Quantile regression heteroscedasticity test

Source: R/quantile_regression_joint.R · Help: ?performQuantileRegressionTest

Tests equality of regression slopes across two or more conditional quantiles using the joint Wald-type test implemented by quantreg::anova.rqs(). This avoids treating quantile-specific coefficient estimates from the same sample as independent.

Usage

performQuantileRegressionTest(
  model,
  data = NULL,
  taus = c(0.25, 0.75),
  se_type = c("nid", "ker"),
  iid = TRUE
)

Arguments

model

A fitted stats::lm object describing the mean structure whose conditional quantile slopes are to be compared.

data

The data frame model was fitted to, or NULL, the default. The response, design and offset are read from the fit itself, so a model that transforms its variables, has an offset() term, or takes its offset from a column of the data needs no data, and the model frame of model is accepted as data (since 0.16.0; before, such models needed data). Data that are given are matched to the observations of the fit: rows with row names of their own by name, in any order; rows that only number themselves (a tibble, a data.table, a data frame whose row names were reset) by the row numbers of the fit, or in the order of the fit when they hold one row per observation. The rows the fit dropped may be there or not. Data whose rows cannot be shown to hold the observations of the fit are refused; see test_results.

taus

Numeric vector containing at least two distinct quantiles in (0, 1). Defaults to c(0.25, 0.75).

se_type

Standard-error method used by quantreg::anova.rqs(). Supported values are "nid" and "ker".

iid

Logical indicating whether the conditional densities are assumed identical when computing the joint test. Passed to quantreg::anova.rqs(). Defaults to TRUE.

Value

An object of class htest containing the F-like joint test statistic, numerator and denominator degrees of freedom, p-value, fitted quantiles, and quantile-specific slope estimates.

Details

Under a pure location-shift model with homoskedastic errors, slopes are equal across quantiles. Systematic slope differences are therefore evidence against that location-shift/homoskedastic specification. The test should not be read as a universal test for every possible form of heteroscedasticity.

Weighted fits are refused, although the resampling tests accept them. On a fit with weights \(w_i\) the null hypothesis is that the Pearson residuals \(\sqrt{w_i}\, e_i\) have constant variance, which is a statement about the regression of \(\sqrt{w_i}\, y_i\) on \(\sqrt{w_i}\, x_i\). The quantile regressions this test fits are of \(y\) on \(x\), where the errors have variance \(\sigma^2 / w_i\) by assumption, so their slopes differ across quantiles even when the weights are right. Fitting them on the weighted scale does not by itself repair that: there the conditional quantiles are the regression plus a constant, and the constant is not a column of the weighted regressors. Adding a column of ones would restore the location-shift form, but that version of the test needs its own derivation and its own size study, and has neither.

Supported models

It accepts an unweighted lm and an unweighted glm with the Gaussian family and the identity link and, since 0.16.0, a rank-deficient one: quantreg::rq() refuses a singular design, so the quantile regressions are fitted to the columns that were not aliased, which gives the value of the same model fitted without them. Since 0.16.0 it also accepts a model with an offset: the quantile regressions are fitted to the response of the fit minus its offset, on the design of the fit, which is the model with the offset, since the offset shifts every conditional quantile as it shifts the mean; before, they were refitted from the formula, which left the offset out. Since 0.16.0 it also accepts a model with a single regressor, such as lm(y ~ x); before, it stopped with "subscript out of bounds" on one, and runHeteroTests() returned its fallback, the rank permutation test, under the name quantile_regression. It refuses a weighted fit, a fitWLS() fit among them, as it refits the model without the weights, a glm of another family or link, a survey::svyglm() fit, a coefficient that is not finite for another reason, and data that lack rows or variables of the fit. It refuses a multivariate lm ("needs a model with one response"), and, since 0.16.0, an mgcv::gam() fit with penalised smooth terms, which is not the least-squares fit of its design, the basis of its smooths, that the test fits again. Since 0.17.0 a gam that is the least-squares fit of its design, as one whose smooth terms are all fixed (s(x, fx = TRUE)), is tested on that design and gives the value of the lm() of it; before, it was refused as a penalised one. A gam without smooth terms is tested as the Gaussian glm it is (decisions 37 and 38 of the design of the model context). Which model each test accepts, and why, is set out in test_results, under Supported models.

References

Koenker, R., & Bassett, G. (1982). Robust tests for heteroscedasticity based on regression quantiles. Econometrica, 50(1), 43-61.

Koenker, R. (2005). Quantile Regression. Cambridge University Press.

Examples

if (requireNamespace("quantreg", quietly = TRUE)) {
  # The test needs at least 40 observations, so mtcars (32) is too small.
  model <- lm(stations ~ mag + depth, data = quakes)
  performQuantileRegressionTest(model, quakes)
}