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

[Patrick Burns]

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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