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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 [135], 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.