Key Points of Heteroscedasticity (2021)

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The following are some key points about heteroscedasticity. These key points are about the definition, example, properties, assumptions, and tests for the detection of heteroscedasticity (detection of hetero in short).

One important assumption of Regression is that the

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.

Heteroscedasticity

The disturbance term of OLS regression $u_i$ should be homoscedastic. By Homo, we mean equal, and scedastic means spread or scatter.
By hetero, we mean unequal.
Heteroscedasticity means that the conditional variance of $Y_i$ (i.e., $var(u_i))$ conditional upon the given $X_i$ does not remain the same regardless of the values taken by the variable $X$.
In case of heteroscedasticity $E(u_i^2)=\sigma_i^2=var(u_i^2)$, where $i=1,2,\cdots, n$.
In case of Homoscedasticity $E(u_i^2)=\sigma^2=var(u_i^2)$, where $i=1,2,\cdots, n$
Homoscedasticity means that the conditional variance of $Y_i$ (i.e. $var(u_i))$ conditional upon the given $X_i$ remains the same regardless of the values taken by the variable $X$.
The error terms are heteroscedastic, when the scatter of the errors is different, varying depending on the value of one or more of the explanatory variables.
Heteroscedasticity is a systematic change in the scatteredness of the residuals over the range of measured values
The presence of outliers may be due to (i) The presence of outliers in the data, (ii) incorrect functional form of the regression model, (iii) incorrect transformation of the data, and (iv) missing observations with different measures of scale.
The presence of hetero does not destroy the unbiasedness and consistency of OLS estimators.
Hetero is more common in cross-section data than time-series data.
Hetero may affect the variance and standard errors of the OLS estimates.
The standard errors of OLS estimates are biased in the case of hetero.
Statistical inferences (confidence intervals and hypothesis testing) of estimated regression coefficients are no longer valid.
The OLS estimators are no longer BLUE as they are no longer efficient in the presence of hetero.
The regression predictions are inefficient in the case of hetero.
The usual OLS method assigns equal weights to each observation.
In GLS the weight assigned to each observation is inversely proportional to $\sigma_i$.
In GLS a weighted sum of squares is minimized with weight $w_i=\frac{1}{\sigma_i^2}$.
In GLS each squared residual is weighted by the inverse of $Var(u_i|X_i)$
GLS estimates are BLUE.
Heteroscedasticity can be detected by plotting an estimated $u_i^2$ against $\hat{Y}_i$.
Plotting $u_i^2$ against $\hat{Y}_i$, if no systematic pattern exists then there is no hetero.
In the case of prior information about $\sigma_i^2$, one may use WLS.
If $\sigma_i^2$ is unknown, one may proceed with heteroscedastic corrected standard errors (that are also called robust standard errors).
Drawing inferences in the presence of hetero (or if hetero is suspected) may be very misleading.