In the previous models (QP1, QP2, QP3) we used the original time-series data to calculate the variance-covariance matrix. In this model we will use the original data directly. By noticing that

sum((i,j),x(i)covar(i,j)x(j)) = sum((i,j), 1/n x(i)x(j) sum(t,(d(i,t)-mu(i))(d(j,t)-mu(j)))) = 1/n sum(t,sum(i,x(i)(d(i,t)-mu(i)))sum(j,x(j)(d(j,t)-mu(j)))) = 1/n sum(t, w(t)*w(t)) where w(t) = sum(k,x(k)(d(k,t)-mu(k))) mu(k) = 1/T sum(t,d(k,t))

we see that we end up with a separable objective function
which consists of just a sum of squares. The matrix *d*
contains the data.

As an aside, this exercise showed that a covariance matrix is
positive semi-definite, as we have proved that ** x'Qx
>= 0** for all

Especially if the number of instruments *n* is larger
than the number of observations *T*, the original data
occupies much less space as the covariance matrix. The first is
an *(n * T)* matrix, while the covariance matrix is *(n *
n) *.