# 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?

• I think you generally use return, or log diffs for stocks and probably differences in yield data for bonds/swaps. – Attack68 Aug 20 '18 at 9:09

## 1 Answer

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.