Cointegrated

Cointegration is a concept in time-series analysis, especially in econometrics, describing a situation where a set of non-stationary series move together in such a way that a linear combination of them is stationary. When individual series are integrated of order one (I(1)), they wander over time, but if there exists a combination that is I(0) (stationary), the series are said to be cointegrated. This implies a long-run equilibrium relationship among the variables.

In practice, cointegration indicates that although each series may drift in the short run, deviations from their long-run relationship are temporary and tend to revert to the equilibrium over time. A common framework to model this behavior is the error correction model, where the short-run dynamics respond to the previous period’s deviation from the long-run relationship. For a vector of variables X_t, cointegration implies the existence of a vector β such that β'X_t is stationary. The speed at which the system returns to equilibrium is captured by the error-correction term.

Tests and methods developed to identify cointegration include the Engle-Granger two-step method, which tests the stationarity of the residuals from a regression between the variables, and the Johansen test, which uses a vector autoregression (VAR) framework to determine the number of cointegrating relationships. Panel cointegration extends these ideas to many cross-sectional units.

Applications of cointegration are common in finance and economics, such as modeling long-run relationships between asset prices, interest rates, exchange rates, and macroeconomic aggregates. Limitations include sensitivity to structural breaks, model specification, and the assumption of linear, stable relationships over time.