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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.

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The first r columns of the loading matrix gives you $\alpha$.

You can also get all the coefficients for the VECM easily with the cajorls() function:

vecm <- cajorls(rates.co)
vecm$rlm$coefficients 
vecm$beta

The VECM in matrix form is: $\Delta X_{t} = \alpha(\beta' X_{t-1} - \mu) + \Gamma_{1} \Delta X_{t-1} + \epsilon_{t}$ where $\alpha$ is the first four columns of the loading matrix and indicates how quickly the rates converge to the equilibrium relationship. $\beta$ contains the weights on the rates in the co-integrating relationships while the constant terms for the relationships go in $\mu$. $\Gamma_{1}$ is not show in the ca.jo summary output but can be shown with ca.jorls().

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  • $\begingroup$ Welcome to our site. This question remains because it has a statistical component: "how do I write my final equation for the ECM in a matrix form?" Would you have an answer to that? $\endgroup$ – whuber Feb 13 at 18:31
  • $\begingroup$ The ECM in matrix form is: $\Delta X_{t} = \alpha (\beta' X_{t-1} - \mu) + \Gamma_{1} \Delta X_{t-1} + \epsilon_{t}$ where $\alpha$ is the first 4 columns of the loading matrix and indicates how quickly the rates converge to the equilibrium relationship. $\beta$ contains the weights on the rates of the co-integrating relationships with the constant terms in $\mu$. $\Gamma_{1}$ is not shown in the ca.jo output but can be shown with ca.jorls. $\endgroup$ – Matt P Feb 14 at 13:43
  • $\begingroup$ Please edit your post to include your answer. That will make it much more useful and clear. $\endgroup$ – whuber Feb 14 at 15:36
  • $\begingroup$ Thank you for the answer, gladly I figured it out eventually and your answer matches my result $\endgroup$ – Makina Feb 15 at 10:26

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