I have the following output from the ugarchfit
function of the “rugarch” package:
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
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GARCH Model : fGARCH(1,1)
fGARCH Sub-Model : GARCH
Mean Model : ARFIMA(1,0,1)
Distribution : ged
Optimal Parameters
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Estimate Std. Error t value Pr(>|t|)
mu 0.000000 0.000018 -0.000081 0.999935
ar1 0.027404 0.000626 43.809972 0.000000
ma1 -0.027404 0.000626 -43.805697 0.000000
omega 0.000085 0.000032 2.625465 0.008653
alpha1 0.212609 0.032685 6.504718 0.000000
beta1 0.786385 0.033784 23.276728 0.000000
shape 0.687910 0.025568 26.905141 0.000000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.000000 0.000017 -0.000083 0.999933
ar1 0.027404 0.000297 92.285201 0.000000
ma1 -0.027404 0.000297 -92.271888 0.000000
omega 0.000085 0.000089 0.955087 0.339534
alpha1 0.212609 0.059261 3.587672 0.000334
beta1 0.786385 0.084830 9.270184 0.000000
shape 0.687910 0.051326 13.402719 0.000000
LogLikelihood : 3605.227
Information Criteria
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Akaike -3.6679
Bayes -3.6480
Shibata -3.6679
Hannan-Quinn -3.6606
Weighted Ljung-Box Test on Standardized Residuals
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statistic p-value
Lag[1] 6.375 1.157e-02
Lag[2*(p+q)+(p+q)-1][5] 8.803 1.734e-10
Lag[4*(p+q)+(p+q)-1][9] 12.601 6.015e-04
d.o.f=2
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
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statistic p-value
Lag[1] 0.04776 0.827
Lag[2*(p+q)+(p+q)-1][5] 0.07166 0.999
Lag[4*(p+q)+(p+q)-1][9] 0.12958 1.000
d.o.f=2
Weighted ARCH LM Tests
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Statistic Shape Scale P-Value
ARCH Lag[3] 0.0003708 0.500 2.000 0.9846
ARCH Lag[5] 0.0475514 1.440 1.667 0.9954
ARCH Lag[7] 0.0930759 2.315 1.543 0.9995
Nyblom stability test
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Joint Statistic: 13.2604
Individual Statistics:
mu 2.9981
ar1 4.4239
ma1 4.4239
omega 0.2867
alpha1 1.9896
beta1 0.5997
shape 2.6809
Asymptotic Critical Values (10% 5% 1%)
Joint Statistic: 1.69 1.9 2.35
Individual Statistic: 0.35 0.47 0.75
Sign Bias Test
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t-value prob sig
Sign Bias 1.16912 0.2425
Negative Sign Bias 0.07816 0.9377
Positive Sign Bias 0.05444 0.9566
Joint Effect 1.68722 0.6398
Adjusted Pearson Goodness-of-Fit Test:
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group statistic p-value(g-1)
1 20 31.99 0.0313705
2 30 58.09 0.0010605
3 40 62.06 0.0108538
4 50 86.67 0.0007291
My Question is:
The Box-Ljung test on squared STZRs returns p-values close to 1 which implies independence. While the Box-Ljung test on non-squared STZRs returns p-values close to 0 which would mean serial correlation. But independence should always imply serial uncorrelation. So, how would you justify the p-values of the Box-Ljung test on non-squared STZRs?