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Determine Cointegration Rank of VEC Model

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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 Δyt, without any cointegration relations.

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

  • 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=αβ′. α contains the adjustment speeds, and β 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.

yt=[10.260-0.110.350.12-0.051.15]yt-1+[-0.2-0.1-0.10.6-0.4-0.1-0.02-0.03-0.1]yt-2+εt.

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.

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