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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

1 Answers

One Answer

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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