Question

I start with 4 time series, labelled A, B, C, D.

I generate the following:

• A 4×1 matrix of means.
• A 4×1 matrix of standard deviations.
• A 4×4 correlation matrix, by taking 30 samples from each time series.

What is the Matlab code to generate a 4×1 matrix of random variables, keeping the correlations between the time series intact?

(reason: this is the first stage in a Monte Carlo simulation).

Answer 1

You need just the means vector (call it m) and the covariance matrix (call it C). Note that you can get the covariance matrix from the correlation using the equation C = R - m*m' (or just compute it directly by computing the correlation of the sequences after subtracting their mean).

Then, to get a vector with the covariance C, you generate an IID random vector (say gaussian):

w = randn(4,1)

Then multiply it with the square root of the covariance matrix (call it Q) and add the mean:

v = Q*w + m

You can either use Matlab’s sqrt function to compute sqrt(C) or compute it using the SVD or EIG.

[u,d] = eig(C)

Q = u*sqrt(d)*u'

The covariance of v will be Q*Q' (=C) and the mean will be m

See the wikipedia article for the properties of the covariance matrix.