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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: $$\log\sigma^1_t = f(\theta^1, t)$$ $$\log\sigma^2_t = f(\theta^2, t)$$ $$\rho_t = g(\theta^3, t)$$ 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?

apocalypsis
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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.

Dave
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  • According to Sylverster's criterion, a symmetric matrix is positive definite if and only if all of its eigenvalues are strictly positive, is positive semidefinite iff all ot its eigenvalues are nonnegative, i.e. some eigenvalues may by 0. By another criterion (see Debreu, "Definite and Semidefinite Quadratic Forms", _Econometrica_, 20:295-300) , a symmetric matrix is positive semidefinite iff all principal minors, including diagonal elements (principal minors of order 1), are nonnegative, i.e. they can't be less than 0. – Sergio Jul 22 '20 at 12:45