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## KURTOSIS CALCULATION FOR VOLATILITY MODELS

Both GARCH and SV models can explain excess kurtosis of returns, and I have to spend some time absorbing the known results.  A couple of references are bai-russell-tiao-kurtosis-garch-sv-2001neto-sardy-2009-moment-structure-sv-models

Our model is $\epsilon_t = e^{h_t/2} z_t$ say with $z_t \sim N(0,1)$:

$h_t = \omega + a_1 h_{t-1} + \cdots + a_5 h_{t-5} + u_t$

and we want to compute the kurtosis of $\epsilon_t$, i.e., $k = E( e^{2h_t} ) /E(e^{h_t})^2 -3$.

If all moments $E(h_t^p)$ are known then $k$ can be computed using series expansions for exponentials but this seems to be a difficult problem: the fourth moments of GARCH were obtained by effort here:  fourth-moment-garch-he-terasvirta-1997.

While closed form expression for kurtosis of these models are difficult problems, it is quite easy to compute the kurtosis by Monte Carlo simulations, which tell us that the excess kurtosis of the model over 1899 stocks is 17.8 +/- 1.4 which is thus producing leptokurtic returns (albeit not with the wide distribution of the empirical kurtosis of returns). Here is the code and the results for the Monte Carlo simulations.

library(e1071)

nassets<-dim(P)[2]-1

empkur<-rep(0,nassets)
modkur<-rep(0,nassets)

for (k in 2:(nassets+1)){
px<-P[,k]
px<-px[!is.na(px)]
ret<-diff(log(px))
ret<-ret[!is.na(ret)]
vol<-log(ret^2+1e-6)
dvol<-diff(vol)
fit<-arima(vol,order=c(5,1,0))
arcoef<-fit\$coef[1:5]
mc<-200.
k0<-0
for (j in 1:mc){
x<-arima.sim(n=1000,model=list(order=c(5,1,0),ar=arcoef))
k0<-k0 + (mean(exp(2*x))/mean(exp(x))^2-3)/mc
}
empkur[k]<-kurtosis(ret)
modkur[k]<-k0
}

x<-data.frame(empirical.kurtosis=empkur,model.kurtosis=modkur)
write.csv(x,"kurtosis.csv")


Results:

"","empirical.kurtosis","model.kurtosis"
"2",26.4401155613586,18.7529818191991
"3",17.7423451453114,18.1404263239723
"4",15.4264842600084,17.773693037547
"5",6.77989932126417,18.2782949501775
"6",14.1573635600717,17.1869781219527
"7",456.327010537305,16.7203646675722
"8",305.101841920518,18.2690575432948
"9",86.4851516898707,16.8813043221515
"10",148.007201390213,19.3048397699599
"11",14.240404135734,17.3314166346185
"12",60.7891612670823,17.1839615411393
"13",5.5170069833091,15.7451323047029
"14",40.8353705525658,16.8493310612957
"15",41.7314322038999,19.3491746550855
"16",22.4738010306624,18.0640214746709
"17",2.90280840422309,15.232147198132
"18",13.7643155427433,20.2128492610045
"19",20.6065274849763,17.2830683188961
"20",311.47735920832,18.1870278049535
"21",7.43570657635023,16.3184964411966
"22",546.510516097041,18.1937872764236
"23",123.293630853091,18.6401048471549
"24",13.2386737403284,19.337053331165
"25",27.9542463901887,18.0841383642117
"26",18.0534296396788,17.8162100273013
"27",71.9732939924976,18.1569930347401
"28",610.179242109168,15.7089348242905
"29",18.3618484233274,20.7929833813786
"30",860.655432659562,17.3673905294521
"31",14.6704614820904,18.0161989429543
"32",201.451057653535,15.1887790510557
"33",194.727553438724,18.5080145237017
"34",15.5394793442236,18.1843685360715
"35",185.99956892959,16.4627487825997
"36",3.79231241883557,19.7442653111776
"37",786.764048726459,20.1649890110597
"38",13.8573249545384,19.0321098201975
"39",863.739690187788,20.0648868543416
"40",27.1255415748999,16.1292131667406
"41",6.66475590697811,18.6135808606823
"42",262.327549890896,16.0246738471912
"43",68.6049262595318,16.1681425733801
"44",52.7201851182236,18.2342919137273
"45",1130.30036536613,16.297434463594
"46",8.05005711890433,18.1628323138486
"47",35.2107572230256,18.7471297998356
"48",601.643035328795,17.801569631641
"49",298.63371248673,16.1155777529125
"50",107.968126535306,17.3368624466855
"51",37.5297845974497,18.1905619304773
"52",505.841242263303,18.3534103624197
"53",74.3720690066451,18.9257298323448
"54",15.0029812978896,15.7001569117733
"55",8.25426917691529,17.9278949589763
"56",2590.44832063732,17.5646000840113
"57",823.372346853171,15.4811652634612
"58",4.53837833050591,21.1218900891219
"59",204.589851977049,20.1904641126913
"60",225.053187733636,19.7972281531899
"61",23.721934443037,18.4884107931119
"62",13.0020890211704,18.266944598958
"63",90.6500178093668,17.5209027587785
"64",363.458956619624,17.9928502195812
"65",170.268796857903,17.1035736585524
"66",69.2677890225941,17.748400561456
"67",5.26510159014945,17.8241884419259
"68",43.1259217047267,17.2974246662326
"69",81.2178212238137,18.9425136846353
"70",371.375427329253,18.7738787233426
"71",87.1788394424975,17.1914646075917
"72",336.186940037519,20.1769887937873
"73",392.874747604422,18.6924041971588
"74",8.18180079836719,15.4541910274848
"75",41.265722465468,16.4631344187791
"76",5.45288520300233,16.7161936856214
"77",109.927324634606,18.759668188963
"78",80.8737553979839,19.484797507813
"79",47.6749209401553,18.1145451514662
"80",23.0197444514101,17.2763712812183
"81",262.965581818031,16.9701765246795
"82",20.0610324151596,16.3077172541879
"83",30.0516791892206,18.069030612185
"84",38.0468959562722,16.495685440052
"85",1104.79983563375,17.791894261915
"86",97.3327876108996,18.153362892595
"87",33.5301136212197,18.1677583488307
"88",56.5362828657774,17.079961668346
"89",67.8318737057709,19.9826225901241
"90",17.9568586287405,17.5884789600866
"91",62.9058173576182,16.8864760533548
"92",160.482990103997,16.597352269854
"93",17.3999288007368,21.2274815002786
"94",12.2605395584168,16.870861394776
"95",37.9247479465047,17.918333873584
"96",130.133233085241,21.6477170485067
"97",22.8450654685861,16.922957525833
"98",60.0147531238848,17.6064248882273
"99",13.1268952823193,17.2834451412982
"100",37.9791481233762,18.9533947466444
"101",16.507833814418,16.2546468775442
"102",67.5475422430421,18.6068798765646
"103",9.66808961563533,17.9622727937415
"104",14.4550966668691,16.972619811957
"105",229.286484791371,18.4002268333278
"106",141.019273636185,17.1331560160195
"107",165.245622928388,16.9152424477546
"108",22.1271352564578,18.0684997329376
"109",40.7107202199552,17.1310444364357
"110",18.0181466666526,19.7505814851298
"111",83.3635096273159,15.4484909333615
"112",534.3405542902,16.551723272519
"113",27.1816048999984,17.4827661608702
"114",378.998444770996,18.49755732112
"115",10.7793632707645,17.2381610202856
"116",7.51333685043557,17.6935620774628
"117",12.5571276456222,17.0434299878841
"118",46.0325938959728,20.1087206967856
"119",593.600592395682,17.1180581882815
"120",10.5283973939453,17.4314898833151
"121",24.872224876356,19.6346873535767
"122",4.34752045600809,16.6929751926109
"123",1050.31669304156,19.5976245847742
"124",107.654674028668,17.2875865060761
"125",7.65409822274872,18.9737244774882
"126",6.7374610434134,20.3418112920437
"127",67.618884890287,20.0533736375691
"128",1000.46498549977,16.6827786207506
"129",6.48859974696685,15.5696879380854
"130",177.061200274659,17.0956811065814
"131",19.1865667610179,18.512761248098
"132",199.462967093143,17.8546797987503
"133",42.5230594772706,18.1596975016719
"134",13.1969043422413,16.0308076294345
"135",540.545035867832,16.4166984214589
"136",72.0422891122961,16.5252930258095
"137",81.4008810082622,16.6530162583709
"138",265.231446317467,20.0821580742163
"139",11.8632927158608,17.1370817491115
"140",162.761551047577,17.6376081709659
"141",49.5010828592714,17.8457217009623
"142",25.7322747788015,19.0671169425311
"143",7.99691582422836,19.58335865314
"144",35.6414486209249,17.9592178131647
"145",92.9349194083902,15.8509405010592
"146",81.8553254885984,16.1657468929317
"147",6.67272274924178,19.2688430860861
"148",355.995889836443,17.5914853109589
"149",49.9475874725858,17.4562390567479
"150",275.734477222682,17.9174718754496
"151",23.0643605192318,19.3369746964541
"152",13.1236440687043,17.4057068089789
"153",299.878898297382,17.3345290223628
"154",213.355126217816,17.9315723873561
"155",27.7266493535681,17.0103224548418
"156",25.0555543068913,18.1460386566931
"157",9.07526669392141,19.8017683386239
"158",298.824564145688,18.4453133145509
"159",21.3594262961121,18.5438912718123
"160",106.365499801128,20.6924540003817
"161",309.610166949824,17.97324956875
"162",29.6602190733288,19.8000975153363
"163",66.7053837115715,18.2935393228627
"164",32.452518771302,18.1924663199682
"165",45.3598541602905,18.2548461618266
"166",5.15034644153739,16.126652744685
"167",47.6830443287765,17.3489847768681
"168",78.195457968597,16.983681708226
"169",4.45621037057718,14.8440381718162
"170",467.53025439993,18.1501376511635
"171",10.5376086809857,19.0741275793414
"172",319.844323985103,18.1802742943647
"173",61.3636157199462,16.4513216147112
"174",25.0654575867128,18.8194791465329
"175",198.208361516843,15.766621490208
"176",12.5901884854908,17.9970916247768
"177",12.3638972771135,15.0841163849569
"178",110.454676425179,16.4952135578119
"179",162.724865760688,15.7390470741275
"180",88.4718701788367,17.5290787844372
"181",3.39506038025227,20.151320308316
"182",96.105593183004,17.1264154552002
"183",59.9388456461345,16.8225047067074
"184",241.239544080115,17.9731786407104
"185",319.914853978807,18.828376220993
"186",369.915360534401,17.9084136386598
"187",4.692240695771,17.0194945061468
"188",127.372683750428,18.4572738164838
"189",11.3068151360641,17.1484594150823
"190",40.0921491279415,17.635982211471
"191",13.7292448697246,18.3115005349015
"192",476.896956265125,20.7575893374634
"193",12.6468556383052,18.7886520116126
"194",244.071368029134,19.0003396945314
"195",83.4018824066381,16.3548404083518
"196",42.6958905698459,17.139692242462
"197",23.00103987723,16.5243501172767
"198",6.52852842827238,16.8031224982384
"199",122.701050426078,18.4129512498686
"200",25.6750479813396,18.6633051377318
"201",193.200426977763,18.47633025655
"202",70.4551581936006,18.0720002636823
"203",275.321907937715,17.734565502585
"204",193.408441908558,17.4285600099466
"205",20.2770860505612,17.1847760544668
"206",72.2789964318856,18.345782791849
"207",21.2772325228586,18.1254166879483
"208",12.3734487216739,18.4776128379127
"209",5.30877544176169,17.2164288517274
"210",33.4623471662203,17.239663170867
"211",12.8007326800775,18.3063498455395
"212",53.6194503460461,15.0339599179384
"213",450.486413996644,17.3229647983741
"214",15.1146811040571,22.8850644806016
"215",7.56803793846048,16.3013975912118
"216",8.91476541351557,17.5996391563658
"217",45.3354659899564,17.5787195043191
"218",47.0547139437488,19.8779511567789
"219",47.6454822590271,17.9260756546378
"220",378.906969487597,17.2657966585067
"221",37.5031843089402,15.373398211884
"222",189.849436119462,17.4862708211668
"223",12.9477264389568,18.2164883381414
"224",229.044940323365,20.3622559268587
"225",5.31023977858596,14.543504248524
"226",401.609886767448,19.598984485156
"227",50.1180158102762,16.2713592618804
"228",19.6602493604201,18.9469051287608
"229",343.736271624397,15.086249078932
"230",92.1755595080491,18.2402479906217
"231",133.17944131291,18.7284280433783
"232",98.3564852672153,18.4382159230171
"233",17.9233485983842,16.2614992323084
"234",6.22679661044056,17.2353336031413
"235",573.891849066365,18.3674800863611
"236",19.6572513432751,16.4662785640164
"237",48.4083586448125,17.9112029101469
"238",35.3213359898842,18.1196278894881
"239",164.766174726635,18.5036074872479
"240",32.3933496657026,18.6157607824377
"241",7.10446341289931,19.1428178832935
"242",294.850096301857,17.2599983774381
"243",22.2031392185579,17.7878739849236
"244",32.1939831065011,15.7847564575304
"245",44.413615961301,20.2352260887722
"246",486.313042450769,18.147195006878
"247",15.5064445364473,16.5007791078663
"248",17.1041278537453,17.0955896954606
"249",643.708728215706,19.7849379567738
"250",35.1361090391305,18.7509803745095
"251",10.7083840186922,19.0640521458264
"252",19.6811896256071,17.6990000150484
"253",41.7868941468266,15.8790426134084
"254",12.4086893188979,17.0554328638449
"255",208.950227176,16.5746667324991
"256",55.4216551612767,16.2704830114604
"257",7.34071245818453,18.0795781950958
"258",52.9295029370309,17.1281991214704
"259",210.940214910177,16.4299296864853
"260",4.50848738912854,15.4957349723291
"261",96.5826760125625,16.3161426539726
"262",104.022223175389,16.2910545043106
"263",14.1008360649472,18.4567687588093
"264",47.1132218749778,18.8291769490503
"265",10.2507419975868,18.9015367831136
"266",302.322994876868,17.1320338689614
"267",225.678086569646,16.8224817912896
"268",55.2975011973453,16.9309430617278
"269",7.32450357793931,16.2235596781271
"270",197.189852773362,15.9199599212289
"271",9.27607307624735,17.9421864395493
"272",14.2427165560279,17.6943754438077
"273",73.1792657939403,17.1244263520052
"274",16.3433146752549,18.5839697992971
"275",330.019270029656,17.7882193044289
"276",42.7601111131445,16.8046944315763
"277",24.9135258975409,17.34751271658
"278",14.7387514017351,18.5033872169669
"279",123.535342130554,20.1996866904318
"280",336.45477742824,16.5739826047436
"281",14.8452775416628,18.7065288029873
"282",1599.78412444969,19.2137552088898
"283",10.3214924710851,18.8867937516067
"284",121.189909807,16.5268334304987
"285",36.7696879658506,17.4005864624091
"286",21.53994011633,17.5453178794676
"287",50.2935329114759,17.9520236340507
"288",851.003309406863,17.4198369940989
"289",134.06349559175,16.7907583105875
"290",13.6494922825758,16.5350147671577
"291",22.4823114957662,20.0254153164646
"292",37.2579718816867,18.129825930516
"293",18.9462390426507,17.6510290929714
"294",23.4420098097103,18.2344210353852
"295",20.1998139251947,16.5289307674931
"296",37.0138961471904,19.2703035300832
"297",14.1044353791935,18.0361899327163
"298",123.132785864135,18.5368334080644
"299",624.197471448321,16.8534431647273
"300",63.7317313998025,18.142021934543
"301",38.3276819713607,19.0651328933214
"302",129.375304947576,20.0357197044666
"303",14.77361815787,18.7498913487832
"304",5.46844916412814,16.3840118355842
"305",19.5416209785977,16.6005861714283
"306",75.0354416404014,18.2067106545662
"307",1553.33355602807,16.464422346568
"308",101.848418385273,16.717771240987
"309",26.6371731288495,16.5425781963258
"310",64.4035944681138,20.4980786111866
"311",121.942135326225,17.3664084001419
"312",11.2849804210376,15.7421010149756
"313",32.8282329073772,19.2026181085556
"314",28.3092150915734,20.6377827434365
"315",91.4162058573727,14.7526557782989
"316",8.81491631768526,18.1513210660328
"317",7.04961225974301,15.257667201516
"318",57.4824574201684,15.9317469399432
"319",20.3502689087787,15.6810660893726
"320",20.1216600811681,16.3468598843191
"321",504.427623453957,17.4968906821564
"322",54.5411125200912,16.8727965941535
"323",90.7964245755905,18.9217060938105
"324",6.37991655495264,18.4939815173692
"325",70.4362340801654,22.3504105350114
"326",17.98574500124,18.4804687844621
"327",219.008460402586,17.520208004014
"328",109.093437134699,18.0864593421073
"329",42.3387197487375,20.1284751117302
"330",18.7433668562906,16.6532108096101
"331",144.496341947652,15.8386675911417
"332",257.21861326479,18.4807910665421
"333",45.0981685273425,17.497302756868
"334",277.118996809735,16.1879968980005
"335",66.9388949782871,20.4985248359899
"336",182.040996782676,18.4813655028859
"337",9.94705029673265,16.9965267394377
"338",438.466655510799,18.6310112964054
"339",8.49843347348211,18.5190615768214
"340",37.5860361660362,18.9991830123848
"341",24.2539890437035,20.3189545089775
"342",58.588149555783,18.3372864977718
"343",143.098153100445,17.9415447688752
"344",12.9914770125373,18.9492848981964
"345",12.8454720012312,17.9864632721984
"346",6.97252211174786,18.3262799507588
"347",7.82736180704722,15.9640519182922
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"349",9.38073960477366,19.9631557307585
"350",388.468137696534,17.1111608777934
"351",35.2548273549412,17.912856814241
"352",15.7996594515549,18.3743188007946
"353",394.634128726892,19.1420837801293
"354",365.53509095947,19.0281489735076
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