# Cointegration and long run restrictions

Below is a typical output from a 5 variable VECM model with 4 cointegrating vectors with $r-1$ restrictions on each equation. Therefore, the top part of the output is a diagonal matrix by default. For instance, $ect1$ is $x_1=-0.5574268x_4+21.07965$ and $ect2$ is $x_2=-0.4633047x_4+4.424252$ etc as seen in the output. Let's now say the theory suggests I should model $x_1$ as $$x_1=\beta_1x_2+\beta_2x_3-\beta_3x_4-\beta_4x_5$$ Let's also say I want to give economic meaning to the $\beta's$ as long-run 'elasticities' as the $x's$ are in log form. How will I be able to obtain this equation while accounting for other CVs i.e. $ect2$, $ect3$ and $ect4$.

               ect1          ect2          ect3      ect4
X1      1.000000e+00  0.000000e+00  0.00000000  0.000000e+00
X2      0.000000e+00  1.000000e+00  0.00000000  0.000000e+00
X3      0.000000e+00  0.000000e+00  1.00000000  0.000000e+00
X4      0.000000e+00  0.000000e+00  0.00000000  1.000000e+00
X5      5.574268e-01 -4.633047e-01  0.25466051  2.523022e-02
constant  -2.107965e+01 -4.424252e+00 -4.92108590 -2.599895e+00


Can you please advise what sort of restrictions matrix I should create? I guess, I will do overidentifying restrictions for this, but any suggestion is welcome.

• You start with I have another cointegration related question. To make this meaningful, you could add links to your other question(s) (if they are relevant to this one). Otherwise you may just delete the sentence. – Richard Hardy Oct 9 '15 at 14:59
• @RichardHardy, just changed the post. – mr.rox Oct 9 '15 at 15:01
• Since you ask How will I be able to obtain this equation while accounting for other CVs i.e. ect2, ect3 and ect4., does the name of structural VAR (SVAR) ring a bell? I am trying to understand what you actually are after. (I might be wasting your time since I never really worked with SVAR models, but I am just interested in your question.) – Richard Hardy Oct 12 '15 at 19:23
• @RichardHardy, I need to run VECM as the series are co-integrated. What I need is the long run cointegrating equation, the first one of the four. The output above shows that there are four cointegrating vectors, and the normalization is default identity matrix. I need to reparameterize i.e. put some restrictions on this set of cointegrating equations so that I have the first equation with all the coefficients (as I described above). For this, I need to create a restrictions matrix, but I have failed to do it in a right way. I need help with constructing that matrix. – mr.rox Oct 13 '15 at 10:14