(MMSE Linear Predictor) Let $z$ be a vector of random variables with $E[z] = 0$ and $Var[z] = \Omega$ finite. Partition $z$ into the first element $z_1$ and the rest of the vector $z_2$ and partition $\Omega$ conformably:

$ Var[z] = Var \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} = \begin{bmatrix} \Omega_{11} & \Omega_{12} \\ \Omega_{21} & \Omega_{22} \end{bmatrix} $

(a) Show that the MMSE predictor of $z_1$ that is a linear function of $z_2$ is $E[z_1 \mid z_2] = z_2'\beta$ where $\beta = \Omega_{22}^{-1}\Omega_{21}$.

(b) Show that the forecast error $z_1 - z_2'\beta$ is uncorrelated with every element of $z_2$.

(c) Show that $z_1$ is linearly dependent (with probability equal to one) on $z_2$ if the variance of the forecast error is zero.

(d) Draw an analogy between these results and those of OLS regression.