The relationship between series can be measured by different methods. The most common is to check if both series move in the same way. We’d like to go further, and see if the difference between them is always the same. We call it cointegration.

In many cases, we are interested in expressing one series according to another, or looking for common characteristics from which we can draw conclusions about their behaviour. The most basic method is in establishing **a linear link between the two series**.

The first thing that comes to mind when we want to know if two series maintain a linear relationship is to draw one against the other to see if we get a straight line.

Let’s consider the relationship between the USDCHF-EURUSD and USDCHF-EURGBP crosses. These pairs have **a fairly clear linear relationship**. Looking at the chart above, we can say that either of the series (EURUSD or EURGBP) could be expressed as a linear function of the USDCHF.

But who’s to say that this relationship isn’t fake? It could be pure chance that they behaved in a similar way throughout history… but **that’s not a stationary relationship**. Further, it’s easy to think that the relationship between EURGBP and USDCHF is casual, since the currencies that make up one cross are not contained in the other. However, for the EURUSD – USDCHF pair, a stationary relationship will be expected when the same currency is held at the two crosses. This is where the term cointegration comes in.

From the point of view of economics: we can say that the two series are said** to be cointegrated if they move together over time**, and the distance between them is stable. Hence, cointegration reflects the presence of **a long-run equilibrium** towards which the economic system converges over time. The differences (or term error) in the cointegration equation are interpreted as **the unbalance error** for each particular point in time.

More formally: two non-stationary series are cointegrated if there is a stationary linear combination of these series.

## Stationary series

Stationary series are those in which the variance is constant over time.

The Dickey-Fuller test allows us to distinguish whether a series is stationary or not. The three series that we handle, USDCHF, EURUSD, and EURGBP, are non-stationary of order 1, according to this test. This means that if we take the first difference of the series, we obtain a stationary series, with a constant variance. That is to say,

USDCHF~I(1), EURUSD~I(1), EURGBP~I(1)

Thus, we have the first condition to know whether the relationships we saw previously are stationary or spurious.

## Stationary relationship: cointegration

Now we have to create the linear regression between pairs of crosses and see if this linear combination is stationary:

USDCHFt = a1 + b1 * EURUSDt + ut

USDCHFt = a2 + b2 * EURGBPt + vt

…where Ut and Vt are the residuals of the regression estimate.

To know if these pairs of variables are cointegrated, we must **verify that the residues are stationary**. To do this we use the Dickey-Fuller test again, and it assures us of what we had initially thought; USDCHF and EURUSD are cointegrated (since Ut is stationary, ut ~ I (0)) and their relation is stationary over time, while USDCHF and EURGBP are not, and the relationship is spurious (vt ~ I (d), d> 0).

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