Cointegration

Cointegration is a concept in time series analysis describing a situation where a set of non-stationary series share a common stochastic trend, so that a linear combination of them is stationary. When individual series are integrated of order one (I(1)) but a certain linear combination is I(0), the variables are cointegrated, implying a long-run equilibrium relationship that binds them together despite short-run fluctuations.

Testing and modeling typically proceed in two steps. First, each series is tested for unit roots (for example with the augmented Dickey-Fuller test) to determine if it is I(1). If cointegration is plausible, the Engle-Granger two-step method regresses one variable on another to estimate a long-run relationship and then checks whether the residuals from that regression are stationary. The Johansen approach generalizes to multiple series and uses a vector autoregression (VAR) framework to determine the cointegration rank and the cointegrating vectors via maximum likelihood. When cointegration is present, a vector error correction model (VECM) provides a bounded representation of short-run dynamics that incorporate long-run equilibrium constraints.

Interpretation and use: Cointegration indicates that deviations from the long-run relationship are temporary and will tend to revert to the equilibrium. In finance, cointegrated asset prices can underpin pairs trading and long-run hedging strategies. In economics and econometrics, cointegration supports the existence of stable, long-run relationships among prices, exchange rates, interest rates, and other macroeconomic variables.

Limitations: results can be sensitive to structural breaks, nonlinearity, or regime shifts; cointegration does not imply causality. Nonlinear cointegration and time-varying parameters may require alternative methods or robust testing approaches.