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Assume that two series ($x_1,\dotso,x_n$) and ($y_1,\dotso,y_n$) are linearly correlated.

What is the connection between $y_j-y_i$ and $x_j-x_i$ in terms of Pearson's $r$ and the variance of $x$ any $y$?

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    $\begingroup$ Do you mean linearly correlated? Also, I think the title of the question could be improved. Also, is this a homework question? If so, please add a self-stufy tag and read its Wiki. $\endgroup$ – Richard Hardy May 19 '15 at 11:29
  • $\begingroup$ Edited, it;s not a homework question. any suggestion for the title? $\endgroup$ – Raba Poco May 19 '15 at 11:33
  • $\begingroup$ I would have proposed one if I had a good candidate. But I think neither the old one nor the new one reflects accurately what you are asking about. Regarding the new one, the question is not about measuring the change; it is rather about statistical properties / relations of the changes in two correlated time series. Also, I think more information is needed to be able to answer the question. Nothing is said about the characteristics of the time series. I suppose $(x_1,...,x_n)$ are not i.i.d. If so, it matters what the dependence structure is. $\endgroup$ – Richard Hardy May 19 '15 at 11:39
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As per Richard, your question is not clear. But I would like to share the following link

Correlating volume timeseries

There is very high probability that above post will answer your question and very much similar to what you are trying to ask.

Also adding one more link to marry your understanding with above question

http://astro.phys.au.dk/~hans/tidsserie/Notes_08.pdf

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If $Cov(x_i,y_i)=\sigma_{xy}$, then the covariance of differences is $$Cov(y_j-y_i,x_j-x_i)=E[(y_j-y_i)(x_j-x_i)]-E[y_j-y_i]E[x_j-x_i]=$$ $$=2\sigma_{xy}-Cov(y_j,x_i)-Cov(y_i,x_j)$$

If you know the covariance of intertermporal cross-terms, then you can get the covariance of differences.

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