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 $0.8$.
