BreuschPagan

The Breusch-Pagan test is a diagnostic procedure in econometrics used to detect heteroskedasticity in linear regression models. It tests whether the variance of the regression errors can be explained by the independent variables included in the model, indicating a departure from homoskedasticity.

The standard procedure begins with estimating a linear regression of the form y = Xβ + ε and obtaining the residuals ε̂. These squared residuals ε̂² are then regressed on the same set of explanatory variables (or a specified subset) Z in an auxiliary regression: ε̂² = δ0 + δ1Z1 + ... + δkZk + v. The test statistic is LM = n × R², where n is the sample size and R² is from the auxiliary regression. Under the null hypothesis of homoskedasticity, LM follows a chi-square distribution with degrees of freedom equal to the number of regressors in the auxiliary regression (excluding the constant term). A small p-value leads to rejection of the null.

The test assumes that the auxiliary regression adequately captures the form of potential heteroskedasticity and that the sample is sufficiently large for the chi-square approximation. Variants and extensions exist, including approaches that relax normality assumptions (robust versions such as Koenker’s/Breusch-Pagan-type tests) and methods that incorporate nonlinearly related regressors or squared terms. The Breusch-Pagan test is widely implemented in econometrics software, often via functions that perform the auxiliary regression on squared residuals and report the LM statistic and p-value.