Testing for Heteroskedasticity

Question 1

What is heteroskedasticity?

Answer 1

Heteroskedasticity occurs when the variance of the error term in a regression model is not constant across observations.

Var(ui/Xi) does not equal standard deviation. (sigma squared)

This violates a key OLS assumption leading to inefficient OLS estimators and invalid standard errors, which distort hypothesis testing.

Question 2

Testing Procedure, H0, H1?

Answer 2

Testing Procedure (regression based):
1) Estimate the model and obtain residuals u.
2) Regress squared residuals on one or more explanatory variables, creating an auxiliary regression.

u^2 = a0 + a1Z1 + … + akZk + v

H0: a1= … = ak=0
H1: At least one aj does not equal 0.

Question 3

What are the two tests (Breusch Pagan Test) we can perform for heteroskedasticity?

Answer 3

LM version: n*R^2 ~ Xq^2 (Chi-squared distribution) (where q is the number of explanatory variables in auxiliary regression)

F-test version: Compares the explained sum of squares in the auxiliary regression to its residual sum of squares using an F-test.

F = (ESS/q) / ((RSS/(n-q-q1))

Question 4

Issue & Considerations for BP test

Answer 4

Heteroskedasticity doesn’t bias OLS coefficient, but invalidates standard errors, making f-tests and t-tests unreliable.

If heteroskedasticity is detected use robust standard errors (white test)

This test is sensitive to model specification, meaning the choice of variables in auxiliary regression matters.