# requirements for simulating a covariance matrix

I was wondering, is any positive semidefinite matrix a valid covariance matrix?

My problem is the following. I want to simulate a stochastic covariance matrix where the log-volatility (log of square root of variance) and the correlation are simulated separately according to some stochastic process. If I can ensure that the resulting covariance matrix is at all times positive semidefinite, is it a valid covariance matrix process?

To make things clearer, let’s assume I want to simulate a $$2 times 2$$ covariance matrix process. I would proceed by simulating two log-volatility processes and one correlation process:
$$logsigma^1_t = f(theta^1, t)$$
$$logsigma^2_t = f(theta^2, t)$$
$$rho_t = g(theta^3, t)$$
where the $$theta$$‘s are some parameters. Then, given $$sigma^1_t = e^{f(theta^1, t)}$$, $$sigma^2_t = e^{f(theta^2, t)}$$, $$cv_t = rho_t sigma^1_t sigma^2_t$$, I build the covariance matrix process
$$X_t = left[begin{array}{cccc} (sigma^1_t)^2 & cv_t \ cv_t & (sigma^2_t)^2 \ end{array}right]$$
My question: if by choosing proper $$theta$$, I can ensure that $$X_t$$ is at all times positive semidefinite, is it a valid covariance matrix process?

Cross Validated Asked by apocalypsis on November 21, 2021

The matrix also must be symmetric and not have any diagonal elements less than $$0$$ (I can’t remember if this is assured by the positive semi-definiteness EDIT see Sergio's comment), but then you always have a valid covariance matrix.

It looks like yours meets these requirements!

I have reservations about allowing for an eigenvalue of $$0$$, since that means you have perfect multicollinearity, but I suppose there’s nothing technically incorrect about including measurements in both feet and meters (for instance) in a multivariate distribution.

Answered by Dave on November 21, 2021

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