Why Cointegration Tests Use Nonstationary Price Series

Summary

The document explains why cointegration is tested on price levels rather than on price differences or returns. Cointegration applies when two nonstationary series have a linear combination that is stationary. Since differences and log returns are often stationary already, regressing them and testing the residuals does not establish the usual cointegration relationship between price levels.

It also connects the statistical definition to a possible trading interpretation. A weighted combination of asset prices can represent a portfolio holding units of both assets, whereas a weighted combination of their log returns does not directly define that same tradable spread. This distinction helps keep the test aligned with the object a pairs strategy might trade. The answer is conceptual and offers no dataset, test procedure, or empirical comparison; assumptions about integration order and practical implementation still need separate consideration.

Key ideas

• Cointegration describes nonstationary series whose linear combination is stationary.
• Price differences and log returns are often stationary, so they are not the usual inputs for cointegration analysis.
• A weighted combination of prices can correspond to a portfolio of the underlying assets.
• A return combination does not directly represent that same price spread portfolio.

Tags

# Cointegration on prices or difference of prices


# Cointegration on prices or difference of prices












Is it better to run my cointegration tests on prices or difference of prices? Difference of prices are more likely to be stationary so the results of my regression (which gives me the beta for my cointegration pair) are likely to be more statistically significant. Then doing my stationarity test on the residuals using the results of the regression Will yield better results What do you think?

## Answer by stans (score 2)

https://quant.stackexchange.com/a/41351

Precisely because differences in prices and log-returns are often stationary, cointegration cannot be done for them. By definition, stochastic processes $X(t)$ and $Y(t)$ are cointegrated if neither of them is stationary but there is a linear combination

$ \ \ \ Y(t) + \beta X(t) $

which is stationary... Also, in the trading context a time series

$ \ \ \ \rm{[Logreturn\ on\ asset\ A]}(t) + \beta\ \rm{[Logreturn\ on\ asset\ B]}(t) $

has no meaning because you cannot trade this process. What you can trade is

$ \ \ \ \rm{[Price\ of\ asset\ A]}(t) + \beta\ \rm{[Price\ of\ asset\ B]}(t). $

This corresponds to a simple portfolio with 1 unit of asset A and $\beta$ units of asset B.

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