2 p = 0.8537
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6218 df = 2 p = 0.4445
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3323 df = 2 p = 0.5137
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1422 df = 2 p = 0.3426
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3999 df = 2 p = 0.8188
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8387 df = 2 p = 0.2419
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0848 df = 2 p = 0.9585
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1536 df = 2 p = 0.9261
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.467 df = 2 p = 0.1072
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6024 df = 2 p = 0.1651
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4786 df = 2 p = 0.4774
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3386 df = 2 p = 0.5121
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0001 df = 2 p = 0.3679
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1427 df = 2 p = 0.9311
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4524 df = 2 p = 0.7976
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3027 df = 2 p = 0.8596
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5451 df = 2 p = 0.4618
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.54 df = 2 p = 0.1703
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.838 df = 2 p = 0.6577
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0446 df = 2 p = 0.5931
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.137 df = 2 p = 0.5664
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5563 df = 2 p = 0.2786
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4367 df = 2 p = 0.4876
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9144 df = 2 p = 0.6331
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2254 df = 2 p = 0.3287
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1526 df = 2 p = 0.562
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7359 df = 2 p = 0.2546
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6554 df = 2 p = 0.7206
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3763 df = 2 p = 0.8285
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3343 df = 2 p = 0.3113
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2846 df = 2 p = 0.8673
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1153 df = 2 p = 0.5725
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5712 df = 2 p = 0.4558
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9667 df = 2 p = 0.1376
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7526 df = 2 p = 0.2525
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.691 df = 2 p = 0.4293
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8331 df = 2 p = 0.6593
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6503 df = 2 p = 0.4382
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6068 df = 2 p = 0.7383
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0913 df = 2 p = 0.3515
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6958 df = 2 p = 0.7062
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2617 df = 2 p = 0.5321
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3832 df = 2 p = 0.1117
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3043 df = 2 p = 0.8589
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3743 df = 2 p = 0.3051
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5084 df = 2 p = 0.2853
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1447 df = 2 p = 0.9302
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4334 df = 2 p = 0.8052
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6421 df = 2 p = 0.1619
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9498 df = 2 p = 0.031
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8361 df = 2 p = 0.3993
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4436 df = 2 p = 0.4859
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0034 df = 2 p = 0.3673
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5795 df = 2 p = 0.2753
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6599 df = 2 p = 0.719
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7069 df = 2 p = 0.2583
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1243 df = 2 p = 0.3457
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4626 df = 2 p = 0.1074
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0931 df = 2 p = 0.3511
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.1755 df = 2 p = 0.0277
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7419 df = 2 p = 0.6901
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6213 df = 2 p = 0.733
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5067 df = 2 p = 0.4708
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.729 df = 2 p = 0.2555
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4796 df = 2 p = 0.2894
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7355 df = 2 p = 0.4199
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1404 df = 2 p = 0.208
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0492 df = 2 p = 0.5918
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6132 df = 2 p = 0.1642
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.678 df = 2 p = 0.0964
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2602 df = 2 p = 0.878
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9049 df = 2 p = 0.3858
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5968 df = 2 p = 0.742
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2699 df = 2 p = 0.53
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1057 df = 2 p = 0.9485
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6894 df = 2 p = 0.7084
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7945 df = 2 p = 0.6722
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1963 df = 2 p = 0.3335
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5666 df = 2 p = 0.2771
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7126 df = 2 p = 0.4247
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5259 df = 2 p = 0.7688
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2288 df = 2 p = 0.541
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4224 df = 2 p = 0.4911
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7414 df = 2 p = 0.2539
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3079 df = 2 p = 0.52
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5426 df = 2 p = 0.7624
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1742 df = 2 p = 0.3372
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2032 df = 2 p = 0.9034
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.729 df = 2 p = 0.094
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0596 df = 2 p = 0.5887
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6653 df = 2 p = 0.4349
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9577 df = 2 p = 0.2279
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8144 df = 2 p = 0.4037
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9629 df = 2 p = 0.6179
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4283 df = 2 p = 0.1801
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9143 df = 2 p = 0.6331
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7301 df = 2 p = 0.0346
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8736 df = 2 p = 0.1442
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1643 df = 2 p = 0.9211
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9352 df = 2 p = 0.6265
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4298 df = 2 p = 0.18
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.985 df = 2 p = 0.1364
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9125 df = 2 p = 0.3843
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8775 df = 2 p = 0.6448
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2436 df = 2 p = 0.537
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5166 df = 2 p = 0.7724
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 10.6911 df = 2 p = 0.0048
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0732 df = 2 p = 0.9641
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2078 df = 2 p = 0.122
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2049 df = 2 p = 0.0741
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.769 df = 2 p = 0.2505
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0094 df = 2 p = 0.9953
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.499 df = 2 p = 0.7792
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9797 df = 2 p = 0.3716
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.934 df = 2 p = 0.6269
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2142 df = 2 p = 0.3305
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4465 df = 2 p = 0.7999
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.5288 df = 2 p = 0.0085
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3062 df = 2 p = 0.858
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6305 df = 2 p = 0.7296
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6281 df = 2 p = 0.163
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3092 df = 2 p = 0.3152
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.0054 df = 2 p = 0.0301
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0781 df = 2 p = 0.3538
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8715 df = 2 p = 0.6468
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8852 df = 2 p = 0.6424
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0141 df = 2 p = 0.993
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9488 df = 2 p = 0.6222
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0804 df = 2 p = 0.9606
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8244 df = 2 p = 0.6622
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7113 df = 2 p = 0.425
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8168 df = 2 p = 0.6647
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6908 df = 2 p = 0.4294
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8061 df = 2 p = 0.6683
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8584 df = 2 p = 0.1453
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4687 df = 2 p = 0.291
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6171 df = 2 p = 0.2702
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5353 df = 2 p = 0.2815
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7935 df = 2 p = 0.4079
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3367 df = 2 p = 0.5126
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.047 df = 2 p = 0.9768
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1751 df = 2 p = 0.5557
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.018 df = 2 p = 0.2211
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1944 df = 2 p = 0.3338
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3104 df = 2 p = 0.315
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5659 df = 2 p = 0.7536
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6561 df = 2 p = 0.7203
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0074 df = 2 p = 0.3665
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7201 df = 2 p = 0.6976
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.407 df = 2 p = 0.3001
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.349 df = 2 p = 0.1137
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1955 df = 2 p = 0.5501
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.2805 df = 2 p = 0.0713
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7265 df = 2 p = 0.6954
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8867 df = 2 p = 0.2361
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4564 df = 2 p = 0.4828
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6147 df = 2 p = 0.446
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0487 df = 2 p = 0.1321
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6346 df = 2 p = 0.7281
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6931 df = 2 p = 0.7071
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3036 df = 2 p = 0.8591
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4477 df = 2 p = 0.7994
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8703 df = 2 p = 0.3925
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4365 df = 2 p = 0.04
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0361 df = 2 p = 0.9821
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3692 df = 2 p = 0.8315
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4855 df = 2 p = 0.4758
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0382 df = 2 p = 0.5951
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.3531 df = 2 p = 0.0093
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6773 df = 2 p = 0.2622
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2221 df = 2 p = 0.1997
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1914 df = 2 p = 0.5512
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7965 df = 2 p = 0.6715
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3335 df = 2 p = 0.5134
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1209 df = 2 p = 0.21
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5362 df = 2 p = 0.4639
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0053 df = 2 p = 0.6049
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6782 df = 2 p = 0.7124
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6498 df = 2 p = 0.7226
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3607 df = 2 p = 0.835
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5784 df = 2 p = 0.0373
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.549 df = 2 p = 0.4609
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3524 df = 2 p = 0.8385
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4198 df = 2 p = 0.4917
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3844 df = 2 p = 0.3035
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7168 df = 2 p = 0.1559
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4588 df = 2 p = 0.4822
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7573 df = 2 p = 0.0927
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3283 df = 2 p = 0.1148
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.428 df = 2 p = 0.8073
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.656 df = 2 p = 0.1607
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2328 df = 2 p = 0.3275
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2625 df = 2 p = 0.3226
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4131 df = 2 p = 0.4933
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.686 df = 2 p = 0.2611
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2138 df = 2 p = 0.8986
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0575 df = 2 p = 0.5893
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5931 df = 2 p = 0.1659
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5751 df = 2 p = 0.7501
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1856 df = 2 p = 0.3353
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8309 df = 2 p = 0.6601
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7024 df = 2 p = 0.035
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6832 df = 2 p = 0.7106
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2657 df = 2 p = 0.3221
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2833 df = 2 p = 0.8679
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4436 df = 2 p = 0.8011
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7918 df = 2 p = 0.6731
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2465 df = 2 p = 0.884
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3606 df = 2 p = 0.835
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2472 df = 2 p = 0.8837
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2083 df = 2 p = 0.3315
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6679 df = 2 p = 0.4343
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0152 df = 2 p = 0.6019
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.96 df = 2 p = 0.3753
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8532 df = 2 p = 0.0883
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1797 df = 2 p = 0.9141
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1231 df = 2 p = 0.9403
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0956 df = 2 p = 0.3507
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.503 df = 2 p = 0.7776
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6003 df = 2 p = 0.1002
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7351 df = 2 p = 0.42
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1704 df = 2 p = 0.1243
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9935 df = 2 p = 0.6085
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5538 df = 2 p = 0.4598
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0179 df = 2 p = 0.2211
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3597 df = 2 p = 0.8354
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6773 df = 2 p = 0.7127
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1265 df = 2 p = 0.2095
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2996 df = 2 p = 0.5222
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2949 df = 2 p = 0.5234
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8885 df = 2 p = 0.389
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8119 df = 2 p = 0.2451
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0407 df = 2 p = 0.9798
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0175 df = 2 p = 0.9913
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.458 df = 2 p = 0.7953
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.1447 df = 2 p = 0.5642
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2862 df = 2 p = 0.1934
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9199 df = 2 p = 0.2323
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4887 df = 2 p = 0.7832
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2066 df = 2 p = 0.547
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7962 df = 2 p = 0.4073
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5433 df = 2 p = 0.7621
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8449 df = 2 p = 0.3975
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.952 df = 2 p = 0.1386
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3774 df = 2 p = 0.3046
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8521 df = 2 p = 0.6531
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3977 df = 2 p = 0.4971
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4821 df = 2 p = 0.1063
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7924 df = 2 p = 0.4081
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0018 df = 2 p = 0.1352
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6271 df = 2 p = 0.1631
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.0131 df = 2 p = 0.0015
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5421 df = 2 p = 0.7626
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9646 df = 2 p = 0.6174
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.175 df = 2 p = 0.3371
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.246 df = 2 p = 0.8843
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3559 df = 2 p = 0.837
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8093 df = 2 p = 0.6672
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9933 df = 2 p = 0.2239
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3472 df = 2 p = 0.8406
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.407 df = 2 p = 0.0149
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5291 df = 2 p = 0.4656
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3641 df = 2 p = 0.8336
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3925 df = 2 p = 0.3023
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3356 df = 2 p = 0.311
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7454 df = 2 p = 0.4178
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4535 df = 2 p = 0.1779
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7386 df = 2 p = 0.4192
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9725 df = 2 p = 0.6149
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3177 df = 2 p = 0.8531
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8764 df = 2 p = 0.2374
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5353 df = 2 p = 0.7652
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1434 df = 2 p = 0.9308
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9106 df = 2 p = 0.6343
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3762 df = 2 p = 0.8285
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1729 df = 2 p = 0.1241
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.4462 df = 2 p = 0.4853
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1012 df = 2 p = 0.9506
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1217 df = 2 p = 0.9409
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2645 df = 2 p = 0.8761
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3767 df = 2 p = 0.0412
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.786 df = 2 p = 0.1506
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1276 df = 2 p = 0.9382
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1021 df = 2 p = 0.3496
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2065 df = 2 p = 0.2012
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.1527 df = 2 p = 0.0761
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.7884 df = 2 p = 0.0075
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1354 df = 2 p = 0.9345
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.1198 df = 2 p = 0.2102
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9485 df = 2 p = 0.3775
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0842 df = 2 p = 0.5815
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.1516 df = 2 p = 0.0462
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5604 df = 2 p = 0.1686
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1006 df = 2 p = 0.951
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.528 df = 2 p = 0.2825
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.932 df = 2 p = 0.6275
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5146 df = 2 p = 0.2844
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2697 df = 2 p = 0.53
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.5059 df = 2 p = 0.7765
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1033 df = 2 p = 0.9497
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.025 df = 2 p = 0.599
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4466 df = 2 p = 0.7999
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6472 df = 2 p = 0.7235
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.6375 df = 2 p = 0.0362
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4317 df = 2 p = 0.2965
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0787 df = 2 p = 0.9614
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4663 df = 2 p = 0.2914
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.3665 df = 2 p = 0.0683
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6791 df = 2 p = 0.7121
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6159 df = 2 p = 0.735
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.7984 df = 2 p = 0.6709
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4204 df = 2 p = 0.1097
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2847 df = 2 p = 0.8673
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3087 df = 2 p = 0.5198
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.2626 df = 2 p = 0.877
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.4728 df = 2 p = 0.7895
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8754 df = 2 p = 0.6455
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.266 df = 2 p = 0.8755
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5587 df = 2 p = 0.4587
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6339 df = 2 p = 0.4418
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8639 df = 2 p = 0.6492
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8726 df = 2 p = 0.6464
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.351 df = 2 p = 0.3087
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0172 df = 2 p = 0.9914
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0965 df = 2 p = 0.3505
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6308 df = 2 p = 0.7295
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.0883 df = 2 p = 0.9568
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6118 df = 2 p = 0.4467
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1313 df = 2 p = 0.9364
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6873 df = 2 p = 0.4301
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.3575 df = 2 p = 0.8363
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1087 df = 2 p = 0.3484
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8554 df = 2 p = 0.652
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9536 df = 2 p = 0.1385
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.625 df = 2 p = 0.7316
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.6808 df = 2 p = 0.7115
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.187 df = 2 p = 0.335
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9719 df = 2 p = 0.6151
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0746 df = 2 p = 0.3544
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.748 df = 2 p = 0.688
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0845 df = 2 p = 0.5814
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4799 df = 2 p = 0.1065
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8919 df = 2 p = 0.2355
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.8462 df = 2 p = 0.655
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4184 df = 2 p = 0.2984
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.468 df = 2 p = 0.065
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1943 df = 2 p = 0.9074
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5919 df = 2 p = 0.4512
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2537 df = 2 p = 0.3241
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4419 df = 2 p = 0.0399
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0393 df = 2 p = 0.5947
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2974 df = 2 p = 0.1923
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3254 df = 2 p = 0.3126
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8509 df = 2 p = 0.3964
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3722 df = 2 p = 0.5035
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.011 df = 2 p = 0.6032
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.0092 df = 2 p = 0.2221
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.912 df = 2 p = 0.3844
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.0467 df = 2 p = 0.5925
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.639 df = 2 p = 0.2673
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.1734 df = 2 p = 0.9169
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4641 df = 2 p = 0.2917
#> 
#> 	Bootstrap White test
#> 
#> data:  mod
#> X-squared = 13.801, B = 499, p-value = 0.002
#> 

# Enable parallel processing when supported by the operating system
if (.Platform$OS.type != "windows") {
  performWhiteTestBootstrap(mod, sim, B = 199, parallel = TRUE)
}
#> [INFO] Running Bootstrap White test
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.8014 df = 2 p = 0.001
#> 
#> 	Bootstrap White test
#> 
#> data:  mod
#> X-squared = 13.801, B = 199, p-value = 0.01
#> 
# }