Test for heteroscedasticity using squared residuals

Summary

Setup

  • The LRM with random sampling

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

  • All explanatory variables are exogenous.
  • The error terms may or may not be homoscedastic.
  • We want to test the hypothesis “the errors are homoscedastic”.

Procedure

  • Estimate the LRM using OLS
  • Save the OLS residuals
  • Specify an auxiliary regression where your dependent variable is \(e_i^2\) .
  • Various choices for the explanatory variables in the auxiliary regression (an intercept is always included):
    • White: use all the \(x\) -variables, the square of all the \(x\) -variables and all interaction variables.
    • Breusch-Pagan: use any collection of variables that you suspect affect \(σ_i^2\) . This could include variable not in the original regression.
  • Estimate the parameters in the auxiliary regression using OLS.
  • Test the null hypothesis that all parameters in the auxiliary regression, except the intercept, are zero using either an \(F\) -test or an \(LM\) -test. Both are, under certain assumptions, approximately correct when \(n\) is large.
  • \(LM\) -test: The test statistic is \(nR^2\) which is approximately \(χ_r^2\) under the null when \(n\) is large ( \(r\) is the number restriction in the null hypothesis).

White’s test, example

  • Setup:

\[y_i=β_1+β_2x_{2i}+β_3x_{3i}+ε_i , i=1,…,n\]

  • Auxiliary regression:

\[e_i^2=γ_1+γ_2x_{2i}+γ_3x_{3i}+γ_4x_{2i}^2+γ_5x_{3i}^2+γ_6x_{2i}x_{3i}+ν_i, i=1,…,n\]

  • Null hypothesis:

\[H_0: γ_2=0, γ_3=0, γ_4=0, γ_5=0, γ_6=0\]

  • Find \(R^2\) in the auxiliary regression. Under the null, use either

\[F= \frac{R^2/5}{\left( 1-R^2 \right)/(n-6)}∼F_{5,n-6}\]

  • or

\[χ^2=nR^2∼χ_5^2\]

  • both are approximately valid under mild assumptions. The first one is called the Wald test while the second one is called the LM test.