1 Mar 2011 01:42

ARCH and GARCH

Hi, everyone,
I used the data in the attachment to run ARCH (2) and GARCH (0,2), i assumed they would give me the same result but they did not.
I have the result showing in below:
ARCH (2)
arch 2 re const

Model 1: WLS (ARCH), using observations 3-7837 (n = 7835)
Dependent variable: re
Variable used as weight: 1/sigma

coefficient   std. error    t-ratio    p-value
----------------------------------------------------------
const      0.000515703   0.000109897    4.693    2.74e-06  ***

alpha(0)   8.11556e-05   6.06157e-06   13.39     1.96e-40  ***
alpha(1)   0.225945      0.0109590     20.62     5.15e-92  ***
alpha(2)   0.243682      0.0109590     22.24     2.85e-106 ***

Statistics based on the weighted data:

Sum squared resid    6676.296   S.E. of regression   0.923158
Log-likelihood      −10490.44   Akaike criterion     20982.87
Schwarz criterion    20989.84   Hannan-Quinn         20985.26

Statistics based on the original data:

Mean dependent var   0.000487   S.D. dependent var   0.012371
Sum squared resid    1.198934   S.E. of regression   0.012371

GARCH ( 0,2)
Model 2: GARCH, using observations 1-7837
Dependent variable: re
Standard errors based on Hessian

coefficient   std. error    t-ratio    p-value
----------------------------------------------------------
const      0.00104076    9.69109e-05    10.74    6.65e-27  ***

alpha(0)   4.16523e-05   1.26856e-06    32.83    1.91e-236 ***
alpha(1)   0.400826      0.0222482      18.02    1.46e-72  ***
alpha(2)   0.437669      0.0224141      19.53    6.53e-85  ***

Mean dependent var   0.000486   S.D. dependent var   0.012370
Log-likelihood       25033.25   Akaike criterion    −50056.50
Schwarz criterion   −50021.67   Hannan-Quinn        −50044.57

Unconditional error variance = 0.000257901
Likelihood ratio test for (G)ARCH terms:
Chi-square(2) = 3457.91 [0]

the Code i used is
open nasdaq.dat
arch 2 re const
garch 0 2; re

Can anyone help me explain why they are not the same?
Attachment (nasdaq.dat): application/octet-stream, 141 KiB
<div>
Hi, everyone, <br>&nbsp;&nbsp; I used the data in the attachment to run ARCH (2) and GARCH (0,2), i assumed they would give me the same result but they did not. <br>&nbsp;I have the result showing in below:<br>ARCH (2)<br>arch 2 re const<br><br>Model 1: WLS (ARCH), using observations 3-7837 (n = 7835)<br>Dependent variable: re<br>Variable used as weight: 1/sigma<br><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; coefficient&nbsp;&nbsp; std. error&nbsp;&nbsp;&nbsp; t-ratio&nbsp;&nbsp;&nbsp; p-value <br>&nbsp; ----------------------------------------------------------<br>&nbsp; const&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.000515703&nbsp;&nbsp; 0.000109897&nbsp;&nbsp;&nbsp; 4.693&nbsp;&nbsp;&nbsp; 2.74e-06&nbsp; ***<br><br>&nbsp; alpha(0)&nbsp;&nbsp; 8.11556e-05&nbsp;&nbsp; 6.06157e-06&nbsp;&nbsp; 13.39&nbsp;&nbsp;&nbsp;&nbsp; 1.96e-40&nbsp; ***<br>&nbsp; alpha(1)&nbsp;&nbsp; 0.225945&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0109590&nbsp;&nbsp;&nbsp;&nbsp; 20.62&nbsp;&nbsp;&nbsp;&nbsp; 5.15e-92&nbsp; ***<br>&nbsp; alpha(2)&nbsp;&nbsp; 0.243682&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0109590&nbsp;&nbsp;&nbsp;&nbsp; 22.24&nbsp;&nbsp;&nbsp;&nbsp; 2.85e-106 ***<br><br>Statistics based on the weighted data:<br><br>Sum squared resid&nbsp;&nbsp;&nbsp; 6676.296&nbsp;&nbsp; S.E. of regression&nbsp;&nbsp; 0.923158<br>R-squared&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.000000&nbsp;&nbsp; Adjusted R-squared&nbsp;&nbsp; 0.000000<br>Log-likelihood&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &minus;10490.44&nbsp;&nbsp; Akaike criterion&nbsp;&nbsp;&nbsp;&nbsp; 20982.87<br>Schwarz criterion&nbsp;&nbsp;&nbsp; 20989.84&nbsp;&nbsp; Hannan-Quinn&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 20985.26<br><br>Statistics based on the original data:<br><br>Mean dependent var&nbsp;&nbsp; 0.000487&nbsp;&nbsp; S.D. dependent var&nbsp;&nbsp; 0.012371<br>Sum squared resid&nbsp;&nbsp;&nbsp; 1.198934&nbsp;&nbsp; S.E. of regression&nbsp;&nbsp; 0.012371<br><br><br>GARCH ( 0,2)<br>Model 2: GARCH, using observations 1-7837<br>Dependent variable: re<br>Standard errors based on Hessian<br><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; coefficient&nbsp;&nbsp; std. error&nbsp;&nbsp;&nbsp; t-ratio&nbsp;&nbsp;&nbsp; p-value <br>&nbsp; ----------------------------------------------------------<br>&nbsp; const&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.00104076&nbsp;&nbsp;&nbsp; 9.69109e-05&nbsp;&nbsp;&nbsp; 10.74&nbsp;&nbsp;&nbsp; 6.65e-27&nbsp; ***<br><br>&nbsp; alpha(0)&nbsp;&nbsp; 4.16523e-05&nbsp;&nbsp; 1.26856e-06&nbsp;&nbsp;&nbsp; 32.83&nbsp;&nbsp;&nbsp; 1.91e-236 ***<br>&nbsp; alpha(1)&nbsp;&nbsp; 0.400826&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0222482&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 18.02&nbsp;&nbsp;&nbsp; 1.46e-72&nbsp; ***<br>&nbsp; alpha(2)&nbsp;&nbsp; 0.437669&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0224141&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 19.53&nbsp;&nbsp;&nbsp; 6.53e-85&nbsp; ***<br><br>Mean dependent var&nbsp;&nbsp; 0.000486&nbsp;&nbsp; S.D. dependent var&nbsp;&nbsp; 0.012370<br>Log-likelihood&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 25033.25&nbsp;&nbsp; Akaike criterion&nbsp;&nbsp;&nbsp; &minus;50056.50<br>Schwarz criterion&nbsp;&nbsp; &minus;50021.67&nbsp;&nbsp; Hannan-Quinn&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &minus;50044.57<br><br>Unconditional error variance = 0.000257901<br>Likelihood ratio test for (G)ARCH terms:<br>&nbsp; Chi-square(2) = 3457.91 [0]<br><br><br><br>the Code i used is <br>open nasdaq.dat<br>arch 2 re const<br>garch 0 2; re <br><br><br>Can anyone help me explain why they are not the same?<br>
</div>

1 Mar 2011 03:13

Re: ARCH and GARCH

On Tue, 1 Mar 2011, zhuhongming wrote:

>  I used the data in the attachment to run ARCH (2) and GARCH
> (0,2), i assumed they would give me the same result but they did
> not.

They don't give the same results because (a) the estimators are
completely different and (b) the model is misspecified. gretl's
"arch" command is not much recommended other than for pedagogical
purposes; it's a rather basic feasible generalized least squares
thing (as explained in the help text), while the "garch" command
does the job properly via Maximum Likelihood.

In some cases the estimates may be fairly close, but I suspect
that misspecification here has something to do with the big gap
between the sets of alpha estimates.

I say the model is misspecified because if you run a GARCH(1,1)
you'll see that it dominates GARCH(0,2): the likelihood is much
higher.

open nasdaq.dat
setobs 1 1 --special
garch 0 2 ; re const
ll02 = $lnl garch 1 1 ; re const lldiff =$lnl - ll02

Allin Cottrell

1 Mar 2011 03:55

Re: ARCH and GARCH

So should I just use garch (0 2) for arch (2) ? And assume the garch (0 2) is the right one?
Sent on the Sprint® Now Network from my BlackBerry®

-----Original Message-----
From: Allin Cottrell <cottrell@...>
Sender: gretl-users-bounces@...
Date: Mon, 28 Feb 2011 21:13:16
To: Gretl list<gretl-users@...>
Subject: Re: [Gretl-users] ARCH and GARCH

On Tue, 1 Mar 2011, zhuhongming wrote:

>  I used the data in the attachment to run ARCH (2) and GARCH
> (0,2), i assumed they would give me the same result but they did
> not.

They don't give the same results because (a) the estimators are
completely different and (b) the model is misspecified. gretl's
"arch" command is not much recommended other than for pedagogical
purposes; it's a rather basic feasible generalized least squares
thing (as explained in the help text), while the "garch" command
does the job properly via Maximum Likelihood.

In some cases the estimates may be fairly close, but I suspect
that misspecification here has something to do with the big gap
between the sets of alpha estimates.

I say the model is misspecified because if you run a GARCH(1,1)
you'll see that it dominates GARCH(0,2): the likelihood is much
higher.

open nasdaq.dat
setobs 1 1 --special
garch 0 2 ; re const
ll02 = $lnl garch 1 1 ; re const lldiff =$lnl - ll02

Allin Cottrell
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Gretl-users mailing list
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1 Mar 2011 04:51

Re: ARCH and GARCH

On Tue, 1 Mar 2011 ajzhmkaven@... wrote:

> So should I just use garch (0 2) for arch (2) ? And assume the
> garch (0 2) is the right one?

You are better using the garch command to estimate an ARCH(2)
model, but you are better still using the right model for the
data, which is surely not ARCH(2) for your NASDAQ returns. Neither
estimator will give "right" results for a wrong specification.

Allin Cottrell

brief back-story:

> On Tue, 1 Mar 2011, zhuhongming wrote:
>
> >  I used the data in the attachment to run ARCH (2) and GARCH
> > (0,2), i assumed they would give me the same result but they did
> > not.
>
> They don't give the same results because (a) the estimators are
> completely different and (b) the model is misspecified [...]

1 Mar 2011 05:40

(no subject)

a_n_i_u_s_k_a-/E1597aS9LT0CCvOHzKKcA@public.gmane.org

<div>
<div>
<div>a_n_i_u_s_k_a@...<br>
</div>
</div>
<br>

&nbsp;</div>

1 Mar 2011 07:51

Dear all:

The formulation of the Schwarz Bayesian Information Criterion is  -2  × L(θ)+ k log n   in gretl users guide.

But the gretl output of ARIMA seems cauculated by the  formulation of    -2  × L(θ)+ k ln n  .

Is it log or ln ?

Thanks a lot

<div>
Dear all:<br>&nbsp;<br>The formulation of the Schwarz Bayesian&nbsp;Information Criterion is &nbsp;-2&nbsp;&nbsp;<span lang="EN-US">&times; L(<span>&theta;</span>)+ k log n&nbsp;&nbsp; in gretl users&nbsp;guide.</span><br>&nbsp;<br>But the gretl output of ARIMA seems cauculated by the&nbsp; formulation of&nbsp;&nbsp;&nbsp;&nbsp;-2&nbsp;&nbsp;<span lang="EN-US">&times; L(<span>&theta;</span>)+ k ln n&nbsp; .</span><br><span lang="EN-US"></span>&nbsp;<br><span lang="EN-US">Is it log or ln ?</span><br><span lang="EN-US"></span>&nbsp;<br><span lang="EN-US">Thanks a lot</span><br><span lang="EN-US"></span>&nbsp;<br><span lang="EN-US"></span>&nbsp;<br>
</div>

1 Mar 2011 08:32

Box-Pierce Q statistic

Dear all:

The Q statistic to test if the series is white noise in gretl is Box-Pierce Q statistic, is it ?
But I can not find its formulation in gretl.
Is the formulation of  Q statistic the same as the reference of Box & Pierce (1970) ?

Thanks a lot

<div>
Dear all:<br>
&nbsp;<br>
The Q statistic to test if the series is white noise in gretl is Box-Pierce Q statistic, is it ?<br>
But I can not find its formulation in gretl.<br>
Is the formulation of&nbsp; Q statistic the same as the reference of Box &amp; Pierce (1970) ?<br>
&nbsp;<br>
&nbsp;Thanks a lot<br>
&nbsp;<br>
</div>

1 Mar 2011 10:40

Re: Box-Pierce Q statistic

El 01/03/11 08:32, Sam Sam escribió:
> Dear all:
>
> The Q statistic to test if the series is white noise in gretl is
> Box-Pierce Q statistic, is it ?

If you are talking about what appears in the ouput text window of the
correlogram. I think although in the help it is said to be the
Box-Pierce Q statistic, really it is the Ljung-Box statistic

If you apply an ols regression to a time series, you may test for
autocorrelation via the "test" menu. In such a case the output text
window shows a Q' estatistic, correctly described as Ljung-Box

> But I can not find its formulation in gretl.
> Is the formulation of Q statistic the same as the reference of Box &
> Pierce (1970) ?

I think gretl is using in both places the Ljung-Box statistic:
(in latex format)

Q'= T(T+2)\sum_{j=1}^M (r_j)^2/(T-j)

being M the number of coeficients you want to test, T the number of
observations and r_j the sample autocorrelation coeficient of order j.
Q' has a chi-square asymptotic distribution with M-p-q degrees of freedom.

It is asymptotically equivalent to the Box-Pierce statistic but has an
smaller bias in small samples. So I think we should always prefer to use
Ljung-Box Q' instead of Box-Pierce Q.

--

--
Ignacio Diaz-Emparanza
UPV/EHU Avda. Lehendakari Aguirre, 83 | 48015 BILBAO
T.: +34 946013732 | F.: +34 946013754
www.ea3.ehu.es

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1 Mar 2011 10:51

El 01/03/11 07:51, Sam Sam escribió:
> Dear all:
>
> The formulation of the Schwarz Bayesian Information Criterion is -2
> ×/L(θ)+ k //log n in /gretl users guide.
>
> But the gretl output of ARIMA seems cauculated by the formulation of -2
> ×/L(θ)+ k //ln n ./
> //
> /Is it log or ln ?/
> //
> /Thanks a lot/
> //
> //
>

When we talk about "likelihood" in econometrics we usually refer to
Gaussian likelihood. In such a context, using natural logs (or neperian
logs) has the advantage of simplifying the likelihood function. So, by
default, when we write 'log' we are talking about natural logs. If you
want to be more precise, you may write it 'ln', but it is very common in
the econometrics literature to write it as 'log.

In short, they are the same: natural logs.

--

--
Ignacio Diaz-Emparanza
UPV/EHU Avda. Lehendakari Aguirre, 83 | 48015 BILBAO
T.: +34 946013732 | F.: +34 946013754
www.ea3.ehu.es

_______________________________________________
Gretl-users mailing list
Gretl-users <at> lists.wfu.edu
http://lists.wfu.edu/mailman/listinfo/gretl-users
1 Mar 2011 11:41

Box-Pierce Q statistic

I am much more confused.
I estimate ARIMA by conditional maximum likelihood and choose the option /residual correlogram/
The Q statistic is Ljung-Box statistic, not Box-Pierce Q statistic. Is it ?
I do not understand. Because I choose the option /help/  and it shows it is Box-Pierce Q stastisitc.
<div>
I am much more confused.&nbsp;<br>
&nbsp;I estimate ARIMA by conditional maximum likelihood and choose the option /residual correlogram/<br>
The Q statistic is Ljung-Box statistic, not Box-Pierce Q statistic. Is it ?<br>
I do not understand. Because I choose the option /help/&nbsp; and it shows it is Box-Pierce Q stastisitc.<br>
</div>
`

Gmane