Simulating Correlated Time Series with Autocorrelation
Summary
The document explains how to combine serial dependence within each simulated series with correlation across series. Its suggested approach is to fit a univariate autoregressive model to each series, then estimate the cross-series covariance from the resulting residuals. Those residuals should have the serial dependence removed, so their covariance can represent the remaining contemporaneous relationship without mixing in autocorrelation.
This is framed as a restricted vector autoregression and is offered as a modeling workflow rather than a worked numerical example. The main caution is to avoid estimating cross-series dependence from the original autocorrelated observations, since that covariance may conflate serial and cross-sectional structure. The document does not specify how to select autoregressive orders, validate the fitted models, or transform the fitted structure into a particular simulation algorithm.
Key ideas
- Fit autoregressive models to individual series before estimating their shared residual covariance.
- Use residuals to estimate cross-series dependence after serial dependence has been removed.
- The proposed setup is a restricted form of a vector autoregressive model.
- Covariance estimated from the original autocorrelated series may mix distinct dependence effects.
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# Correlated random variables with additional autocorrelation - multi dimensional Cholesky?
# Correlated random variables with additional autocorrelation - multi dimensional Cholesky?
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?
## Answer by John (score 2)
https://quant.stackexchange.com/a/21101
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.