Breusch-Pagan Test vs. White Test: Choosing the Right Heteroscedasticity Test

📚 Quick Study Guide

• 📊 Heteroscedasticity: Refers to the unequal variance of errors in a regression model. This violates OLS assumptions.
• 🧪 Breusch-Pagan Test: Assumes the error variance is related to a set of independent variables. It's a parametric test.
• 📝 Breusch-Pagan Test Steps:
  1. Fit the OLS regression model: $Y = X\beta + \epsilon$
  2. Calculate the squared residuals: $\hat{\epsilon}^2$
  3. Regress the squared residuals on the independent variables: $\hat{\epsilon}^2 = Z\gamma + v$, where $Z$ can be the same as $X$ or a subset.
  4. Compute the test statistic: $LM = \frac{1}{2}SSR$, where $SSR$ is the explained sum of squares from the auxiliary regression. Under the null hypothesis of homoscedasticity, $LM \sim \chi^2(p)$, where $p$ is the number of regressors in $Z$.
• 💡 White Test: A more general test that doesn't require specifying which variables are related to the error variance. It's non-parametric.
• 📈 White Test Steps:
  1. Fit the OLS regression model: $Y = X\beta + \epsilon$
  2. Calculate the squared residuals: $\hat{\epsilon}^2$
  3. Regress the squared residuals on the original independent variables, their squares, and cross-products: $\hat{\epsilon}^2 = Z\gamma + v$, where $Z$ includes $X$, $X^2$, and all cross-products of $X$.
  4. Compute the test statistic: $nR^2$, where $n$ is the sample size and $R^2$ is the coefficient of determination from the auxiliary regression. Under the null hypothesis of homoscedasticity, $nR^2 \sim \chi^2(p)$, where $p$ is the number of regressors in $Z$.
• 🤔 Choosing Between the Tests: Use Breusch-Pagan if you have a specific hypothesis about which variables cause heteroscedasticity. Use White if you don't have a specific hypothesis or suspect a more complex form of heteroscedasticity.

Practice Quiz

1. Which of the following assumptions is tested by both the Breusch-Pagan and White tests?
   1. A) Normality of errors
   2. B) Homoscedasticity
   3. C) Autocorrelation
   4. D) Multicollinearity
2. The Breusch-Pagan test is a ________ test, while the White test is generally considered ________.
   1. A) non-parametric, parametric
   2. B) parametric, non-parametric
   3. C) linear, non-linear
   4. D) non-linear, linear
3. In the auxiliary regression for the Breusch-Pagan test, what is the dependent variable?
   1. A) The independent variables
   2. B) The original dependent variable
   3. C) The squared residuals from the original regression
   4. D) The predicted values from the original regression
4. What is a key difference in the auxiliary regression between the Breusch-Pagan and White tests?
   1. A) The White test always includes squared terms and cross-products of the independent variables.
   2. B) The Breusch-Pagan test always includes squared terms and cross-products of the independent variables.
   3. C) The Breusch-Pagan test uses the original dependent variable as a regressor.
   4. D) The White test uses the original dependent variable as a regressor.
5. If the test statistic for the Breusch-Pagan or White test is significant, what should you conclude?
   1. A) There is no heteroscedasticity.
   2. B) There is evidence of heteroscedasticity.
   3. C) The model is perfectly specified.
   4. D) The sample size is too small.
6. Which test is more suitable when you suspect that heteroscedasticity is caused by a complex and unspecified relationship among the variables?
   1. A) Breusch-Pagan test
   2. B) White test
   3. C) Goldfeld-Quandt test
   4. D) Durbin-Watson test
7. Under the null hypothesis of homoscedasticity, the test statistic for the White test follows a chi-squared distribution. What determines the degrees of freedom for this distribution?
   1. A) The number of observations.
   2. B) The number of independent variables in the original regression.
   3. C) The number of regressors in the auxiliary regression.
   4. D) The number of squared residuals.