The pose is about a general discussion and an introduction to heteroscedasticity.
Introduction Heteroscedasticity and Homoscedasticity
The term heteroscedasticity refers to the violation of the assumption of homoscedasticity in linear regression models (LRM). In the case of heteroscedasticity, the errors have unequal variances for different levels of the regressors, which leads to biased and inefficient estimators of the regression coefficients. The disturbances $u_i$ in the Classical Linear Regression Model (CLRM) appearing in the population regression function should be homoscedastic; that is they all have the same variance.
In short words, heteroscedasticity means different (or unequal), and the Greek word Skodastic means spread (or scatter). Homoscedasticity means equal spread and heteroscedasticity means unequal spread.
Effect on the Var-Cov Matrix of the Error Terms:
The Var-Cov matrix of errors is
$$E(uu’) = E(u_i^2)=Var(u_i^)=\begin{pmatrix}
\sigma^2 & 0 & \cdots & 0\\ 0 & \sigma^2 & \vdots & 0\\ \vdots & \vdots & \vdots & \vdots\\ 0&0&\ddots &\sigma^2
\end{pmatrix}=\sigma^2 I_n,$$
where $I_n$ is an $n\times n$ identity matrix.
In the presence of heteroscedasticity, the Var-Cov matrix of the residuals will no longer be constant.
$$E(uu’)= E(u_i^2)=Var(u_i^)==\begin{pmatrix}
\sigma_1^2 & 0 & 0 & \cdots & 0 \\0 & \sigma^2_2 & 0 & \cdots & 0 \\ 0 & 0 & \sigma^2_3 & \cdots & 0 \\ 0 & 0 & 0 &\ddots & \sigma_n^2
\end{pmatrix}$$
The Var-Cov matrix of the OLS estimators $\hat{\beta}$ is
\begin{align*}
Cov(\hat{\beta}) &= E\left[(\hat{\beta}-\beta)(\hat{\beta}-\beta)’ \right]\\
&=E\left[[(X’X)^{-1}X’u][(X’X)^{-1}X’u]’ \right]\\
&=E\left[(X’X)^{-1}X’uu’X(X’X)^{-1} \right]\\
&=(X’X)^{-1}X’E(uu’)X(X’X)^{-1}\\
&=(X’X)^{-1}X’\Omega X (X’X)^{-1}
\end{align*}
The following are questions when we are concerned with heteroscedasticity:
- What is the nature of heteroscedasticity?
- What are the consequences of heteroscedasticity?
- How does one detect/ Test Heteroscedasticity?
- What are the remedial measures of heteroscedasticity?
That’s all about some basic introduction to heteroscedasticity.