White Test of Heteroscedasticity Detection (2022)

The post is about the White test of heteroscedasticity.

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

White test of Heteroscedasticity

White test (Halbert White, 1980) proposed a test that is very similar to that by Breusch-Pagen. The White test of Heteroscedasticity is general because it does 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 the White test of heteroscedasticity 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 the set of original regressors, the cross-products of the regressors, and the squared regressors.

The step-by-step procedure for performing the White test of Heteroscedasticity is as follows:

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

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

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

1. Now run the following regression model to obtain squared residuals from original regression on the original set of the independent variable, the square value of independent variables, and the cross-product(s) of the independent variable(s) such as
   \[Y_i=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_1^2+\beta_4X_2^2+\beta_5X_1X_2 \tag{2}\]
2. Find the $R^2$ statistics from the auxiliary regression in step 2.
   You can also use the higher power regressors such as the cube. Also, note that there will be a 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 the number of regressors in equation (2)
4. If the calculated chi-square value obtained in (3) is greater than the critical chi-square value at the chosen level of significance, reject the hypothesis of homoscedasticity in favor of heteroscedasticity.

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 are 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 the cross-product is introduced in the model, then it is a test of both heteroscedasticity and specification bias.

By employing the White test of heteroscedasticity, one can gain valuable insights about the presence of heteroscedasticity and decide on appropriate corrective measures (like Weighted Least Squares (WLS)) if necessary to ensure reliable standard errors and hypothesis tests in your regression analysis.

Summary

The White test of heteroscedasticity is a flexible approach that can be used to detect various patterns of heteroscedasticity. This test indicates the presence of heteroscedasticity but it does not pinpoint the specific cause (like model misspecification). The White test is relatively easy to implement in statistical software.

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

Click Links to learn more about Tests of Heteroscedasticity: Regression Residuals Plot, Bruesch-Pagan Test, Goldfeld-Quandt Test

See the Numerical Example of the White Test of Heteroscedasticity

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