[Gretl-devel] VECM restrictions and scaling

[Sven Schreiber]

Hi,
thanks very much for the new snapshots with T-c and restricted exogenous
variables. This makes it possible to compare directly to PcGive's
results and thus testing is quite easy.

I've done some testing with a real-world case; it's a 6-dim Vecm with a
restricted step dummy in the equilibrium (cointegration) relations.

The unrestricted estimates (cointegration rank r=4) are exactly
identical, including the standard errors. The only difference is that
gretl prints "Determinante der Kovarianzmatrix = 1.614282e-015"
(determinant of the covariance matrix), whereas pcgive reports
"-T/2log|Omega|     1873.29374". Now I guess these numbers are not
supposed to mean the same thing, but nevertheless a result of 1e-15
strikes me as strange. Maybe it's actually the last improvement of the
convergence algorithm or something like that?

---------------------------

Next, with some overidentifying restrictions on beta (both exclusion and
 more general restrictions, partly also applied to the restricted
exogenous variable, and apart from normalization), gretl's results are
not so good (I also append pcgive's results for comparison):

<gretl>
Rank of Jacobian = 34, number of free parameters = 34
Model is fully identified
Based on Jacobian, df = 2
Switching algorithm: 29552 iterations
 -(T/2)log|Omega| = 1873.9556, lldiff = 3.99749e-011

Unrestricted loglikelihood (lu) = 932.72411
Restricted loglikelihood (lr) = 928.94253
2 * (lu - lr) = 7.56317
P(Chi-Square(2) > 7.56317 = 0.0227865
</gretl>

<pcgive>
log-likelihood      932.00515  -T/2log|Omega|     1877.01821
no. of observations       111  no. of parameters         100
rank of long-run matrix     4  no. long-run restrictions   2
beta is identified

LR test of restrictions: Chi^2(2) =   1.4379 [0.4873]

Switching (scaled linear) using analytical derivatives (eps1=0.0001;
eps2=0.005):
Weak convergence
</pcgive>

Adding some restrictions on alpha, I sometimes manage to hit gretl's
bound of 50000 iterations :-)

--------------------

So I tried out something more modest: removing the restricted exogenous
variable and imposing some just-identifying but somewhat unusual
restrictions.

Then PcGive reports:
<pcgive>
log-likelihood     910.764932  -T/2log|Omega|       1855.778
no. of observations       111  no. of parameters          98
rank of long-run matrix     4  no. long-run restrictions   0
beta is identified
No restrictions imposed

Switching (scaled linear) using analytical derivatives (eps1=0.0001;
eps2=0.005):
Strong convergence
</pcgive>

But gretl's output is:
<gretl>
Rank of Jacobian = 32, number of free parameters = 32
Model is fully identified
Based on Jacobian, df = 0
Switching algorithm: 2319 iterations
 -(T/2)log|Omega| = 1855.6699, lldiff = 3.99838e-011

Unrestricted loglikelihood (lu) = 910.76493
Restricted loglikelihood (lr) = 910.65685
</gretl>

So gretl seems to have some problems with "rotations" of the
cointegration space, which in theory (and confirmed by pcgive) leave the
likelihood unaffected.

Let me know if some other details or setups would help you. I'll also
try to think about what other approaches could help.

Cheers,
sven