Heteroscedasticity in Regression Analysis

1. CHAPTER 5 REGRESSION DIAGNOSTIC II: HETEROSCEDASTICITY Damodar Gujarati Econometrics by Example

2. HETEROSCEDASTICITY • One of the assumptions of the classical linear regression (CLRM) is that the variance of ui, the error term, is constant, or homoscedastic. • Reasons are many, including: • The presence of outliers in the data • Incorrect functional form of the regression model • Incorrect transformation of data • Mixing observations with different measures of scale (such as mixing high-income households with low-income households) Damodar Gujarati Econometrics by Example

3. CONSEQUENCES • If heteroscedasticity exists, several consequences ensue: • The OLS estimators are still unbiased and consistent, yet the estimators are less efficient, making statistical inference less reliable (i.e., the estimated t values may not be reliable). • Thus, estimators are not best linear unbiased estimators (BLUE); they are simply linear unbiased estimators (LUE). • In the presence of heteroscedasticity, the BLUE estimators are provided by the method of weighted least squares (WLS). Damodar Gujarati Econometrics by Example

4. DETECTION OF HETEROSCEDASTICITY • Graph histogram of squared residuals • Graph squared residuals against predicted Y • Breusch-Pagan (BP) Test • White’s Test of Heteroscedasticity • Other tests such as Park, Glejser, Spearman’s rank correlation, and Goldfeld-Quandt tests of heteroscedasticity Damodar Gujarati Econometrics by Example

5. BREUSCH-PAGAN (BP) TEST • Estimate the OLS regression, and obtain the squared OLS residuals from this regression. • Regress the square residuals on the k regressors included in the model. • You can choose other regressors also that might have some bearing on the error variance. • The null hypothesis here is that the error variance is homoscedastic – that is, all the slope coefficients are simultaneously equal to zero. • Use the F statistic from this regression with (k-1) and (n-k) in the numerator and denominator df, respectively, to test this hypothesis. • If the computed F statisticis statistically significant, we can reject the hypothesis of homoscedasticity. If it is not, we may not reject the null hypothesis. Damodar Gujarati Econometrics by Example

6. WHITE’S TEST OF HETEROSCEDASTICITY • Regress the squared residuals on the regressors, the squared terms of these regressors, and the pair-wise cross-product term of each regressor. • Obtain the R2 value from this regression and multiply it by the number of observations. • Under the null hypothesis that there is homoscedasticity, this product follows the Chi-square distribution with df equal to the number of coefficients estimated. • The White test is more general and more flexible than the BP test. Damodar Gujarati Econometrics by Example

7. REMEDIAL MEASURES • What should we do if we detect heteroscedasticity? • Use method of Weighted Least Squares (WLS) • Divide each observation by the (heteroscedastic) σi and estimate the transformed model by OLS (yet true variance is rarely known) • If the true error variance is proportional to the square of one of the regressors, we can divide both sides of the equation by that variable and run the transformed regression • Take natural log of dependent variable • Use White’s heteroscedasticity-consistent standard errors or robust standard errors • Valid in large samples Damodar Gujarati Econometrics by Example