# Creating correlated time series data

Consider some time series $(x_t)_{t=1}^T$. Are there any popular methods for creating a new time series $(y_t)_{t=1}^T$ (or even better a whole family of time series), such that $y_t$ is correlated with $x_t$? Specifically, I want a method such that

• The correlation can be chosen to be either positive or negative.

• If positive: when $x_t$ increases, $y_t$ is also more likely (but not guaranteed) to increase.

• If negative: when $x_t$ increases, $y_t$ is more likely (but not guaranteed) to decrease.
• The degree of correlation should be adjustable.

Example. Given $(x_t)_{t=1}^T$, we define the relative increase by $x_{t+1} = x_t(1+r_t)$ or $r_t = \frac{x_{t+1}-x_t}{x_t}$. Then I wish to create e.g. a thousand new time series whose relative increase have correlations with $r_t$ ranging from e.g. $-0.8$ to $0.8$.

• did you search for cholesky factorization method already? Nov 1, 2016 at 14:55
• @Aksakal No, I have never heard of using Cholesky factorization for an application like this. I have only used it for solving linear systems. If you have anything that I can read, I'll be glad. I'll also try and look it up myself.
– Eff
Nov 1, 2016 at 15:02
• there are many discussions such as this one: stats.stackexchange.com/questions/38856/… Nov 1, 2016 at 15:04
• @Aksakal Thank you. I will try to look into it, and I'll see if it solves my problem.
– Eff
Nov 1, 2016 at 15:06
• Also consider arima.sim in R Nov 1, 2016 at 15:18