Test the functional form of the error variance

Source: R/performVarianceFormTest.R · Help: ?performVarianceFormTest

Tests whether the error variance follows a stated parametric function of stated variables, \(\sigma_i^2 = h(z_i; \gamma)\). Every other test in the package has constant variance as its null hypothesis, so it can say that the variance moves with some variables but not whether an exponential or a power function of them describes it. Here the variance function is the null hypothesis, and rejecting it means that the chosen shape is inadequate.

Usage

performVarianceFormTest(
  model,
  data = NULL,
  var_formula = NULL,
  form = c("exponential", "power"),
  against = "squares"
)

Arguments

model

A fitted stats::lm object for the mean equation, or the result of fitWLS(). In the second case the variance function that was fitted is the one tested, and var_formula and form must be left at their defaults.

data

Optional base::data.frame holding the variables named in var_formula or against. It is needed only when they are not in the model frame, and is matched to the fitted rows by row name.

var_formula

One-sided formula naming the variance regressors \(z_i\), for example ~ x. NULL, the default, uses the regressors of model.

form

Character scalar giving the variance function under the null: "exponential" (the default) or "power". They are defined in the section on variance functions of fitWLS().

against

The terms the variance function is tested against. The default, "squares", adds the squares and pairwise products of the variance regressors, on the scale the form uses (logarithms for the power form). A one-sided formula such as ~ w or ~ I(x^3) names the added terms instead, which tests the variance function against a variable it leaves out or against a particular departure.

Value

An object of class htest with the F statistic, its numerator and denominator degrees of freedom, and the p-value for the null hypothesis that the variance function is correctly specified. estimate holds the fitted coefficients of the variance function: slopes of the log-variance for the exponential form and exponents for the power form. The element variance_function records the form, the variance regressors and the added terms that entered the test.

Details

The procedure has three steps.

1. The variance function is fitted to the residuals of model and the model is refitted by weighted least squares, exactly as fitWLS() does.
2. The squared standardized residuals \(r_i = \tilde{e}_i^2 / \hat\sigma_i^2\) of the weighted fit are formed. If the variance function is right their mean does not depend on anything.
3. \(r_i\) is regressed on a constant, on \(g_i = \hat\sigma_i^{-2}\, \partial \sigma_i^2 / \partial \gamma\), and on the added terms \(a_i\). The statistic is the F test that the coefficients on \(a_i\) are zero.

For the exponential form \(g_i\) is \(z_i\), and for the power form it is \(\log z_i\). The regression is the score direction of the variance function extended by the added terms, so the test is a studentized Lagrange multiplier test of the stated form against that extension.

Step 3 keeps \(g_i\) in the regression for a reason. Estimating \(\gamma\) fits the variance along \(g_i\), so those directions say little about the form: a Breusch–Pagan or Koenker test of the weighted fit on the same variance regressors rejects a wrong form about as often as the right one. The information is in the terms outside the fitted variance function. Conditioning on \(g_i\) is also what makes the null distribution free of the error in \(\hat\gamma\) to first order, which is the regression-based construction of Wooldridge (1990, 1991).

Interpretation and caveats

A rejection says that the added terms move the variance beyond what the stated function allows. A non-rejection says only that those terms do not; it does not establish the form. Two functions that are close over the observed range of \(z_i\) cannot be told apart, and the test will then accept both: a variance that is linear in a regressor is accepted as exponential or as a power when the regressor varies over a short range. Running the test for several forms is a comparison of fits, and choosing the form with the largest p-value is not a test.

The test judges the shape, not the fitted coefficients. Because it conditions on \(g_i\), it does not respond to weights that are off along the variance regressors themselves. A test of constant variance on the weighted fit is sensitive to that, with the reservations about its level described under weighted fits in performKoenkerTest().

The F reference distribution assumes that the standardized errors have a constant fourth moment, the assumption behind the Koenker test. It does not assume normality.

With \(q\) variance regressors the default adds up to \(q (q + 1) / 2\) terms. When that is large relative to the sample, name the variance regressors with var_formula or the added terms with against.

Validation

No reference implementation exists, so the statistic is checked against a reconstruction from its definition to 1e-8 (tests/testthat/test-variance-form.R) and by simulation. Over 24 designs in which the form under test was right (both forms, constant and heteroscedastic variances, 50 to 400 observations, Gaussian and \(t_5\) errors, 5000 replications each) the rejection rate at the 5% level ran from 4.2% to 5.7%. Power against a log-variance that is quadratic in a regressor was 98% at 150 observations. Against the other log-linear form it was 22% at 150 observations and 74% at 400. See inst/validation/README.md.

References

Wooldridge, J. M. (1990). A unified approach to robust, regression-based specification tests. Econometric Theory, 6(1), 17–43. doi:10.1017/S0266466600004898

Wooldridge, J. M. (1991). On the application of robust, regression-based diagnostics to models of conditional means and conditional variances. Journal of Econometrics, 47(1), 5–46. doi:10.1016/0304-4076(91)90076-P

Harvey, A. C. (1976). Estimating regression models with multiplicative heteroscedasticity. Econometrica, 44(3), 461–465. doi:10.2307/1913974

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

Carroll, R. J., & Ruppert, D. (1988). Transformation and Weighting in Regression. Chapman & Hall. Chapter 3 treats the estimation and checking of variance functions.

See also

fitWLS() fits the variance functions tested here. performHarveyTest() and performParkTest() test constant variance against the exponential and the power form respectively.

Examples

# Standard deviation proportional to x, so the variance is a power of x
set.seed(2026)
n <- 300
sim <- data.frame(x = runif(n, 1, 5))
sim$y <- 1 + 2 * sim$x + sim$x * rnorm(n)
mod <- lm(y ~ x, data = sim)

# The power form is the true one; estimate is the fitted exponent
performVarianceFormTest(mod, form = "power")

# The exponential form in x is not
performVarianceFormTest(mod, form = "exponential")

# Test the variance function of a weighted fit directly
wls <- fitWLS(mod, form = "power")
performVarianceFormTest(wls)

# Test a variance function against one particular added term
performVarianceFormTest(mod, form = "exponential", against = ~ log(x))