Interpretation and finding the long run relationship in vector error correction model for 4 variables and 2 cointegration vectors

[Jaimin Lee]

I tried to do cointegration test for 4 variables (y, x, z, w) 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+δ₁z+ δ₂w+constant=0 and the other cointegration vector is x+γ₁z+ γ₂w+constant=0.

How to find the long run relationship between y and x? When there are two cointegration relationships between 4 variables, can we long run relationship among four variables?


Jaimin Lee