VECM - Johansen test and derivation of Cointegration Ecuation.

[John Riveros]

Greetings again Stata Users. I'm here with a probably dumb question, but stills a question.

How to derive from VECM regression using vec var1 var2 the cointegration equation?. (also yes, i've checked non stationary of log_pib_real and log_gp in levels, stationary in first differences, Cointegration test using Johansen at lags(7) Johansen results at the end of the VECM follows as:

Johansen normalization restriction imposed
            
beta                   Coef.   Std. Err.      z    P>z    [95% Conf.    Interval]
            
_ce1         
log_pib_real           1          .        .    .    .    .
log_gp            -.6103451    .111848    -5.46    0.000    -.8295631    -.3911271
_cons             -14.44061          .        .    .    .    .

I'm thinking the cointegration ecuation is:

log_pib_real = 14.44061 + 0.6103451*log_gp 

but i'm not sure of it. I need some confirmation over here!!!!

Also i'm putting the VECM as well downhere.

vec log_pib_real log_gp, lags(7) trend(constant) rank(1)

Vector error-correction model

Sample:  1997 - 2017                            Number of obs     =         21
                                                AIC               =  -7.425222
Log likelihood =  106.9648                      HQIC              =  -7.112176
Det(Sigma_ml)  =  1.29e-07                      SBIC              =  -5.982786

Equation           Parms      RMSE     R-sq      chi2     P>chi2
----------------------------------------------------------------
D_log_pib_real       14     .052562   0.7921   26.67022   0.0212
D_log_gp             14     .065114   0.8071   29.27988   0.0096
----------------------------------------------------------------

--------------------------------------------------------------------------------
               |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
D_log_pib_real |
          _ce1 |
           L1. |  -.6154371   .2616498    -2.35   0.019    -1.128261    -.102613
               |
  log_pib_real |
           LD. |  -.2222438   .3793556    -0.59   0.558    -.9657671    .5212795
          L2D. |  -.0054909   .3307023    -0.02   0.987    -.6536554    .6426737
          L3D. |   .2496612   .3029083     0.82   0.410    -.3440281    .8433504
          L4D. |   .4396373   .3784581     1.16   0.245    -.3021269    1.181402
          L5D. |   .4052015   .3444006     1.18   0.239    -.2698113    1.080214
          L6D. |   .5882393   .4339737     1.36   0.175    -.2623336    1.438812
               |
        log_gp |
           LD. |   .3232428   .3215655     1.01   0.315     -.307014    .9534997
          L2D. |   .0584609    .255566     0.23   0.819    -.4424392     .559361
          L3D. |  -.2284644   .1991852    -1.15   0.251    -.6188601    .1619314
          L4D. |  -.5250229   .2567672    -2.04   0.041    -1.028277   -.0217684
          L5D. |  -.6069792   .3440342    -1.76   0.078    -1.281274    .0673155
          L6D. |  -.2024935   .2530504    -0.80   0.424    -.6984632    .2934761
               |
         _cons |  -.0279986   .0513774    -0.54   0.586    -.1286964    .0726992
---------------+----------------------------------------------------------------
D_log_gp       |
          _ce1 |
           L1. |   -.493292   .3241343    -1.52   0.128    -1.128583    .1419996
               |
  log_pib_real |
           LD. |   .2004946   .4699494     0.43   0.670    -.7205892    1.121578
          L2D. |  -.3279454   .4096772    -0.80   0.423    -1.130898    .4750071
          L3D. |  -.4937168   .3752456    -1.32   0.188    -1.229185    .2417511
          L4D. |   .5011307   .4688375     1.07   0.285    -.4177739    1.420035
          L5D. |    .098285   .4266468     0.23   0.818    -.7379273    .9344974
          L6D. |   .5788976   .5376108     1.08   0.282    -.4748001    1.632595
               |
        log_gp |
           LD. |  -.1155646   .3983584    -0.29   0.772    -.8963328    .6652036
          L2D. |   .1093424   .3165976     0.35   0.730    -.5111775    .7298622
          L3D. |   .2351203   .2467525     0.95   0.341    -.2485057    .7187462
          L4D. |  -.1739218   .3180857    -0.55   0.585    -.7973583    .4495146
          L5D. |  -.5374406   .4261929    -1.26   0.207    -1.372763    .2978821
          L6D. |  -.5157345   .3134812    -1.65   0.100    -1.130146    .0986774
               |
         _cons |   .0349314   .0636468     0.55   0.583     -.089814    .1596768
--------------------------------------------------------------------------------

Cointegrating equations

Equation           Parms    chi2     P>chi2
-------------------------------------------
_ce1                  1   29.77794   0.0000
-------------------------------------------

Identification:  beta is exactly identified

                 Johansen normalization restriction imposed
------------------------------------------------------------------------------
        beta |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_ce1         |
log_pib_real |          1          .        .       .            .           .
      log_gp |  -.6103451    .111848    -5.46   0.000    -.8295631   -.3911271
       _cons |  -14.44061          .        .       .            .           .
------------------------------------------------------------------------------

A final question, since there are not significant coefficients of short-run causality. The only causality above is that log_gp (logarithm of public speding) causes positive long-term causality over log of GDP (log_pib) or is it backwards. ?

[John Riveros]

Well, i've just found in page 6 of the help vec command the example i was looking for. ( https://www.stata.com/manuals13/tsvec.pdf )

cointegration ecuation in my case is

log_pib_real -0.6103451*log_gp -14.44061 = 0 (i assume it equals 0).