Cointegration

Recall that the levels variables in the ECM entered the estimating equation in a special way: they entered combined into a single entity that captured the extent to which the system is out of equilibrium. It could be that even though these levels variables are individually I(1) (a variable which must be differenced once to become stationary), this special combination of them is I(0). If this is the case, their entry into the estimating equation will not create spurious (false, although seeming to be genuine) results.

This possibility does not seem unreasonable. A nonstationary variable will tend to wander extensively (that is what makes it nonstationary), but some pairs of nonstationary variables can be expected to wander in such a way that they do not drift too far apart. Thanks to dis-equilibrium forces that will tend to keep them together. Some examples are short and long term interests rate, prices and wages, household income and expenditures, imports and exports, spot and future prices of a commodity, and exchange rates determined in different markets. Such variables are said to be Cointegrated: although individually they are I(1), a particular linear combination of them is I(0). The cointegrating combination is interpreted as an equilibrium relationship, since it can be shown that variables in the error-correction term in an ECM must be cointegrated, and vice versa, that cointegrated variables must have an ECM representation. This is why economists have shown such interests in the concept of cointegration - it provides a formal framework for testing and for estimating long-run (equilibrium) relationships among economic variables.

One important implication of all this is that differencing is not the only means of eliminating unit roots. Consequently, if the data are found to have unit roots, before differencing (and thereby losing all the long-run information in the data) a researcher should test for cointegration; if a cointegrating relationship can be found, this should be exploited by undertaking estimation in an ECM framework.