Consider a p-th order model:
Where:
yt. k-dimensional vector of non-stationary variables.
xt. d-dimensional vector of exogenous variables.
et. k-dimensional vector of random components.
The model can be represented as follows:
Where:
The key Granger theorem states that if a matrix P has incomplete rank r<k, there are kxr matrices α and β, each having the rank r, such as P = α · βT, the series βT is stationary, and each column of the matrix β is a cointegration vector, r - is the number of cointegration relations. Elements of the matrix α are called smoothing parameters of error correction model.
If there are k endogenous variables, each of which contains a unit root, there can be zero to k-1 linearly independent cointegration relations. If there are no cointegration relations, a standard time series analysis can be applied to the series in the first differences. On the contrary, if the system includes one cointegration equation, one linear combination of endogenous variables βTyt-1 must be added to each equation of the system. After multiplying by the equation coefficient (that is, the smoothing parameter α) the user gets the output component α · βT · yt-1, that is a component of error correction. Each next cointegration equation adds an additional error correction component with a unique linear combination of parameters.
If there are k cointegration relations, neither of the series has a unit root, and the model can be described without taking the differences.
The analyzed series can have a non-zero mean or a trend. Cointegration equations may also contain a constant and a trend. The following model types are used most frequently:
| Series y | Cointegration equations | Model |
| No trend | No constant |  |
| No trend | Constant available |  |
| Linear trend | Constant available |  |
| Linear trend | Linear trend |  |
| Quadratic trend | Linear trend |  |
α' - the matrix that comes from the ratio αT · α, = 0
Within this scheme, two parameters can vary when creating a model. One can fix model type and vary rank. Or on the contrary, fix model rank and select the best fitting model form. Economic adequacy of the model, as well as statistical criteria, needs to be taken into account when building a model. Pay attention to normalized cointegration equation to ensure that they meet your expectations on the nature of the analyzed process.
The model can also be generalized:
See also:
Library of Methods and Models | Cointegrated Processes | Error Correction Model | Vector Error Correction Model | ISmErrorCorrectionModel