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I am examining the relationship between the S&P 500 and the Industrial Production Index. Computing the correlation between these these variables yield vastly different results if expressed in percentage changes as opposed to using the index approach (i.e. choosing an index year and multiplying the percentage changes).

Percentage changes CORR (S&P500, IndustrialProduction) = -0.006460759

Index levels CORR (S&P500, IndustrialProduction) = 0.890445169

How does this make sense?

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The linked to answer does explain it all, but in brief because one set are stationary processes and the others are not.

Correlation as a measures gives us the normalized degree of co-movement between process residuals, which assumes stationary processes. With non-constant mean term (ie, non-stationary processes), there's no way to parse out and relate which portion of the movement is based on the drift and which is based on the residual. Same goes for regressions (ie, you can't regress a price time series against a predictor variable; you need to use returns).

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