# 5 – M2L6 09 Cointegration V6

Let’s add more detail to describe the relationship between a pair of stocks. Recall that some time series such as stock prices are integrated of order one. By this, I mean that if we take the time difference of this time series, we can get a stationary series. When a series is stationary, it is also integrated of order zero. Another way to get a stationery process is to find some linear combination of a pair of stocks. For example, if stock X and stock Y have an economic link, then if we subtract one from the other, we may get a time series that is stationary. More generally, we multiply X by hedge ratio using regression. Then we take Y minus the estimate of Y based on the hedge ratio and the value of X. This difference is the spread. If the spread is stationary, then it is integrated of order zero. Also if the spread is stationary, then we say that the original series of X and Y are cointegrated. The hedge ratio is also called the coefficient of cointegration. Note that cointegration is not the same as correlation. Correlation is the covariance re-scaled to range from negative one to positive one. A strong positive correlation means that when stock A moves up, stock B moves up at the same time and vice versa. However, if the magnitude of movement in one stock is greater than the other, then they can still drift apart even though they are correlated. In other words, two correlated stocks may not be cointegrated. Cointegration means that, over a range of days, the relative increase in A is matched by a relative increase in B. Let’s imagine that we use $100 to buy stock A and$100 to buy stock B. Assuming these are cointegrated, the value of both positions would remain similar over time. So keep in mind that a pair of stocks may be correlated but not cointegrated. Similarly, a pair of stocks may be cointegrated but not correlated. For pairs trading, we want to find pairs of stocks that are cointegrated. The method for checking if two series are cointegrated is called the Engle-Granger Test. The Engle-Granger Test involves two steps that we saw previously. One, find the hedge ratio using regression. Two, calculate the spread and test if the spread is stationary. If the spread is stationary, then the two series are cointegrated. To check if the spread is stationary, we use the Augmented Dickey-FullerTest. The Augmented Dickey-Fuller Test outputs a P-value. If the P-value is small, say, 0.05 or less, then we can assume that the spread is stationary. Therefore, we can assume that the two are cointegrated.