White test for Heteroskedasticity detection

One of important assumption of Regression is that the variance of Error Term is constant across observations. If the error have constant variance, then the errors are called homoscedastic, otherwise heteroscedastic. In case of heteroscedastic errors (non-constant variance), the standard estimation methods becomes inefficient. Typically, to assess the assumption of homoscedasticity, residuals are plotted.

White’s test for Heteroskedasticity

White test (Halbert White, 1980) proposed a test which is vary similar to that by Breusch-Pagen. White test for Heteroskedasticity is general because it do not rely on the normality assumptions and it is also easy to implement. Because of the generality of White’s test, it may identify the specification bias too. Both White’s test and the Breusch-Pagan test are based on the residuals of the fitted model.

To test the assumption of homoscedasticity, one can use auxiliary regression analysis by regressing the squared residuals from the original model on set of original regressors, the cross-products of the regressors and the squared regressors.

Step by step procedure or perform White test for Heteroskedasticity is as follows:

Consider the following Linear Regression Model (assume there are two independent variable)
\[Y_i=\beta_0+\beta_1X_{1i}+\beta_1X_{2i}+e_i \tag{1} \]

For given data, estimate the regression model and obtain the residuals $e_i$’s.

  1. Now run the following regression model to obtain squared residuals from original regression on the original set of independent variable, square value of independent variables and the cross-product(s) of the independent variable(s) such as
    \[Y_i=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_1^2+\beta_4+X_2^2+\beta_5X_1X_2 \tag{2}\]
  2. Find the $R^2$ statistics from the auxiliary regression in step 2.
    You can also use higher power of regressors such as cube. Also note that there will be constant term in equation (2) even though the original regression model (1)may or may not have the constant term.
  3. Test the statistical significance of \[n \times R^2\sim\chi^2_{df}\tag{3},\] under the null hypothesis of homoscedasticity or no heteroscedasticity, where df is number of regressors in equation (2)
  4. If calculated chi-square value obtained in (3) is greater than the critical chi-square value at chosen level of significance, reject the hypothesis of homoscedasticity in favour of heteroscedasticity.

Note that the regression of residuals can take linear or non-linear functional form.

For several independent variables (regressors) model, introducing all the regressors, their square or higher terms and their cross products, consume degrees of freedom.

In cases where the White test statistics is statistically significant, heteroscedasticity may not necessarily be the cause, but specification errors. In other words, “The white test can be a test of heteroscedasticity or specification error or both. If no cross product terms are introduced in the White test procedure, then this is a pure test of pure heteroscedasticity.
If cross product are introduced in model, then it is a test of both heteroscedasticity and specification bias.

References

  • H. White (1980), “A heteroscedasticity Consistent Covariance Matrix Estimator and a Direct Test of Heteroscedasticity”, Econometrica, Vol. 48, pp. 817-818.
  • https://en.wikipedia.org/wiki/White_test

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