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## RETURN KURTOSIS EXPGARCH(1,1) VERSUS EXPGARCH(1,1)+AGGVOL

Normal model for expGARCH(1,1) or expGARCH(1,1)+AggVol produces kurtosis that is too small.  Recall that these models are

$r_t = e^{h_t/2} z_t$

where $z_t \sim N(0,1)$.  An empirical approach to solve the problem is to assume the model with some Student-t distributed $z_t$.  The approach that is coarse but sensible is to estimate the parameter (df) for the t-distribution from the raw returns and check the corresponding kurtosis for our two models by Monte Carlo simulations.  This coarse result produces more reasonable kurtosis figures which are still quite innacurate but at least the order of magnitudes are sane comparisons.  We then do our train-test sequence for different stocks comparing the actual kurtosis versus kurtosis produced by the expGARCH(1,1) and expGARCH(1,1)+AggVol with iid t samples with the measured df.  Averaging the difference in errors versus actual kurtosis gives us a coarse measure of which model is capturing kurtosis of actual returns better in-sample.

As an example, we see below that the kurtosis figures $k_{G11}-k_{act}$ and $k_{G11AV}-k_{act}$ with the average of difference of absolute values of these statistics so that negative values imply that expGARCH(1,1) with t-distribution produces better match to actual kurtosis than expGARCH(1,1)+AggVol with t-distribution.

```Return kurtosis 15.3291056496
Kurtosis errors:  -3.47780449225 -0.609265872865
Return kurtosis 17.9497940216
Kurtosis errors:  0.318873311637 -11.1676913635
Return kurtosis 10.6588967775
Kurtosis errors:  -0.980294038525 -2.47608614042
Return kurtosis 12.052305201
Kurtosis errors:  -2.70003353438 4.21989694377
Return kurtosis 11.082836272
Kurtosis errors:  -0.758373791903 1.14740951543
Return kurtosis 11.7122002785
Kurtosis errors:  1.32542540808 7.97038366158
Return kurtosis 11.2723067977
Kurtosis errors:  -1.84254159079 -2.05518402196
Stock:  ICUI MSEdiff= -0.164785748195 MADdiff= -0.0160051519559 Ktdiff= -1.22593961898
Return kurtosis 1.48307130039
Kurtosis errors:  0.606827065855 1.35062809441
Return kurtosis 1.38838129542
Kurtosis errors:  1.01335394892 1.66287232558
Return kurtosis 1.69570845626
Kurtosis errors:  1.437650289 1.44564623289
Return kurtosis 5.38295125445
Kurtosis errors:  -1.64668610603 -0.953608412253
Return kurtosis 5.43389673511
Kurtosis errors:  -2.01509072715 -2.11202355531
Return kurtosis 5.88565171898
Kurtosis errors:  -3.27660494573 -2.98917149895
/usr/local/lib/python2.7/dist-packages/statsmodels/base/model.py:466: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
"Check mle_retvals", ConvergenceWarning)
Return kurtosis 6.96174362141
Kurtosis errors:  -2.83707925003 -0.145942026844
Return kurtosis 7.97505704098
Kurtosis errors:  -4.07065811659 -3.26756482916
Return kurtosis 3.58177817784
Kurtosis errors:  0.987293266732 2.56661929572
Return kurtosis 4.3279205944
Kurtosis errors:  0.932303319562 0.0152847120161
Return kurtosis 7.07791806239
Kurtosis errors:  15.9064703415 12.8874795048
Return kurtosis 5.57050162813
Kurtosis errors:  15.5426267905 22.2695912142
Return kurtosis 5.30253366531
Kurtosis errors:  19.407402562 24.8042730047
Return kurtosis 6.29071437512
Kurtosis errors:  10.1341009126 6.8420076044
Return kurtosis 6.22361096698
Kurtosis errors:  11.8497221083 9.20718493176
Stock:  IGT MSEdiff= -0.255149007843 MADdiff= -0.0306925067578 Ktdiff= -0.0570684995019
Return kurtosis 4.96996791147
Kurtosis errors:  -1.253432014 -1.84963677862
Return kurtosis 5.12186082831
Kurtosis errors:  -2.28901936008 -1.93142443842
Return kurtosis 5.06945364721
Kurtosis errors:  -2.44449873301 -1.8965254885
Return kurtosis 6.04551925638
Kurtosis errors:  1.24899983295 -1.46685713835
Return kurtosis 4.66073714546
Kurtosis errors:  0.731476356112 0.557463117038
Return kurtosis 5.6642679976
Kurtosis errors:  3.79363763279 -0.785012271605
Return kurtosis 6.15250537911
Kurtosis errors:  -0.470219605714 0.146696709322
Return kurtosis 8.08473890543
Kurtosis errors:  1.04678066727 -1.30515611859
Return kurtosis 11.1345923831
Kurtosis errors:  -3.42893107288 3.7673201671
Return kurtosis 17.5579125445
Kurtosis errors:  -0.689887565952 -9.5180379366
Return kurtosis 11.9425463156
Kurtosis errors:  0.629488218004 6.6915876266
Return kurtosis 11.2101475087
Kurtosis errors:  0.00958366139585 0.961445151118
Return kurtosis 8.99144444912
Kurtosis errors:  2.86121289838 4.31980084856
Return kurtosis 7.49361932937
Kurtosis errors:  2.19149790585 2.63358857811
Return kurtosis 8.23040807359
Kurtosis errors:  17.422256209 23.7197377732
Stock:  SIAL MSEdiff= -0.0683413331826 MADdiff= -0.0158784483717 Ktdiff= -1.40262456055
Return kurtosis 89.7999685105
Kurtosis errors:  -73.8290640417 -62.8165049394
Return kurtosis 98.6424639968
Kurtosis errors:  -85.5365258978 -85.4191026868
Return kurtosis 21.8181165515
Kurtosis errors:  -13.8233486712 -14.8285422796
Return kurtosis 32.5085153874
Kurtosis errors:  -24.5493060271 -24.8713672643
Return kurtosis 36.982531621
Kurtosis errors:  -28.5161620511 -29.5776511597
Return kurtosis 25.5434617149
Kurtosis errors:  -18.1271659185 -17.1391154879
Return kurtosis 34.0242325043
Kurtosis errors:  -17.1814196088 -16.8910371655
Return kurtosis 26.2933396651
Kurtosis errors:  -12.2220531039 -12.4332327494
Return kurtosis 22.4040254237
Kurtosis errors:  -16.0818266127 -13.3647457045
Return kurtosis 20.5924293597
Kurtosis errors:  -12.2060365752 -13.6900971216
Return kurtosis 18.736310113
Kurtosis errors:  -13.1827109119 -12.6884540429
Return kurtosis 14.9729333883
Kurtosis errors:  -9.99941347209 -8.87996218436
Return kurtosis 18.6222941601
Kurtosis errors:  -7.71325504745 -14.0854998034
Return kurtosis 21.2488027596
Kurtosis errors:  -12.1749917469 -12.841831395
Return kurtosis 20.0401775694
Kurtosis errors:  -13.1728705487 -13.1307832742
Stock:  KND MSEdiff= -0.100984744698 MADdiff= -0.014521918976 Ktdiff= 0.377214865096
Return kurtosis 8.80696857657
Kurtosis errors:  -3.71474672867 -3.71702131559
Return kurtosis 8.2098202032
Kurtosis errors:  -3.59710559419 -2.45249792684
Return kurtosis 8.46868392892
Kurtosis errors:  -4.38794273271 -4.8177337628
Return kurtosis 1.6338239479
Kurtosis errors:  1.6222276346 1.76383457979
Return kurtosis 1.14625053175
Kurtosis errors:  1.16506039967 1.22096477927
Return kurtosis 1.60051078942
Kurtosis errors:  0.159341826809 1.41456478598
Return kurtosis 1.99407699856
Kurtosis errors:  -0.14564407314 1.61463105204
Return kurtosis 12.5977635741
Kurtosis errors:  -10.1084203979 -9.3183342063
Return kurtosis 11.1895516232
Kurtosis errors:  -8.87562907385 -8.75072795896
Return kurtosis 120.670824015
Kurtosis errors:  -118.09524902 -117.886128544
Return kurtosis 112.882140617
Kurtosis errors:  -109.642242589 -109.905227115
```
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