What is the square of a matrix?

Question

For my Econometrics course I have the following problem. I have a $x_i^\prime$ which is a $1$ x $K$ vector of explanatory variables and $i=1,\dots,N$ and the regression model is $y_i = x_i^\prime \beta$.

Now I need to compute the variance of $x_iu_i$. They answer sheets starts with $V[x_i u_i] = E[u_i^2 x_i x_i^\prime]$.

I do not see how one leads to another. I come this far: \begin{align} V[x_i u_i] &= E[(x_iu_i-E[x_iu_i])^2]\\ &= E[(x_iu_i)^2]\\ &= ... \end{align}

The second equality sign follows from $E[x_iu_i] =0$, but I cannot see how you get from the second equation to $E[u_i^2 x_i x_i^\prime]$. Can anyone explain?

Answer 1

Hint:

$$E[(x_iu_i - E[x_iu_i])^2] = E\left[(x_iu_i)^2 - 2 E[x_iu_i] x_iu_i + (E[x_iu_i])^2\right]$$

Now, use the rules:

• $E[a + b] = E[a] + E[b]$
• $E[\alpha a] = \alpha E[a]$.