# Cointegration vs combination of returns

Hi Quantitative Finance,

I understand that there are a wealth of pairs trading models out there. Recently, it got me thinking as to why we go through the trouble to find cointegrated pairs while we can simply combine their returns to achieve stationarity. Consider case 1 and 2:

Case 1, cointegration.

Assume $Y_t$ and $X_t$ are $I(1)$ and find $\beta$ such that

$$Y_t +\beta X_t$$

is $I(0)$. Then you can predict $Y_t +\beta X_t$ using AR. Assume series is OU and derive $\beta$ as the value that gives the highest log-likelihood.

Case 2, combination of returns.

Assume $Y_t$ and $X_t$ are $I(1)$ and consider

$$Y'_t +\beta X'_t$$

where $Y'_t$ and $X'_t$ are the returns. The key is, what is $\beta$ which could depend on the correlation between both variables or on risk management, how much $X_t$ hedges out a particular risk factor of $Y_t$

Based on my experience, I find that for case 1, it's harder to find cointegrated pairs and a particular $\beta$ will only gives stationarity for at most one month, thus requiring the $\beta$ to be estimated again. For case 2, it's difficult to get the length of the returns; too short (1h) and you're ignoring prior history (2 days) which could constitute the mean reverting relationship.

Is case 1 superior to case 2? Can it be shown that case 2 doesn't have the constant spread property that case 1 is suppose to have?

Cheers, Donny

• Could you please define what you denoted as $I(n), n \in \{0,1\}$ please? Jul 26 '17 at 7:28
• $I(n),n\in{0,1}$ means integrated with order $n$ Jul 28 '17 at 2:18