Comprehensive Guide to Heteroscedasticity Testing¶

Introduction¶

This comprehensive guide covers the theory, implementation, and practical application of heteroscedasticity tests in the heteroTests package.

Theoretical Background¶

What is Heteroscedasticity?¶

Heteroscedasticity occurs when the variance of the error terms in a regression model changes with the level of one or more predictors. This violates the constant variance assumption of ordinary least squares and can lead to biased standard errors.

When Does It Matter?¶

Non-constant variance implies inefficient parameter estimates and may invalidate hypothesis tests that rely on the usual standard errors. It is therefore important to detect and correct for heteroscedasticity before drawing substantive conclusions from a model.

Test Catalog¶

Regression-Based Tests¶

White's Test¶

White's test regresses the squared residuals on the original regressors, their squares, and cross-products. The resulting \(R^2\) multiplied by the sample size follows a chi-squared distribution under the null of homoscedasticity.

Breusch-Pagan Test¶

The Breusch–Pagan test regresses the squared residuals on the original regressors only. A significant regression indicates that the error variance is related to the predictors.

Graphical Diagnostics¶

The package includes convenient functions for residual vs fitted plots, scale–location plots, QQ plots with envelopes, and leverage diagnostics to help visualise patterns in the residuals.

Practical Examples¶

Example 1: Financial Time Series¶

set.seed(1)
x <- rnorm(250)
y <- 0.5 + 0.3 * x + rnorm(250, sd = abs(x))
dat <- data.frame(x, y)
mod <- lm(y ~ x, dat)
performWhiteTest(mod, dat)
#> Warning: Residual outliers detected at rows 24 (|z| > 5). Inspect leverage
#> before running White.
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 70.9198 df = 2 p = 0
#>
#>  White's test for heteroscedasticity
#>
#> data:  mod
#> X-squared = 70.92, df = 2, p-value = 3.981e-16
#> alternative hypothesis: heteroscedasticity present

Example 2: Cross-Sectional Data¶

data(mtcars)
fit <- lm(mpg ~ wt + hp, data = mtcars)
performBPTest(fit, mtcars)
#> [INFO] Running Breusch-Pagan test
#>
#>  Breusch-Pagan test for heteroscedasticity
#>
#> data:  mpg ~ wt + hp
#> X-squared = 1.0268, df = 2, p-value = 0.5985

Example 3: Panel Data¶

dat <- simulate_hetero(n = 250, beta0 = 1, beta1 = 2, sigma_func = sigma_linear)
fit <- lm(y ~ x, data = dat)
performSzroeterTest(fit, dat, order_by = "x")
#> [INFO] Running Szroeter test
#>
#>  Szroeter test for ordered heteroscedasticity
#>
#> data:  fit
#> Q = 9.062, n = 250, p-value < 2.2e-16
#> alternative hypothesis: variance increases with 'x'
#> sample estimates:
#>        h
#> 183.9943

Remediation Strategies¶

Common remedies include transforming the response (e.g. log or square root), modelling the variance via weighted least squares, or using heteroscedasticity-consistent standard errors.

Comparison with Other Packages¶

Functions in the car and lmtest packages implement many of the same diagnostics. heteroTests aims to provide a unified interface and a few additional tools such as automated remediation suggestions.

References¶

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838.

Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47, 1287–1294.