Perform Harvey test for multiplicative heteroscedasticity

Harvey's (1976) Lagrange multiplier test, which regresses log(residuals^2) on a set of variance regressors.

Usage

performHarveyTest(model, auxiliary = c("regressors", "fitted"),
                  studentize = FALSE)

Details

Harvey (1976) models the error variance as \(\sigma_i^2 = \exp(z_i^\top \gamma)\) and tests \(\gamma = 0\) through the auxiliary regression of \(\log \hat{e}_i^2\) on \(z_i\). Under the null the auxiliary error behaves like a centred \(\log \chi^2_1\) variate with variance \(\pi^2 / 2 \approx 4.9348\), so the classical statistic is \(\mathrm{ESS} / (\pi^2 / 2)\), asymptotically chi-square with \(q\) degrees of freedom.

Setting studentize = TRUE replaces the fixed constant \(\pi^2 / 2\) with the auxiliary residual mean square and refers the overall F statistic to an F distribution. The two forms are asymptotically equivalent under normal errors; the studentized form is more reliable when normality is doubtful, because \(\pi^2 / 2\) is the null variance of \(\log \hat{e}^2\) only for Gaussian errors.

Degrees of freedom are taken from the realised rank of the auxiliary design, so a rank-deficient variance model is not credited with degrees of freedom for aliased columns.

Arguments

model: an object of class lm.
auxiliary: character scalar choosing the variance regressors. "regressors" (the default) uses the model's own explanatory variables, the specification in Harvey (1976). "fitted" uses the fitted values and their square, which was the behaviour of releases before 0.7.0.
studentize: logical; if FALSE (the default) the chi-square statistic \(\mathrm{ESS} / (\pi^2 / 2)\) is reported, otherwise the auxiliary regression F statistic.

Value

An object of class htest containing the test statistic, its degrees of freedom and the p-value.

Validation

The default statistic reproduces an independent reconstruction of Harvey (1976) to within 1e-8; see tests/testthat/test-pass-a-reference.R. Simulated size and power are recorded in inst/validation/pass-a-size-power.csv.

References

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

Greene, W. H. (2018). Econometric Analysis (8th ed.). Pearson. Section 9.5 derives the \(\mathrm{ESS} / 4.9348\) form of the statistic.

Examples

 data(mtcars)
 m <- lm(mpg ~ wt + qsec, data = mtcars)
 performHarveyTest(m)
#> [INFO] Running Harvey test
#> 
#> 	Harvey test for multiplicative heteroscedasticity
#> 
#> data:  mpg ~ wt + qsec; variance regressors: model regressors
#> X-squared = 2.4457, df = 2, p-value = 0.2944
#> alternative hypothesis: error variance is a multiplicative function of the variance regressors
#> 

 # Studentized variant, which does not assume normal errors
 performHarveyTest(m, studentize = TRUE)
#> [INFO] Running Harvey test
#> 
#> 	Harvey test for multiplicative heteroscedasticity (studentized)
#> 
#> data:  mpg ~ wt + qsec; variance regressors: model regressors
#> F = 0.86808, df1 = 2, df2 = 29, p-value = 0.4304
#> alternative hypothesis: error variance is a multiplicative function of the variance regressors
#> 

 # Variance driven by the conditional mean rather than by the regressors
 performHarveyTest(m, auxiliary = "fitted")
#> [INFO] Running Harvey test
#> 
#> 	Harvey test for multiplicative heteroscedasticity
#> 
#> data:  mpg ~ wt + qsec; variance regressors: fitted values and their square
#> X-squared = 4.653, df = 2, p-value = 0.09764
#> alternative hypothesis: error variance is a multiplicative function of the variance regressors
#>