Studentized Breusch–Pagan test (deprecated)¶

Source: R/additional_tests.R · Help: ?performStudentizedBPTest

performStudentizedBPTest() is deprecated since 0.14.0: use performKoenkerTest(), which computes the same statistic. It still works and warns, and returns what it returned before except on a fit returned by fitWLS(). The model frame of that fit holds the weights of the model fitWLS() refitted (none, for an unweighted model), not the estimated weights, and before 0.15.0 performStudentizedBPTest() read them from there, so it computed the statistic of the unweighted fit. Since 0.15.0 it reads the weights of the fit, as performKoenkerTest() does, and the two agree. On a model with an offset they still differ: performStudentizedBPTest() drops the offset, which is to be fixed in 0.16.0, and performKoenkerTest() is the correct one.

Usage¶

performStudentizedBPTest(model, data = NULL)

Arguments¶

model: A fitted stats::lm object providing the residuals and design matrix for the auxiliary regression.
data: A base::data.frame containing the variables used to fit model. It must include all observations referenced by the model object. When NULL, the default, the model frame of model is used. It holds the terms of the model rather than its variables (log(x) rather than x), so a model that transforms a variable needs data, as does a model with an offset() term, and so may a fit whose weights or offset argument is a column of the data, which the model frame holds as (weights) or (offset).

Value¶

An object of class htest reporting the chi-squared statistic and p-value for the null hypothesis of constant error variance.

Details¶

Computes Koenker's (1981) studentized form of the Breusch–Pagan Lagrange multiplier statistic, \(n R^2\) from the regression of the squared residuals on the regressors. The classical Breusch–Pagan statistic assumes normal errors and this one does not: in the simulations of inst/validation/ it holds its level under \(t_5\) errors, where the classical statistic rejects far too often.

Following Koenker (1981) and the implementation in lmtest::bptest(), the procedure fits an auxiliary regression of \(e_i^2 - \hat{\sigma}^2\) on the regressors from the original model (including the intercept), where \(e_i\) denotes the residuals and \(\hat{\sigma}^2 = \sum e_i^2 / n\) the mean of their squares. With \(\hat{g}_i\) the fitted values of that regression, the statistic \(T = n \sum \hat{g}_i^2 / \sum (e_i^2 - \hat{\sigma}^2)^2\) is its \(n R^2\). Under homoskedasticity it is asymptotically chi-squared with degrees of freedom equal to the number of regressors beyond the intercept, whether or not the errors are normal. On a weighted fit \(e_i\) is the Pearson residual \(\sqrt{w_i}\) times the raw residual and the auxiliary regression is unweighted; see the section on weighted fits in performKoenkerTest(). This departs from lmtest::bptest(), which keeps the raw residuals and weights the auxiliary regression. The implementation shares the validation helpers used across the package to ensure that: (i) the model and data satisfy minimum sample-size thresholds via rvalidateModelInputs() and rvalidateDataInputs(), (ii) missing values are handled by rhandleMissingValues(), and (iii) the requirements registered for this test in rvalidateTestRequirements() are met.

References¶

Koenker, R. (1981). A note on studentizing a test for heteroscedasticity. Journal of Econometrics, 17(1), 107–112. doi:10.1016/0304-4076(81)90062-2

Davidson, R., & MacKinnon, J. G. (2004). Econometric Theory and Methods. Oxford University Press. Section 7.5 derives the test as \(n R^2\) from the regression of the squared residuals on the variance regressors.

Examples¶

data(mtcars)
mod <- lm(mpg ~ wt + qsec, data = mtcars)
# Deprecated: warns, and returns the statistic of performKoenkerTest()
suppressWarnings(performStudentizedBPTest(mod, mtcars))
performKoenkerTest(mod, mtcars)