I have 4 time series variables, y, x, z, w,, which are filtered by tsmooth command.
I tried to do cointegration test for these 4 variables using vecrank command and found that there are 2 cointegration relationships. I tried to do vector error correction model using vce command.
I got two cointegration vectors below. One cointegration vector is y-2.888z-9.519w+constant=0 and the other cointegration vector is x+1.231z+1.467w+constant=0.

Array

My question is how to calculate the long-run relationship between y and x. I think three methods. The first is to insert z=-(x+1.467w+constant)/1.231 in the second cointegration (_ce2) to the first cointegration equation (_ce1) and get the relationships like this, y=-2.345x+6.077w+constant.

The second is to insert w=-(x+1.231z+constant)/1.467 in the second cointegration (_ce2) to the first cointegration equation (_ce1) and get the relationships like this, y=-6.487x-5.099z+constant.

The third is to insert z=-(x+1.467w+constant)/1.231 and w=-(x+1.231z+constant)/1.467 in the second cointegration (_ce2) to the first cointegration equation (_ce1) and get the relationships like this, y=-8.833x—7.987z-3.441w+constant.

Which one is correct?
In the first cointegration vector, y=2.888z+9.519w+constant, there is positive relationship between y and z and y and w. However, in the third method, their relationship signs are changed. How do I interpret their relationship?

Please explain the long -run relationships among four variables with more than 2 cointegration relationships.

Jaimin Lee