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

[markleeds at verizon.net]

>From: michal miklovic <mmiklovic at yahoo.com>
>Date: 2007/12/27 Thu PM 12:21:11 CST
>To: r-sig-finance at stat.math.ethz.ch
>Subject: [R-SIG-Finance] ljung-box tests in arma and garch models

I'm replying privately because I don't want
to get abused by the geniuses on this list
in the case that I'm totally wrong but I
think you'd have to look at the derivation of the
Q statistic to know what the right df is
and i'm sure it's derived in the original
paper.

I think Box-Leung wrote a paper on
the derivation of the statistic in
the early 70's but I forget the journal.
Possibly biometrika but i can't recall.
Just google Box-Ljung and it will probably
shoot up. You're best bet
is to get the original paper.

But, here's my unofficial 2 cents  that you can
take with a grain of salt. I used
to know this stuff but it's blurry
so that's why I say take it with a grain of salt.

Conceptually, the df used should be the  number of
observations that go into the estimate of
whatever the Q statistic is trying to estimate.
Generally, I don''t the number of parameters estimates estimation
during parameter estimation should come into
play as far as what df are used in looking
up the p-value for Q. The Box test
just used all of the observation that
went into Q. Then, I think
Ljeung came along and figured out
that for small samples, you could
correct the df to get better convergence
to whatever asympototic assumption is
being made in the derivation. I foreget
what correction he/she made but it's
in any decent time series book.

Clearly, the first p+q values in the series
go unestimated but the residuals considered at
for the calculation of Q should start at
whatever the non-NA residual of the series  is ?
When the lag is on the horizontal axis
in acf plot, that denotes the number
of lags between two values in the series
and what the acf estimate was for that lag distance.
So, there's no need to not start at lag 1.
1 just represents the correlation between
the values that were were one lag apart.

yes, in the calculation of estimates and
residuals, whatever number of data points
have to be skipped but this has nothing
to do with lag(p+q) in the acf plot or
the calculation of Q().

I may not be understanding your question 
and hopefully someone else will respond with
their take on it.






















> 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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