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Cointegration is an econometric technique for testing the correlation between non-stationary time series variables. If two or more series are themselves non-stationary, but a linear combination of them is stationary, then the series are said to be cointegrated. For instance, a stock market index and the price of its associated futures contract move through time, each roughly following a random walk. Testing the hypothesis that there is a statistically significant connection between the futures price and the spot price could now be done by finding a cointegrating vector. (If such a vector has a low order of integration it can signify an equilibrium relationship between the original series, which are said to be cointegrated of an order below one.)

Before the 1980s many economists used linear regressions on (de-trended) non-stationary time series data, which Clive Granger and others showed to be a dangerous approach, that could produce spurious correlation. His 1987 paper with Robert Engle^[1], formalized the cointegrating vector approach, and coined the term. For his contribution to the technique's development Clive Granger shared the 2003 Nobel Memorial Prize.

It is often said that cointegration is a means of valid hypothesis testing between two variables having unit roots (Integrated of order one).

What does this mean? A series is said to be integrated of order d if one can get a stationary series by "differencing" the term d times. For example, suppose a stock price is 5 on Monday, 6 on Tuesday, 7 on Wednesday, and 8 again on Thursday. One differences that series by turning it into a series of daily price increments. In this case, if we difference just once we get 1 ... 1 ...1. So we get a stationary series by differencing it once, which means our original series was integrated of order one. The usual procedure for hypothesis testing between non-stationary variables was run OLS regressions on data which had initially been differenced. Although this method is correct in large samples, cointegration provides more poweful tools when the data sets are limited, as most time-series are.

The two main most common methods for testing for cointegration:
1. The Engle-Granger two-step method.
2. The Johansen procedure.

In practise, cointegration is used for such series in typical econometric tests, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other second differencing effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.

↑  Co-integration and error correction: Representation, estimation and testing, (1987) Econometrica 55(2): 251-276.