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## FUNDAMENTAL PROBLEM OF VOLATILITY FORECASTING

From my current point of view, the fundamental problem of univariate volatility forecasting is the enigma of volatility having clear long memory with information at lags in the thousands of days while the top performing univariate volatility forecasting is still GARCH(1,1). This is a delicate problem for which I have some new tools and ideas and I am confident that this problem can be solved independently of the recent result I have obtained that GARCH(1,1) can be improved by using aggregate volatility as an external regressor. The last result is important is showing that there can be universal improvements over GARCH(1,1), but the annoying problem of having a long memory process producing the best out-of-sample performance by throwing away the long memory information is yet unsolved. The new tools are the following: exact models for the autocorrelations of volatility, which is the improvement I obtained to the Ding-Granger-Engle 1993 model of autocorrelations. Autocorrelations of volatility are well-fit by Mittag-Leffler functions.

## OUT-OF-SAMPLE UNIVERSAL FORECAST IMPROVEMENT OVER EXPONENTIAL GARCH(1,1) USING AGGREGATE VOLATILITY WITH RETURN KURTOSIS ESTIMATES

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zulfikar.ahmed@gmail.com

Attachments3:08 PM (0 minutes ago)

to jfrenkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira

The exponential GARCH(1,1) specification for returns is

return_t = exp( h_t/2 ) z_t

with h_t following an autoregressive process. We have found out-of-sample improvements over this sort of model with using the aggregate market volatility as an external regressor for h_t. The results are attached in the file â€‹aggvol-prediction-with-kurtosis.csv for the historical data and code also attached.

â€‹We consider the problem of kurtosis of returns which is a measure of fat-tails. When z_t above is normally distributed, neither model gives the high kurtosis observed in returns; in fact this is an important problem in finance because it is often claimed that GARCH models and Stochastic Volatility models can ‘explain the kurtosis’. At least in this example, we did not find a compelling match of the model kurtosis to actual return kurtosis if we take z_t to be normal. However, we did find much better return kurtosis from the models assuming z_t is iid Student-t.

Using the aggregate volatility in the model does improve the MSE/MAD out-of-sample for volatility prediction, but as the attached results show, the aggregate volatility model kurtosis is for these data slightly worse at explaining actual return kurtosis than the exponential GARCH(1,1) model.

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GXP,nan,nan,-0.708970977775
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C,-0.62589226535,-0.0747980688102,-1.70448322629
PGNX,-0.0463267497128,-0.00529239557406,-0.665463499742
AIV,-0.533804178518,-0.0662020707812,2.62969053832
ALEX,-0.268453387077,-0.0298244244,-0.194327500663
DELL,-0.260064907416,-0.0313392425299,0.397948713033
CMG,-0.500743761721,-0.0483522107766,-0.0619458566169
REGN,-0.395357452205,-0.0472966946719,-0.787469508666
CUTR,0.00178824858754,0.00187162995734,1.01958660811
GGG,-0.0796773490858,-0.00901277666169,0.320092917244
FCBC,-0.283567188894,-0.0329931282865,-2.86650609528
ACN,-0.305986568617,-0.0331415478696,0.313996963234
GNW,-0.318565981869,-0.0408888595355,-0.00060834783434
WDR,-0.709930979054,-0.0727763731635,1.13751962079
ACTG,-0.24071261153,-0.0274965906161,-0.279686201151
HRL,-0.218339473241,-0.0243589740693,0.707229661253
CDNS,-0.054664789781,-0.00925372080402,0.447766462061
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HILL,-0.39358464478,-0.0472710544633,-1.68768065222
HLF,-0.699931567254,-0.088218343406,-0.603724857309
MBFI,-0.257277210666,-0.0293762543125,2.01785009884
AZO,-0.482045969826,-0.0502082414163,0.0630878658928
DVN,-0.279171486265,-0.0324806904197,-0.107805419766
SYKE,-0.254640733237,-0.0280980899048,-2.03934910645
SIMG,-0.271373011504,-0.0300486260755,-0.286277053491
FUL,-0.319516433338,-0.0389869733742,-1.65745173035
PRA,-0.231513656438,-0.0221121387184,2.204435339
MCRL,-0.662635104518,-0.0808290051653,-0.221083911339
O,-0.612042724529,-0.0709097118162,2.79986722837
CPWR,-0.318336115973,-0.0377810678251,-0.541524893573
DVA,-0.179828683707,-0.0185914551477,-0.030527930362
ORB,-0.106209544293,-0.0111215169831,-0.833093958589
OPLK,-0.420476333377,-0.0404710626916,0.541771062476
DTE,-0.276528078714,-0.03187643368,0.0518366873077
NFLX,-0.126514423001,-0.0160814854573,-1.33452896769
SF,-0.00341497366003,-0.000476049195127,-13.0235875138
MYL,0.990312735353,0.151964365692,4.64433580218
OLN,-0.280338508384,-0.0315277403181,1.34327737239
ABG,-0.652764201431,-0.0767407620227,1.58357427735
PBY,-0.392952248657,-0.0524118649386,-1.59485963452
HL,-0.057873799762,-0.00654254241374,1.54283250093
UPS,-1.05558563274,-0.112451048268,-0.857548176907
EXPD,-0.154499862999,-0.0178668073795,0.121984913812
GBX,-0.0563248994212,-0.0080485484259,1.4689515767
ITW,-0.606207320562,-0.0739873180243,-0.633643514969
VIVO,-0.182613224304,-0.0187732446534,2.4421688366
NILE,-0.365835224332,-0.0435900221177,-0.365729618294
GBNK,-0.480743975555,-0.0638888276343,0.161120594869
PPG,-0.489084370462,-0.0584513440137,-2.35507420509
USMO,-0.38114644157,-0.043639597322,-0.308529313224
LGND,0.010247557839,-0.0014880924088,-1.3361238074
AYR,-0.863444395326,-0.10127109832,-0.320845957862
XRM,-0.298834266022,-0.0341934377305,-0.686813488974
ISIS,-0.0320120652645,-0.00212720158086,0.697459284565
IBM,-0.699987970947,-0.0889387557403,1.03642636037
MLHR,-0.252047366771,-0.0286811282197,-0.760906612927
GNTX,-0.123798957736,-0.0169766565294,2.13369779756
EPAY,-0.384741932566,-0.0409554142244,1.33035511777
GPS,-0.29368550473,-0.0346786549626,1.19007045664
JMBA,-0.902985077564,-0.108008293129,1.31041435928
TRV,-0.399093724452,-0.0477442470272,-1.8136444021
KLAC,-0.482329098848,-0.0557969686898,1.06852625156
PFG,-0.23614689732,-0.0296196368892,3.74404743364
STRA,-0.105637575072,-0.0111993165861,0.882506775425
FRO,-0.353292534367,-0.0511230207463,-0.213192780039
IBKC,-0.554257765755,-0.0625190303626,-2.20929863416
AMSF,-0.313612588199,-0.0360663573922,0.208554900709
AIMC,-1.68434077369,-0.223695520751,-0.335883763629
IIVI,-0.306599508465,-0.032268672406,-2.6293856121
CY,-0.549076367741,-0.0636346987801,-1.03709935884
BAX,-0.623927964085,-0.070450677299,-61.9187381743
PSUN,0.102170009556,0.0105378207289,0.10380309147
LYV,0.0600311200109,0.000636226567923,0.155750180288
SCOR,-0.144465425348,-0.0160276595824,0.29712219623
CL,-0.348831177374,-0.0413211732021,1.60323991207
DRC,-0.277980505585,-0.0395470129601,-1.22688446482
ESV,-0.325823866065,-0.0398747358317,1.42605804288
WRB,-0.204872685271,-0.0218823661161,1.21052396322
VAR,-0.193710702679,-0.0200612966222,0.0839964285233
SKT,-0.182638208542,-0.0215574519623,-2.40135349701
VNDA,-0.657081340316,-0.0659846703587,2.07235556589
UIS,-0.420590094375,-0.0479091247613,-5.68986069266
GSAT,-0.36768296353,-0.0373458368806,0.289531874663
AVT,-0.244926945192,-0.029334002484,0.623015609687
TCB,-0.0983722615893,-0.0129968200085,0.767975369629
PPL,-0.427036886788,-0.0514465397715,-0.549817643251
SGEN,-0.258799464406,-0.0300710545522,-0.491370482515
DNB,0.0981420715864,0.00298527894541,1.17816553111
ODP,-0.762616577054,-0.100359851486,4.33138038406
TSN,0.0208660368436,0.0034369489879,-1.24961173232



â€‹
DatedHistPx.csv.gz
â€‹â€‹
3 Attachments
Preview attachment aggvol-prediction-with-kurtosis.csv
[Comma Separated Data]
Preview attachment garch11-kurtosis.py
[Text]
Preview attachment DatedHistPx.csv.gz
[Compressed Archive]
Not virus scanned

## 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.Â  Recall that these models are

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

where $z_t \sim N(0,1)$.Â  An empirical approach to solve the problem is to assume the model with some Student-t distributed $z_t$.Â  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.Â  This coarse result produces more reasonable kurtosis figures which are still quite innacurate but at least the order of magnitudes are sane comparisons.Â  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.Â  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


## PETER CLARK’S SUBORDINATED PROCESS FOR STOCKS

Trading time being random is due to Peter Clark (1971) Peter-Clark-SubordinatedProcess1971. I had not noticed that he had precise formulae for the kurtosis effects on price changes in terms of statistics of the random time directing process.Â  So this is an interesting exercise:Â  we know that the renewal times of volatility shocks in the equities is fractional Poisson so an exercise is to update Clark’s analysis for this case including some analysis of kurtosis.

If the directing process is $T(t)$ with $z=\Delta T(t)$ with mean $\alpha$, then the kurtosis of the price process $\Delta X(T(t))$ is

$k_{\Delta X(T(t))} =Â \frac{3 (\alpha^2 + Var(z))}{\alpha^2}$

according to Clark’s Corollary 4.1.

Instead of looking at the kurtosis of price changes directly, probably the best thing to do is to consider the model

$r_t = e^{h_t/2} \epsilon_t$

and consider Clark’s formula as it applies to $h_t$, because we know that the volatility shock distribution is regular.Â  This is a synthesis of approaches to describing the kurtosis of price changes.Â  Rather than ‘stochastic volatility is the answer’ or ‘random times is the answer’ we have a synthesis because it turns out that the volatility series has fractional Poisson features and excess kurtosis.

## BREAKTHROUGH IMPROVEMENT IN OUT-OF-SAMPLE PERFORMANCE OF VOLATILITY FORECASTING OVER GARCH(1,1)

Inbox
x
zulfikar.ahmed@gmail.com

Attachments5:37 AM (24 minutes ago)

to jfrenkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira

In order to consolidate our breakthrough for out-of-sample forecasting performance over (exponential) GARCH(1,1)â€‹ here are results for 872 stocks. The out-of-sample MSE improvement in this sample of stocks is -0.367 with standard error 0.03.

â€‹Our model here is simply exponential GARCH(1,1) with an external regressor that is the average log(return^2) of ~1900 stocks. Thus algorithmically it is simple. I am reporting these results as the runs complete. I want to ensure that it is very very clear that this is a universal result. Please feel free to examine the code and let me know of any errors. I use an AR(5) model for the external regressor aggregate volatility that I forecast and use this in the forecast of the individual stock volatility. The test periods are long determined by cutting up the time series into blocks and using a series to train and test on a single block depending on the data availability. The focus here is to provide universality in improvement over (exponential) GARCH(1,1) rather than provide a precise forecasting algorithm for use.

PRE,-0.31135612636,-0.0358209716849
IRBT,-2.04456614071,-0.262406747071
G,-0.415592494987,-0.0410662678252
ICUI,-0.164785748195,-0.0160051519559
IGT,-0.255149007843,-0.0306925067578
SIAL,-0.0683413331826,-0.0158784483717
KND,-0.100984744698,-0.014521918976
ASH,-0.256073193706,-0.0311308139461
ABMD,-0.20305946292,-0.0278662179978
WSM,-0.192651130727,-0.0247252237014
RTEC,-0.485853919873,-0.0554500487653
ELNK,-0.246727867935,-0.0277235992015
CVO,0.0336239698175,-0.00233686292701
GOV,-0.252188479774,-0.0308140885013
RRGB,-0.207786536188,-0.0221921387626
OEH,-0.27084069879,-0.033427833751
SIRO,-1.17052480768,-0.145326115153
FFIN,-0.425727142967,-0.0506846135863
PBCT,-0.643677635806,-0.0709072850523
SRI,-0.462729567991,-0.0505749593891
MM,0.0567851213402,-0.00431864522968
SWKS,-0.267610707889,-0.0330479313145
TOL,-0.419396896295,-0.0460293582077
HBI,-0.146933201106,-0.0168434519422
MCRI,-0.283826018828,-0.0306601489675
WABC,-0.309362100574,-0.040946064124
ITRI,-0.171745334426,-0.0181676159193
APH,-0.152701790865,-0.0177153020109
DF,0.0837623565872,0.0148754594786
HTZ,-0.705319572038,-0.0880959319421
KNL,-0.450551486144,-0.0581198966654
ACGL,-0.030661247546,-0.00141535089293
SCSS,0.12901575063,0.0135184283877
MRK,-0.484125340187,-0.0587912203095
BKD,0.0658563742821,0.00305157099516
TITN,-0.430050538511,-0.052421384628
FNSR,-0.04500010456,-0.00426515258386
CR,-0.345910031676,-0.0409282594165
UEIC,-0.215831110451,-0.022412085673
VOD,-0.237071768594,-0.0267948209294
OC,-0.50085147354,-0.0633371246599
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LPNT,-0.0519482871157,-0.00664064323788
LOPE,-0.270297440705,-0.0326350215693
DUK,-1.02345689184,-0.101794655024
DPS,-0.122632562945,-0.0135591517278
CPS,0.0324781099145,-0.0054217843748
VGR,-0.147821437174,-0.0156021971491
MLNK,-0.563013993821,-0.0681291170682
TEN,-0.222644279208,-0.0251151672179
MMSI,0.0914754579245,0.0117490469879
PPS,-0.364012044991,-0.0452180980327
LO,-0.30318772216,-0.0337832272397
CUB,-0.0635695149421,-0.0090552143754
JBT,-0.37634337713,-0.0342388741736
HI,-0.428000148649,-0.0482668668137
STAA,-0.0473144418619,-0.00619317012907
PCL,-0.312721879758,-0.0372369652799
LOW,-0.478838362886,-0.057048912064
LYTS,0.0280401554473,0.0016947860203
WLK,-0.565788165556,-0.0643557825455
SEE,-0.211039340821,-0.0246276952883
UHAL,-0.345638765902,-0.0413263969487
NUS,-0.66900619561,-0.079342846558
FNB,-0.391840878852,-0.0456811285747
EV,-0.471723638843,-0.0563784663691
MDT,-0.243436486238,-0.0286541934532
ARW,-0.910058705877,-0.115737584383
QLGC,-0.368831881351,-0.0414244564038
MANT,-0.449148491113,-0.0474422863092
TRMK,-0.284200649441,-0.0325854435108
MCHP,-0.402974600217,-0.0467291965519
SUI,-0.343384043163,-0.0401865590915
TMK,-0.450203739621,-0.0533246459499
PKI,-0.51139297303,-0.0559813942854
CHH,-0.318905128379,-0.0373727416054

## RELIABLE OUT-OF-SAMPLE VOLATILITY FORECASTING IMPROVEMENT OVER GARCH(1,1)

MSE improvement out-of-sample over an exponential GARCH specification by a model that uses aggregate volatility as a regressor -0.38 se 0.0017 and MAD -0.041 se 0.00008. The table below lists out-of-sample improvement of predicting log-volatility. Our benchmark is the exponential GARCH(1,1) model:
$\log(\sigma_t^2) = \omega + \alpha \log(\sigma^2_{t-1}) + \beta \log(\epsilon_{t-1}^2)$
versus
$\log(\sigma^2_t) = \omega + \alpha \log(\sigma^2_{t-1}) + \beta \log(\epsilon_{t-1}^2) + \gamma AV_{t-1}$
where $AV_{t}$ is defined a the average over 1899 stocks of log-return-squared.

Stock,MSEImpOverGARCH11,MADImpOverGARCH11
PRE,-0.31135612636,-0.0358209716849
IRBT,-2.04456614071,-0.262406747071
G,-0.415592494987,-0.0410662678252
ICUI,-0.164785748195,-0.0160051519559
IGT,-0.255149007843,-0.0306925067578
SIAL,-0.0683413331826,-0.0158784483717
KND,-0.100984744698,-0.014521918976
ASH,-0.256073193706,-0.0311308139461
ABMD,-0.20305946292,-0.0278662179978
WSM,-0.192651130727,-0.0247252237014
RTEC,-0.485853919873,-0.0554500487653
ELNK,-0.246727867935,-0.0277235992015
CVO,0.0336239698175,-0.00233686292701
GOV,-0.252188479774,-0.0308140885013
RRGB,-0.207786536188,-0.0221921387626
OEH,-0.27084069879,-0.033427833751
SIRO,-1.17052480768,-0.145326115153
FFIN,-0.425727142967,-0.0506846135863
PBCT,-0.643677635806,-0.0709072850523
SRI,-0.462729567991,-0.0505749593891
MM,0.0567851213402,-0.00431864522968
SWKS,-0.267610707889,-0.0330479313145
TOL,-0.419396896295,-0.0460293582077
HBI,-0.146933201106,-0.0168434519422
MCRI,-0.283826018828,-0.0306601489675
WABC,-0.309362100574,-0.040946064124
ITRI,-0.171745334426,-0.0181676159193
APH,-0.152701790865,-0.0177153020109
DF,0.0837623565872,0.0148754594786
HTZ,-0.705319572038,-0.0880959319421
KNL,-0.450551486144,-0.0581198966654
ACGL,-0.030661247546,-0.00141535089293
SCSS,0.12901575063,0.0135184283877
MRK,-0.484125340187,-0.0587912203095
BKD,0.0658563742821,0.00305157099516
TITN,-0.430050538511,-0.052421384628
FNSR,-0.04500010456,-0.00426515258386
CR,-0.345910031676,-0.0409282594165
UEIC,-0.215831110451,-0.022412085673
VOD,-0.237071768594,-0.0267948209294
OC,-0.50085147354,-0.0633371246599
NTRI,0.184411690422,0.0171985103255
CNA,-0.363319518774,-0.041002889982
SYY,-0.299716790748,-0.0366508772261
CTL,-0.320195528911,-0.0325234896997
NJR,-0.470877690165,-0.0526652388219
UNFI,-0.225772533114,-0.0234696049094
VIAB,-0.572899281702,-0.0598872358483
EXLS,-0.735702003025,-0.0856340395238
YRCW,-0.629064691758,-0.0643079838347
STAR,0.090673468249,0.00843662766921
CHTR,-0.250862759022,-0.0310642265717
AFAM,0.00221102207088,9.65637952502e-05
PSEM,-0.359946543478,-0.0438619384357
PHH,-0.202202027858,-0.0177063105992
PVA,-0.131423629804,-0.0170859682627
WY,-0.590503701791,-0.0720683375219
GCI,-0.264686702802,-0.0292336768461
RGLD,-0.178542064599,-0.0194842620423
CRK,-0.117833041678,-0.0143840242898
SGMO,-0.089688159276,-0.0118119385796
S,-0.224769224469,-0.0249553979476
ORCL,-0.490100136954,-0.0555340716645
ABM,-0.222283545272,-0.0241112705924
CBSH,-0.404740139896,-0.0453165087654
AHS,-0.269205473077,-0.0316392747418
VSH,-0.449316140562,-0.0537683976635
MLM,-0.314527251969,-0.0357481530058
OREX,-0.142662394274,-0.0250411831321
KKD,-0.455758764594,-0.0570958704544
FRP,-0.289397111418,-0.030256728725
AGO,-0.886543771357,-0.0962321384809
SWN,-0.1557720406,-0.0177333923283
PLCM,-0.388076179822,-0.0464177077098
MDC,-0.387089922392,-0.0334146995038
DD,-0.653953528088,-0.0820846188124
DOX,-0.36235667105,-0.0364465692961
CDI,-0.194181724912,-0.0180027861516
TRGT,-1.72586243447,-0.201287585819
VICR,-0.283014531087,-0.0319767428888
TEG,-0.334623483891,-0.0362603430403
EOG,-0.300830492858,-0.0346096312968
HTH,-0.534089836046,-0.0592169053762
WSO,-0.246632598797,-0.027533165882
NYNY,-0.00376108985235,-0.000213591413925
WYN,-1.81222560431,-0.223645626384
LANC,-0.453544723688,-0.0499146274648
AON,-0.277997332445,-0.0294436354986
SCHN,-0.419892875733,-0.0510104807544
DHI,-0.119062871822,-0.0150679443852
CRY,0.0535059542321,0.0067519365463
VDSI,-0.409962748991,-0.044199456813
ETR,-0.17438029283,-0.020401812394
AVA,-0.278751399003,-0.0305232431344
MCO,-0.663856110635,-0.0738455236329
FELE,-0.260110060732,-0.0318911548189
RHT,-0.268088120363,-0.0289583343464
SAH,-0.104322667826,-0.0132421369081
RST,-0.0282302772123,-0.000465947961805
JACK,-0.231908294351,-0.0254699364717
LMT,-0.517156872622,-0.0587819550402
VSAT,-0.566965193274,-0.0637404643354
VVC,-0.335248944897,-0.0362804002686
SE,-1.10316341463,-0.131469635886
GTS,-0.603935468546,-0.0732036843562
OB,-0.543451349068,-0.0563176836781
AZPN,-0.127315221669,-0.017122681956
HAS,-0.531828638101,-0.057095771785
ALOG,-0.162030653598,-0.0174154434022
OMEX,-0.177186482101,-0.0214373100182
CNS,-0.740963720073,-0.0978569100974
MO,-0.460744458537,-0.0542028255724
ELY,-0.0225634906503,-0.0012586295515
SFY,-0.242486176838,-0.0293123903361
RSH,-0.330832079022,-0.0429830764788
SNV,-0.542184043166,-0.0585230000898
IVR,-0.281297662716,-0.0264886939018
CMC,-0.191338847369,-0.0245942320846
HIL,-0.291674014221,-0.0361703274783
UNH,-0.814072616429,-0.096108195746
ZQK,-0.341173021872,-0.0386805281378
DRH,-0.493823062594,-0.0618076139966
REXI,-0.209106215809,-0.0224245392898
PH,-0.26850999784,-0.0336005860773
ARO,-0.429725037795,-0.0505462362594
SXT,-0.123158547934,-0.0132171416962
ZEP,-0.138944104574,-0.0170786525115
SBUX,-0.163384340133,-0.0171580647174
WRE,-0.600267090044,-0.0706438206571
WNC,-0.0945128751657,-0.0123521634278
VSI,-0.165587377175,-0.0171797552931
CHK,-0.579495878244,-0.070942502315
STI,-0.560006837188,-0.0721519291903
CLF,-0.506284770803,-0.062346654075
TRAK,-0.251150478127,-0.0283442971884
AJG,-0.373390912088,-0.040447927977
BDSI,-0.441383315252,-0.051152179332
HRB,-0.219979091056,-0.030262845956
WR,-0.46125966704,-0.0509330679273
TTEC,-0.114543889585,-0.0128719989437
BZH,-0.129460086294,-0.0148422261652
SNA,-0.584186895823,-0.0630772365072
OI,-0.42524755167,-0.0433158592158
INWK,-0.475222538001,-0.051147222838
WRES,-0.636603173121,-0.0750629499119
PCLN,-0.285565076424,-0.0337228607487
SAFM,-0.158682678468,-0.0191041195614
GTXI,-0.153007325065,-0.0187706975242
TFX,-0.680851861615,-0.0732908183871
AMD,-0.221778329775,-0.027598648552
CLW,-0.279533667708,-0.035598714278
BEAV,-0.319424259751,-0.0349280444407
ALXN,-0.107407502538,-0.0159504953315
ROSE,-0.255792595187,-0.035966509936
RJF,-0.388382018844,-0.0445700510564
BHE,-0.741937279159,-0.0955326307483
ANR,-0.495436371526,-0.070622752259
TGT,-0.276745372389,-0.0348916409444
UMPQ,-0.384400573977,-0.0472655762548
EMR,-0.480924665139,-0.0569022707834
BKE,-0.620080061214,-0.0615375975012
HIBB,-0.259071136449,-0.0295582631869
AF,-0.605293003023,-0.0731113120269
MPW,-0.822843358248,-0.092225083056
AMP,-0.942345066764,-0.126940500016
EHTH,-0.178643370108,-0.0186413077589
BSX,-0.0597387623316,-0.00645710634302
AL,-0.641839096378,-0.0791071811859
WHR,-0.347858576718,-0.041373935135
PAY,-0.549019961163,-0.0588230537848
BK,-0.377553560665,-0.0451188178961
AXP,-0.589285244343,-0.0774819176467
BRKL,-0.932819697749,-0.104709551754
CYNO,-0.331987150308,-0.0383982183188
ROG,-0.075929569914,-0.0112549474914
HEES,0.0298982400542,0.00349362063072
ONB,-0.316525639495,-0.0376561667945
ORN,-0.364132568719,-0.0429555481332
IMMU,-0.118016937837,-0.0133597765461
TSS,-0.329383920388,-0.0349596060337
OSIR,0.0890648965531,0.0128135557258
ZUMZ,-0.355341309198,-0.0470436404811
ARQL,-0.125234640845,-0.0139728597681
BBG,-0.366085734804,-0.0436520681047
IVC,-0.378257953022,-0.0422094672317
CREE,-0.33150524876,-0.037311668849
CBT,-1.20216504386,-0.156799527927
AXL,-0.125537212378,-0.0150300934291
GFIG,-0.697385330061,-0.0789023286667
AVP,-0.562651836499,-0.0643737687082
LTM,-0.80882490171,-0.0844174286618
SPW,-0.216410991777,-0.0238243382033
KOPN,-0.160029679751,-0.0210461026776
SWC,-0.363725228066,-0.0432575755386
PKD,-0.239976829258,-0.0296183710732
KFRC,-0.576117252982,-0.0660303148009
CCMP,-0.677205188528,-0.0789917773619
TPX,0.0605865892932,0.00696176408764
WGL,-0.51859582926,-0.0618959450226
APP,-0.116412350679,-0.0149950404552
CRZO,-0.188891691028,-0.0246263507669
FSLR,-0.274044082541,-0.0338791638476
POWI,-0.527025512911,-0.065169897432
CWST,-0.131374489818,-0.0150015666434
NTAP,-0.670920165421,-0.0846859113174
JDSU,-0.600495860598,-0.0732971491261
FOSL,-0.467498379552,-0.054374058242
HPQ,-0.306678356729,-0.0411096936393
CRL,-0.116351674898,-0.0121486553105
HUM,-0.214591781599,-0.0261014597792
CLNE,-0.062748704615,-0.0068044600167
FMD,-0.25730124436,-0.0295716405644
PNM,-0.110224038155,-0.0122643442391
SYNA,0.0905860296369,0.013530151048
WAG,-0.614658323859,-0.0725318531488
FFBC,-0.261591223745,-0.0293684839714
ASPS,-0.111227310126,-0.0161995913988
MEG,-0.323454234363,-0.0358561765808
AMT,-0.235132334248,-0.0276578506827
RAIL,-0.214513188828,-0.0217313171331
SO,-0.742878902199,-0.0852890561919
ODFL,-0.00403255067683,-0.0034342186972
INFA,-0.233758040713,-0.028211392814
PLXS,-0.263702864087,-0.0337770402938
XNPT,0.0190433586421,-0.0140260789947
AKRX,0.655681232131,0.075386611216
NWE,-1.03003260889,-0.124523186389
MCY,-0.472096349558,-0.052818033217
ANH,-0.649391560337,-0.0685325777502
WU,-0.909709368445,-0.0952862758905
BXC,-0.520233389445,-0.0568685040205
TE,-0.195800870808,-0.0235028005445
RGEN,-0.121261589032,-0.0128380004305
BMR,-1.04695212503,-0.120142240786
ED,-0.534031290064,-0.0596841342814
ISRG,-1.02546988433,-0.120658777927
EBS,-0.0180856005822,0.00539306073032
AVID,-0.0894123306976,-0.00901603308445
VRSN,-0.459616449244,-0.0479940586404
IMGN,-0.178275711983,-0.0208806837045
COST,0.019306239685,2.7932427452e-05
RVBD,-0.327418372026,-0.0407751029188
TNK,-0.196773460881,-0.02317759035
TJX,-0.454537159284,-0.0479353578034
WTI,-0.68366900147,-0.0887034123256
RT,-0.142444903086,-0.0168805385904
NEWP,-0.502367355716,-0.0586250089965
ESS,-0.415448238896,-0.0485116799883
AGCO,-0.141896389414,-0.0187078338528
IPHS,-0.197859883664,-0.0189715382855
XRAY,-0.138057923814,-0.0164168887934
LBTYA,-0.884379220575,-0.0972281417265
TWGP,-1.0031920951,-0.132583872961
ACC,-0.736186308963,-0.0901669991354
AFG,-0.628238801387,-0.0733909452859
KALU,-0.916588576679,-0.109988700263
CHFC,-0.381219961708,-0.0451182482789
QSII,-0.23422885218,-0.0240533366481
BXP,-0.366709954177,-0.0456846993945
SIX,-0.320274335709,-0.0334584541134
UPL,-0.246673310187,-0.0311702618561
IART,-0.642805450112,-0.0775540799219
NLS,-0.169761971893,-0.0176397134125
CTB,-0.056223002523,0.000258265570407
ATO,-0.407067554483,-0.0411654788329
CERS,-0.148205821403,-0.0184970483599
AIG,-0.207587419008,-0.0207713905098
DTV,-0.599139942309,-0.0722079156068
CAM,-0.362195614856,-0.0447148271902
ARIA,-0.0206848818931,-0.00654368671976
GPOR,nan,nan
OSIS,-0.0668554968325,-0.00843207828571
NSC,-0.763778622996,-0.0964078464591
JCOM,-0.483659741596,-0.0507781046925
CENX,-0.50363525023,-0.0631425908161
INDB,-0.368513848189,-0.0427000531164
ESL,-0.223778625459,-0.0270258211442
HZO,-0.402736199202,-0.0467042642673
CLH,-0.174316324888,-0.0167474436135
ANN,-0.533816903349,-0.0645335692552
WGO,-0.262942575614,-0.0306371387063
CVG,-0.768024683235,-0.0834829862995
CNC,-0.133277292127,-0.0144345089455
GXP,nan,nan
MNKD,-0.100301660548,-0.00638053029397
MNI,-0.0328707272341,-0.00456547868165
OCR,0.0913488500738,0.0104160833143
FARO,-0.10869814483,-0.0129518818994
RBCN,-0.15684032178,-0.0197292899286
PEGA,-0.0236061116117,-0.00375973998267
MOS,-0.0729564169173,-0.00991570211619
C,-0.62589226535,-0.0747980688102
PGNX,-0.0463267497128,-0.00529239557406
AIV,-0.533804178518,-0.0662020707812
ALEX,-0.268453387077,-0.0298244244
DELL,-0.260064907416,-0.0313392425299
CMG,-0.500743761721,-0.0483522107766
REGN,-0.395357452205,-0.0472966946719
CUTR,0.00178824858754,0.00187162995734
GGG,-0.0796773490858,-0.00901277666169
FCBC,-0.283567188894,-0.0329931282865
ACN,-0.305986568617,-0.0331415478696
GNW,-0.318565981869,-0.0408888595355
WDR,-0.709930979054,-0.0727763731635
ACTG,-0.24071261153,-0.0274965906161
HRL,-0.218339473241,-0.0243589740693
CDNS,-0.054664789781,-0.00925372080402
VLO,-0.0825977225177,-0.00978660442514
HILL,-0.39358464478,-0.0472710544633
HLF,-0.699931567254,-0.088218343406
MBFI,-0.257277210666,-0.0293762543125
AZO,-0.482045969826,-0.0502082414163
DVN,-0.279171486265,-0.0324806904197
SYKE,-0.254640733237,-0.0280980899048
SIMG,-0.271373011504,-0.0300486260755
FUL,-0.319516433338,-0.0389869733742
PRA,-0.231513656438,-0.0221121387184
MCRL,-0.662635104518,-0.0808290051653
O,-0.612042724529,-0.0709097118162
CPWR,-0.318336115973,-0.0377810678251
DVA,-0.179828683707,-0.0185914551477
ORB,-0.106209544293,-0.0111215169831
OPLK,-0.420476333377,-0.0404710626916
DTE,-0.276528078714,-0.03187643368
NFLX,-0.126514423001,-0.0160814854573
SF,-0.00341497366003,-0.000476049195127
MYL,0.990312735353,0.151964365692
OLN,-0.280338508384,-0.0315277403181
ABG,-0.652764201431,-0.0767407620227
PBY,-0.392952248657,-0.0524118649386
HL,-0.057873799762,-0.00654254241374
UPS,-1.05558563274,-0.112451048268
EXPD,-0.154499862999,-0.0178668073795
GBX,-0.0563248994212,-0.0080485484259
ITW,-0.606207320562,-0.0739873180243
VIVO,-0.182613224304,-0.0187732446534
NILE,-0.365835224332,-0.0435900221177
GBNK,-0.480743975555,-0.0638888276343
PPG,-0.489084370462,-0.0584513440137
USMO,-0.38114644157,-0.043639597322
LGND,0.010247557839,-0.0014880924088
AYR,-0.863444395326,-0.10127109832
XRM,-0.298834266022,-0.0341934377305
ISIS,-0.0320120652645,-0.00212720158086
IBM,-0.699987970947,-0.0889387557403
MLHR,-0.252047366771,-0.0286811282197
GNTX,-0.123798957736,-0.0169766565294
EPAY,-0.384741932566,-0.0409554142244
GPS,-0.29368550473,-0.0346786549626
JMBA,-0.902985077564,-0.108008293129
TRV,-0.399093724452,-0.0477442470272
KLAC,-0.482329098848,-0.0557969686898
PFG,-0.23614689732,-0.0296196368892
STRA,-0.105637575072,-0.0111993165861
FRO,-0.353292534367,-0.0511230207463
IBKC,-0.554257765755,-0.0625190303626
AMSF,-0.313612588199,-0.0360663573922
AIMC,-1.68434077369,-0.223695520751
IIVI,-0.306599508465,-0.032268672406
CY,-0.549076367741,-0.0636346987801
BAX,-0.623927964085,-0.070450677299
PSUN,0.102170009556,0.0105378207289
LYV,0.0600311200109,0.000636226567923
SCOR,-0.144465425348,-0.0160276595824
CL,-0.348831177374,-0.0413211732021
DRC,-0.277980505585,-0.0395470129601
ESV,-0.325823866065,-0.0398747358317
WRB,-0.204872685271,-0.0218823661161
VAR,-0.193710702679,-0.0200612966222
SKT,-0.182638208542,-0.0215574519623
VNDA,-0.657081340316,-0.0659846703587
UIS,-0.420590094375,-0.0479091247613
GSAT,-0.36768296353,-0.0373458368806
AVT,-0.244926945192,-0.029334002484
TCB,-0.0983722615893,-0.0129968200085
PPL,-0.427036886788,-0.0514465397715
SGEN,-0.258799464406,-0.0300710545522
DNB,0.0981420715864,0.00298527894541
ODP,-0.762616577054,-0.100359851486
TSN,0.0208660368436,0.0034369489879
TDC,0.189007077314,0.0229053483818
BDN,-0.219480000526,-0.0284958459618
AVB,-0.254420982476,-0.0290126047041
TWI,-0.067136337555,-0.00722385420411
CVS,-0.463117920601,-0.0518982222529
CCOI,-0.280106005055,-0.0336452396581
UNF,-0.231633067922,-0.0214227867503
BABY,-0.284067757823,-0.0310683479452
AGII,-0.237070739205,-0.028609966402
HRS,-0.336675006582,-0.03848921319
PNRA,-0.191440950849,-0.0188835159625
ALJ,-0.173518703204,-0.0225569817672
AEO,-0.12758661262,-0.0106796208272
DHR,-0.524340801652,-0.0619370703398
MHGC,-0.478176372622,-0.0586108145688
MON,-0.0171325108556,-0.00253889873718
LUV,-0.554165464718,-0.0704599101121
SFLY,0.0362429230099,0.00293627836123
LIOX,-0.117725735537,-0.016062144968
GS,-0.924402230895,-0.11131129089
FCN,-0.874379910442,-0.0945403192559
FCX,-0.241858706982,-0.0310629038127
OCN,-0.602944461265,-0.0684075643724
HGR,-0.107683980165,-0.0134876824636
PKG,-1.0232123126,-0.122915948135
HBAN,-0.264752256406,-0.0299219593676
HLIT,-0.711052256534,-0.0766205618535
OMER,-27.5447410237,-0.945891684373
ENDP,-0.0938466563093,-0.0141073099519
ULTR,-0.683166758281,-0.0828385091344
ORA,-0.372688527906,-0.0443501227384
RTIX,-0.0388601997608,-0.00366255766174
SLAB,-0.102446760771,-0.0114487468851
WWW,-0.0258615470995,-0.00280650150868
RYN,-0.27776353298,-0.0295161467204
ITG,-0.00377427820249,0.00716847937842
HSNI,-0.497266303444,-0.0499644478281
IRM,-0.200728876283,-0.0236834775705
V,-1.00892317353,-0.144247433034
TRR,-0.173902582594,-0.0180587940318
REV,-0.459476428336,-0.0479645110133
TECD,-0.257826728212,-0.02793324735
GLW,-0.422945663098,-0.050414416516
BEN,-0.556451784217,-0.0695145449887
HGG,0.0865915580796,0.00544763313387
ENZ,-0.226769486961,-0.0276677508873
ORLY,-0.462048437901,-0.0520389508106
TESO,-0.337004866506,-0.0393359736457
EAC,-0.122150832339,-0.0136064730644
ENS,-0.0436027988163,-0.0113420778349
DISH,-0.241862523239,-0.0257509939174
ALL,-0.758431153097,-0.0793032749612
GLRE,-0.487860544465,-0.0549545276277
VMC,-0.294879781217,-0.0318963354821
PRKR,-0.0834002711237,-0.00899078444154
CRUS,0.253342905641,0.0293607767384
HSIC,0.0835786959759,0.0134485485515
CYBX,-0.115761417828,-0.0121307553223
TECH,-0.503371746236,-0.0547862909894
GTLS,-0.984848636733,-0.122491382357
ROIAK,-0.0976487617204,-0.008516307148
NPO,-0.586115868622,-0.0675003496636
TSRA,-0.320629167568,-0.0377417481989
CKP,-0.315559787637,-0.0345749397741
GY,-0.492092214876,-0.0570705485002
ANV,0.203051218654,0.0219603809468
PLAB,-0.436148827634,-0.0527142205792
DSPG,-0.548502322999,-0.0613690503362
LG,-0.472399214037,-0.052756108355
GCO,-0.177790296378,-0.0193411294147
SHLM,-0.272989761413,-0.0323876254269
COHR,-0.25765160921,-0.0295717501066
JBLU,-0.227163135838,-0.027647040228
BIOS,0.258135632934,0.0297248262656
EW,-0.204080708452,-0.0209917498742
SPLS,-0.172431102119,-0.0203517100956
EXR,-0.888761391276,-0.112105987079
BKH,-0.209956724272,-0.0242029182976
CSE,-0.111480317358,-0.0137865649811
FMBI,-0.206667888796,-0.0269358850682
MCK,-0.290887262648,-0.0334677352391
UNM,-0.552715030253,-0.0624088816196
FTI,-0.465586402706,-0.054538570274
TRW,-0.620114381237,-0.0712283104933
CSH,-0.0511291835485,-0.0114983913009
TSCO,-0.24126886232,-0.0279643270286
GME,-0.185107025055,-0.0228554410334
PTEN,-0.328388528918,-0.0406344118826
HNT,-0.264811516911,-0.0287629549103
WAL,-1.49103474066,-0.176410329763
WRLD,-0.215132970009,-0.0285442335841
SCS,-0.399446263481,-0.0437040857071
PCAR,-0.361225669807,-0.0468351457624
BRLI,-0.0308737382267,-0.00431835952211
AMAT,-0.193459441484,-0.0236414053322
IMN,-0.320239588516,-0.0406500564307
BXS,-0.112030198057,-0.0114507020352
FHN,-0.673492031255,-0.0719527666673
DPZ,-0.372110550573,-0.0418485056673
MNTA,-0.56846382492,-0.0690117833957
MWW,-0.587294577129,-0.0541155731008
SMRT,-0.207587071086,-0.0225750552989
SPPI,-0.112827855555,-0.011433384647
HMN,-0.318992402706,-0.0343732096956
CSCO,-0.334215358644,-0.0361453906033
CECO,-0.151085328462,-0.0159567243205
MCGC,-0.309279833834,-0.0348181383724
AVX,-0.836439154084,-0.0893149902539
EL,-0.332526975682,-0.0389248665067
FCF,-0.151045045172,-0.0200145981829
ACW,-0.693794232077,-0.0815382544959
IVZ,-0.847799435553,-0.0993705210251
AEE,-0.712202846113,-0.0804474640355
CVBF,-0.147030116493,-0.019145283532
WERN,-0.314275960955,-0.0342862137966
HE,-0.307964873986,-0.0355147890914
SLM,-0.28940294766,-0.0317453830768
MSM,-0.332698166428,-0.0369726975414
AAP,-0.436044943876,-0.0497618785394
BOKF,-0.0953135806747,-0.00615205191012
PLT,-0.175681066433,-0.0208751361816
SVU,-0.252961495508,-0.0312781947021
ATI,-0.518793495061,-0.0624845066052
AYI,-1.08442119341,-0.127765194725
HMSY,-0.286192494776,-0.0314545928643
IPCC,-1.02096545876,-0.116989639117
CPT,-0.443966763905,-0.0504959560612
WFC,-0.467374102648,-0.0574436787135
AOI,-0.0569356644122,-0.00634419596723
DYN,-0.00322162371594,-0.00444582050624
THOR,-0.276298346309,-0.0278042060584
WOR,-0.420864356774,-0.0496186366706
OHI,-0.177474164769,-0.0187326567852
ISCA,0.990349125046,0.160539746386
FICO,0.0938970907905,0.0111022938976
WAB,-0.467702214734,-0.0531925052684
TKR,-0.478465780063,-0.0551908225456
D,-0.147634493592,-0.0169581228997
UNT,0.0266452706581,-0.00143419611515
NOG,-0.301032869001,-0.0371851120846
PBI,-0.357206668693,-0.04293189952
CFN,-0.540810239892,-0.0550752685275
BANR,-0.0943962673479,-0.0133869422416
FAST,0.181387813009,0.0174941804535
CHS,0.0459244555132,0.0051412300961
GT,-0.413881884758,-0.0500480072349
NX,-0.625528388939,-0.0698149449224
PJC,-0.330724267538,-0.0462480380068
HCBK,-1.00211929732,-0.120559983732
PICO,-0.227669686819,-0.0262928527718
COLM,-0.388940767233,-0.040988553078
EME,-0.485020645797,-0.0566131420749
BP,-0.0369164861184,-0.00492490446477
ISSC,-0.110105646514,-0.0111252886281
JKHY,-0.42596368588,-0.0472158369631
PII,-0.428898611167,-0.0503283909636
NATI,-0.461168400566,-0.0543060544219
BTH,-0.246979185782,-0.0223391883981
BKMU,-0.423904216889,-0.0533938579669
CPSI,-0.350577216608,-0.0450914308916
PETS,-0.232270724213,-0.027069791106
PGR,-0.305045924944,-0.0327526935659
N,-0.807524719673,-0.0950434511025
EVR,-0.6844939188,-0.0841800712496
CA,-0.479898199693,-0.0515554117036
JWN,-0.259060481545,-0.0311676955886
SPR,-0.740896215409,-0.0926355291438
HBHC,-0.60873796437,-0.0717970174005
NFX,-0.146084434898,-0.0166823032854
QGEN,-0.152877263992,-0.0151239265245
CHRW,-0.44993788551,-0.0517130686802
BOH,-0.471579666364,-0.0531929007248
COCO,0.0193116999035,0.000899563344703
CAMP,1.11142992518,0.0839309740243
MINI,-0.450799814549,-0.0536597600542
LGF,-0.184116571545,-0.0192348572266
GERN,-0.188296725278,-0.0273467397776
URI,-0.287641765428,-0.0303829350859
CHD,-0.247766779953,-0.0262007149376
SGI,-0.256186199904,-0.0297281784744
KFY,-0.623359740331,-0.0741093863578
PFS,-0.801756366304,-0.106365019096
MOH,-0.452907760762,-0.0559291919875
OMC,-0.388297191997,-0.0485382546593
CONN,-0.302890758586,-0.0340898696257
NRG,-0.479315713072,-0.0517942083609
FINL,-0.230690618284,-0.0256799134615
DDR,-0.298075099637,-0.0398443176324
MA,-0.479312525392,-0.0533201958918
NANO,0.0364228589263,0.00276242772875
APOG,-0.185112155667,-0.0210253962786
MDU,-0.305549451157,-0.0368411603913
NU,-0.216913217998,-0.0256145393897
CINF,-0.65387763554,-0.0816961982253
ROST,-0.14199580327,-0.0152665769321
SBGI,-0.303578922083,-0.0328982687199
UFCS,-0.202632673237,-0.0235399552559
DAKT,-0.17927657133,-0.0180111536177
ARUN,-0.200072669029,-0.0330631451177
AWK,-0.327768579587,-0.0342033001566
L,-0.485548513551,-0.0585124661427
UCTT,-0.272648266141,-0.034874425984
VLCCF,-0.138145597757,-0.0202696838038
NWPX,-0.184781185617,-0.0206984902326
CSC,-0.38764543723,-0.039869203544
MOSY,0.0416964380541,0.00446293779243
EPR,-0.667207145551,-0.0711531105132
DRIV,-0.558278001794,-0.0643321510983
DISCK,-0.465912709843,-0.0554780673581
INTC,0.293845670589,0.0276410385774
WDC,-0.21396058469,-0.0249602381582
BEE,-0.129072564986,-0.0253181744443
CB,-0.516981004324,-0.0529106686865
FNFG,-1.55142767998,-0.182161328831
PMCS,-0.363909015828,-0.0393747405946
STX,-0.304540533085,-0.0342753639527
DRI,-0.131542459612,-0.0093645560355
ACOR,-0.116182281427,-0.0166084017055
GTI,-0.236738240418,-0.0330536612401
OSTK,-0.415551602941,-0.0534116940464
CACB,-0.241348030084,-0.0288672819006
OFC,-0.0609503739712,-0.00460651504785
ISLE,-0.204258073101,-0.0234657526212
WMGI,-0.223825008936,-0.0172264193965
FEIC,-0.497373452412,-0.0580111712064
EFII,-0.303077048108,-0.0282580836754
RGS,-0.4066393754,-0.0428145590241
RTI,-0.223596262314,-0.0269275619347
OKSB,-0.18420175776,-0.0215144395473
JNJ,-0.628186365775,-0.0747875484276
FSYS,-0.0759631761123,-0.00698326734631
SM,-0.342730334512,-0.0411390816694
THG,-0.591297285488,-0.0680479891549
APEI,-0.478466918755,-0.0561983952029
UIL,-0.361869727766,-0.0421370989015
OGE,-0.358335918659,-0.0409337544095
RECN,-0.271528525074,-0.0374650568123
MTD,-0.429725591152,-0.044058001027
CFFN,-0.227383984524,-0.0241330006828
SBCF,-0.132286221327,-0.0135880785025
TIBX,-0.533829780182,-0.0650174976689
SSP,-0.208460598391,-0.0222596105914
COH,0.0310177861907,-0.00820623709534
AAPL,-0.324023276876,-0.0405217769285
POR,-0.553904898892,-0.0572176672952
FTK,-0.351068060692,-0.0447287713867
BWLD,-0.0145011895695,0.000526563714229
UDR,-0.282668610121,-0.0309453799678
SCHL,-0.109089254238,-0.00928743236298
WBMD,-0.37780196686,-0.0394315851077
MYE,-0.230315037239,-0.0267774353434
LL,-0.321094272159,-0.0318355734561
EBAY,-0.804052263557,-0.100816215718
ATR,-0.279946064081,-0.0303135296043
RE,-0.604655944063,-0.0679279082525
PAG,-0.142078356064,-0.016724799204
PEP,-0.537095893526,-0.0629962159648
SYMC,-0.307174365515,-0.0374925832136


Code:

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import statsmodels.api as sm
from scipy import optimize
import re

def acf(series):
numpy=np
n = len(series)
data = numpy.asarray(series)
mean = numpy.mean(data)
c0 = numpy.sum((data - mean) ** 2) / float(n)

def r(h):
acf_lag = ((data[:n - h] - mean) * (data[h:] - mean)).sum() / float(n) / c0
return round(acf_lag, 3)
x = numpy.arange(n) # Avoiding lag 0 calculation
acf_coeffs = map(r, x)
return acf_coeffs

def getGlobalAggregateVolatility():

ntimes=P.shape[0]
nassets=P.shape[1]-2

aggvol=np.zeros((ntimes,))
numnonna=np.zeros((ntimes,))

for k in range(2,nassets):
r=np.diff(np.log(P.ix[:,k]))
for j in range(2,ntimes):
if not np.isnan(r[j-1]):
v=np.log(r[j-1]**2+1e-6)
if abs(v)0:
aggvol[j]=aggvol[j]/numnonna[j]

daggvol=np.diff(aggvol)
n=len(daggvol)
#globalaggvol=pd.Series(P.index[-n:],daggvol)
globalaggvol=daggvol
return (globalaggvol)

aggvol=getGlobalAggregateVolatility()

def GARCH11_logL(param, r,xreg=None):
if xreg is None:
omega, alpha, beta= param
else:
omega, alpha, beta, gamma = param

n = len(r)
r = r - np.mean(r)
s = np.ones((n,))*0.01
s[2] = np.log(np.var(r[0:3]))
for i in range(3, n):
s[i] = omega + alpha*np.log(r[i-1]**2+1e-6) + beta*(s[i-1])
if xreg is not None:
s[i] += gamma*xreg[i-1]
logL = ((0.5*np.exp(s/2) + 0.5*r**2/np.exp(s)).sum())
return logL

def estimate_garch11xreg(r,xreg):
R = res['x']
print("omega = %.6f\nbeta  = %.6f\nalpha = %.6f\ngamma = %.6f\n" % (R[0], R[1], R[2],R[3]))
return R

def estimate_garch11(r):
res = optimize.minimize(GARCH11_logL,np.array([.1,.1,.1]), args=(r,None),
R = res['x']
print("omega = %.6f\nbeta  = %.6f\nalpha = %.6f\n" % (R[0], R[2], R[1]))
return R
def residuals_garch11(param,r,xreg=None):
if xreg is None:
omega, alpha, beta= param
else:
omega, alpha, beta, gamma = param
n = len(r)
s = np.ones((n,))*0.01
s[2] = np.log(np.var(r[0:3]))
for i in range(3, n):
s[i] = omega + alpha*np.log(r[i-1]**2+1e-6) + beta*(s[i-1])
if xreg is not None:
s[i] += gamma*xreg[i-1]
return np.log(r**2+1e-6) - s

ntimes=P.shape[0]
nassets=P.shape[1]-2
for j in range(2,nassets+2):

stock =P.columns[j]
m = re.match(r'WIKI.([A-Z]+) - Adj. Close', P.columns[j])
if m is not None:
stock = m.group(1)

r=np.diff(np.log(P.ix[:,j]))

av = []
rx = []
dates = []
for l in range(len(r)):
if not np.isnan(r[l]):
dates.extend([P.index[l]])
av.extend([aggvol[l]])
rx.extend([r[l]])
av = np.array(av)
r = np.array(rx)

#print(np.mean(av))
#print(np.mean(r))

total_len = len(r)
block_size = int(total_len*0.05)

MSE_diff = []
RMSE_diff = []
for k in range(5,20):

train_dates = dates[block_size*(k-5):block_size*k]
test_dates = dates[block_size*k:block_size*(k+1)]
rtrain = r[block_size*(k-5):block_size*k]
rtest  = r[block_size*k:block_size*(k+1)]
avtrain = av[block_size*(k-5):block_size*k]

avARMA = sm.tsa.ARMA(avtrain,(5,0)).fit(disp=0)

avtest = avARMA.predict(start=0,end=block_size)

#avtest = av[block_size*k:block_size*(k+1)]

train_len = len(rtrain)
test_len = len(rtest)

theta1 = estimate_garch11(rtrain)
resid1 = residuals_garch11(theta1,rtrain)
resid1os = residuals_garch11(theta1,rtest)
theta2 = estimate_garch11xreg(rtrain,avtrain)
resid2 = residuals_garch11(theta2,rtrain,avtrain)
resid2os = residuals_garch11(theta2,rtest,avtest)

"""
print('In-sample')
print 'Dates:', train_dates[0], train_dates[-1]
print 'Sample length:', train_len
print 'MSE GARCH:', np.mean(resid1**2)
print 'MSE GARCH+aggvol:', np.mean(resid2**2)
print 'RMSE GARCH:', np.sqrt(np.mean(resid1**2))
print 'RMSE GARCH+aggvol:', np.sqrt(np.mean(resid2**2))

print('Out-sample')
print 'Dates:', test_dates[0], test_dates[-1]
print 'MSE GARCH:', np.mean(resid1os**2)
print 'MSE GARCH+aggvol:', np.mean(resid2os**2)
print 'RMSE GARCH:', np.sqrt(np.mean(resid1os**2))
print 'RMSE GARCH+aggvol:', np.sqrt(np.mean(resid2os**2))
"""
MSE_diff.extend([np.mean(resid2os**2)-np.mean(resid1os**2)])
RMSE_diff.extend([np.sqrt(np.mean(resid2os**2))-np.sqrt(np.mean(resid1os**2))])

summary_MSE_diff = np.mean(np.array(MSE_diff))
summary_RMSE_diff = np.mean(np.array(RMSE_diff))

#plt.plot(acf(resid1))
#plt.plot(acf(resid2))
#plt.title('Residual acf GARCH and GARCH+aggvol')
#plt.show()



## FIT IMPROVEMENT STATISTICS OF GARCH WITH EXTERNAL REGRESSOR AGGREGATE VOLATILITY

This is a typical example of a stock where GARCH(1,1) fit is improved by addition of aggregate volatility as external regressor.

Loglikelihood: Model 1 (7828.6) Model 2 (7832.7)

But Ljung-Box statistics also show slight improvements toward uncorrelated residuals.

[1] "Stock:  OEH"
[1] "model1"

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model	: eGARCH(1,1)
Mean Model	: ARFIMA(1,0,1)
Distribution	: norm

Optimal Parameters
------------------------------------
Estimate  Std. Error     t value Pr(>|t|)
mu      0.000200    0.000397  5.0389e-01  0.61433
ar1     0.294100    0.025826  1.1388e+01  0.00000
ma1    -0.201580    0.025466 -7.9158e+00  0.00000
omega  -0.042524    0.001039 -4.0915e+01  0.00000
alpha1 -0.052686    0.004754 -1.1081e+01  0.00000
beta1   0.993677    0.000005  1.9141e+05  0.00000
gamma1  0.055457    0.001874  2.9593e+01  0.00000

Robust Standard Errors:
Estimate  Std. Error     t value Pr(>|t|)
mu      0.000200    0.000519     0.38581 0.699640
ar1     0.294100    0.012892    22.81319 0.000000
ma1    -0.201580    0.015113   -13.33784 0.000000
omega  -0.042524    0.001893   -22.46280 0.000000
alpha1 -0.052686    0.010502    -5.01655 0.000001
beta1   0.993677    0.000018 54543.45910 0.000000
gamma1  0.055457    0.003535    15.68783 0.000000

LogLikelihood : 7828.589

Information Criteria
------------------------------------

Akaike       -4.4952
Bayes        -4.4828
Shibata      -4.4952
Hannan-Quinn -4.4907

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1]                    0.03168 0.85874
Lag[2*(p+q)+(p+q)-1][5]   4.09768 0.05171
Lag[4*(p+q)+(p+q)-1][9]   9.79919 0.01122
d.o.f=2
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic   p-value
Lag[1]                      9.111 2.541e-03
Lag[2*(p+q)+(p+q)-1][5]    29.419 4.609e-08
Lag[4*(p+q)+(p+q)-1][9]    41.969 2.013e-10
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale   P-Value
ARCH Lag[3]    0.2171 0.500 2.000 6.413e-01
ARCH Lag[5]   28.9211 1.440 1.667 1.181e-07
ARCH Lag[7]   31.8728 2.315 1.543 5.172e-08

Nyblom stability test
------------------------------------
Joint Statistic:  0.9779
Individual Statistics:
mu     0.05281
ar1    0.22028
ma1    0.21496
omega  0.03583
alpha1 0.07878
beta1  0.03372
gamma1 0.03592

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
------------------------------------
t-value    prob sig
Sign Bias            1.022 0.30675
Negative Sign Bias   2.316 0.02060  **
Positive Sign Bias   1.541 0.12329
Joint Effect         8.638 0.03452  **

------------------------------------
group statistic p-value(g-1)
1    20     184.2    4.584e-29
2    30     193.8    2.699e-26
3    40     215.4    2.857e-26
4    50     235.7    3.102e-26

Elapsed time : 1.381735

[1] "model2"

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model	: eGARCH(1,1)
Mean Model	: ARFIMA(1,0,1)
Distribution	: norm

Optimal Parameters
------------------------------------
Estimate  Std. Error     t value Pr(>|t|)
mu      0.000180    0.000297  6.0673e-01 0.544028
ar1     0.284376    0.025201  1.1284e+01 0.000000
ma1    -0.191546    0.024881 -7.6986e+00 0.000000
omega  -0.034393    0.000912 -3.7725e+01 0.000000
alpha1 -0.052627    0.004681 -1.1243e+01 0.000000
beta1   0.994796    0.000003  3.1656e+05 0.000000
gamma1  0.045328    0.003880  1.1683e+01 0.000000
vxreg1  0.021053    0.006909  3.0474e+00 0.002308

Robust Standard Errors:
Estimate  Std. Error     t value Pr(>|t|)
mu      0.000180    0.000316  5.6922e-01 0.569209
ar1     0.284376    0.017694  1.6072e+01 0.000000
ma1    -0.191546    0.009615 -1.9921e+01 0.000000
omega  -0.034393    0.001556 -2.2105e+01 0.000000
alpha1 -0.052627    0.014974 -3.5145e+00 0.000441
beta1   0.994796    0.000008  1.2690e+05 0.000000
gamma1  0.045328    0.012510  3.6233e+00 0.000291
vxreg1  0.021053    0.015404  1.3667e+00 0.171719

LogLikelihood : 7832.756

Information Criteria
------------------------------------

Akaike       -4.4970
Bayes        -4.4828
Shibata      -4.4970
Hannan-Quinn -4.4919

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic  p-value
Lag[1]                   0.002608 0.959269
Lag[2*(p+q)+(p+q)-1][5]  4.326456 0.027582
Lag[4*(p+q)+(p+q)-1][9] 10.677264 0.004678
d.o.f=2
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic   p-value
Lag[1]                       9.66 1.883e-03
Lag[2*(p+q)+(p+q)-1][5]     32.83 5.013e-09
Lag[4*(p+q)+(p+q)-1][9]     47.41 5.682e-12
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale   P-Value
ARCH Lag[3]    0.3833 0.500 2.000 5.358e-01
ARCH Lag[5]   33.4985 1.440 1.667 8.059e-09
ARCH Lag[7]   37.0384 2.315 1.543 2.196e-09

Nyblom stability test
------------------------------------
Joint Statistic:  1.4117
Individual Statistics:
mu     0.03548
ar1    0.18923
ma1    0.18499
omega  0.08700
alpha1 0.08519
beta1  0.07871
gamma1 0.02958
vxreg1 0.08641

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:     	 1.89 2.11 2.59
Individual Statistic:	 0.35 0.47 0.75

Sign Bias Test
------------------------------------
t-value    prob sig
Sign Bias            1.086 0.27746
Negative Sign Bias   2.394 0.01673  **
Positive Sign Bias   1.552 0.12084
Joint Effect         9.119 0.02775  **

------------------------------------
group statistic p-value(g-1)
1    20     179.8    3.434e-28
2    30     189.3    1.844e-25
3    40     216.9    1.566e-26
4    50     246.9    3.407e-28

Elapsed time : 1.359669