Performs the Breusch-Pagan test for heteroscedasticity in linear regression models. Tests the null hypothesis that the error variance is constant (homoscedasticity) against the alternative that the error variance depends on the fitted values.
Arguments
- model
A fitted linear model object (from
lm()).
Value
An object of class "htest" with components:
- statistic
the value of the chi-squared test statistic.
- parameter
degrees of freedom.
- p.value
the p-value of the test.
- method
a character string indicating the test performed.
- data.name
a character string giving the name of the model.
Details
Implements the Koenker variant of the Breusch-Pagan test, which regresses the squared raw residuals on the fitted values. The test statistic is:
$$BP = n \cdot R^2$$
where:
\(n\) = sample size
\(R^2\) = coefficient of determination from auxiliary regression of \(e_i^2\) on \(\hat{y}_i\)
Under the null hypothesis of homoscedasticity, the test statistic follows a chi-squared distribution: \(BP \sim \chi^2(k-1)\) where \(k\) is the number of parameters in the model (including intercept).
Large values of the test statistic (small p-values) provide evidence against homoscedasticity.
References
Breusch, T. S., and Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287-1294. DOI: 10.2307/1911963
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
Examples
# Example with homoscedastic errors
set.seed(123)
x <- runif(100)
y <- 2 + 3 * x + rnorm(100, sd = 1)
model1 <- lm(y ~ x)
bp.test(model1) # Should not reject (p > 0.05)
#>
#> Breusch-Pagan test for heteroscedasticity
#>
#> data: model1
#> BP = 0.025212, df = 1, p-value = 0.8738
#>
# Example with heteroscedastic errors (variance increases with x)
set.seed(456)
x <- runif(100)
y <- 2 + 3 * x + rnorm(100, sd = 0.5 + 2 * x)
model2 <- lm(y ~ x)
bp.test(model2) # Should reject (p < 0.05)
#>
#> Breusch-Pagan test for heteroscedasticity
#>
#> data: model2
#> BP = 20.435, df = 1, p-value = 6.169e-06
#>