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The ‘hiddenness’ of the latent volatility in a stochastic volatility process is a $\log(\chi_1^2)$ variable with fixed mean -1.2704 and fixed variance 4.93. Therefore latent volatility is hidden in these models by a non-normal but assumed iid noise. To the extent these models have real validity, the latent volatilities are recoverable by wavelet thresholding; moreover compared to general applications of wavelet thresholding where one makes effort to determine the noise level from the data, here the situation is clearer. The efficacy of wavelet thresholding in this case would therefore have enormous benefits since otherwise extremely complicated Bayesian schemes need to be employed in order to ferret out the latent volatilities. Even a simple AR(1) process for latent volatilities have a complicated structure, while the truth is that that latent volatilities are far from AR(1): they are processes that are Mittag-Leffler renewal process events with long memory also fit by Mittag-Leffler functions.

In order to test our hypothesis that in fact a simple wavelet denoising to extract the latent volatility can be competitive with and perhaps even superior to the MCMC approach, consider taking training periods of a reasonable size such as 100 days and we extract the volatilities using an MCMC package ‘stochvol’ and use wavelet denoising for a reasonable fixed parameter, fit an ARFIMA to the resulting volatilities and then predict these, do a Monte Carlo simulation of adding iid chi-squared noise to both the predictions and then compute average MAD error versus the out-sample noisy log-volatility that is observed.

Using 1087 train-test pairs, the results are:

T-test result t-stat=2.42 and p-value=0.016

So the wavelet denoising strategy for extraction of latent volatilities is significantly better using out-of-sample prediction.

library(wavethresh)
library(stochvol)
library(forecast)

wt<-function(signal,noiseSigma){
thresh<-noiseSigma
N<-2^ceiling(log2(length(signal)))
x<-rep(0,N)
x[1:length(signal)]<-signal
wx<-wd(x)
wxt<-threshold(wx,type="soft",policy="manual",value=thresh,levels=3:nlevelsWT(wx)-1)
y<-wr(wxt)
y0<-y[1:length(signal)]
return(y0)
}

for (asset in 20:200){
px<-P[,asset]
r<-diff(log(as.numeric(px)))
r<-r[!is.na(r)]

N<-length(r)
for (h in 1:floor(N/100)){
train = r[(1+(h-1)*100): (h*100)]
train[is.na(train)]<-0
if (mean(train)==0){
next
}
nv.train = log(train^2+1e-6)

test = as.numeric(r[(h*100) : (h*100 + 9)])
test[is.na(test)]<-0
actual<-log(test^2+1e-6)

res<-svsample( train,priormu=c(0,100),priorphi=c(5,1.5),priorsigma=1)
mcmcHiddenStates = summary(res)$latent wdHiddenStates = wt(nv.train,sqrt(4.8)) mcmcFit<-arfima( as.numeric(mcmcHiddenStates[,1])) wdFit<-arfima(wdHiddenStates) mcmcPredict<-forecast(mcmcFit,h=10)$mean
wdPredict<-forecast(wdFit,h=10)\$mean

TestMC<-100
mcmcErrors<-rep(0,TestMC)
wdErrors<-rep(0,TestMC)

for (k in 1:TestMC){
rv<-rnorm(10)
pMCMC<-mcmcPredict+rv^2
pWD<-wdPredict+rv^2
if (length(pMCMC) != length(actual)){
print('Error')
print(length(pMCMC))
print(length(actual))
}
}
}

}


And the output

1.47896919817366,1.53248987309949
2.83134369625743,3.86431538614593
2.22895999588572,1.40247372568584
2.34252618969963,2.15804769583239
3.25909378642937,4.38088950179745
2.41330502593728,2.31134370058289
2.24838893541102,2.25575027659032
3.85539793988091,2.99265547081671
3.35081039285777,2.5533099151687
1.81885708053182,1.5508493788577
3.54055405394335,3.70704652684816
1.5928481873372,2.18907047262303
2.50951199562457,2.13347535962542
2.14772772826057,1.7808555398865
1.89003288607082,4.65735057557713
2.46037872354161,2.47163780189726
2.27919280212255,1.82577453903522
3.28063928950084,2.56364020678337
2.57362853012357,2.45623615120248
2.06106542754225,2.63069332100817
4.15748464416032,4.28248885277097
3.25427048165303,2.57394802318358
3.28853133856835,2.45767842140874
5.1312899281532,3.97040653313302
2.89111480475571,2.0263611884266
3.05491981532926,2.18247027417255
1.80712696614414,1.69822097329974
2.99602605818544,1.49668623113045
6.07747629305919,6.43177734846413
3.63189506504397,2.99611077828403
1.61106734901743,1.72525970298342
2.56544985343049,2.1765260096596
3.01746834653955,2.43408886046404
1.90558179208973,1.49294059017111
2.23816009420754,3.03779945362919
3.58658946155424,2.94362481315334
2.73675131723866,1.99229095393638
3.73547410346201,2.90993723915632
1.68403123256198,1.56643459025857
4.02912676538347,3.66429799404696
6.45349701037754,5.39419446854369
1.7196279590705,3.3497488450682
3.02535554871762,4.02637853337132
2.85514825950295,3.35974901417167
6.02780814757445,5.99757549266201
3.05057606730386,3.57284345925419
5.40045828323327,4.93225485077024
6.65901349478913,5.88914889705797
5.02703438595377,4.6771864479822
6.31908475362921,5.37812263224521
5.64558118569884,5.14140311509076
7.19290658487827,4.39964047511048
5.36007587126559,4.17246910932878
6.27464290346361,3.13667732259998
2.48351429415608,2.01845821201449
2.404256524966,2.41847928132544
2.66733239648828,2.48753472789971
3.90546054304951,3.61557133697651
8.94081587636452,2.70992160715697
2.98173294473157,2.48740036026424
6.4128357883718,4.79698148223911
4.35834023944629,5.25167456030447
5.10308385074464,4.18131057718019
6.00690570904748,4.78189920458488
2.7792580841779,2.2147969572506
4.29437097127112,3.89151945718489
2.74066430454632,2.69013225821135
2.31841528961568,2.96685443685653
1.15457652304712,1.36952925354556
2.9133423760939,2.90107958768886
2.73017170204275,3.0055882716364
3.06088638353385,1.94974683926638
4.83708872682408,4.40960422949736
2.50399327737463,2.43581437409861
4.04901967569868,2.56499015563667
7.61560350031261,4.18136508817036
3.27014730206875,2.61660702525406
3.50815034616226,3.12362502353043
2.09892451134452,2.08502722803942
2.30151439005615,2.06178464154152
2.31960239454384,2.28103432808001
2.81311911383918,2.92793018890476
1.99792676848548,1.87833422183702
2.74859491015392,2.45148778083546
4.23624000314221,3.35544350864707
2.60848728708755,3.10567416261325
2.27909328479296,2.49721803146902
4.30268752174684,5.05035816523427
3.1952775883174,2.44885432551067
3.643470677864,3.20015663351038
3.61179634299827,3.16581715809971
3.49614847932579,1.91129169306089
2.0734661945972,1.95559187487068
1.74800571110831,2.22507010226532
3.1263919115188,2.73212839051686
3.56511023626617,2.93555057359447
6.58895952179747,6.24514884436676
1.68827194125021,1.60866370248529
2.75497260832765,2.98502450201231
3.17716181605832,2.57389179433954
2.2874259153494,2.4535965608119
3.5806076837657,3.36147225687832
1.31236522073045,1.3470546219149
1.80600048786272,2.31939067964755
2.34775511939048,2.15815052812565
2.1890147708196,2.52984575352792
1.66320240210117,1.70715492648706
1.42192103709853,1.53046658139293
2.92417656406398,2.16121979823263
3.22697052198493,3.44833150754706
2.11345405679157,1.61862366919121
2.32918472309045,2.54697261112172
2.47231662842446,2.22812416078116
3.25558069059303,4.48276621913833
3.47628899035327,2.98356085706936
3.40783015287105,2.83137228635851
2.40567449349608,2.53145401944539
4.59613391268616,3.17184748601106
2.62045922645174,2.32647569936611
2.2559676551243,2.39222915449781
2.30773282311643,2.17948973925447
3.28195944490531,3.60873986244784
2.35806096612573,2.91672477049403
2.5856365472369,3.53211556523322
1.49392003572666,3.51879564854815
5.5101632576974,5.5519831413872
1.43874114530149,1.59154112273297
2.66918071106503,2.77384191548264
3.69465553253131,3.7660936813822
5.91734552853196,5.35170254531402
5.45738386932877,4.93119009803274
2.28292268423337,2.54467801325201
2.12149523629975,1.84515035209491
4.12101931548107,3.76070695296071
2.93758313695375,2.91937717012724
1.99492832802114,1.60025254165133
2.00949269658656,2.42335520321734
3.34293650287524,3.54958506024335
2.79170020605621,4.07415872108913
4.18812648255588,3.61331365582349
2.17035689672186,1.49895739600081
3.05538295273537,3.48599807223708
3.11802415568591,3.37261137015334
3.98179642900029,3.50914571737772
2.16558491843661,2.28607144403419
2.536522869343,2.64662714718374
2.37240544868388,2.38431849924459
2.66489472180521,3.05400768906907
1.61319181671731,1.91185645457852
1.94678287457765,1.83055525092355
2.1204045701288,2.31404357346926
1.94053356483478,2.63154620182698
1.77203850605227,1.58728741583759
2.52809283473495,2.20966603089728
2.36971180745729,2.0181101035147
1.45187100648005,2.12789630022156
4.0726959355702,6.7417222770994
6.47079348901226,6.24363355264461
1.52225967767624,2.46889633587448
2.52810423481896,2.39485569713026
4.26963004267459,3.93744788150771
3.65694211385627,3.06888778595032
2.03625963644321,1.84343257589442
1.93192910466474,3.68538742603919
1.97853899764817,1.73832898680558
1.32559656132237,1.32311313887091
2.45017831755948,1.8202483475671
2.41332308055724,2.21768930012699
1.78757855029856,1.79171559427785
3.79942982815602,4.31504843987264
4.25613935502731,3.45666561831428
2.01502426266838,1.98609845342725
2.60821875678778,2.47901477417601
3.47154302855466,2.56918664081698
2.154861134236,1.69493658824206
1.84395537431805,1.62188230403303
2.15643692656461,2.22848631681616
0.830048045577212,0.555371241521139
2.77409565491871,2.93625598287427
4.53096835770181,3.88268199234647
1.73235805420186,1.66325758372559
2.67375888866422,3.60028285734727
1.86981543743995,1.97181238907453
4.02241239573579,4.49010194895259
4.40736296423539,2.9613335042534
4.03055870650466,4.26750464287278
3.69557594606406,2.58434895205789
2.78168283479071,3.39277991355058
3.99666806035856,4.40599400582455
4.21674213348585,3.64628023128394
3.23630815881643,2.35833747035611
2.25585241678889,2.34076635342595
6.12206726637331,5.35253869728921
2.66882500298433,4.63392820092581
4.38132217738336,3.65490714938356
3.69056287211142,3.64262105673265
1.86108436449406,1.86083078867473
1.57444751851129,2.61488641739102
1.93091507221429,2.82705602575858
1.6811043389075,1.6572150859393
2.52222059109866,2.24638363051184
1.69351423763751,1.98703957825019
2.50619287681743,2.47437628079814
3.29540425080904,2.80010167518228
2.01065218264204,1.9978828149863
1.582629636184,3.23711396488724
4.12062894476099,3.64804905502572
2.52877264827943,2.26426127494502
2.50715946161537,2.69495190937501
2.95951501872303,2.74660520385552
1.48097667081893,1.57933979772214
2.51062770994443,2.35957073727342
1.43145480167695,2.37144725082913
2.80501026663544,3.01672256277706
2.62690262161634,3.41478702410781
2.2404580085211,2.19958965244802
1.97148945966706,2.50836277339432
1.94506375336177,3.13375001252038
2.03591439976491,2.89066768327183
1.95274002061521,8.64918292072036
2.77058402433988,2.17603774010222
5.08830904429739,5.27766342866582
3.86915213849657,2.68498392760378
2.88370313333846,2.50830911727422
1.65147185177558,1.72198632180178
3.11094741449213,2.7837760025481
3.41090201356458,2.79023665681981
2.35319239102626,2.05721165813088
3.21105714017193,3.14108238540999
1.39392441913091,1.65290771607204
2.90655562487831,3.2877024190415
1.42490535078785,1.6501543619209
1.3577771351886,1.25413445467594
7.50777861814651,7.0536322067628
1.85572201047714,2.03946994798274
4.59270403164154,2.86991601176775
2.39240364672098,2.07684042865528
4.77486429619296,6.03404786178146
3.35139055035186,2.8401218371483
2.4259556466565,2.40040537195455
3.49158033074596,2.57750383900116
2.06963922038477,2.0469230726472
2.53628442864519,2.57375478410754
1.96982176320651,2.0025350686939
3.06889294473978,3.38211010477709
2.30593506650299,3.29085775370575
5.93466892510843,5.18724817034949
1.93703455569978,2.07542717362355
2.71166236021821,2.85451218018173
3.34762935380111,3.03057015934289
2.96117372266529,3.02519886971418
2.4928237334147,2.81991422579186
3.6271413203701,3.04825056596724
2.80087211464019,2.31396531318258
3.77078758093179,3.47876621189097
4.26967522051374,4.02141944271046
4.61507323041307,4.97023197117376
3.90522916948397,2.65655241442214
1.58524886690219,1.72466406821389
6.24342464044254,5.39812198260807
2.42837251265512,1.74965533506633
1.87722531722124,1.76085864934322
1.74106557607517,1.59928598152403
4.61417829401929,3.74059211129816
4.45856981573798,4.52661841774844
4.05420571424447,2.40476375519856
2.56378654891378,1.92175345678096
2.18748081172188,1.46961152434916
2.88847876718415,2.58219785614154
3.25876565458503,2.96469400355307
1.86456128800601,1.91565027958908
2.00298519449146,2.09522036463646
3.53298938065011,4.55276900922015
5.03983400727809,4.35310435079836
2.34953917123294,2.35378752640089
1.99044140077205,2.10794327828046
3.49369302995368,3.46669767563601
3.70182477884759,3.36438917912098
1.99031532934505,1.39013526260388
1.75986299302368,1.88304706889502
2.32349174703106,1.87151388291648
4.46752973308831,4.41844690305598
3.91126357640502,3.59915670499911
2.50451440914921,1.89409277409554
3.48686440509345,4.04414028572705
1.59173453499115,1.23509102166682
1.47000352251464,1.51141489358082
1.85164025468718,1.65829327700502
2.55738518990527,2.38231350733313
5.13533636259732,4.49042437344373
4.15380790242539,3.38753571930079
1.74124380061773,1.73784507087716
2.44311401725119,2.64618225505332
2.9806484739449,3.1136843742112
2.94330078625799,2.17402966285467
3.15128720555895,2.9291552168846
1.17026515594751,0.931157012055869
3.04550203217785,2.49464243808175
3.53667756676872,3.28885088868343
2.01673746479512,2.0135201645347
4.81139085944405,4.25567251801367
3.37182774530472,3.56653643695824
1.61807131869526,1.49678319606526
2.26221891980924,1.87291041937125
4.2288390958702,3.17209671822244
3.122162452215,1.59837507950032
5.75852163867321,5.19170304184994
4.45194178612744,3.22559319518118
2.90451291665791,2.49961336002958
4.35628779806356,4.4965974249577
3.21062515356784,4.3916215069514
5.26391930671542,4.88648565218079
4.60102100851032,6.2638327596687
4.96346229256352,4.27817038185787
6.13808061798978,5.0799095318636
6.1221612988362,5.31860020801423
3.69627979933583,4.06488984184503
6.67964922484997,6.17548728248105
5.56744237994958,5.17342292247547
4.05561858128231,3.78383299545324
7.43906402573334,6.97918784063581
4.19092161659621,2.24661564992099
4.05915495963671,3.0690774298103
2.49028747482259,1.13421019121094
4.51719172183188,3.83242292551366
3.15104666090991,2.19921419323639
1.4871874177378,1.80393344633194
1.64673864555095,2.40073188205244
2.69840909528726,2.85215598740246
3.35939706913054,2.44079539437039
3.23070600999296,2.17347434635421
2.71310838442565,2.16241200581226
3.08622269006314,2.93106915690073
2.39112453358314,2.0999215589507
2.28847562862574,2.07051408531573
4.15874543737101,3.88814285752032
3.41828274195684,2.9962825795621
3.33596709013642,2.81525293307169
1.26521563623329,1.46158020940093
2.29793675688804,2.28086912505162
1.95989816005673,1.71928074005869
1.79666748097412,1.76366301807643
1.89225010718829,1.98985872823967
4.01250936851209,3.76371244957796
4.94371586782316,5.30310983048937
2.02876167685723,1.59815728878755
6.17729618327234,4.60797190547929
2.46123498198058,2.69883719701521
3.23432241183099,3.22197851290559
3.66067886008236,3.84028780667792
2.54388971712908,1.81509416650564
3.00015525193796,2.56604809204359
3.11653407422523,2.977256504224
3.13713088145394,3.88089143564743
1.12627902826424,0.751011919167401
1.86668708449708,1.62884705249726
2.4014356870633,2.96173241235666
1.82259697353404,1.64259369720653
5.96815193430678,4.62076655577998
1.92665485261063,1.53460233869318
4.36062638609736,3.9422093420751
2.58871773476229,2.55754995913076
1.98946613886637,2.06427488130629
2.42004591456544,2.68424200492419
2.4122662342569,2.64686902479001
2.33370171256776,2.25980906052577
2.71696778845336,2.21916198012439
2.72615518461027,1.7903348221601
2.29894649308644,2.1916812934429
3.29911850405164,3.51379250933281
1.69101430976879,1.5739032312271
3.55781712686167,3.06442661040772
2.34452643640097,1.94059383347553
3.51935282103947,4.08681375743433
2.27204821645305,2.65771142354259
1.9817838653618,2.19793099916201
2.49632086029051,2.58860908769207
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