## What is Cointegration?

Cointegration is a statistical method used to test the correlation between two or more non-stationary time series in the long-run or for a specified time period. The method helps in identifying long-run parameters or equilibrium for two or more sets of variables. It helps in determining the scenarios wherein two or more stationary time series are cointegrated in such a way that they cannot depart much from the equilibrium in the long-run.

### Explanation

- The method is used to determine the sensitivity of two or more variables to the same set of conditions or parameters in a period of time.
- Let us understand the method with the help of a graph. The prices of two commodities A and B are shown on the graph. We can infer that these are perfectly co integrated commodities in terms of price as the difference between the prices of both the commodities has remained the same for decades. Though this is a hypothetical example, it perfectly explains the cointegration of two non- stationary time series.

### History

- Earlier Linear regression was being used as a statistical method to find the relation between two or more time-series. Granger and Newbold, British economists argued against the use of linear regression as a technique for analyzing time series for a specified period of time. As per them, using linear regression sometimes produce false correlation due to the impact of other factors.
- In 1987, Granger and Engle published a paper on this topic where they established the concept of the cointegration of non-stationary time series to find the correlations between them. They established the fact that two or more non-stationary time series are cointegrated in such a way, that they can move much from equilibrium. The two economists were awarded the Nobel memorial prize in economic sciences for their revolutionary work.

### Examples of Cointegration

- Cointegration as correlation does not measure whether two or more time-series data or variables move together in the long-run, while it measures whether the difference between their means remains constant or not.
- So that means that two random variables completely different from each other can have one common trend that combines them in the long-run if this happens, variables are said to be cointegrated.
- Now let’s take the example of Cointegration in pair trading. In pair trading, a trader purchases two cointegrated stocks, Stock A at the long position and Stock B in the short position. The Trader was unsure about the direction of price for both the stocks but was sure that Stock A’s position will definitely be better than stock B.
- Now let us say that the prices of both the stocks go down, the trader will still make a profit as long as stock A’s position is better than stock B if both the stocks were equally weighted at the time of purchase.

### Methods of Cointegration

The three main methods are explained below:

#### #1 – Engle-Granger Two-Step Method

This method is based on testing the residuals created based on static regression for the presence of unit roots i.e. if two non-stationary time series are cointegrated, the result will confirm the stationary characteristic of residuals. There are some limitations with this method because if there are two or more non-stationary variables, the method will reflect two or more cointegrated relationship and also, the method is a single equation model. Some of these limitations have been addressed in recent times tests like Johansen’s and Philip-Ouliari’s test.

#### #2 – Johansen Test

Johansen test is used for testing Cointegration between several time-series data at a time. This test overcomes the limitation of an incorrect test result for more than two time series of Engle-Granger method. This test is subject to asymptotic properties i.e. it takes a large sample size because a small sample size would give incorrect or false results. There are two further bifurcations of Johansen Test i.e. Trace test and Maximum Eigenvalue test.

#### #3 – Philip-Ouliaris Test

This test proves that when residual-based unit root test is applied on time series; the cointegrated residuals give asymptotic distribution instead of Dickey-Fuller distribution. The resulted asymptotic distributions are known as Philip-Ouliaris distributions.

### Condition of Cointegration

The Cointegration test is based on the logic that more than two-time series variables have some similar deterministic trends that can be combined over a period of time. this is the utmost condition for all cointegration testing for non-stationary time series variables that they should be integrated in the same order or they should have a similar identifiable trend that can define a correlation between them. So that they should not deviate much from the average parameter in the short-run and in the long run they should be reverting to the trend.

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