[R-SIG-Finance] ljung-box tests in arma and garch models
Spencer Graves
spencer.graves at pdf.com
Sun Dec 30 17:01:40 CET 2007
Hi, Michal and Patrick:
PATRICK:
In your 2002 paper on the "Robustness of the Ljung-Box Test and
its Rank Equivalent"
(http://www.burns-stat.com/pages/Working/ljungbox.pdf), do you consider
using m-g degrees of freedom, where m = number of lags and g = number
of parameters estimated (ignoring an intercept)? I didn't read every
word, but I only saw you using 'm' degrees of freedom, and I did not
notice a comment on this issue.
Your Exhibit 3 (p. 7) presents a histogram of the "Distribution of
the 50-lag Ljung-Box p-vallue under the Gaussian distribution with 100
observations". It looks to me like a Beta(a, b) distribution, with a <
b < 1 but with both a and b fairly close to 1. The excess of p-values
in the lower tail suggests to me that the real degrees of freedom for a
reference chi-square should in this case be slightly greater than 50.
Your Exhibit 10 shows a comparable histogram for the "Distribution of
the Ljung-Box 15 lag p-value for the square of a t with 4 degrees of
freedom with 10,000 observations." This looks to me like a Beta(a, b)
distribution with b < a < 1 but with many fewer p-values near 0 than
near 1. This in turn suggests to me that the degrees of freedom of the
reference chi-square test would be less than 15 in this case. Apart
from this question, your power curves, Exhibits 14-22 provide rather
persuasive support for your recommended use of the rank equivalent to
the traditional Ljung-Box.
MICHAL:
Thanks very much for your further comments on this. The standard
asymptotic theory would support Enders' and Tsay's usage of m-g degrees
of freedom, with m = number of lags and g = number of parameters
estimated, apart from an intercept -- PROVIDED the parameters were
estimated using to minimize the Ljung-Box statistic. However, the
parameters are typically estimated to maximize a likelihood. The effect
of this would likely be to understate the p-value, which we generally
want to avoid.
However, we never want to use these statistics infinite sample
sizes and degrees of freedom. Therefore, the asymptotic theory is only
a guideline, preferably with some adjustment for finite sample sizes and
degrees of freedom. Therefore, it is wise to evaluate the adequacy of
the asymptotics with appropriate simulations. These may have been
done; I have not researched the literature on this, apart from Burns
(2002). If anyone knows of other relevant simulations, I'd like to hear
about them.
By the way, Tsay's second edition (2005, p. 44) includes a similar
comment: "For an AR(p) model, the Ljung-Box statistic Q(m) follows
asymptotically a chi-square distribution with m-g degrees of freedom,
where g denotes the number of AR coefficients used in the model." This
is similar to but different from your quote from the first edition.
Best Wishes,
Spencer Graves
michal miklovic 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 <mailto: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
> <http://www.mrs.umn.edu/%7Esungurea/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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