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I'm using the URCA package in R to test for cointegration by Johansen's method. Can anyone tell me what the weights (loading matrix) tells me or what I would use it for? I understand what the critical value, test statistic and eigenvectors are for but confused by the loading matrix.

Here's an example I am working with:

library(urca)
da=read.table("m-bnd.txt")
aaa=da[,4]
baa=da[,5]

x=cbind(aaa,baa)
colnames(x) <- c("aaa","baa")
coeig=ca.jo(x,ecdet="const",type="eigen",K=7,spec="transitory")

summary(coeig)

The output is:

Test type: maximal eigenvalue statistic (lambda max),
without linear trend and constant in cointegration
Eigenvalues (lambda): [1] 3.963354e-02 5.378046e-03 1.853392e-19

Values of teststatistic and critical values of test:

test 10pct 5pct 1pct
r <= 1 | 3.25 7.52 9.24 12.97
r = 0 | 24.35 13.75 15.67 20.20

Eigenvectors, normalised to first column:
(These are the cointegration relations)
aaa.l1 baa.l1 constant
aaa.l1 1.00000000 1.0000000 1.0000000
baa.l1 -0.89285894 0.6561369 -0.1770799
constant 0.05492912 -13.9106270 4.6994175

Weights W:
(This is the loading matrix)

aaa.l1 baa.l1 constant
aaa.d -0.006761805 -0.003055965 1.480155e-19
baa.d 0.066353797 -0.002469648 -1.113343e-17
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Loading matrix is the adjustment matrix ($α$ matrix). The elements of $α$ determine the speed of adjustment to the long-run equilibrium.

Please see p.4 of this article to understand the relationship between adjustment matrix and cointegrating vector.

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  • $\begingroup$ So how would I actually use the adjustment matrix in the example above to determine the the time it takes to reach long run equilibrium ? If the cointegrating relationship is y = .05492+ aaa-.8929*bbb, then how would I use W to determine the time ? $\endgroup$
    – user1408304
    Aug 25, 2013 at 17:05
  • $\begingroup$ Please refer to the Time Series Analysis by James Hamilton for details on adjustment matrix. $\endgroup$
    – Metrics
    Aug 25, 2013 at 17:37
  • $\begingroup$ Anywhere else I could look for an example besides buying a $70 book ? $\endgroup$
    – user1408304
    Aug 25, 2013 at 18:23
  • $\begingroup$ Of course, you can look at here to understand that.In particular, try to relate α with A in Section "Tests on Adjustment Speeds" $\endgroup$
    – Metrics
    Aug 25, 2013 at 18:30
  • $\begingroup$ Still having trouble understanding how I take the loading matrix and determine the number of time periods required to adjust to the long run average if my cointegrating relationship is :y = .05492+ aaa-.8929*bbb. Assuming the loading matrix and example as shown above. $\endgroup$
    – user1408304
    Sep 22, 2013 at 17:40

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