Hi,

First, I would like to thank Patrick and Spencer for their comments and suggestions.

Second,
I did a literature search on the computation of degrees of freedom for
the Ljung-Box Q-statistic when testing residuals from an arma model. I
do not mean an optimum number of lags for the ACF or the LB Q-statistic
but I tried to find an answer to the question: how do I determine
degrees of freedom for a given LB Q-statistic from an arma(p,q) model?
Enders
states the following in Applied Econometric Time Series (2nd edition,
2004, Wiley & Sons) on pp. 68 - 69: "The Box-Pierce and Ljung-Box
Q-statistics also serve as a check to see if the residuals from an
estimated arma(p,q) model behave as a white noise process. However,
when the s correlations from an estimated arma(p,q) model are formed,
the degrees of freedom are reduced by the number of estimated
coefficients. Hence, using the residuals of an arma(p,q) model, Q has a
chi-squared [distribution] with s - p - q degrees of freedom."
Tsay
states the following in Analysis of Financial Time Series (1st edition,
2002, Wiley & Sons) on p. 52: "The Ljung-Box statistics of the
residuals can be used to check the adequacy of a fitted model. If the
model is correctly specified, then Q(m) follows asymptotically a
chi-squared distribution with m - g degrees of freedom, where g denotes
the number of parameters used in the model."

The two above
quotations are in line with mine and Spencer's opinions. Considering
what the books say, I would suggest that the computation of the degrees
of freedom and, consequently, p-values could be altered in the next
release of fArma and fGarch.

I did not find any exact
formulations concerning the computation of degrees of freedom for the
LB Q-statistics when testing squared standardised residuals from an
estimated garch model.

Best regards

Michal Miklovic



----- Original Message ----
From: Patrick Burns <patrick@burns-stat.com>
To: Spencer Graves <spencer.graves@pdf.com>
Cc: michal miklovic <mmiklovic@yahoo.com>; r-sig-finance@stat.math.ethz.ch
Sent: Friday, December 28, 2007 11:21:33 AM
Subject: Re: [R-SIG-Finance] ljung-box tests in arma and garch models


I heartily agree with Spencer that a simulation is the
way to answer the question.  However, my intuition is
the opposite of Spencer's regarding what the answer
will be.

The Burns Statistics working paper on Ljung-Box tests
makes it clear that using rank tests for testing the garch
adequacy will be much more important than messing with
the degrees of freedom.


Patrick Burns
patrick@burns-stat.com
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")

Spencer Graves wrote:

>Dear Michal:
>
>      The best way to check something like this is to do a simulation,
>tailored to your application.  If you do such, I'd like to hear the
>results.
>
>      Absent that, my gut reaction is to agree with you.  The
 chi-square
>distribution with k degrees of freedom is defined as distribution of
 the
>sum of squares of k independent N(0, 1) variates
>(http://en.wikipedia.org/wiki/Chi-square_distribution).  In 1900, Karl
>Pearson published "On the criterion that a given system of deviations
>from the probable in the case of a correlated system of variables is
>such that it can be reasonably supposed to have arisen from random
>sampling", Philosophical magazine, t.50
>(http://fr.wikipedia.org/wiki/Karl_Pearson).  In this test, Pearson
>assumed that the sums of squares of k N(0, 1) variates, independent or
>not, would follow a chi-square(k).  R. A. Fisher determined that the
>number of degrees of freedom should be reduced by the number of
>parameters estimated
>(http://www.mrs.umn.edu/~sungurea/introstat/history/w98/RAFisher.html).
>This led to a feud that continued after Pearson died.
>
>      The "Box-Pierce" and "Ljung-Box" tests are both available in
>'Box.test{stats}' and discussed in Tsay (2005) Analysis of "financial
>Time Series (Wiley, p. 27), which includes a comment that, "Simulation
>studies suggest that the choice of" the number of lags included in the
>Ljung-Box statistic should be roughly log(number of observations) for
>"better power performance."
>
>       Based on this, the "FinTS" package includes a function "ARIMA"
>that calls "arima", computes Box.test on the residuals and adjusts the
>number of degrees of freedom to match the examples in Tsay (2005).  I
>haven't looked at this in depth, but it would seem to conform with
>Eviews, etc., and not with fArma, etc., as you mentioned.
>
>      I haven't done a substantive literature search on this, but if
>anyone has evidence bearing on this issue beyond the original
 Ljung-Box
>paper, I'd like to know.
>
>      Hope this helps.
>      Spencer Graves
>
>michal miklovic wrote:
>  
>
>> Hi,
>>
>>I would like to ask/clarify how should degrees of freedom (and
 p-values) for the Ljung-Box Q-statistics in arma and garch models be
 computed. The reason for the question is that I have encountered two different
 approaches. Let us say we have an arma(p,q) garch(m,n) model. The two
 approaches are as follows:
>>
>>1) In R and fArma and fGarch packages, the arma and garch orders are
 disregarded in the computation of degrees of freedom for the Ljung-Box
 (LB) Q-statistics. In other words, regardless of p, q, m and n, the LB
 Q-statistic computed from the first x autocorrelations of (squared)
 standardised residuals has x degrees of freedom. Given the statistic and
 degrees of freedom, the corresponding p-value is computed.
>>
>>2) In EViews, TSP and other statistical software, the LB Q-statistic
 computed from the first x autocorrelations of standardised residuals
 has (x - (p+q)) degrees of freedom. Degrees of freedom and p-values are
 not computed for the first (p+q) LB Q-statistics. A similar method is
 applied to squared standardised residuals: the LB Q-statistic computed
 from the first x autocorrelations
>>of squared standardised residuals has (x - (m+n)) degrees of freedom.
>>Degrees of freedom and p-values are not computed for the first (m+n)
 LB
>>Q-statistics.
>>
>>I think the second approach is better because the first (p+q) orders
 in standardised residuals and the first (m+n) orders in squared
 standardised residuals should not exhibit any pattern and higher orders should
 be checked for any remaining arma and garch structures. Am I right or
 wrong?
>>
>>Thanks for answers and suggestions.
>>
>>Best regards
>>
>>Michal Miklovic
>>
>>
>>
>>
>>
>>    
  ____________________________________________________________________________________
>>Be a better friend, newshound, and 
>>
>>
>>    [[alternative HTML version deleted]]
>>
>>_______________________________________________
>>R-SIG-Finance@stat.math.ethz.ch mailing list
>>https://stat.ethz.ch/mailman/listinfo/r-sig-finance
>>-- Subscriber-posting only. 
>>-- If you want to post, subscribe first.
>>
>>    
>>
>
>_______________________________________________
>R-SIG-Finance@stat.math.ethz.ch mailing list
>https://stat.ethz.ch/mailman/listinfo/r-sig-finance
>-- Subscriber-posting only. 
>-- If you want to post, subscribe first.
>
>
>  
>






      ____________________________________________________________________________________
Never miss a thing.  Make Yahoo your home page. 

	[[alternative HTML version deleted]]

