ljung-box tests in arma and garch models
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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