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.
