Concerning mle, I like to test and check results independently and I came across
a problem.

I can't send all details because this is research in progress and my colleague
did not give green light to do so. But I'll describe what is going on as best as I can.

I enlarged IGARCH process with additional variable call it gamma1
so that alpha+beta+gamma1=1, and I did 3D plot in Mathematica for
log like. function and found maximum to be at

beta=0.912565
gamma1=0.0106716

I rechecked it with the FORTRAN and result was the same,
since this was the only maximum. Then I run modified
script from the mentioned example in the guide. Gretl produced
the following result for initial gamma1=0.01

for --robust --lbfgs options:

             estimate    std. error   t-ratio    p-value
-------------------------------------------------------
beta       0,916221    0,0350035    26,18     5,10e-151 ***
gamma1     0,0175130   0,0164981     1,062    0,2885

----------

for --lbfgs only:

Tolerance = 1e-014
Function evaluations: 16
Evaluations of gradient: 16

Model 3: ML, using observations 1-15073
ll = check ? -0.5*(log(h) + (ep^2)/h) : NA
Standard errors based on Outer Products matrix

             estimate    std. error    t-ratio    p-value
--------------------------------------------------------
beta       0,916221    0,00147268    622,1     0,0000    ***
gamma1     0,0175130   0,000790365    22,16    8,71e-109 ***

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

with default mle options

Tolerance = 1e-014
Function evaluations: 81
Evaluations of gradient: 14

Model 5: ML, using observations 1-15073
ll = check ? -0.5*(log(h) + (ep^2)/h) : NA
Standard errors based on Outer Products matrix

              estimate    std. error    t-ratio    p-value
---------------------------------------------------------
beta       0,932768     0,00109127    854,8     0,0000    ***
gamma1     0,00513404   0,000340124    15,09    1,76e-051 ***

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

if I changed initial gamma1 results varied a lot. Imposed conditions were

scalar check = (gamma1>0) && (beta>0) && ((beta+gamma1)<1)

Can anyone supply explicit code to determine GARCH(1,1) coefficients
with different method (OLS, MLE via R ...) for the simple example from
the user guide.

open djclose
series y = 100*ldiff(djclose)
scalar mu = 0.0
scalar omega = 1
scalar alpha = 0.4
scalar beta = 0.0
mle ll = check ? (-0.5*(log(h) + (e²)/h)) : NA
series e = y - mu
series h = var(y)
series h = omega + alpha*(e(-1))² + beta*h(-1)
scalar check = (alpha>0) && (beta>0)
params mu omega alpha beta
end mle

Thanks really a lot, any help is appreciated since
I really like gretl as a research tool.

All the best,
Davor

On 30.11.2009. 8:58, Riccardo (Jack) Lucchetti wrote:

On Sun, 29 Nov 2009, Allin Cottrell wrote:

;-) This is a subtle bug (if it's a bug; I'm not sure) with an
easy workaround. With the initialization

scalar alpha = 0.4
scalar beta = 0

The "check" condition -- alpha>0 && beta>0 -- is violated on the
first iteration. Therefore the formula for "ll" comes down to

ll = NA

This generates a scalar value, which is not allowed in the mle
context. The fix is to initialize such that the check is
satisfied on the first round, e.g.

scalar beta = 0.001

Not really relevant here, but the constraints on alpha and beta in a garch(1,1) model are not exactly the same. Alpha needs to be strictly positive (otherwise the model is unidentified), but beta can be 0, so initialising it at 0 is perfectly legitimate. The check should be written as (alpha>0) && (beta>=0). Sorry for being pedantic.

Riccardo (Jack) Lucchetti
Dipartimento di Economia
Università Politecnica delle Marche

r.lucchetti@univpm.it
http://www.econ.univpm.it/lucchetti
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