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 x.
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) .