# 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? – JejeBelfort Jul 26 '17 at 7:28
• $I(n),n\in{0,1}$ means integrated with order $n$ – Donny Lee Jul 28 '17 at 2:18

The reason we look for cointegration is to find some long-term relationship between levels of prices. When you use the returns, you are seeking a relationship between first differences, which are usually not integrated.

In your case 1, your model helps to describe what will happen between the two prices. In the cases of oil prices, for instance, Brent and WTI are essentially the same product. If their prices differ too strongly and their cointegrated series is far from the mean, we can expect it to mean-revert at some point in the future, whether that's now or before the end of the world.

In case 2, you are looking for a way of modeling an instantaneous change in returns from one asset due to changes in another -- this is by it's nature more short term. As Richard points out in his excellent comment below, this omits a variable and may not be an effective test.

Case 1 is your best option, as it's unlikely that first differences of a financial time series are cointegrated.

• First differences are not usually integrated of order 1, i.e. I(1); rather, they are usually not integrated, i.e. I(0). Also, if the original variables are truly integrated and cointegrated, then running a regression on their first differences will have an omitted variable problem (omitting the error correction term) and thus will yield subpar performance. – Richard Hardy Oct 10 '17 at 7:10

In practice, of course, you use the model that will perform better for your purpose. Formally, cointegration is first and absolutely a long run relationship, and more data must give you a better estimate of \beta. If it does not, then case 1 should in principle be rejected. If case 1 is appropriate , it is very powerful in the long run. Working with case 2 may offer more from a RM standpoint, as the consequences of working with case 1 when it is wrong seem pretty dire. This is great "model risk" compounding your more easily quantified "market risk".