1
$\begingroup$

One of the assumptions of linear regression is that $E(ee') = \sigma^2 * I $, where I is the identity matrix and sigma squared is the variance of residuals. Why is $E(ee')$ a matrix, though? $ee'$ is the dot product of the residuals so it is a 1x1 vector. How do we get a matrix out of a 1x1 vector?

$\endgroup$
3
$\begingroup$

You obtain this by using matrix algebra. $\epsilon=(\epsilon_1,\dots,\epsilon_T)^\top$ is a $T \times 1$ column vector. Now look at: \begin{align} \epsilon\cdot \epsilon^\top=\begin{pmatrix} \epsilon_1 \\ \vdots \\\epsilon_T\end{pmatrix}\cdot \begin{pmatrix} \epsilon_1 & \dots & \epsilon_T\end{pmatrix}=\begin{pmatrix} \epsilon_1^2 & \dots & \epsilon_1\epsilon_T \\ \vdots & \dots & \vdots \\ \epsilon_T\epsilon_1 & \dots & \epsilon_T^2 \end{pmatrix} \end{align}

This is a $T \times T$ matrix.

Regards

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.