CHAPTER 5
REGRESSION DIAGNOSTIC II:
HETEROSCEDASTICITY
Damodar Gujarati
Econometrics by Example
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 lowincome households)
Damodar Gujarati
Econometrics by Example
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
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
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 statistic is 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
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
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