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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4Try also the similar questions on this site, e.g. ["Does correlation assume stationarity of data?"](http://stats.stackexchange.com/questions/7376/does-correlation-assume-stationarity-of-data?rq=1). – Richard Hardy Apr 23 '16 at 21:00
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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? – rrrrr Apr 28 '16 at 03:57
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In terms of returns, I would say. – Richard Hardy Apr 28 '16 at 05:21
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Thanks. If we knew the initial price and then the period-by-period returns, couldn't we construct the prices at any subsequent time? This makes me think that we could say something about the correlation of prices, but I'm not sure if we can aay this without the additional assumption of knowing the starting points... – rrrrr Apr 28 '16 at 13:11
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1Even 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. – Richard Hardy Apr 28 '16 at 13:16
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Should we consider the question answered? As of now it appears as a mildly upvoted, unanswered one. Or should we close it as a duplicate of "Does correlation assume stationarity of data?" (see my first comment)? Or should I post a brief answer summarizing the essence in the comments? – Richard Hardy Apr 30 '16 at 16:28
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1Thanks 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? – rrrrr May 02 '16 at 03:25