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For my thesis I'm currently generating several time series of random numbers, so far so good. Now I realized some autocorrelation in the series as well and don't really know how to cope with it. Can I use the Cholesky factorization to generate random numbers with auto-correlation and then afterwards use the Cholesky decomposition again for simulating with the overall correlation structure between the different time series? Because I'm uncertain whether that destroys the autocorrelation I previously created?

Or put differently, I'm currently doing this for $n$ variables:

$$x_{t,1} = x_{t,0} \exp(my+std\cdot rv_1)$$ $$y_{t,1} = y_{t,0} \exp(my+std(p\cdot rv_1+(1-p^2)^{0.5}rv_2))$$

Now those are correlated just fine, but how do I insert the autocorrelation without harming the cross series correlation? Or is it unaffected when I change the random variables?

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To produce autocorrelated correlated random variables, you would want to first generate the correlated random variables and then add the relevant autocorrelation terms.

You ask about adding the autocorrelation without harming the correlation terms. You need to think carefully about the steps taken to estimate the terms used in the simulation. You need to first estimate the appropriate univariate AR(p) models, get the residuals, then fit the covariance matrix to the residuals. This makes it like a restricted VAR(p) model. You cannot use the covariance matrix of the original time series (i.e. not the residuals) because it presumably has autocorrelation in it. The residuals should (if you have done things correctly) should have had the autocorrelation removed.

See here for more details on a more general way to think about it.

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