Cointegration
Cointegration is a statistical property of a collection (X1, X2, ..., Xk) of time series variables. First, all of the series must be integrated of order 1 (see Order of integration). Next, if a linear combination of this collection is integrated of order zero, then the collection is said to be co-integrated. Formally, if (X,Y,Z) are each integrated of order 1, and there exist coefficients a,b,c such that aX + bY + cZ is integrated of order 0, then X, Y, and Z are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends—either deterministic or stochastic. In an influential paper, Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends—these are also called unit root processes, or processes integrated of order 1—I(1). They also showed that unit root processes have non-standard statistical properties, so that conventional econometric theory methods do not apply to them.
If two or more series are individually integrated (in the time series sense) but some linear combination of them has a lower order of integration, then the series are said to be cointegrated. A common example is where the individual series are first-order integrated (I(1)) but some (cointegrating) vector of coefficients exists to form a stationary linear combination of them. For instance, a 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 the existence of a cointegrated combination of the two series.
The first to introduce and analyse the concept of spurious—or nonsense—correlations was Udne Yule in 1926. Before the 1980s many economists used linear regressions on (de-trended) non-stationary time series data, which Nobel laureate Clive Granger and Paul Newbold showed to be a dangerous approach that could produce spurious correlation, since standard detrending techniques can result in data that are still non-stationary. Granger's 1987 paper with Robert Engle formalized the cointegrating vector approach, and coined the term.
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