4
$\begingroup$

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?

Improve this question
$\endgroup$
7
  • 4
    $\begingroup$ Try also the similar questions on this site, e.g. "Does correlation assume stationarity of data?". $\endgroup$
    – Richard Hardy
    Commented Apr 23, 2016 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
    Commented Apr 28, 2016 at 3:57
  • $\begingroup$ In terms of returns, I would say. $\endgroup$
    – Richard Hardy
    Commented Apr 28, 2016 at 5:21
  • 1
    $\begingroup$ 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
    Commented Apr 28, 2016 at 13:16
  • 1
    $\begingroup$ 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
    Commented May 2, 2016 at 3:25

0

Reset to default