I am starting with statistics and I am facing an issue with an ARMA-eGARCH model.
Im having an issue with the interpretation of the gamma coefficient from the fit of ARMA(1,2)-eGARCH(2,1) using the rugarch package.
As I know the gamma parameter is the leverage and when gamma is negative it means that the model has leverage effect, but the problem is here in this model I have two gamma parameters and one is negative but the other is positive.
How can I interpret this case?
Here is the model fit.
Thanks for your help!
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
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GARCH Model : eGARCH(2,1)
Mean Model : ARFIMA(1,0,2)
Distribution : std
Optimal Parameters
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Estimate Std. Error t value Pr(>|t|)
mu 0.002335 0.000586 3.9845e+00 0.000068
ar1 0.984861 0.004176 2.3583e+02 0.000000
ma1 -1.039860 0.000004 -2.6259e+05 0.000000
ma2 0.062858 0.002236 2.8106e+01 0.000000
omega -0.027390 0.009831 -2.7860e+00 0.005336
alpha1 -0.036038 0.048493 -7.4316e-01 0.457383
alpha2 0.092501 0.048862 1.8931e+00 0.058344
beta1 0.995196 0.001996 4.9852e+02 0.000000
gamma1 0.345047 0.061117 5.6457e+00 0.000000
gamma2 -0.093125 0.058722 -1.5859e+00 0.112770
shape 2.533085 0.146629 1.7276e+01 0.000000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.002335 0.000570 4.1000e+00 0.000041
ar1 0.984861 0.004516 2.1809e+02 0.000000
ma1 -1.039860 0.000003 -4.0642e+05 0.000000
ma2 0.062858 0.002186 2.8754e+01 0.000000
omega -0.027390 0.015389 -1.7799e+00 0.075091
alpha1 -0.036038 0.048691 -7.4014e-01 0.459217
alpha2 0.092501 0.047530 1.9462e+00 0.051636
beta1 0.995196 0.002408 4.1322e+02 0.000000
gamma1 0.345047 0.058689 5.8792e+00 0.000000
gamma2 -0.093125 0.057076 -1.6316e+00 0.102767
shape 2.533085 0.150352 1.6848e+01 0.000000
LogLikelihood : 5007.332
Information Criteria
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Akaike -4.1004
Bayes -4.0742
Shibata -4.1004
Hannan-Quinn -4.0909
Weighted Ljung-Box Test on Standardized Residuals
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statistic p-value
Lag[1] 13.66 2.196e-04
Lag[2*(p+q)+(p+q)-1][8] 21.16 0.000e+00
Lag[4*(p+q)+(p+q)-1][14] 27.59 7.616e-10
d.o.f=3
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1] 0.01507 0.9023
Lag[2*(p+q)+(p+q)-1][8] 3.28506 0.6332
Lag[4*(p+q)+(p+q)-1][14] 4.72452 0.8028
d.o.f=3
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[4] 0.2978 0.500 2.000 0.5853
ARCH Lag[6] 4.7944 1.461 1.711 0.1259
ARCH Lag[8] 5.0893 2.368 1.583 0.2395
Nyblom stability test
------------------------------------
Joint Statistic: 4.1596
Individual Statistics:
mu 0.16397
ar1 0.07774
ma1 0.06603
ma2 0.05386
omega 0.57411
alpha1 1.27031
alpha2 1.05386
beta1 0.56587
gamma1 0.73976
gamma2 0.33343
shape 0.15746
Asymptotic Critical Values (10% 5% 1%)
Joint Statistic: 2.49 2.75 3.27
Individual Statistic: 0.35 0.47 0.75
Sign Bias Test
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t-value prob sig
Sign Bias 0.9524 0.3410
Negative Sign Bias 1.4660 0.1428
Positive Sign Bias 0.6865 0.4925
Joint Effect 3.3397 0.3422
Adjusted Pearson Goodness-of-Fit Test:
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group statistic p-value(g-1)
1 20 24.02 0.1955
2 30 33.45 0.2599
3 40 45.94 0.2067
4 50 51.08 0.3919
Elapsed time : 0.820385