Why Correlations of Returns Usually Beat Correlations of Price Levels
Summary
The document compares correlations calculated from levels of two economic series with correlations calculated from their period-to-period changes. The example involving a Federal Reserve balance sheet measure and producer prices shows that the estimated relationship changes substantially when the data are differenced. The responses advise that returns or first differences are usually more appropriate when the research question concerns portfolio returns or short-term co-movement.
The rationale is that price levels often trend and may be nonstationary, creating misleading correlations between unrelated series. A level series also accumulates past changes: early returns affect many later observations, while recent returns affect fewer, which can distort the apparent relationship. Pearson correlation describes co-variation around means, making it poorly suited to trending levels when those levels are not stationary. The document’s guidance is conditional rather than universal: use a transformation appropriate to the variables and question, and consider stationarity. It offers no formal tests or causal interpretation, and first differences are suggested for the example rather than established as the right transformation in every application.
Key ideas
- Correlations of price or economic levels can be misleading when the series trend or are nonstationary.
- Returns or first differences are usually preferable when the question concerns portfolio performance or changes over time.
- Accumulated price levels weight earlier changes across many later observations, complicating correlation interpretation.
- The appropriate transformation depends on the research question and properties of the series.
- Correlation describes association and does not by itself establish causation.
Tags
Full text
# Correlation between prices or returns? # Correlation between prices or returns? If you are interested in determining whether there is a correlation between the Federal Reserve Balance Sheet and PPI, would you calculate the correlation between values (prices) or period-to-period change (returns)? I've massaged both data sets to be of equal length and same date range and have labeled them WWW (WRESCRT) and PPP (PPIACO). Passing them into R we get the following: ``` > cor(WWW, PPP) [1] 0.7879144 ``` Then applying the Delt() function: ``` > PPP.d <- Delt(PPP) ``` Then applying the na.locf() function: ``` PPP.D <- na.locf(PPP.d, na.rm=TRUE) ``` Then passing it through cor() again: ``` > cor(WWW.D, PPP.D) [1] -0.406858 ``` So, bottom line is that it matters. NOTE: To view how I created the data view http://snipt.org/wmkpo. Warning: it needs refactoring but good news is that it's only 27 iines. ## Answer by ldnquant (score 17) https://quant.stackexchange.com/a/490 Short answer, you want to use the correlation of returns, since you're typically interested in the returns on your portfolio, rather than the absolute levels. Also, correlations on price series have very strange properties. If you think about a time series of prices, you could write it out as [P0,P1,P2,...,PN], or [P0,P0+R1,P0+R1+R2,...,P0+R1+...+RN], where Ri = Pi-P(i-1). Written this way you can see that the first return R1, contributes to every entry in the series, whereas the last only contributes to one. This gives the early values in the correlation of prices more weight than they should have. See the answers in this thread for some more details. ## Answer by Owe Jessen (score 10) https://quant.stackexchange.com/a/491 It depends, usually you would want to measure correlation between variables that are both stationary, else you would always be able to measure a correlation in the case of variables developing with a trend, even if they are unrelated. In this case I would guess that you should use first differences. ## Answer by Federico Caccia (score 5) https://quant.stackexchange.com/a/40548 There is a good explanation here. Summarizing, what we are computing with Pearson correlations are relations between deviations respect to the means, which is no meaningful using prices. So, you should calculate Pearson correlations using returns.
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