# Johansen cointegration analysis, cointegrating vectors and identification

I want to analyse the long run relation between, let's say 5 variables.

Scenario 1. Below is the output from a Johansen test on the five variables:

          test 10pct  5pct  1pct
r <= 4 |  3.23  6.50  8.18 11.65
r <= 3 |  8.88 15.66 17.95 23.52
r <= 2 | 18.03 28.71 31.52 37.22
r <= 1 | 35.73 45.23 48.28 55.43
r = 0  | 70.70 66.49 70.60 78.87

Eigenvectors, normalised to first column:
(These are the cointegration relations)

y.l1      x1.l1      x2.l1      x3.l1      x4.l1
y.l1   1.0000000  1.0000000   1.000000  1.0000000  1.0000000
x1.l1  2.3748202  0.0301862  11.683827 -1.8732409 -0.9878609
x2.l1 -2.9807221  1.7189828  14.803906 -2.2577727  1.0170132
x3.l1  0.5113696 -0.5399476 -13.691106  4.6226096 -0.3082810
x4.l1 -1.0472717 -2.1741134   7.600424  0.5206757  0.2049654


The Johansen test shows that I may have at least one cointegrating vecto. So I picked the first cointegrating vector $(1,2.37, -2.98, 0.511, -1.04)$ for further analysis and this equation, in scalar form is $y_{t-1}=-2.37x_{1,t-1}+ 2.98x_{2,t-1}- 0.511x_{3,t-1} + 1.04x_{4,t-1}$

Scenario 2. Let's say, I apply Johansen method on a different set of 5 series, and got 3 cointegrating relations:

           test 10pct  5pct  1pct
r <= 4 |   2.04  6.50  8.18 11.65
r <= 3 |  10.45 15.66 17.95 23.52
r <= 2 |  25.76 28.71 31.52 37.22
r <= 1 |  53.93 45.23 48.28 55.43
r = 0  | 102.66 66.49 70.60 78.87

Eigenvectors, normalised to first column:
(These are the cointegration relations)

y.l1     x1.l1      x2.l1     x3.l1       x4.l1
y.l1   1.000000000  1.000000  1.0000000  1.000000  1.00000000
x1.l1  0.033617469  3.127830 -0.8996560 -1.463796 -0.67601225
x2.l1  0.006907967 -2.858281  1.1413846 -1.701980  0.87209175
x3.l1 -0.003253034 -1.315143 -0.6540034  1.439399 -0.77830137
x4.l1 -1.010257005 -1.818696 -2.2384992 -2.270676 -0.04830754


For further analysis, I must now impose identifying restrictions on the three of these vectors. However, I am only interested in one of the cointegrating vectors which can be formulated as $(1,\beta_1, \beta_2, \beta_3, \beta_4)$. Can I pick one of the above cointegrating vectors for further economic analysis of the coefficients without imposing further identifying restrictions? Let's say I picked the first one and in scalar form it is $y_{t-1}=-0.003x_{1,t-1}- 0.0069x_{2,t-1} + 0.003x_{3,t-1}+ 1.01x_{4,t-1}$. I must say, I am not after a VECM model, but the long run equation only.

This was easy to do in the first scenario above where I had only 1 cointegrating vector and the normalisation was on the first variable. In the second scenario, there are three vectors and I must impose identifying restrictions for VECM analysis. But I do not want the VECM and hence limit the estimation to determining the cointegration rank and extracting the long run equation of interest.