I have a sequence of normally distributed random variables. Let's call it $S_1$. I want to generate 4 more series, each of which has its own correlation with $S_1$ and its own variance. Let's call them $S_2$, $S_3$, $S_4$ and $S_5$.

If I use a covariance matrix, I have to designate the covariances between $S_1$ and each of the other series, which is fine. But I also have to designate covariances between each of the other series (e.g. between $S_2$ and $S_3$). I will therefore end up with a correlation between $S_2$ and $S_3$, and there is no reason why there should be one.

How can I do generate $S_2$ to $S_5$ without forcing a correlation between them?

  • $\begingroup$ You can try and put $0$'s in the entries for correlation between $S_i$ and $S_j$ for $i, j \ge 2$ and $i\ne j$. Just make sure the non-zero correlations you choose are such that the matrix is positive semi-definite (so is a valid covariance matrix). $\endgroup$ – Minus One-Twelfth Mar 7 at 21:49
  • $\begingroup$ Then I am forcing a correlation of zero, and there is no more reason for the correlation to be zero than there is for it to match its historic correlation. $\endgroup$ – Paella1 Mar 7 at 22:35
  • $\begingroup$ I thought that's what you wanted by reading your post at first (like the part that you wanted to generate values without forcing a correlation, I interpreted this requiring zero correlation), but perhaps I misunderstood your goal. What is your goal? You want the correlations to be random as well or something? In that case, you could first just imagine generating two variables $X$ and $Y$ with your desired properties. If you can figure this out, it should help you do your case of generating several variables. $\endgroup$ – Minus One-Twelfth Mar 7 at 22:38
  • $\begingroup$ I thought it's what I wanted too, but now I'm not sure. I’m predicting which of stocks A & B will perform best over time. Returns of both have + correlation with M, and there’s reason to believe both are influenced by M. They also have a small +r with each other. There may be causation between A&B, but the correlation may be due only to shared correlation with M. I’m thinking that forcing the historic correlation between A&B is akin to assuming causation. So I want a simulation for A that correlates with M, & an independent simulation for B that correlates with the same M. $\endgroup$ – Paella1 Mar 7 at 23:32
  • $\begingroup$ If you want $A$ statistically independent of $B$, then the correlation of $A$ and $B$ must be $0$. $\endgroup$ – Minus One-Twelfth Mar 7 at 23:48

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