OLS Estimation in the Presence of Heteroscedasticity (2020)

OLS Estimation Method is a widely used method in regression analysis for the estimation of the parameters used in a linear regression model. However, when heteroscedasticity exists (which refers to the situation where the variance of the error terms is not constant across observations) the assumptions of OLS may be violated. This violation leads to biased and inefficient parameter estimates, as well as unreliable hypothesis tests and confidence intervals. For more information see Consequences of Heteroscedasticity.

For the OLS Estimation in the presence of heteroscedasticity, consider the two-variable model

\begin{align*}
Y_i &= \beta_1 +\beta_2X_i + u_i\\
\hat{\beta}_2&=\frac{\sum x_i y_i}{\sum x_i^2}\\
Var(\hat{\beta}_2)&= \frac{\sum x_i^2\, \sigma_i^2}{(\sum x_i^2)^2}
\end{align*}

OLS Estimation in the Presence of Heteroscedasticity

OLS Estimation in the Presence of Heteroscedasticity, the variance of the OLS estimator will be

$Var(\hat{\beta}_2)$ under the assumption of homoscedasticity is $Var(\hat{\beta}_2)=\frac{\sigma^2}{\sum x_i^2}$. If $\sigma_i^2=\sigma^2$ the both $Var(\hat{\beta}_2)$ will be same.

OLS Estimation in the Presence of Heteroscedasticity (2020)

Note that in the case of heteroscedasticity, the OLS estimators

$\hat{\beta_2}$ is BLUE if the assumptions of the classical model, including homoscedasticity, hold.
To establish the unbiasedness of $\hat{\beta}_2$, it is not necessary for the disturbances ($u_i$) to be homoscedastic.
The variance of $u_i$, homoscedasticity, or heteroscedasticity plays no part in the determination of the unbiasedness property.
$\hat{\beta}_2$ will be a consistent estimator despite heteroscedasticity.
With the increase of sample size indefinitely, the $\hat{\beta}_2$ (estimated $\beta_2$) converges to its true value.
$\hat{\beta}_2$ is asymptotically normally distributed.

OLS Estimation in the presence of Heteroscedasticity

Overall, addressing the existence of heteroscedasticity in regression analysis is crucial to ensure the validity and reliability of the estimated parameters and inference results. Various methods and techniques are available to account for heteroscedasticity and obtain accurate estimates in regression analysis. For more details see Tests of Heteroscedasticity.

The best approach to address heteroscedasticity depends on the specific situation and the characteristics of the data being studied. The general guidelines are:

For mild heteroscedasticity, robust standard errors might be sufficient.
If the form of heteroscedasticity is known and the assumptions are comfortable, consider WLS or GLS.
Data transformation can be a simple solution, but weigh the benefits against the potential drawbacks of interpreting the transformed coefficients.

Remember that The OLS estimates remain unbiased under heteroscedasticity, however, addressing it can improve the efficiency and reliability of regression analysis, leading to more robust and interpretable results.