Determine Cointegration Rank of VEC Model

This example shows how to convert an n-dimensional VAR model to a VEC model, and then compute and interpret the cointegration rank of the resulting VEC model.

The rank of the error-correction coefficient matrix, C, determines the cointegration rank. If rank(C) is:

• Zero, then the converted VEC(p) model is a stationary VAR(p - 1) model in terms of $$\Delta y_t$$, without any cointegration relations.

• n, then the VAR(p) model is stable in terms of $$y_t$$.

• The integer r such that $$0<r<n$$, then there are $$r$$ cointegrating relations. That is, there are $$r$$ linear combinations that comprise stationary series. You can factor the error-correction term into the two n-by- r matrices $$C=\alpha {\beta }^{\prime }$$. $$\alpha$$ contains the adjustment speeds, and $$\beta$$ the cointegration matrix. This factorization is not unique.

For more details, see Cointegration and Error Correction and [154], Chapter 6.3.

Consider the following VAR(2) model.

$$y_t=\left[\begin{array}{ccc}1& 0.26& 0\\ -0.1& 1& 0.35\\ 0.12& -0.05& 1.15\end{array}\right]y_{t-1}+\left[\begin{array}{ccc}-0.2& -0.1& -0.1\\ 0.6& -0.4& -0.1\\ -0.02& -0.03& -0.1\end{array}\right]y_{t-2}+{\epsilon }_t.$$

Create the variables A1 and A2 for the autoregressive coefficients. Pack the matrices into a cell vector.

A1 = [1 0.26 0; -0.1 1 0.35; 0.12 -0.5 1.15];
A2 = [-0.2 -0.1 -0.1; 0.6 -0.4 -0.1; -0.02 -0.03 -0.1];
Var = {A1 A2};

Compute the autoregressive and error-correction coefficient matrices of the equivalent VEC model.

[Vec,C] = var2vec(Var);

Because the degree of the VAR model is 2, the resulting VEC model has degree $$q=2-1$$. Hence, Vec is a one-dimensional cell array containing the autoregressive coefficient matrix.

Determine the cointegration rank by computing the rank of the error-correction coefficient matrix C.

r = rank(C)

r = 
2

The cointegrating rank is 2. This result suggests that there are two independent linear combinations of the three variables that are stationary.