# Compute the cross-correlation between the time series $X_t = X_{t-1}+w_t$ and $Y_t = \theta X_t+u_t$

Compute the cross-correlation between the time series

$$X_t = X_{t-1}+w_t$$ $$Y_t = \theta X_t+u_t$$

where $$w_t$$ and $$u_t$$ are white noise processes independent of each other.

I have problem when calculating the covariance (the numerator of cross-correlation); here's my step: $$E[(x_t -\mu _{x_t})(y_t-\mu_{y_t})]$$ $$=E[(x_{t-1}+w_t-0)(\theta x_t+u_t+0)]$$ $$=E[x_{t-1}\theta x_t+x_{t-1}u_t+w_t \theta x_t+w_tu_t]$$ $$=0+0+0+0$$ $$=0$$ ??

• Why did the means $\mu_{x_t}$ and $\mu_{y_t}$ disappear in the first equality and mysteriously reappear in the second equality? Why did you expand $x_t$ in terms of $x_{t-1}$ in that equality?
– whuber
Feb 3, 2020 at 19:11
• @whuber it should be u, typo (edited). And i just expand it like a quadratic formula. Feb 3, 2020 at 19:17
• But that's not what you have done. The way the means slip in and out of your equations makes it impossible to follow what you're doing and makes one (reasonably) suspect your work. Thus, your algebra is the first thing you should check: you need to get that right before you worry about anything else.
– whuber
Feb 3, 2020 at 20:28

Hint: For some strange reason, even though you are computing the cross-correlation at time $$t$$, you have decided to use the recursive equation to replace $$X_t$$ with $$X_{t-1}$$. Instead of doing that, try looking at time $$t$$, by taking $$\mathbb{Cov}(X_t,Y_t) = \mathbb{E}(X_t \cdot (\theta X_t + u_t)).$$