I want to generated $N$ time series, each with length of $N_t$, i.e., I am considering $N_t$ time instants. The generated time series must have an auto correlation of $\rho_a$, i.e., the correlation between every two successive samples taken at $t$ and $t-1$ must have a correlation of $\rho_a$. Also, the cross correlation between every two time series $i$ and $j$ must be $\rho_{ij}$. Neglecting the cross correlation between the different time series, this is a straightforward and it can be done by auto regressive model AR(1). For each of the N time series, I do the following (assume 3 time series to be generated). I generate the time samples according to $X_t=\rho_a X_{t-1}+u_t$, $Y_t=\rho_a Y_{t-1}+v_t$ and $Z_t=\rho_a Z_{t-1}+\epsilon_t$ where $u_t,v_t,\epsilon_t$ are $N(0,1)$. However, considering the cross correlation, this is not applicable as the generated time series does not have the predefined cross correlation. Note that a similar question addresses a similar problem but it is only valid for two time series (not multiple time series). Here is the link :How to simulate two correlated AR(1) time series?

  • $\begingroup$ The answer you refer to is written in a way to make it clear how it generalizes to multiple time series. $\endgroup$ – whuber Jul 28 '20 at 13:54
  • $\begingroup$ Thanks @whuber. I do not understand how we got the equation for the covariance matrix which is written after this sentence (If we denote by Σ the covariance matrix of Zt and Q the covariance matrix of εt then it can be checked that). Any related topic I can google so I understand the math behind it. $\endgroup$ – mohamed Jul 28 '20 at 15:47
  • $\begingroup$ It's unclear what your $\Sigma$ and $Q$ are because your statement of the problem may be ambiguous. For instance, do you really use the same $\epsilon$ for $X_t,$ $Y_t,$ and $Z_t$? Do you use the same $\epsilon$ for all $t$? Most likely not, despite what you write. $\endgroup$ – whuber Jul 28 '20 at 16:09
  • $\begingroup$ @whuber. I edited the question per your comment. I hope it is clear now $\endgroup$ – mohamed Jul 28 '20 at 16:43
  • $\begingroup$ Presumably all the $u_t,$ $v_t,$ and $\epsilon_t$ are independent, and therefore uncorrelated: this gives the matrix $Q$. You have specified the matrix $\Sigma$ via $\rho_{ij}.$ It looks like you're ready to apply the referenced thread :-). $\endgroup$ – whuber Jul 28 '20 at 18:22

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