# Normal Random Variables that all correlate with a single time series but not necessarily with each other

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?

• 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). – Minus One-Twelfth Mar 7 at 21:49
• 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. – Paella1 Mar 7 at 22:35
• 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. – Minus One-Twelfth Mar 7 at 22:38
• 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. – Paella1 Mar 7 at 23:32
• If you want $A$ statistically independent of $B$, then the correlation of $A$ and $B$ must be $0$. – Minus One-Twelfth Mar 7 at 23:48