Question

1. Show that the covariance (and also correlation) matrix $ ho = egin{bmatrix} 1 & 0.63 & 0.45 \ 0.63 & 1 & 0.35 \ 0.45 & 0.35 & 1 end{bmatrix}$ for the $p = 3$ standardized random variables $Z_1$, $Z_2$, and $Z_3$ can be generated by the $m = 1$ factor model $Z_1 = 0.9F_1 + epsilon_1$, $Z_2 = 0.7F_1 + epsilon_2$, $Z_3 = 0.5F_1 + epsilon_3$, where $Var(F_1) = 1$, $Cov(epsilon, F_1) = [0, 0, 0]^prime$, and $Psi = Var(epsilon) = egin{bmatrix} 0.19 & 0 & 0 \ 0 & 0.51 & 0 \ 0 & 0 & 0.75 end{bmatrix}$. That is, write $ ho$ in the form $ ho = LL^prime + Psi$.

          1. Show that the covariance (and also correlation) matrix
$
ho =  egin{bmatrix} 1 & 0.63 & 0.45 \ 0.63 & 1 & 0.35 \ 0.45 & 0.35 & 1 end{bmatrix}$
for the $p = 3$ standardized random variables $Z_1$, $Z_2$, and $Z_3$ can be generated by the $m = 1$
factor model
$Z_1 = 0.9F_1 + epsilon_1$,
$Z_2 = 0.7F_1 + epsilon_2$,
$Z_3 = 0.5F_1 + epsilon_3$,
where $Var(F_1) = 1$, $Cov(epsilon, F_1) = [0, 0, 0]^prime$, and
$Psi = Var(epsilon) =  egin{bmatrix} 0.19 & 0 & 0 \ 0 & 0.51 & 0 \ 0 & 0 & 0.75 end{bmatrix}$.
That is, write $
ho$ in the form $
ho = LL^prime + Psi$.

1. Show that the covariance (and also correlation) matrix
ho = eginbmatrix 1 0.63 0.45 0.63 1 0.35 0.45 0.35 1 endbmatrix
for the p = 3 standardized random variables Z1, Z2, and Z3 can be generated by the m = 1
factor model
Z1 = 0.9F1 + epsilon1,
Z2 = 0.7F1 + epsilon2,
Z3 = 0.5F1 + epsilon3,
where Var(F1) = 1, Cov(epsilon, F1) = [0, 0, 0]^prime, and
Psi = Var(epsilon) = eginbmatrix 0.19 0 0 0 0.51 0 0 0 0.75 endbmatrix.
That is, write ho in the form ho = LL^prime + Psi.

Added by Samuel C.

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