[R-SIG-Finance] ljung-box tests in arma and garch models

Sun Dec 30 19:54:23 CET 2007

For a proof that the appropriate degrees of freedom is s-p-q see
Brockwell and Davis (1990), Time Series: Theory and Methods, 2nd
Edition, Springer, page 310.

John Frain

On 30/12/2007, michal miklovic <mmiklovic at yahoo.com> wrote:
> 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 at burns-stat.com>
> To: Spencer Graves <spencer.graves at pdf.com>
> Cc: michal miklovic <mmiklovic at yahoo.com>; r-sig-finance at 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 at 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
> >>
> >>
> >>
> >>
> >>
> >>
>   ____________________________________________________________________________________
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> >>
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-- 
John C Frain
Trinity College Dublin
Dublin 2
Ireland
www.tcd.ie/Economics/staff/frainj/home.htm
mailto:frainj at tcd.ie
mailto:frainj at gmail.com