Cointegration

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Cointegration is an econometric property of 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 testing for 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 for correctly testing hypotheses concerning the relationship between two variables having unit roots (i.e. integrated of order one).

What does this mean? A series is said to be "integrated of order d" if one can obtain a stationary series by "differencing" the series 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. (This series is actually trend stationary, so should be de-trended rather than differenced). We obtained a stationary series by differencing it just once, which means our original series is integrated of order one.

The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run Ordinary Least Squares (OLS) regressions on data which had initially been differenced. Although this method is correct in large samples, cointegration provides more powerful tools when the data sets are of limited length, as most economic time-series are.

The two main methods for testing for cointegration are:

The Engle-Granger two-step method.
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-difference effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.

However, these tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the long-run relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress, economic crises, changes in the people’s preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account Gregory and Hansen (1996) have introduced tests for cointegration with one unknown structural break and Hatemi-J (2007) has introduced tests for cointegration with two unknown breaks.