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I have the following time series: $$y_t = \mu_t + \sigma_tx_t$$

I want to get the covariance from this time series. How would one proceed with this? I have found the following formula: $$cov(x, y) = E[(x - Ex)(y - Ey)]$$

However, I am clueless on how to implement this formula to my series.

Many thanks.

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  • $\begingroup$ Covariance is estimated from two variables, x and y. You only have one, y. Are you searching for autocorrelation? $\endgroup$
    – user2974951
    Commented Sep 12, 2019 at 8:19

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You've random processes instead of variables, in which you can calculate the covariance between samples at specific times, i.e. $\operatorname{cov}(x_{t_1},y_{t_2})$, and this is a function of $t_1,t_2$. We have a name for it, as also pointed out in the comments: cross-covariance. In case of joint stationarity, this function becomes univariate and depends only on $t_1-t_2$.

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