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I'm a beginner at Econometrics, and I'm trying to learn the main econometric techniques in R. My doubt is on how to normalize the cointegration matrix to ensure that the coefficient of the cointegration relations is 1 from the output of the ca.jo function for Johansen procedure.

I simulated a multivariate process with 4 series, which 2 are cointegration relations and 2 are common trend:

sigma<-diag(0.5,4)
u<-mvrnorm(n=250,mu=c(0,0,0,0),sigma)
ut1<-arima.sim(n=250,list(order=c(1,0,0),ar=c(0.3)),innov=u[,1])
ut2<-arima.sim(n=250,list(order=c(1,0,0),ar=c(0.5)),innov=u[,2])
y2<-cumsum(u[,3]) ##trend (random walk)
y3<-cumsum(u[,4]) ##trend (random walk)
y1<-0.2*y2+ut1  ##cointegration relation
y4<-0.8*y3+ut2  ##cointegration relation

{ts.plot(y1,y2,y3,y4,col=c(1:4),main="Simulazione di un sistema con 2 relazioni di cointegrazione")
legend("topleft",legend=c("y1-cointegrata1","y2-common_trend1","y3-common_trend2","cointegrata2"),pch=15,col=c(1:4))}

Then for testing the presence of 0.2 and 0.8 of my relations in the cointegration matrix, I use ca.jo, this is the output of interest:

xx<-ca.jo(cbind(y1,y2,y3,y4),type="eigen",ecdet="none",K=2,spec="transitory")

xx@Vorg
           y1.l1       y2.l1       y3.l1        y4.l1
y1.l1  1.4948016  0.36787213 -0.02461994 -0.040606696
y2.l1 -0.2951314 -0.07362746  0.05016916  0.362012720
y3.l1 -0.1873146  1.07028673  0.59184042  0.021477244
y4.l1  0.2217581 -1.40606285 -0.05629814  0.009461934

As I understand, now I have to normalize the parameters for the cointegration relations that are y1 and y4, I do in this way:

X<-xx@Vorg
beta.y1 <- X[, 1]/X[1, 1]
beta.y1
     y1.l1      y2.l1      y3.l1      y4.l1 
 1.0000000 -0.1974385 -0.1253107  0.1483528

As I expected for y2.l1 there's a value really close to -0.2...reasoning in the same way for y4 I expected to find "my" -0.8:

beta.y4 <- X[, 4]/X[4, 4]
beta.y4
    y1.l1     y2.l1     y3.l1     y4.l1 
-4.291585 38.259906  2.269858  1.000000 

Instead there isn't -0.8 -- I find it is like this:

beta.y4 <- X[, 2]/X[4, 2]
beta.y4
      y1.l1       y2.l1       y3.l1       y4.l1 
-0.26163278  0.05236427 -0.76119409  1.00000000

-0.76 for y3.l1. I cannot understand why for the first relation, I divided the first column of the matrix for the element [1,1], while for the second relation I have to divided the second column (that is the common trend) and not the fourth.

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    $\begingroup$ Good question! Apparently, the ordering of the eigenvectors in xx@Vorg is different from what you expect. Maybe there is a reason for it? Perhaps they are ordered by the size of the corresponding eigenvalues and the corresponding eigenvalues of these two vectors are the largest (smallest?). Is there some reason behind that? (By the way, xx@V should give you the matrix of eigenvectors normalised with respect to the first variable.) I look forward to a good answer. If you find one by yourself, please post it. Thanks! $\endgroup$ – Richard Hardy Apr 23 '15 at 18:47
  • $\begingroup$ That 's what I thought too. Probably the function automatically puts the found cointegration relations as the first columns of the matrix. $\endgroup$ – Giacomo Rosaspina Apr 23 '15 at 21:30

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