# What does it mean to have k cointegration relationships? (example with some R code)

I'm working with the ca.jo function in R in order to evaluate cointegration among 4 economic variables (GDP, m1, bank interest rates and currency US dollar-mexican peso) but I still don't have a clear idea on how to correctly interpret the output.

####################
# Johansen-Procedure #
######################

Test type: trace statistic , with linear trend

Eigenvalues (lambda):
[1] 0.70039082 0.45734841 0.19752107 0.04859328

Values of teststatistic and critical values of test:

test 10pct  5pct  1pct
r <= 3 |  1.74  6.50  8.18 11.65
r <= 2 |  9.45 15.66 17.95 23.52
r <= 1 | 30.84 28.71 31.52 37.22
r = 0  | 73.02 45.23 48.28 55.43

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

diff.tf.tiie..l4 diff.tf.tipo_cambio..l4 diff.tf.logpib..l4 diff.tf.logm1..l4
diff.tf.tiie..l4                1.000000                1.000000             1.0000        1.00000000
diff.tf.tipo_cambio..l4        -3.616626               -2.639547          -117.1233       -0.04544856
diff.tf.logpib..l4           -391.182869               10.919719         -3348.6089      -16.51841837
diff.tf.logm1..l4              98.884942                7.698052          8735.8534        1.46417115


Basically, we can say we reject null for r=0 and r<=1 and therefore conclude we have 1 cointegration vector. So I have two questions:

1) What does "one cointegration vector" exactly mean? That, for example, all variables are stationary independent and don't share a common trend in the long-term?

2) Does it make sense to construct a new stationary serie using the coefficients from the first column of eigenvectors? If yes, how this can be useful in order to understand the relationship among variables?

new_serie = 1.000000*var1   -3.616626*var2  -391.182869* var3 + 98.884942*var4


I read many posts in this site (some of them below) but I'm still a lot confused. Thanks in advance for any advice and suggestion

Cointegration structure

clarification Johansen cointegration matlab