cointegrate

Cointegration is a statistical property of a set of non-stationary time series. If the series are integrated of order one, I(1), they are cointegrated when there exists a linear combination of them that is stationary, I(0). In this case the series share a long-run equilibrium relationship, so while each series may wander in the short run, their movements do not drift apart indefinitely. Note: economists more commonly speak of cointegration; cointegrate is used as a verb to describe the action of forming a cointegrating relationship.

Testing for cointegration typically involves two steps or multivariate methods. First, assess unit roots with tests such as the Augmented Dickey-Fuller or Phillips-Perron tests to determine whether the series are I(1). Then conduct a cointegration test: the Engle-Granger two-step method for two variables or the Johansen test for multiple variables. Engle-Granger uses an OLS regression of one variable on others and tests whether the regression residuals are stationary. Johansen uses a likelihood framework to identify the number of independent cointegrating relationships.

If cointegrated, models can incorporate an error-correction mechanism. A vector error-correction model (VECM) includes terms that drive short-run dynamics toward the long-run equilibrium, as implied by the cointegrating relationships. This contrasts with estimating differences only, which would discard long-run information.

Applications span economics and finance, including price relationships, interest rates, and pair trading strategies. Cautions include sensitivity to structural breaks, nonlinearity, and sample size, which can affect test validity and interpretation. The concept emerged in the late 1980s with Engle and Granger, and is central to modern time-series econometrics.