Realised Volatility Forecasts for Stock Index Futures Using the HAR Models with Bayesian Approaches *

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1 DOI /s Volume 18, Number 1 Mrch 016 C h i n A c c o u n i n g n d F i n n c e R e v i e w 中国会计与财务研究 016 年 3 月第 18 卷第 1 期 Relised Voliliy Forecss for Sock Index Fuures Using he HAR Models wih Byesin Approches * Jiwen Luo 1 nd Lngnn Chen Received 5 h of Mrch 015 Acceped 6 h of November 015 The Auhor(s) 016. This ricle is published wih open ccess by The Hong Kong Polyechnic Universiy Absrc We invesige he relised voliliy (RV) forecss for he shor, mid, nd long erm by developing he HAR models wih Byesin pproches nd employing he high-frequency d of he Chin Sock Index 300 (CSI300) fuure for he period from 16 April 010 o 1 My 014. We lso evlue he performnces of compeing models for boh in-smple forecss nd ou-of-smple forecss. We find h he proposed HAR-ype models wih Byesin pproches cpure he ime-vrying properies of prmeers nd predicor ses. We lso find h he HAR-ype models wih Byesin pproches hve superior forecs performnce for boh in-smple forecss nd ou-of-smple forecss s compred wih he benchmrk HAR-ype models. Keywords: Relised Voliliy Forecs, Sock Index Fuures, HAR Model, Byesin Approches, Time-vrying 基于 HAR 模型和贝叶斯方法股指期货已实现波动率的预测 罗嘉雯陈浪南中山大学岭南学院 摘要本文以中国股指期货 010 年 4 月 16 日到 014 年 5 月 1 日的高频数据为研究样本, 通过构建结合贝叶斯方法的 HAR 模型对中国股指期货的已实现波动率进行短期 中期和长期的预测 同时, 本文对不同模型的样本内和样本外预测进行评价 结果表明 : 结合贝叶斯方法的 HAR 模型能够更好地体现参数和预测变量集的时变特征 此外, 与原模型相比, 结合贝叶斯方法的 HAR 模型具有更好地样本内和样本外预测效果 关键词 : 已实现波动率的预测 股指期货 HAR 模型 贝叶斯方法 时变特征 * We grefully cknowledge he vluble suggesions mde by he Execuive Edior of he Chin Accouning nd Finnce Review nd he nonymous referees. This reserch is suppored by he Chin Socil Science Foundion under Grn No. 14ZDA00, he Humniies nd Socil Sciences Foundion of Chinese Minisry of Educion under Grn No. 14YJA7900. 作者衷心感谢 中国财务与会计研究 主编和匿名评审人的建议 本文为国家社科基金重大课题 (14ZDA00) 教育部规划基金项目(14YJA7900) 及广东软科学(013B ) 资助成果之一 1 Jiwen Luo, Ph.D cndide, Lingnn College, Sun Y-sen Universiy, P. R. Chin; emil: jiluo@163.com. Lngnn Chen, corresponding uhor, Professor of Finnce nd Economics, Lingnn College, Sun Y-sen Universiy, P. R. Chin; emil: lnscln@mil.sysu.edu.cn.

2 Relised Voliliy Forecss for Sock Index Fuures 3 I. Inroducion Accure forecs of voliliy is cenrl for sse pricing, porfolio selecion, nd risk mngemen. The relised voliliy (RV) proposed by Andersen nd Bollerslev (1998) is one of he mos populr ex pos mesures of reurn voliliy nd is widely used in voliliy forecss. In conrs o he rdiionl voliliy mesures such s squre reurn, condiionl vrince wih (generlised) uoregressive condiionl heeroskedsiciy ((G)ARCH-ype) models, nd sochsic voliliy (SV), RV is more efficien s i is consruced wih high-frequency d h re more informive. In ddiion o RV, number of relised mesures of high-frequency d hve been developed in he recen lierure, such s he relised bi-power vriion (BPV) of Brndorff-Neilsen nd Shephrd (004), he hreshold bi-power vriion (TBPV) of Corsi e l. (010), nd he semi-vriion of Brndorff-Nielsen e l. (010). For RV forecss, Koopmn e l. (005) consruc he SV-RV model nd he uoregressive frcionlly inegred moving verge (ARFIMA)-RV model by incorporing RV ino he convenionl SV models nd he ARFIMA models, respecively. Engle nd Gllo (006) presen he muliplicive error model (MEM) h includes he RV in he GARCH equion. The heerogeneous uoregressive regression (HAR) models proposed by Corsi (009) re recen developmen in modelling RV. The HAR model for RV forecs is consruced by incorporing he ps dily, weekly, nd monhly voliliy componens ino he AR model bsed on he heerogeneous mrke hypohesis nd he HARCH model of Müller e l. (1997). The HAR model s rcble esimion, flexible srucure, nd superior forecs performnce hve resuled in severl exended models. Andersen e l. (007) find h he disconinuous pr in RV conribues lile o he voliliy forecs. They sepre RV ino coninuous componen (C) nd disconinuous componen (jumps, J) nd consruc he HAR-CJ model by incorporing hem ino convenionl HAR model. As n exension, Corsi e l. (010) sugges HAR-TCJ model by inroducing hreshold in esiming he vriions nd jumps, which provides more ccure forecs of he discree pr in RV. More recenly, Pon nd Shepprd (015) consruc he HAR-ΔJ model by incorporing signed jump vriion in he HAR model, where he RV is decomposed ino good voliliy nd bd voliliy so h he leverge effec in voliliy forecss cn be ken ino ccoun. Moreover, Corsi nd Renò (01) propose he LHAR model by considering he leverge effec wih long-rnge dependence in he HAR model. Bndi nd Renò (01) provide nonprmeric esimion in he coninuous-ime sochsic voliliy model wih boh jumps in reurns nd vrince, mking i fesible o idenify he ime-vrying leverge effecs in he HAR model. However, mos of he forecs models bove re bsed on consn prmeers nd invrible predicor ses. Due o exogenous shocks, such s policy chnges nd finncil crises, voliliy usully exhibis differen perns during differen periods. The prmeers

3 4 Luo nd Chen in he RV forecs models migh chnge over ime. In ddiion, here migh be model risk ssocied wih specifying single model wih invrible predicors over ime s here is considerble model unceriny. In oher words, he rnking of predicbiliy for predicor cn vry over he forecs horizons. Thus, forecsing voliliy wihou king ino ccoun he ime-vrying properies of prmeers nd predicors will resul in bis. More recenly, Byesin pproches hve been employed o provide more flexible wy of voliliy forecs. Wrigh (008) proposes Byesin model verging (BMA) pproch o ccoun for ps nd fuure srucurl breks. Liu nd Mheu (009) uilise he BMA pproch o forecs RV nd sugges h compred wih he benchmrk models, he BMA pproch improves boh poin forecss nd densiy forecss. Rfery e l. (010) improve he BMA pproch by inroducing forgeing prmeers ino he esimion nd lso propose new ypes of BMA pproches: dynmic model verging (DMA) nd dynmic model selecion (DMS). Koop nd Korobilis (01) use DMA nd DMS o forecs inflion for boh one-sep nd muli-sep hed forecss. To cpure he ime-vrying properies of prmeers nd predicor ses, we develop HAR-ype models wih he Byesin pproches nd use he proposed models o forecs he RV of sock index fuures for one-sep nd muli-sep hed forecss. We lso evlue he in-smple forecs performnces on he bsis of he Mincer-Zrnowiz regression (MZ-R ), he men squre error (MSE) loss funcion, he qusi-likelihood (QLIKE) loss funcion, nd he sum of log prediced likelihood nd he ou-of-smple forecs performnces on he bsis of he loss funcion of Pon (011) nd he model confidence se (MCS) of Hnsen e l. (011) mong he compeing models. The reminder of his pper is orgnised s follows: secion II develops he HAR-ype models wih Byesin pproches; secion III describes he high-frequency d nd sisics; secion IV presens he in-smple forecs nd ou-of-smple forecs resuls; nd secion V concludes he pper. II. Mehodologies.1 Relised Voliliy Relised voliliy is defined s RV 1 1 ( ) r j, j1, (1) r where δ is he smple frequency of he RV nd j, is he 5-minue frequency reurns clculed by r j, 100 (log P j log P( j1) ). Brndorff-Neilsen nd Shephrd (004) presen jump-robus mesure of RV clled he relised BPV o obin robus esime of jumps. BPV is clculed s

4 Relised Voliliy Forecss for Sock Index Fuures 5 1 1( ) 1 j, ( j1), j BPV r r, () where 1 1 E( ) is he men of he bsolue vlue of rndom vlue Z, Z~N(0,1). When δ 0, ( ) 1 BPV 1 sds, where s is diffusive càdlàg process bsed on he d generion process of price s deiled in Appendix A. Corsi e l. (010) propose n lernive esime of he inegred powers of voliliy clled he hreshold bi-power vriion (TBPV) which provides less bised esimes regrding he sndrd muli-power vriion of he coninuous qudric vriion in finie smples. TBPV is clculed s 1 ( ) 1 ( j1), j, ( j1), ( j1) j, j j TBPV r r r r (3) where I{ } is he indicion funcion, j cv j, c ϑ is he hreshold-djus consn, nd Vˆj is he non-prmeer recursive filer for clculing he pril vrince. On he bsis of Corsi e l. (010), we se c ϑ = 3. We clcule he C_ Z nd C_ TZ sisics of Brndorff-Neilsen nd Shephrd (006) nd Corsi e l. (010), respecively. ˆ C _ C_ T ( RV BPV ) RV TriPV ( ) 5) mx(1, ) BPV ( RV TBPV ) RV TTriPV ( ) 5) mx(1, ) TBPV, where TriPV r( j1 k), j3 k1 nd ( j 1 k), ( j 1 k), ( j 1 k) j3 k TTriPV r r Boh C_ Z nd C_ TZ follow sndrd norml disribuion. The RV is divided ino wo componens: coninuous componen nd disconinuous componen (jumps). ˆ J I( C _ Z ) ( RV BPV), Cˆ ˆ RV J ˆ TJ I( C _ TZ ) ( RV TBPV ), TCˆ RV TJˆ (4) Brndorff-Nielsen e l. (010) use he semi-vriion o sepre he RV ino posiive pr nd negive pr by considering leverge effec: 1 j, j, j1 RS r I r 0, 1 j, j, j1 RS r I r 0, (5) where RV RS RS. Pon nd Shepprd (015) define he signed jump vriion s: J RS RS.

5 6 Luo nd Chen. HAR-ype Models The relised mesures re defined s RM, 1 h h RM j h j1, where RM represens relised mesures such s RV, C, J, TC, TJ, nd BPV in (6)-(9). RM,1, RM,5, nd RM, re he verge esimes of he ps 1, 5, nd rding dys, corresponding o he dily, weekly, nd monhly mesures of RV, respecively. The proposed forecs models re bsed on four ypes of HAR models, nmely he rdiionl HAR models (Corsi, 009), he HAR-CJ model (Andersen e l., 007), he HAR-TCJ model (Corsi e l., 010), nd he HAR-ΔJ model (Pon nd Shepprd, 015), which re defined s follows: HAR: RV 0 drv,1 wrv,5 mrv, u HAR-CJ: RV 0 dc,1 wc,5 mc, djj,1 wjj,5 mjj, u HAR-TCJ: RV 0 dtc,1 wtc,5 mtc, djtj,1 wjtj,5 mjtj, u HAR-ΔJ: RV 0 J J,1 dbpv,1 wrv,5 mrv, u (6) (7) (8) (9).3 HAR-ype Models wih Byesin Approches Assume h X is predicor se wih m predicors nd h we hve K sub-models which re chrcerised by hving differen combinions of predicors, where K = m. 3 We ( k ) denoe he predicors in ech sub-model s X, where k = 1,,,K. Consider se spce forecs model wih he ime-vrying coefficiens: ( ) where boh k nd ( k) ( k) ~ N(0, Q ) (1) ( ) RV X ( k) ( k) ( k) ( k) ( k) ( k) 1, (10) ( k ) ( k) ( k) re normlly disribued: ~ N(0, H ) nd L 1,,, K is he model pplied in period, RV RV,, ' 1 RV is he informion. Denoe h ',, ' is se of coefficiens, nd se ime. When m is lrge, he number of sub-models K is lrge nd i is burdensome o employ he full Byesin pproch for compuion. Thus, he pplicion of Klmn filer helps o reduce he compuion worklod. Bsed on Rfery e l. (010), wo prmeers, λ nd α, which re boh forgeing fcors, re involved in he esimion models. λ nd α re se slighly less hn 1 such h, 0.95,1. The role of forgeing fcors is o reduce he weigh of informion for more disn period s we mke observions ime j in he ps, suggesing grdul evoluion of coefficiens. For exmple,if λ = 0.99, he observions ime 0 hed receive pproximely 80% s much weigh s hose for he ls period, while if λ = 0.95, he observions ime 0 hed receive nerly 35% s much weigh s hose for he ls period. In ddiion, he only disincion beween he DMA-HAR-ype model nd 3 For he specil cse where no predicor is incorpored in he forecs model, K = m.

6 Relised Voliliy Forecss for Sock Index Fuures 7 he BMA-HAR-ype model is h we se λ = α = 1 for he ler, suggesing h he ps informion is ssigned equl weigh nd he effec of forgeing fcors is elimined. For one-model cse, he upding process for coefficiens is shown in (B3)-(B4) in Appendix B. For muli-model cse, we denoe L k if he k-h sub-model M k is involved in ime. The upding process in he muli-model cse is similr o h in he one-model cse. By denoing Q, he evoluion of coefficiens is s follows: L k, RV ~ N( ˆ, ) (11) 1 ( k) ( k) L k, RV ~ N( ˆ, ) (1) 1 ( k) ( k) 1 1 L k, RV ~ N( ˆ, ) (13) ( k) ( k) (11) nd (13) re respecively he condiionl disribuions of nd ( k ), which re pproximed by norml disribuions, nd (1) is he prmeer predicion equion. The upding of ˆ nd 1 1 in (11)-(13) is deiled in (B5) nd (B6) in Appendix B. 1 By incorporing he forgeing prmeer λ, we hve Q ( 1) 1 1, 0 < λ < 1, where ( k ) (14) The recursive forecs of RV in he sub-model M k cn be done on he bsis of he predicive disribuion vi Klmn filering: RV RV ~ N( X ˆ, X X '), (15) 1 ( k) ( k) ( k) ( k) ( k) ( k) 1 1 ( ) where H k is n imporn middle prmeer in he DMA pproch. The esimion of he DMA pproch is deiled in Appendix B. Rfery e l. (010) inroduce noher forgeing fcor α, which hs he sme funcion k s λ, in he upding funcion of he inclusion probbiliy of ech sub-model 1,. The upding procedure for he inclusion probbiliies of he sub-models is s follows: 1 1, k 1, k K 1 1, l l1 (16), k 1 1, kpk( RV RV ) K 1 1, lpl RV RV l1, (17) ( ) -1 where pl( RV RV ) is he predicion densiy for he l-h model condiionl on he previous informion defined in (15). The derivion of 1, k is deiled in Appendix B. As he inclusion probbiliies 1, k of he sub-models evolve over ime, he predicors cn be considered s he vribles wih heir ime-vrying weighed verge inclusion probbiliies. Thus, we cn ssign more weigh o he predicors wih higher inclusion

7 8 Luo nd Chen probbiliies. Wihin he BMA nd DMA, we obin he forecs vlues of RV by clculing he weighed verge of ll he sub-models forecs vlues, while he DMS proceeds by selecing he single model wih he highes vlue of inclusion probbiliies ech poin in ime nd simply using i for forecs. The d-sep hed forecs of RV is compued by he ( ) weighed verge of he esimed RV ˆ k in ech sub-model wih he inclusion probbiliy. d 1 d, k DMS ( kˆ) ˆ ( kˆ) d1 d RV X K K DMA/ BMA ˆ ( k ) ( k ) ˆ ( k ) d1 d, k d 1 d, k d1d k1 k1 RV RV X, where, k ˆ : mx 1,,,, 1,, d d k d1 d,1 d 1 d, l l (18) In his pper, we consruc 1 combinion models by incorporing he DMA, DMS, nd BMA pproches ino he four HAR-ype models respecively. Regrding he four HAR-ype models (HAR model, HAR-CJ model, HAR-TCJ model, nd HAR-ΔJ model), he predicor ses X ( k ) in (10) re RV,1, RV,5, RV,, C,1, C,5, C,, J,1, J,5, J,, TC,1, TC,5, TC,, TJ,1, TJ,5, TJ,, nd J,1, BPV,1, RV,5, RV,, respecively. Considering he long memory propery in he serils, we specify wek forgeing fcors Performnce Evluion We use he MZ-R, he MSE loss funcion, he QLIKE loss funcion, nd he sum of log prediced likelihood o evlue he in-smple forecs performnces. The MSE loss funcion nd he QLIKE loss funcion re wo specil cses of loss funcions in Pon (011) which re robus o he presence of noise in he voliliy proxy. We lso use he loss funcions nd he MCS o evlue he ou-of-smple forecs performnces. Pon (011) proposes clss of loss funcions h re robus o he noise of he voliliy proxy nd re homogeneous of degree b+: ( RV ) ( RV ) ) ( RV ) ( RV RV ), b 1, T b b b1 True, Forecs, Forecs, True, Forecs, 1 1( b1)( b) ( b1) T 1 RVTrue, ( True,, Forecs,, ) ( Forecs, True, True, ln ), b 1 RV 11 Forecs, T 1 RVTrue, RVTrue, L RV RV b RV RV RV ( ln 1), b 1 1 RVForecs, RVForecs, On he bsis of Pon (011), when b = 0 nd b = -, he loss funcions degenere o wo convenionl loss funcions: he MSE nd he QLIKE, respecively. In his pper, we lso use he homogeneous robus loss funcion when b = -1 nd he posiive robus loss funcion when b = 1 o evlue nd compre he vrious forecs models, where he homogeneous robus loss penlises he under-predicion of voliliy more hevily while he (19)

8 Relised Voliliy Forecss for Sock Index Fuures 9 posiive robus loss penlises he over-predicion of voliliy more hevily. In ddiion, we employ he MCS developed by Hnsen e l. (011) o es he significnce of forecs performnces mong vrious compeing models. The MCS procedure ims selecing se of models wih he bes forecs performnces from se of cndide forecs models M0 M i, i 1,,M. By defining he relive performnce vrible dij, Li, Lj,, for ll i, j M 0 nd ij Ed ( ij, ), where L is specified loss rue forecs funcion Li, L( RV, RV ), model i M is preferred o model j M if ij 0. The se of superior models is defined s M * {i M 0 : ij 0 for ll j M 0 }. The null hypohesis in he MCS pproch is H 0, M : ij 0 for ll i, j M, where M M 0. Iniilly, le Mˆ 1 M 0. The MCS lgorihm is consruced wih n equivlence es nd n elimining rule s follows: Sep 1: Tes H 0, M on he bsis of n equivlence es he significnce level of α. Hnsen e l. (011) presen hree differen ess of sisics for he equl predicive ccurcy (EPA) hypohesis. We employ wo of hem: he rnge sisics T R nd he semi-qudric sisics T SQ. Boh ypes of EPA es sisics re bsed on he following -sisics: ij d ij vr( ˆ d ) ij for i, j M, (0) 1 N N 1 d where ij dij,, N is he lengh of he forecs period, nd vr( ˆ d ) is n esime of vr( d ij ) obined by using he sionry block boosrp of Poliis nd Romno (1994). The -sisics ij provide scled informion on he verge difference in he forecs quliy of model i nd model j. The rnge sisics T R nd he semi-qudric sisics T SQ re given by ij T R mx mx ij i, jm i, jm d ij vr( ˆ d ) ij T SQ ( d ) ij ij i, jm i, jm ˆ dij vr( ) (1) Sep : If H is cceped, se * 0, M Mˆ ˆ 1 M 1 ; oherwise, use he eliminion rule o remove model from M ˆ 1 nd go bck o Sep 1. * In generl, given he significnce level α is fixed ech sep, M ˆ 1 conins he bes forecs models from M 0 wih (1 α) confidence. III. D We employ he 5-minue d of he Chin Sock Index 300 (CSI 300) fuure from 16 April 010 (he firs dy when he CSI 300 fuure ws lised on he mrke) o 1 My 014,

9 30 Luo nd Chen giving ol of 991 rding dys; he ol number of observions is 53,514. The d re from he Wind Dbse. The inrdy rding period of he CSI 300 fuure is from 9:15 unil 15:15, nd he ol number of inrdy observions is 54. We firs clcule he inrdy 5-minue log reurn r. The relised voliliy RV is obined by ccumuling he inrdy reurns. In he recen lierure, some scholrs propose he overnigh RV esimor (Ahoniemi nd Lnne, 013; Koopmn e l., 005; Mrens, 00) so h he RV cn be scled wih he off-rding hours informion. There re hree resons why we only dop he rding hour d o compue he relised vrinces. Firs, lhough he incorporion of overnigh reurn yields more complee nlysis, i will increse he model complexiy. In fc, he exclusion of overnigh reurns is he common prcice in he HAR-ype model (Andersen e l., 007; Corsi e l., 010; Bndi nd Reno, 01; Xu nd Perron, 014; ec.). Second, Hnsen nd Lunde (006) demonsre h he overnigh reurn is fr more volile hn he inrdy 5-minue reurns nd will bring exr noise. Tsiks (008) suggess h rding hour reurns hve differen d genering process from overnigh reurns. Third, s he overnigh voliliy is proporionl o he dily voliliy, i migh no hve n effec on he overll resuls. We hen clcule he relised BPV of Brndorff-Neilsen nd Shephrd (004) nd he TBPV of Corsi e l. (010). We furher obin he coninuous componens nd jumps by employing wo ypes of sisics, C_ Z nd C_ TZ, which re deiled in Pr. Finlly, we clcule he relised semi-vriion RS + nd RS nd derive he signed jump componen J on he bsis of Brndorff-Nielsen e l. (010) nd Pon nd Shepprd (015). Tble 1 presens he descripive sisics of he d, including he inrdy verge reurn r, he relised voliliy RV, he relised bi-power vriion BPV, he hreshold bi-power vriion TBPV, nd he relised semi-vriions RS + nd RS. As shown in Tble 1, ll he relised esimors exhibi he feures of high pek nd f il. The JB sisics of ll he vribles significnly rejec he null hypohesis of being normlly disribued. The resuls of he Ljung-Box sisics indice h ll he vribles exhibi significn uocorrelion nd ll he relised esimors hve he propery of long memory. Moreover, he significnce levels of he ADF sisics indice h ll he vribles belong o sionry ime series. Figure 1 shows he perns of he clculed coninuous componens (C nd TC) nd he jump componens (J nd TJ) of he RV by employing wo ypes of sisics: C_ Z nd C_ TZ. The firs hree grphs in Figure 1 show h he coninuous componens (C) depred from he RV exhibi smooher pern hn he RV, nd he hreshold coninuous componen (TC) hs n even smooher pern hn he coninuous componens (C). The ls hree grphs in Figure 1 indice h ll he jump componens, including he jump wih or wihou he hreshold effec (J nd TJ) s well s he signed jump componens, exhibi he feures of voliliy clusering nd jumps in priculr periods.

10 Relised Voliliy Forecss for Sock Index Fuures 31 Tble 1 Summry Sisics Vrible R RV BPV TBPV RS - RS + Men -9.13e Sndrd Deviion Skewness Kurosis JB sisic *** *** 7300 *** *** *** 1585 *** Ljung-Box,Q(5) *** *** *** *** *** *** Ljung-Box,Q(10) ** *** *** *** *** 50.1 *** Ljung-Box,Q(0) *** *** *** *** *** ADF *** *** *** *** *** *** Noe: Tble 1 presens he summry sisics of he vribles. r is he inrdy verge reurn, RV is he relised voliliy, BPV is he relised bi-power vriion, TBPV is he hreshold bi-power vriion, nd RS + (RS ) is he posiive (negive) relised semi-vriion. *** denoes significnce level of 1%, ** denoes significnce level of 5%, nd * denoes significnce level of 10%. Figure 1 Coninuous Componens nd Jump Componens of he Relised Voliliy 5 RV Apr10 Apr11 Apr1 Apr13 My C Apr10 Apr11 Apr1 Apr13 My14

11 3 Luo nd Chen 14 TC Apr10 Apr11 Apr1 Apr13 My14 7 J Apr10 Apr11 Apr1 Apr13 My TJ Apr10 Apr11 Apr1 Apr13 My J Apr10 Apr11 Apr1 Apr13 My14 Noe: Figure 1 shows he ime-vrying chrcerisics of he mjor vribles. RV is he relised voliliy, C is he coninuous componen, TC is he hreshold coninuous componen, J is he jump componen, TJ is he hreshold jump componen, nd dj is he sign-jump componen.

12 Relised Voliliy Forecss for Sock Index Fuures 33 IV. Relised Voliliy Forecss nd Evluion 4.1 In-Smple Forecss Using he HAR-ype Models We conduc he in-smple forecss for differen forecs seps by employing he four HAR models menioned bove nd compre he forecs performnces in erms of hree ypes of sisics: he MZ-R, 4 he MSE, nd he QLIKE loss funcion. The coefficiens of he HAR models re esimed by ordinry les squre, nd he -sisics re clculed wih Newey-Wes HAC (Heeroskedsiciy nd Auocorrelion Consisen). As shown in Tble, lmos ll he coefficiens of he HAR-ΔJ nd he HAR models re significn he 10% confidence inervl boh for one-sep nd muli-sep hed forecss, indicing he good in-smple forecs performnces of hese wo models. The HAR-CJ model hs significn coefficiens for ll he coninuous componens for one-sep nd five-sep hed forecss, while ll he jump componens re insignificn for ll forecs seps. Excep for he lgged weekly voliliy for he one-sep hed forecs nd he lgged dily nd weekly voliliies for he five-sep hed forecs, ll he oher coefficiens in he HAR-TCJ model re insignificn. From he resuls of he hree ypes of sisics, he MZ-R, he MSE, nd he QLIKE, he HAR-ΔJ model performs bes while he HAR model performs wors for ll forecs seps for he in-smple forecs. The HAR-TCJ model rnks second for he one-sep nd -sep hed forecss, while he HAR-CJ rnks second for he five-sep hed forecs. Tble In-Smple Forecs Resuls Using he HAR-ype Models h = 1 Models HAR-ΔJ HAR HAR-CJ HAR-TCJ coefficien -vlue coefficien -vlue coefficien vlue coefficien -vlue *** *** *** *** d ** *** * w *** *** *** *** m ** ** * dj wj mj J ** MZ-R MSE QLIKE The Mincer Zrnowiz-R is he R-squred of he rue vlues of he dependen vrible gins is forecs vlues in he regression.

13 34 Luo nd Chen h = *** *** *** *** 4.31 d ** ** * * w *** *** *** * m * * * dj wj mj J ** MZ-R MSE QLIKE h = *** *** *** *** d w ** ** m dj wj mj J ** MZ-R MSE QLIKE Noe: Tble presens he esimed coefficiens s well s he -sisics for ech predicor in differen models. The in-smple forecs performnces re evlued on he bsis of he MZ-R, he MSE, nd he QLIKE. *** denoes significnce level of 1%, ** denoes significnce level of 5%, nd * denoes significnce level of 10%. h = 1 is he one-sep-hed forecs, h = 5 is he five-sep hed forecs, nd h = is he -sep hed forecs. The regression models re s follows: HAR: RV RV,1 RV,5 RV, u 0 d w m HAR-CJ: RV 0 dc,1 wc,5 mc, djj,1 wjj,5 mjj, u HAR-TCJ: RV 0 dtc,1 wtc,5 mtc, djtj,1 wjtj,5 mjtj, u HAR-ΔJ: RV 0 JJ,1 dbpv,1 wrv,5 mrv, u 4. In-Smple Forecss Using he HAR-ype Models wih Byesin Approches We combine he HAR-ype models wih he DMA, he DMS, nd he BMA pproches nd use hem o forecs he RV for boh one-sep nd muli-sep hed forecss, including he shor-erm forecs h 1, he mid-erm forecs h 5, nd he long-erm forecs h.

14 Relised Voliliy Forecss for Sock Index Fuures 35 Wih Byesin pproches, he predicors nd he coefficiens in he HAR-ype models re no longer fixed bu ime-vrying, nd hus he disurbnces from unknown shocks in he RV series cn be elimined. The ol number of predicors (including consn) for he HAR-ΔJ model nd he HAR model re 5 nd 4 respecively, nd for boh he HAR-CJ model nd he HAR-TCJ model, he ol number of predicors is 7. Therefore, he number of sub-models equls 5 3 for he HAR-ΔJ model, 4 16 for he HAR model, nd 7 18 for boh he HAR-CJ model nd he HAR-TCJ model. We obin he ime-vrying size of he predicors se in ech model by clculing he weighed verge of he number of predicors in ech sub-model on he bsis of he Byesin weighs: DMS E( Size ) K DMA/ BMA ( ) 1, ksizek k 1 E Size Sizekˆ, kˆ : ˆ 1, mx k 1,1,, 1, l, l 1,,, where Size k is he number of predicors in ech sub-model M k nd 1, k is he Byesin probbiliy h ech sub-model is included in he model. Tble 3 shows he sisics of he ime-vrying sizes of predicor ses in vrious Byesin models for differen forecs seps. The ime-vrying sizes in he DMA nd BMA models re he sum of he weighed sizes, while he size of DMS model is equl o he size of he predicors in he sub-model wih he highes inclusion probbiliy. We divide he smple period ino four sub-periods, nd he lengh of ech sub-period is round 1 yer. We compue he mens nd sndrd deviions of he ime-vrying expeced sizes in ech sub-period, where he men is he posiion indicor of sizes nd he sndrd deviion scles he flucuion of sizes. Tble 3 clerly indices h he sizes of he predicor ses in vrious Byesin HAR-ype models exhibi significn ime-vrying perns. The verge sizes of he predicors included in he Byesin HAR-ΔJ models re round 3 o 4 for he shor-erm forecs,.5 o 3.5 for he mid-erm forecs, nd o 3 for he long-erm forecs. The verge sizes of he predicor ses of he Byesin HAR models re round.6 o 3. for he shor-erm forecs, o 3.5 for he mid-erm forecs, nd o 3.8 for he long-erm forecs. For he Byesin HAR-CJ models nd HAR-TCJ models, here re bou 3 o 5 predicors included in he shor-erm forecs models,.8 o 4.6 predicors included in he mid-erm forecs models, nd.8 o 4 predicors included in he long-erm forecs models. In Tble 3, we lso presen he mximum numbers of predicors llowed in he Byesin models (or Mx_N), which re lso he numbers of predicors in he fixed-prmeer HAR-ype models. The differences beween he mens nd he Mx_N reflec he fc h he bd performnce predicors wih smller weighs re incorpored ino he Byesin HAR-ype models over he forecs horizons. In conclusion, here re more predicors included in he shor-erm forecs model, while fewer predicors re included in he mid-erm nd long-erm forecs models for ll models excep for he Byesin HAR-CJ models. In ddiion, he sndrd deviions of sizes decrese over ime in mos Byesin models, indicing less flucuion of sizes during he ler smple period.

15 36 Luo nd Chen Tble 3 Sisics for he Time-Vrying Sizes of Predicor Ses in Vrious HAR-ype Models wih Byesin Approches Sr-April 011 My 011- April 01 My 01- April 013 My 013-End Whole period men sd men sd men sd men sd Mx_N h = 1 DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ h = 5 DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ h = DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ Noe: Tble 3 presens he men nd sndrd deviions (or sd) of ime-vrying sizes in Byesin HAR-ype models for one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss. Tble 3 lso presens he mximum number of predicors llowed in he model, or Mx_N. The 4-yer smple period is divided ino four sub-periods: April 010-April 011, My 011-April 01, My 01-April 013, nd My 013-April 014.

16 Relised Voliliy Forecss for Sock Index Fuures 37 Furher, we nlyse he poserior inclusion probbiliy () of differen predicors in he DMA-HAR-ype models nd he BMA-HAR-ype models. The poserior inclusion probbiliies of he sub-models in DMS-HAR-ype models re he sme s hose of he DMA-HAR-ype models. For he DMS pproch, we only selec he sub-model wih he highes probbiliy ech ime so h here is no need o compue he weighed of ech predicor. The of he i-h predicor wihin he DMA nd BMA pproches is defined s K k i, 1, k k k 1 E ( ) IX ( sub_ M), where 1, k is he Byesin probbiliy h he k-h sub-model is included in he model nd I( ) is he indicion funcion wheher he i-h predicor is included in he k-h sub-model ( sub _ M k ). A greer inclusion probbiliy indices sronger explnory power of his predicor: h is, i conins more informion in he forecs. Bsed on Koop e l. (01), he predicor is considered s good one for period if is inclusion probbiliy is greer hn 0.5. Tble 4 nd Tble 5 show he sisics of he ime-vrying of predicors in he DMA-HAR-ype nd he BMA-HAR-ype models, respecively. We presen he mens nd sndrd deviions of poserior inclusion probbiliies for ech predicor for he shor-erm, mid-erm, nd long-erm forecss. In ddiion, we compue he proporion of high poserior inclusion probbiliies (p(>0.5)), which is equl o he ccumulion of periods when he poserior inclusion probbiliies re greer hn 0.5 divided by he whole smple period. We consider predicor o be good one in he forecs model if eiher he men of is is greer hn 0.5 or he proporion of is high poserior inclusion probbiliies (>0.5) is higher hn 0.5, mening h he periods when his predicor is good one domine he whole forecs period. As shown in Tble 4, when h = 1, he dily, weekly, nd monhly voliliy vribles in four DMA-HAR-ype models re he good predicors, wih heir proporions of high poserior inclusion probbiliies being greer hn 0.5. When h = 5, he dily nd weekly voliliy vribles re he good predicors for ll DMA-HAR-ype models nd he monhly jump vribles re he good predicors for boh he DMA-HAR-CJ model nd he DMA-HAR-TCJ model. When h =, he monhly voliliy vribles re he good predicors for ll DMA-HAR-ype models nd he dily jump vribles re he good predicors for boh he DMA-HAR-CJ model nd he DMA-HAR-TCJ model. As presened in Tble 5, when h = 1, he dily nd monhly voliliy vribles nd he signed jump fcor ΔJ re he good predicors for he BMA-HAR-ΔJ model nd he monhly voliliy is he only good predicor for he BMA-HAR model. The dily voliliy vribles nd he dily nd monhly jump vribles re he good predicors for boh he BMA-HAR-CJ model nd he BMA-HAR-TCJ model, bu he weekly voliliy vrible is

17 38 Luo nd Chen Tble 4 Time-Vrying Poserior Inclusion Probbiliies of Predicors for he DMA-HAR-ype Models Models DMA-HAR-ΔJ DMA-HAR DMA-HAR-CJ DMA-HAR-TCJ men sd p(>0.5) men sd p(>0.5) men sd p(>0.5) men sd p(>0.5) h = 1 d w m dj wj mj J h = 5 d w m dj wj mj J h = d w m dj wj mj J Noe: Tble 4 presens he mens nd sndrd deviions of he ime-vrying poserior inclusion probbiliies of differen predicors in DMA-HAR-ype models for one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss. p(>0.5) is he proporion of he poserior probbiliies h re higher hn 0.5 for ech predicor. d, w, nd m re he poserior inclusion probbiliies of he dily, weekly, nd monhly voliliy vribles, respecively; dj, wj, nd mj re he poserior inclusion probbiliies of he dily, weekly, nd monhly jump vribles, respecively; nd J is he poserior inclusion probbiliy of he signed jump vrible in he DMA-HAR-ΔJ model. The good predicors re denoed in bold. he good predicor only for he BMA-HAR-TCJ model. When h = 5, he dily nd weekly voliliy vribles re he good predicors for boh he BMA-HAR-ΔJ model nd he BMA-HAR model. The weekly voliliy vrible nd he monhly jump vrible re he good predicors for he BMA-HAR-CJ model, while he weekly voliliy vrible nd he weekly jump vrible re he good predicors for he BMA-HAR-TCJ model. When h =, he monhly voliliy vrible nd he signed jump fcor ΔJ re he good predicors for he

18 Relised Voliliy Forecss for Sock Index Fuures 39 BMA-HAR-ΔJ model nd ll vribles in he BMA-HAR model re good predicors. The monhly voliliy vribles nd he dily jump vribles re he good predicors for boh he BMA-HAR-CJ model nd he BMA-HAR-TCJ model. Tble 5 Time-Vrying Poserior Inclusion Probbiliies of Predicors for he BMA-HAR-ype Models Models BMA-HAR-ΔJ BMA-HAR BMA-HAR-CJ BMA-HAR-TCJ men sd p(>0.5) men sd p(>0.5) men sd p(>0.5) men sd p(>0.5) h = 1 d w m dj wj mj J h = 5 d w m dj wj mj J h = d w m dj wj mj J Noe: Tble 5 presens he mens nd sndrd deviions (sd) of he ime-vrying poserior inclusion probbiliies for differen predicors in BMA-HAR-ype models for one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss. p(>0.5) is he proporion of he poserior probbiliies h is greer hn 0.5 for ech predicor. is he poserior inclusion probbiliy. d, w, nd m re he poserior inclusion probbiliies of he dily, weekly, nd monhly voliliy vribles, respecively; dj, wj, nd mj re he poserior inclusion probbiliies of he dily, weekly, nd monhly jump vribles, respecively; nd J is he poserior inclusion probbiliy of he signed jump vrible in BMA-HAR-ΔJ model. The good predicors re denoed in bold.

19 40 Luo nd Chen Tble 6 Time-Vrying Coefficiens for Vrious HAR-ype Models wih Byesin Approches Models HAR-ΔJ HAR HAR-CJ HAR-TCJ DMA BMA DMA BMA DMA BMA DMA BMA h = 1 d men 0.181* 0.353** 0.073* * * 0.470** * 0.173** sd w men 0.855** ** * 0.450** 0.187* ** ** sd m men * sd dj men ** ** sd wj men sd mj men sd ΔJ men ** ** sd h = 5 d men * * * sd w men 0.660** 0.563** ** ** 0.496** ** * ** sd m men sd dj men * sd wj men sd mj men * * * sd ΔJ men * sd h = d men sd w men sd m men * * sd dj men ** * sd wj men sd mj men sd ΔJ men * ** sd Noe: Tble 6 shows he mens nd sndrd deviions (sd) of he ime-vrying coefficiens for vrious HAR-ype models wih Byesin pproches for one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss. d, w, nd m re he coefficiens for he dily, weekly, nd monhly voliliy vribles; dj, wj, nd mj re he coefficiens for he dily, weekly, nd monhly jump vribles; nd J is he coefficien for he signed jump fcor in he HAR-ΔJ model. We mrk he predicors wih men/sd greer hn 1 wih wo srs nd he predicors wih men/sd greer hn 0.5 wih one sr.

20 Relised Voliliy Forecss for Sock Index Fuures 41 Tble 6 shows he verge mens nd sndrd deviions of expeced coefficiens over he smple period. Wihin he DMS pproch, only he sub-model wih he highes is employed in he forecs model for ech period, which indices h he coefficiens for predicors re ime-discree rher hn ime-coninuous. 5 Therefore, we only lis he sisics of ime-vrying coefficiens in he DMA-HAR-ype models nd he BMA-HAR-ype models. The coefficien of ech predicor wihin he DMA nd BMA pproches re compued by K DMA/ BMA ˆ ( k ) 1, k k 1 E( ˆ ) As shown in Tble 6, he mens nd sndrd deviions sugges significn ime-vrying chrcerisics of coefficiens in he HAR models wih Byesin pproches. The bsolue rio of he poserior men gins he sndrd deviion ( men / sd ) of coefficien in he Byesin models is used s crierion of sble predicbiliy. A high men / sd mens he sble predicbiliy of predicor over ime nd vice vers. We mrk he predicors wih men / sd greer hn 1 wih wo srs nd he predicors wih men / sd greer hn 0.5 wih one sr. The resuls in Tble 6 show h men / sd decreses wih he increse in forecs seps, indicing he weker predicbiliy of predicors in he longer erm forecs. Tble 6 lso shows h when h = 1, he dily nd weekly voliliy vribles, he dily jump vrible, nd he signed jump vrible ΔJ hve sble predicbiliy. When h = 5, he weekly voliliy vribles nd he monhly jump vribles hve sble predicbiliy. When h =, he monhly voliliy vrible, he dily jump vribles, nd he signed jump vrible ΔJ hve sble predicbiliy. Moreover, he coefficiens in he Byesin HAR-ype models re smller hn hose in he fixed prmeer models, indicing h ll he predicors wihin he DMA nd BMA pproches hve ime-vrying predicbiliy so h hey cn be incorpored ino he forecs models on he bsis of heir forecs performnces over ime. Tble 7 presens he resuls of he in-smple forecs performnces for 1 HAR-ype models wih Byesin pproches bsed on hree ypes of sisics: he MZ-R, he MSE loss funcion, nd he QLIKE loss funcion. The in-smple resuls re bsed on he ls 800 forecs smples (bou 80% of he smple size) excluding some ouliers in he iniil ierions of he Klmn filer. As shown in Tble 7, ll ypes of sisics sugges h compred wih he oher Byesin models, he DMS-HAR-ΔJ model, he DMS-HAR-CJ model, nd he DMS-HAR-TCJ model hve he superior performnces for he shor-erm forecs nd he mid-erm forecs. Among hese models, he DMS-HAR-CJ model. 5 Time-discree mens h he predicors do no hve he poserior weighed verge coefficiens for ll periods (he DMS-HAR-ype models). On he conrry, ime-coninuous mens h ll he predicors hve he poserior weighed verge coefficiens ech period (he DMA-HAR-ype nd he BMA-HAR-ype models).

21 4 Luo nd Chen performs bes for boh he shor-erm forecs nd he mid-erm forecs. For he long-erm forecs, ll he evluion pproches sugges h he BMA-HAR-TCJ model performs bes. In ddiion, he HAR-ype models wih Byesin pproches perform beer hn heir originl HAR-ype models for he in-smple forecs. Tble 7 In-Smple Forecs Comprisons mong Vrious Forecs Models h = 1 h = 5 h = MZ-R MSE QLIKE MZ-R MSE QLIKE MZ-R MSE QLIKE Byesin models DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ Benchmrk models HAR-ΔJ HAR HAR-CJ HAR-TCJ Noe: Tble 7 presens he in-smple forecs comprisons mong he vrious forecs models for he one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss. The in-smple forecs performnces re evlued on he bsis of he MZ-R, he MSE loss funcion, he QLIKE loss funcion, nd he sum of log prediced likelihood. The HAR-ype models re used s he benchmrk models. 4.3 Ou-of-Smple Forecss We divide he whole smple ino wo prs: he in-smple pr h conins he firs 491 observions from 16 April 010 o 3 April 01, nd he ou-of-smple pr h conins he ls 500 observions from 4 April 01 o 1 My 014. We employ he recursive forecs o obin he shor-erm (h = 1), mid-erm (h = 5), nd long-erm (h = ) ou-of-smple forecs vlues corresponding o one-dy, one-week, nd one-monh hed forecss. Pon s (011) loss funcions re uilised o compre he forecs performnces of vrious compeing models. We employ four loss funcions: he MSE nd he QLIKE loss funcions when b = 0 nd b = - respecively nd he homogeneous robus loss funcion nd he posiive robus loss funcion when b = -1 nd b = 1 respecively. We obin 1 cndide models wih ime-vrying prmeers nd vrible predicor ses by combining he

22 Relised Voliliy Forecss for Sock Index Fuures 43 HAR-ype models wih he hree Byesin pproches, nd he convenionl HAR-ype models re used s he benchmrk models. The resuls re repored in Tble 8. Tble 8 Lis of he Forecs Models Rnked by he Loss Funcions h = 1 h = 5 h = h = 1 h = 5 h = Models vlue rnk vlue rnk vlue rnk vlue rnk vlue rnk vlue rnk MSE loss funcion (b = 0) QLIKE loss funcion (b = -) DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ HAR-ΔJ HAR HAR-CJ HAR-TCJ Homogeneous robus loss funcion Posiive robus loss funcion (b = 1) (b = -1) DMA-HAR-ΔJ DMS-HAR-ΔJ BMA-HAR-ΔJ DMA-HAR DMS-HAR BMA-HAR DMA-HAR-CJ DMS-HAR-CJ BMA-HAR-CJ DMA-HAR-TCJ DMS-HAR-TCJ BMA-HAR-TCJ HAR-ΔJ HAR HAR-CJ HAR-TCJ Noe: Tble 8 compres he ou-of-smple performnces mong he vrious forecs models for one-sep (h = 1), five-sep (h = 5), nd -sep (h = ) hed forecss on he bsis of he MSE, he QLIKE, he homogeneous robus loss funcion, nd he posiive robus loss funcion. The op hree rnked models re in bold.

23 44 Luo nd Chen Tble 8 shows lis of forecs models rnked by Pon s (011) loss funcions; he lis suggess h he HAR-ype models wih he Byesin pproches perform beer hn he corresponding HAR models. According o he resuls of he four loss funcions, he forecs performnces of he combinion models re 0 o 50 per cen higher hn hose of he corresponding HAR models. When h = 1, he DMS-HAR-TCJ model, he DMA-HAR-ΔJ model, nd he DMS-HAR-ΔJ model re rnked he firs hree models mong he compeing models on he bsis of he MSE, he QLIKE, nd he posiive loss funcions. When h = 5, ll he loss funcions sugges h he firs hree models re he DMS-HAR-CJ model, he DMS-HAR-ΔJ model, nd he DMS-HAR-TCJ. On he bsis of ll he loss funcions, when h =, he firs hree models re he BMA-HAR-TCJ, he DMS-HAR-CJ, nd he BMA-HAR-ΔJ. In generl, ll he loss funcions sugges h he DMS-HAR-TCJ model, he DMS-HAR-CJ model, nd he BMA-HAR-TCJ model perform bes in he shor erm, he miderm, nd he long erm, respecively, improving heir benchmrk HAR-ype models by 41.9%, 46.6%, nd 45.7% respecively on he bsis of he MSE nd 44.7%, 41.1%, nd 9.5% respecively on he bsis of he QLIKE. We lso compre ll he forecs models on he bsis of he MCS es of Hnsen e l. (011). The MSE nd QLIKE re used s he loss funcions of he MCS es. The number of boosrp smples used o obin he p-vlue h he null is rejeced is se 10,000, where he greer he p-vlue of forecs model, he higher he probbiliy h i belongs o he opiml forecs model se. The resuls re summrised in Tble 9. As shown in Tble 9, for he shor-erm forecs models (h = 1), ll he Byesin models re included in he MCS he 10% confidence level ccording o T R sisics nd only he DMS-HAR-TCJ model, he DMA-HAR-ΔJ model, nd he DMA-HAR-ΔJ model re included in he MCS he 10% confidence level wihin boh he MSE nd QLIKE loss funcions. For he mid-erm forecs models (h = 5), ll he Byesin models re included in he MCS he 10% confidence level on he bsis of boh T R nd T SQ wihin he wo loss funcions. Finlly, for he long-erm forecs models (h = ), ll he forecs models re included in he MCS he 10% confidence level. The bes models for he shor-erm forecs nd he long-erm forecs re he DMS-HAR-TCJ model nd he BMA-HAR-TCJ model, respecively. For he mid-erm forecs, he models h perform bes re he DMS-HAR-CJ model bsed on he MSE loss funcion nd he DMS-HAR-TCJ model bsed on he QLIKE loss funcion. The DMS pproch improves he forecs performnces of he HAR-ype models in boh he shor erm nd miderm o he grees exen, while he BMA pproch improves he forecs performnces of he HAR-ype models in he long erm for boh in-smple nd ou-of-smple forecss, s shown in bles 7 hrough 9. Priculrly, he DMS-HAR-CJ model nd he BMA-HAR-TCJ model perform bes for boh he in-smple nd ou-of-smple forecss in he miderm nd long erm, respecively. The DMS-HAR-CJ