Cointegration and Error Correction
Introduction to Cointegration Analysis
Integration and Cointegration
A univariate time series y t is integrated if it can be brought to stationarity through differencing. The number of differences required to achieve stationarity is called the order of integration. Time series of order d are denoted I (d ). Stationary series are denoted I (0).
An n -dimensional time series y t is cointegrated if some linear combination β 1 y 1 t + … + β n y nt of the component variables is stationary. The combination is called a cointegrating relation. and the coefficients β = (β 1. …. β n )′ form a cointegrating vector. Cointegration is usually associated with systems of I (1) variables, since any I (0) variables are trivially cointegrated with other variables using a vector with coefficient 1 on the I (0) component and coefficient 0 on the other components. The idea of cointegration can be generalized to systems of higher-order variables if a linear combination reduces their common order of integration.
Cointegration is distinguished from traditional economic equilibrium, in which a balance of forces produces stable long-term levels in the variables. Cointegrated variables are generally unstable in their levels, but exhibit mean-reverting "spreads" (generalized by the cointegrating relation) that force the variables to move around common stochastic trends. Cointegration is also distinguished from the short-term synchronies of positive covariance, which only measures the tendency to move together at each time step. Modification of the VAR model to include cointegrated variables balances the short-term dynamics of the system with long-term tendencies.
Cointegration and Error Correction
The tendency of cointegrated variables to revert to common stochastic trends is expressed in terms of error-correction. If y t is an n -dimensional time series and β is a cointegrating vector, then the combination β ′y t −1 measures the "error" in the data (the deviation from the stationary mean) at time t −1. The rate at which series "correct" from disequilibrium is represented by a vector α of adjustment speeds. which are incorporated into the VAR model at time t through a multiplicative error-correction term αβ ′y t −1 .
In general, there may be multiple cointegrating relations among the variables in y t . in which case the vectors α and β become matrices A and B. with each column of B representing a specific relation. The error-correction term becomes AB ′y t −1 = C y t −1. Adding the error-correction term to a VAR model in differences produces the vector error-correction (VEC ) model :
Δ y t = C y t − 1 + ∑ i = 1 q B i Δ y t − i + ε t.
If the variables in y t are all I (1), the terms involving
differences are stationary, leaving only the error-correction term to introduce long-term stochastic trends. The rank of the impact matrix C determines the long-term dynamics. If C has full rank, the system y t is stationary in levels. If C has rank 0, the error-correction term disappears, and the system is stationary in differences. These two extremes correspond to standard choices in univariate modeling. In the multivariate case, however, there are intermediate choices, corresponding to reduced ranks between 0 and n. If C is restricted to reduced rank r. then C factors into (nonunique) n -by-r matrices A and B with C = AB ′, and there are r independent cointegrating relations among the variables in y t .
By collecting differences, a VEC(q ) model can be converted to a VAR(p ) model in levels, with p = q +1:
y t = A 1 y t − 1 +. + A p y t − p + ε t.
Conversion between VEC(q ) and VAR(p ) representations of an n -dimensional system are carried out by the functions vectovar and vartovec using the formulas:
A 1 = C + I n + B 1 A i = B i − B i − 1 , i = 2. q A p = − B q > VEC( q ) to VAR( p = q + 1 ) (using v e c t o v a r )
C = ∑ i = 1 p A i − I n B i = − ∑ j = i + 1 p A j > VAR( p ) to VEC( q = p − 1 ) (using v a r t o v e c )
Because of the equivalence of the two representations, a VEC model with a reduced-rank error-correction coefficient is often called a cointegrated VAR model. In particular, cointegrated VAR models can be simulated and forecast using standard VAR techniques.
The Role of Deterministic Terms
The cointegrated VAR model is often augmented with exogenous terms D x :
Δ y t = A B ′ y t − 1 + ∑ i = 1 q B i Δ y t − i + D x + ε t.
Variables in x may include seasonal or interventional dummies, or deterministic terms representing trends in the data. Since the model is expressed in differences ∆y t . constant terms in x represent linear trends in the levels of y t and linear terms represent quadratic trends. In contrast, constant and linear terms in the cointegrating relations have the usual interpretation as intercepts and linear trends, although restricted to the stationary variable formed by the cointegrating relation. Johansen  considers five cases for AB´y t −1 + D x which cover the majority of observed behaviors in macroeconomic systems: