Why Correlations of Index Levels and Percentage Changes Differ

Summary

The document examines why correlations between the S&P 500 and an industrial production index differ sharply when calculated from index levels versus percentage changes. Its explanation centers on stationarity: index levels may be non-stationary, while changes or returns are more suitable for analyses that assume stable statistical properties. A high correlation between trending levels can reflect shared drift rather than a meaningful relationship between the underlying fluctuations.

Correlation summarizes normalized co-movement, but with changing means it cannot separate drift from residual variation. The answer extends the same caution to regression, advising against regressing one price-level series on a predictor without addressing non-stationarity; returns or other suitable transformations are typically used instead. The example illustrates a potential spurious relationship, but the text does not provide tests for stationarity, specify the data frequency, or establish that percentage changes are stationary in every case. Transformation choices should match the series and research question.

Key ideas

• Correlations of index levels can differ greatly from correlations of percentage changes.
• Trending, non-stationary levels can produce apparent co-movement driven by shared drift.
• Correlation is most interpretable when the processes satisfy its stationarity assumptions.
• Regression on price levels can be misleading when the series are non-stationary.
• Returns or other appropriate transformations may help focus analysis on fluctuations rather than drift.

Tags

# Why does computing correlation between index levels vs. percentage changes yield completely different results?


# Why does computing correlation between index levels vs. percentage changes yield completely different results?












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?

## Answer by Chris (score 4, accepted)

https://quant.stackexchange.com/a/44309

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.