# 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

## Related Questions

### ‘Translate’ ANOVA comparison on regression parameters into linear mixed model

1  Asked on August 13, 2020 by laurie

### Uncertainty propagation for the solution of an integral equation

0  Asked on August 12, 2020 by clment-f

### Which test should I use to compare 2 unrelated dichotomous variables?

1  Asked on August 10, 2020 by anna

### Difference in Differences with Multiple Time Periods and Multiple Treatment Periods

1  Asked on August 8, 2020 by john-baker

### ARDL and ECM lags

0  Asked on August 8, 2020 by php-useless

### Combining categorical and continuous features for neural networks

2  Asked on August 5, 2020 by 3michelin

### What statistical analysis to used for kinetic data with multiple groups?

1  Asked on August 5, 2020 by carlos-valenzuela

### In R, why do the p-values from anova() change when you add more predictors?

0  Asked on August 4, 2020 by m-smith

### Random forest after cross validation

1  Asked on August 1, 2020 by steven-niggebrugge

### Grey relation between two datasets?

0  Asked on July 31, 2020 by msilvy

### General procedures for combined feature selection, model tuning, and model selection?

1  Asked on July 31, 2020 by uared1776

### Classification model not working for a large dataset

1  Asked on July 30, 2020 by gabriel-ullmann

### Sigma algebra generated by random variable on a set with generators

0  Asked on July 28, 2020 by gabriel

### What is the seasonal trend lowess model in time series?

0  Asked on July 28, 2020 by christopher-u

### Non seasonal and seasonal parameters of this time-series

0  Asked on July 27, 2020 by statsmonkey

### Extended Cox model and cox.zph

2  Asked on July 25, 2020 by finance