In a purely historical or backward-looking, descriptive context, is it incorrect to naively compute the covariance $$ \mathbf{E}(XY) - \mathbf{E}(X)\mathbf{E}(Y) $$ and Pearson correlation coefficient of two sequences of RVs $(X_i)_{i=1}^n$ and $(Y_i)_{i=1}^n$ before checking whether $X$ and $Y$ are stationary (e.g. if $X$ and $Y$ are stock prices)? If I calculate the naive correlation of these two raw series, is the only correct interpretation "the historical correlation of the prices of $X$ and $Y$ is 0.5"? If I transform these processes to stationary processes by calculating the percent change $\frac{X_t - X_{t-1} } {X_{t-1}}$ for both processes and then compute the standard correlation, what is the interpretation of the Pearson correlation coeff?

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    $\begingroup$ Try also the similar questions on this site, e.g. "Does correlation assume stationarity of data?". $\endgroup$ – Richard Hardy Apr 23 '16 at 21:00
  • $\begingroup$ Thanks for the link -- it helped clarify the mathematics (re convergence to a RV vs a constant) for me. As a quick follow up, if I then calculated the correlation on two stationary %-change (returns) processes, would the interpretation of the linear relationship be in terms of returns or could I say something about the correlation of the raw prices? $\endgroup$ – rrrrr Apr 28 '16 at 3:57
  • $\begingroup$ In terms of returns, I would say. $\endgroup$ – Richard Hardy Apr 28 '16 at 5:21
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    $\begingroup$ Even if the starting points are known (which they are if you have the price data), you can talk about certain statistical properties of returns but not necessarily the same properties of prices. Correlation works for stationary series but not I(1) processes. $\endgroup$ – Richard Hardy Apr 28 '16 at 13:16
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    $\begingroup$ Thanks for the follow up -- sorry, I'm unaware of standard protocol. I'd consider it answered based on the link and your subsequent comments. I can edit the question to include my follow-on and then let you summarize an answer? Open to alternatives. Let me know what you'd prefer? $\endgroup$ – rrrrr May 2 '16 at 3:25