I'm struggling with the interpretation of one of my course examples.
Here is the R output
library(urca)
rates.co <- ca.jo(data_rate, type = "eigen", ecdet = "const", K = 2, spec = 'transitory', season = NULL)
summary(rates.co)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.839734e-01 1.863156e-01 1.390916e-01 7.538804e-02 3.502591e-02
## [6] -5.514428e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 7.56 7.52 9.24 12.97
## r <= 3 | 16.62 13.75 15.67 20.20
## r <= 2 | 31.75 19.77 22.00 26.81
## r <= 1 | 43.71 25.56 28.14 33.24
## r = 0 | 70.82 31.66 34.40 39.79
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## cofi.l1 X1ycmt.l1 X5ycmt.l1 primeRate.l1 X3mTbill.l1
## cofi.l1 1.000000 1.0000000 1.0000000 1.000000 1.00000000
## X1ycmt.l1 -5.016228 2.8868596 2.3696681 1.613084 2.44060485
## X5ycmt.l1 1.447589 -0.9385606 -1.3010429 -1.864387 -3.90107940
## primeRate.l1 1.798685 0.2357253 0.9647242 -3.533503 -2.05938244
## X3mTbill.l1 1.274463 -3.3546732 -3.1195849 2.575369 -0.04422163
## constant -7.582615 -0.2506138 -2.4391828 13.674437 20.63481751
## constant
## cofi.l1 1.0000000
## X1ycmt.l1 -0.9501416
## X5ycmt.l1 0.6813733
## primeRate.l1 -2.7068548
## X3mTbill.l1 2.2016130
## constant 3.7872633
##
## Weights W:
## (This is the loading matrix)
##
## cofi.l1 X1ycmt.l1 X5ycmt.l1 primeRate.l1
## cofi.d -0.03002766 -0.025996453 0.001132855 -0.005679920
## X1ycmt.d -0.02895731 0.034635064 -0.021423063 -0.005731646
## X5ycmt.d -0.02949078 0.000379748 0.041174443 0.018020526
## primeRate.d -0.04390107 0.028939965 -0.066368855 0.007284998
## X3mTbill.d -0.04602132 0.113886433 -0.017776290 -0.011264136
## X3mTbill.l1 constant
## cofi.d -2.078558e-05 4.181518e-17
## X1ycmt.d 9.894523e-03 1.014418e-16
## X5ycmt.d 1.145797e-02 -4.228840e-16
## primeRate.d 5.813215e-04 1.985921e-16
## X3mTbill.d 5.261185e-03 1.341192e-16
library(tsDyn)
vecm<-VECM(data_rate, lag=2-1, include = 'const', estim = 'ML', r=4)
summary(vecm)
## #############
## ###Model VECM
## #############
## Full sample size: 214 End sample size: 212
## Number of variables: 5 Number of estimated slope parameters 50
## AIC -4907.008 BIC -4725.753 SSR 26.22101
## Cointegrating vector (estimated by ML):
## cofi 1ycmt 5ycmt primeRate 3mTbill
## r1 1.000000e+00 -8.326673e-17 0.000000e+00 1.110223e-16 -0.9659268
## r2 -2.081668e-16 1.000000e+00 -1.110223e-16 -5.551115e-17 -1.0839784
## r3 -2.775558e-17 1.110223e-16 1.000000e+00 0.000000e+00 -1.0269771
## r4 5.551115e-17 0.000000e+00 5.551115e-17 1.000000e+00 -0.9514607
##
##
## ECT1 ECT2
## Equation cofi -0.0606(0.0138)*** 0.0690(0.0404).
## Equation 1ycmt -0.0198(0.0517) 0.1848(0.1518)
## Equation 5ycmt 0.0320(0.0623) 0.2752(0.1830)
## Equation primeRate -0.0738(0.0295)* 0.1585(0.0867).
## Equation 3mTbill 0.0397(0.0405) 0.4991(0.1190)***
## ECT3 ECT4
## Equation cofi -0.0097(0.0160) -0.0389(0.0162)*
## Equation 1ycmt -0.0362(0.0604) -0.0501(0.0610)
## Equation 5ycmt -0.1307(0.0727). -0.0836(0.0735)
## Equation primeRate -0.0184(0.0345) -0.1624(0.0348)***
## Equation 3mTbill -0.1295(0.0473)** -0.0363(0.0478)
## Intercept cofi -1
## Equation cofi 0.1527(0.0593)* 0.2211(0.0647)***
## Equation 1ycmt 0.2005(0.2230) -0.0020(0.2434)
## Equation 5ycmt 0.3884(0.2687) -0.2470(0.2933)
## Equation primeRate 0.5904(0.1273)*** 0.1548(0.1390)
## Equation 3mTbill 0.2179(0.1747) 0.1439(0.1907)
## 1ycmt -1 5ycmt -1
## Equation cofi -0.0422(0.0569) 0.0006(0.0319)
## Equation 1ycmt -0.0086(0.2141) 0.2610(0.1200)*
## Equation 5ycmt 0.1253(0.2580) 0.3309(0.1447)*
## Equation primeRate -0.0946(0.1223) 0.0696(0.0686)
## Equation 3mTbill -0.1980(0.1678) 0.1649(0.0941).
## primeRate -1 3mTbill -1
## Equation cofi 0.0630(0.0329). 0.0856(0.0494).
## Equation 1ycmt -0.0642(0.1237) 0.1667(0.1857)
## Equation 5ycmt -0.0211(0.1491) -0.2666(0.2239)
## Equation primeRate -0.0156(0.0707) 0.3743(0.1061)***
## Equation 3mTbill -0.0234(0.0970) 0.3131(0.1456)*
round(vecm$model.specific$coint, 3)
## r1 r2 r3 r4
## cofi 1.000 0.000 0.000 0.000
## 1ycmt 0.000 1.000 0.000 0.000
## 5ycmt 0.000 0.000 1.000 0.000
## primeRate 0.000 0.000 0.000 1.000
## 3mTbill -0.966 -1.084 -1.027 -0.951
So from what I understood, I got 4 cointegrating vectors since my test rejected $r \leq 3$ and accepted $r \leq 4$ at 5% critical level.
At the very end I have 4 cointegrating vectors written in a column, which, from what I understood, form a matrix for $\beta$ in the VECM.
What I do not understand though is how to derive the $\alpha$ matrix from this output or, generally speaking, how do I write my final equation for the ECM in a matrix form.