Journal of Financial Studies Vol.7 No.3 December 1999 (61-94) 61

Transcription

1 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 61 Miigaing Tail-faness, Lepo Kuric and Skewness Problems in VaR Esimaion via Markov Swiching Seings An Empirical Sudy on Major TAIEX Index Reurns Hsiou-Wei Lin 1 Naional Taiwan Universiy Hsiu-Hua Rau Ming-Yuan Li Naional Cheng-Chi Universiy GARCH GARCH 5% 1% 2.5% Absrac This paper serves as one of he firs sudies ha esimae he value a risk (VaR) via a Markov-swiching (MS) model. Specifically, we use a wo-regime MS specificaion, a MS seing wih wo ses of regime mean and regime variance, on TAIEX as well as Taiwan s major indusrial group sock index reurns. We demonsrae ha MS effecively correc non-normaliy problems and oushine boh GARCH and he mixing normal models, wih he former (laer) alernaive being subjec o over- (under-esimaing) he persisence of sock reurn volailiy (hereafer volailiy). As for esimaing he 5% VaR, MS appears o be equally effecive as Bayesian mixing normal and GARCH. In conras, MS significanly ouperforms he wo non-linear alernaives for esimaing VaR wih 1% or 2.5% ail probabiliies. Furhermore, as for he window of learning period on rare evens, we find ha one need o go much farher back o effecively depic he lef as opposed o he righ ail. Keywords: Markov-swiching models, value a risk 1 Darrel DuffieJames D. Hamilon Rual SusmelChang-Jin Kim The Eighh Conference on he Theories and Pracices of Securiy and Financial Markes

2 Journal 62 of Financial Sudies Vol.7 No.3 December 1999 (61-94) Miigaing Tail-faness, Lepo Kuric and Skewness Problems in VaR Esimaion via Markov Swiching Seings An Empirical Sudy on Major TAIEX Index Reurns 1. Inroducion This paper serves as one of he firs sudies ha adop Markov-swiching (MS) models o conrol regime swiches and gain efficiency in esimaing he value a risk (VaR). Specifically, we use a wo-regime MS seing for boh mean and variance parameers of Taiwan Sock Exchange marke index (hereafer TAIEX) reurns as well as major indusrial group sock index reurns including TAIEX-Consrucion (hereafer TAIEX-CONS), TAIEX-Finance (hereafer TAIEX-FIN), TAIEX-Elecronic (hereafer TAIEX-ELEC), and TAIEX-Elecric&Machinery (hereafer TAIEX-EL&MACH). Our empirical findings suppor he noion ha he MS models significanly help miigae ypical problems such as fa ails, Lepo kuric and skewness in VaR esimaion. Mos praciioners urned heir aenion o VaR no earlier han he mid 1990s. The Basle Commiee on Banking Supervision of Bank for Inernaional Selemens (BIS) presened Amendmen o Capial accord o Incorporae Marke Risks in January Mos financial insiuions now diversify heir operaions ou of heir original businesses and acively rade capial marke securiies, foreign currencies and derivaive insrumens. These insiuions risks keep increasing because of increased marke reurn volailiy, new compeiion, and deregulaion. In April 1996, The Basle Commiee furher endorsed he use of he value a risk o measure he banks capial adequacy raios. By he end of 1997, he G10 bank regulaors ook marke risk ino accoun for deermining he risk-based asss. 2 Since January 1999, financial insiuions in Taiwan have been insruced o use VaR o measure heir capial adequacy raios. Financial insiuions have wo alernaives. The firs alernaive, or sandardized risk measuremen proposal, appears o be ad hoc and subjec o srong assumpion of homogeneiy among he banks. The second alernaive, also named he value a 2 The G10 group includes Belgium, Canada, France, Germany, Ialy, Japan, Neherlands, Sweden, Swizerland, Unied Kingdom, Unied Saes and Luxembourg.

3 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 63 mogeneiy among he banks. The second alernaive, also named he value a risk approach, in conras, allows he users o gain forecas accuracy for esimaing marke risks via ailored VaR mehods 3. VaR is also a he cener of he recen ineres in he risk managemen field. 4 I serves o measure he level of marke risks and accordingly, he capial adequacy. Duffie and Pan (1997) furher suggesed ha one could measure defaul risk, credi risk, operaion risks and liquidiy risk via he algorihm of VaR. There has been an exensive lieraure on VaR esimaion. Esrella e al. (1994), Kupiec (1995), Jackson, Maude and Peerraudin (1997) explored banks capial adequacy raios and VaR. Hendricks (1994), Bender (1995), Simons (1996), Fong and Vasicek (1997) invesigaed he srenghs and weaknesses of VaR esimaion measures including parameric, hisorical daa simulaion and Mone Carlo simulaion mehods. Empirical sudies on esimaing VaR for porfolios of TAIEX securiies include Wang and Chen (1998), Liao and Lin (1998), Uong and Chen (1998), Ho and Lin (1999). Many invesmen risk sudies, neverheless, documened ha he hisograms for hisorical reurns mos financial asses deviae from he normal disribuions. Typical non-normaliy properies include Lepo kuric, ail-faness and skewness. Furhermore, here exis problems of win peak, he resul ofen found in he New Taiwan dollar price per U.S. dollar hisograms. The purpose of his sudy is o examine he exen o which compeing non-linear seings including Markov Swiching, GARCH and he mixing normal models help miigaing non-normaliy problems in VaR esimaion. We consider our research quesion a non-rivial issue because (1) VaR is a he cener of he recen ineres for boh researchers and praciioners, and (2) as demonsraed by sudies in prior sudies as well as he empirical secion of his sudy, differen VaR models generae significanly differen VaR esimaes. We recommend he use of Markov Swiching models since we conjecure ha srucural changes of financial and economic variables serve as a driver o he non-normaliy problems. Saisically, an increase in sample size can eliminae measuremen errors bu ineviably gain exposure o srucural changes. Transiional variabiliy in fin a n c i a l v a r i a b l e s m a y 3 This saemen is especially descripive for banks wih large rading accouns. 4 One of he mos frequenly adoped VaR models is he RiskMerics of J. P. Morgan Financial Service Co. Incorporaed. Is web sie is: hp:\\ RiskMerics, which is publicly available on he Inerne, is a represenaive parameric VaR model.

4 Journal 64 of Financial Sudies Vol.7 No.3 December 1999 (61-94) dwarf mos ypic al scenario analyses, sress ess, and risk limi guidelines. Unforunaely, he radiional linear models do no allow heir parameer ses o adjus for srucural changes. This sudy serves as one of he firs papers o demonsrae he effeciveness of MS models in filering srucural changes in reurn volailiy and hus correc he downward bias in esimaed losses encounered by he popular linear models due o ail-faness. 5 This paper also conribues o he conemporary lieraure in he following aspecs. Firs, we documen he relaive srenghs of Markov-swiching, he mixing normal and GARCH models in miigaing non-normaliy problems. Namely, all hese hree alernaives help reduce Lepo kuric and fa-ail problems for reurn disribuions. Neverheless, he mixing normal (GARCH) models appear o over- (under-) reac o Lepo kuric and ail faness. The explanaion is ha he general mixing normal seings disregard informaion for regime persisence, whereas GARCH models over-esimae he persisence of reurn volailiy when here exis srucural changes during he esimaion periods. Accordingly, he prevalen normal seing for he error erm can benefi more from incorporaing he MS model o conrol he srucural changes. Second, we show ha he wo- variance-regime specificaion helps incorporae Lepo kuric and ail-faness, and ha he wo-mean-regime seing helps filering skewness in reurn disribuions. Namely, via endogenizing he sae variable for each period, one may more effecively assign weighs for he wo normal disribuions wih differen variance and mean parameers. Our finding ha he MS models help solve he non-normaliy problems and achieves higher finess han compeing models suppors he noion ha srucural changes are a major variable o non-normaliy. Third, we examine he saisical properies of Taiwan major indices including TAIEX, TAIEX-ELEC, TAIEX-FIN, TAIEX-CONS, TAIEX-EL&MACH. For he reurn disribuions of hese indices, we documen serious (non-rivial) fa ails in he 1% (2.5%) regions for boh lef- and righ-ails bu insignifican ail-faness in he 5% regions. This finding suppors he noion of significan lumps owards boh exremes in he reurn disribuions and is consisen wih he resul of Ho and Lin (1999). Fourh, his sudy compares he relaive accuracy in esimaing VaR for lin- 5 See Simons (1996) and Ho and Lin (1999).

5 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 65 ear and MS models in erms of violaion raes. Our resuls on TAIEX indices show ha, o conservaive invesors, who are ypically he audience wih 1% lef-ailed criical probabiliy, MS models are significanly more accurae. As o he 2.5% lef-ailed region, Markov-swiching model sill performs beer. Bu he differences in accuracy are much less han ha corresponding he 1% region. Furhermore, when he lef-ailed probabiliy is se o be greaer han 5%, he differences in accuracy become rivial. As o he learning window, he seing ha uses he sample of 500 (250) pre-var-day observaions appears o yield a more (less) accurae esimaor. A poenial explanaion is ha he former seing incorporaes a richer informaion se. Fifh, we avoid he srong assumpion of symmeric reurn disribuions in Venkaaraman (1997) and alernaely apply he VaR algorihm on each of he wo ails wih horse races for he compeing models and learning windows. Our resuls for he sample index reurns show ha he forecas accuracy wih MS exceeds ha of he ohers in boh 1% and 2.5% righ-ailed regions in erms of violaion rae. However, wih 5% righ-ailed probabiliies, he difference in accuracy becomes insignifican. The second secion presens he definiion for VaR and evaluaes he compeing models. Secion 3 presens our empirical resuls. Finally, Secion 4 concludes he paper. 2. The Definiion of he VaR and he Seings for Reurn Disribuions 1.The Definiion for he Value a Risk Se he criical probabiliy o be α, hen VaR, he absolue value a risk, is he expeced maximum (wors) loss associaed wih a porfolios over a arge ime horizon, say, one rading day. Namely, VaRis he lower bound for equaion, where R P denoes he daily reurns of a porfolio + f R ) = 1 α VAR ( α P dr P wih a probabiliy disribuion funcion f(r P ). Taking he firms opporuniy coss ino accoun, one can furher derive he relaive value a risk, which is defined as he difference beween he absolue VaR and he expeced reurns. Wih a given informaion se, one can complee he probabiliy disribuion for R P as shown in he following figure and hus idenify VaR:

6 Journal 66 of Financial Sudies Vol.7 No.3 December 1999 (61-94) VaR for a Confidence Inerval of 95% Relaive VaR α = 5% AbsolueVaR VAR 0 2. Specificaions for he reurn disribuions The key o VaR esimaion is o esimae he reurn disribuion based on our exising informaion se. Le R denoe he reurns for some specific porfolio for period. There exis he following five compeing specificaions. (a) Linear Model wih Sandard Gaussian Disribuion (hereafer he Linear Model) R (1) = u0 + σ 0 e In his seing, u 0 and 0 represen mean and sandard deviaion of R, respecively. Also, e, which denoes he reurn shock, follows a sandard Gaussian disribuion. (b) Linear Model wih Suden Disribuion (hereafer Suden Seing) R (2) = u0 + σ 0 e

7 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 67 In his seing, reurn shock e follows a disribuion wih sandard error equal o uniy. Some prior sudies used Suden seings o approximae he securiy reurn disribuions 6, which are ofen subjec o Lepo kuric and ail-faness. Our empirical resuls, neverheless, indicae ha he Suden seing, which is no descripive for (1) disribuions ha are Lepo kuric bu no fa-ailed and (2) disribuions ha are fa ailed bu no Lepo kuric, does no fi mos of Taiwan s indusrial index reurns. Noe ha all TAIEX indusrial reurns have significan skewness. TAIEX-ELEC and TAIEX-EL&MACH (TAIEX-FIN and TAIEX-CONS) reurns have negaive (posiive) skewness 7, consisen wih he noion ha large negaive (posiive) reurns are more frequen han large posiive (negaive) reurns. Neverheless, mos reurn series including TAIEX-ELEC have significan ail-faness bu insignifican kurosis. Accordingly, his sudy excludes he measure of Suden seings. (c) GARCH(p, q) Models wih Sandard Gaussian Disribuion Bollerslev (1986) firs inroduced he GARCH models. The seing for GARCH(p, q) is : ε (3) R σ = u + σ e = σ e = a 0 +, e iid ~ ND(0,1) q 2 p 2 a = + i iε i b i= 1 iσ 1 i The popular ARCH(q) model proposed by Engle (1982) can be viewed as a special case of a GARCH (p, q) wih p = 0. In he GARCH, he condiional variance may be affeced by prior-period condiional variance and error sum of squares. λ, he volailiy persisence measure, can be expressed as: =(a 1 +a 2 + +a q +b 1 +b 2 + +b p ) 6 Please refer o Jorion (1997). 7 The kurosis coefficien for disribuions is 3(-2)/(-4), 4. The skewness coefficien for disribuions is 0. The kurosis of a disribuion decreases wih he degree of freedom. As he degree of freedom approaches infiniy, he disribuion approaches a sandard normal disribuion. Wih finie degree of freedom, a disribuion is boh Lepo kuric and fa-ailed. Namely, he -disribuion is no descripive for disribuions ha are Lepo kuric bu no fa-ailed and disribuions ha are fa ailed bu no Lepo kuric.

8 Journal 68 of Financial Sudies Vol.7 No.3 December 1999 (61-94) (4) (d) Mixure of Normal Disribuions (Mixing-normal) R (5) = u + σ e s s A wo-regime seing incorporaes an unobservable regime variable s = 1, 2, wih corresponding mean reurns of u 1 and u 2, and corresponding sandard deviaions of 1 and 2. Specifically, R is generaed via wo normal disribuions wih differenial means and variances. Also, reurn shock e follows a sandard Gaussian disribuion. In a mixing-normal seing, s follows a sochasic process. These models disregard he persisence of reurn volailiy and assume ha he unobservable regime variables s conforms he following Bernoulli disribuion: P( s = 1) = q, P( s = 2) = 1 q (6) where 0 < q < 1. Hamilon (1991) suggesed ha we inroduce he prior consrain in Bayesian models and rewrie he ML funcion as he follows: log p( r1, r2,..., r (7) T θ) ns ns 2 ( ak / 2)logσ k k= 1 k = 1 2 ( m u ) /(2σ ns k = 1 k k 2 k ) b k /(2σ 2 k ) where {a k },{b k },{c k } and {m k } are ses of consans summarizing an analys s prior belief for u k and k for k = 1,, n s, wih nonnegaive a k, b k, and c k for all k. The proposed esimaor may degenerae o he MLE as a special case for he diffuse prior a k = b k = c k = 0. In his sudy, we adop he seing in Hamilon (1991) and se a i= b i = 0.20, c i = 0.10 and m i = 0, for i = 1 or Venkaaraman (1997) adoped he same seing.

9 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 69 (e) Markov-swiching (MS) Models wih Sandard Gaussian Disribuion R (8) = u + σ e s s A wo-regime MS seing incorporaes unobservable regime variable s = 1, 2. Specifically, R is generaed via wo normal disribuions wih differenial means u 1 and u 2 and differenial variances 1 and 2. Also, reurn shock e follows a sandard Gaussian disribuion. MS models, in conras wih mixing-normal, incorporae Markov process for s and are effecive in conrolling regime swiches and preserving he saisics for persisence in regimes. The wo-regime Markov ransiion probabiliies can be expressed as he follows: p( s p( s (9) = 1 s = 2 s 1 1 = 1) = p 11 = 2) = p, 22, p( s p( s = 2 s 1 = 1 s 1 = 1) = p 12 = 2) = p 21 where p 11 +p 12 =p 21 +p 22 =1. The sochasic process for s is sricly saionary if boh p 11 and p 22 are less han uniy and do no ake on he value of 0 simulaneously. Because when p 11 =1, once he process eners sae 1, i would never reurn o sae 2. 9 Sae variable s conforms o he following AR (1) seing: s =(1-p 11 )+s -1 +v,=-1+p 11 +p 22 (10) The above equaion is a special case of a ypical AR(1) model wih an unusual probabiliy disribuion of he innovaion sequence {v }. As shown by Hamilon (1989), can be inerpreed as a measure of persisence in he forcing process 10. More general, an N-sae Markov chain is said o be reducible if here exiss a way o label he saes 11 such ha he ransiion marix can be wrien in he form: 9 In such a case sae 1 is an absorbing sae and he Markov chain is reducible. 10 Please refer o Cai (1994). 11 Tha is, a way o designae which sae should be named sae 1, which one should be named sae 2,and so on.

10 Journal 70 of Financial Sudies Vol.7 No.3 December 1999 (61-94) B P = 0 C D (11) where B denoes a (K X K) marix, wih 1KN. If P is upper block-riangular, hen so is P m for any m. Hence, once such a process eners a sae j such ha jk, here exiss no probabiliy of is ever reurning o any one of he saes K+1, K+2,... N. 12 Noe ha, conras wih Li and Lin (1999), who adoped a hree-regime SWARCH (Markov-swiching ARCH) model, 13 his sudy selecs a more simplified seing for reurn volailiy. The significance of he SWARCH model is ha i incorporaes MS and ARCH, wih he Markov-swiching specificaions filering ou mos reurn volailiy and he radiional ARCH models and he disribuion for he error erm conrolling he residual reurn volailiy. In conras, his paper focuses on daily reurns, which is less volaile in erms of magniude as opposed o he weekly reurns in Li and Lin (1999). Moreover, wih limied number of prior observaions for he learning process in he models, wo- insead of hree-regime seings may be sufficien. Wih Markov-swiching models ha help filering reurn volailiy, he conribuion of he ARCH algorihm and he Suden seing would be less significan. 3. Why Do We Propose he Use of MS Models for VaR Esimaion? GARCH 14, mixing-normal and MS 15 models are sochasic volailiy seings for non-consan volailiy. A simple example would help illusrae how he sochasic volailiy models help solving he non-normaliy problems. Assume ha he reurn volailiy for series X is non-consan and insead aking he value of eiher σ 1 or σ 2. We may hen skech wo normal disribuions insead for X. As one incorporaes he wo disribuions, he may resemble a reurn disribuion wih kurosis and ail-faness. Moreover, if he mean reurn can be eiher µ 1 or µ 2, oo, hen one may hus miigae skewness and win peak. 12 Please refer o Hamilon (1994). 13 SWARCH models incorporae he MS, ARCH and suden seing. 14 Duffie and Pan (1997) summarized he alernaives as wo ypes of sochasic volailiy models. However, our ess provide evidence as o he significance of he disincion among he hree specificaions. 15 Duffie and Pan (1997) discussed he poenial o apply MS models o VaR esimaion wihou any analysis or empirical ess o discriminae amongs he compeing models. Bu Li and Lin (1999) documened ha, for TAIEX reurns, MS effecively filers ou srucural changes during he esimaion period and hus miigaes he ail-faness problems. Li and Lin (1999) also suggesed ha MS models be applied o VaR esimaion.

11 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 71 Now, le us evaluae he hree compeing models. Firs, mixing normal and MS are boh mixing wo normal disribuions o approximae reurn disribuions wih kurosis and ail-faness. The mixing normal seing is a recipe for drawing fa-ailed reurns by mixing wo normal disribuions 16. In conras, hanks o he Markov-chain propery, MS models use he ransiion probabiliies as he mechanism o incorporae he persisence of reurn volailiy. The noion of volailiy persisence is consisen wih a large probabiliy for a sage wih low (high) volailiy in reurns o be followed by a low (high) volailiy sage. Take he sunspos and earhquakes aciviies as our examples. They boh have persisence in energy absorpion and dissipaion processes. Usually i is scienifically foreseeable ha an earhquake would rigger afer-shocks in he subsequen periods. Specifically, crush moions in he neighboring periods are serially correlaed. MS and mixing-normal boh use a discree sae variable o indicae which disribuion he reurns are drawn from and are boh nonlinear models. The difference beween he wo measures is ha he mixing normal models use Bernouli disribuions o conrol he regime swiching process, whereas MS models use Markov-chain o conrol i. In conras wih MS and GARCH, accordingly, mixing-normal specificaions do no model he sochasic volailiy wih persisence. The mixing-normal seings assume he sae variable for neighboring periods o be independen of one anoher and may no fi mos business cycle variables and perhaps some financial marke variables. If here exiss persisence, when he period reurn is drawn from he firs (he more volaile) disribuion, hen i is more (less) likely ha he period +1 reurn is drawn from he firs (he second) disribuion. Namely, p 11 > 0.5 > p 12, p 22 > 0.5 > p 21. During our empirical process, all esimaes for ransiion probabiliies (p 11 and p 22 ) exceed 50%. 17 The observed saisical propery of our TAIEX samples appears o lend suppors o he noion ha MS ouperforms mixing-normal. Furhermore, consisen wih he noion ha he Markov chain of 16 Duffie and Pan (1997) brough forward he jump diffusion models, which are akin o he mixing normal model. Insead of he radiional Bernoulli disribuion, hey adoped Poisson disribuion o conrol he probabiliy of jump diffusions. Wih simulaed daa, hey furher demonsraed he impac of, he expeced jump frequency per uni ime inerval, and, he expeced jump sandard deviaion (he magniude), on VaR esimaion. They conclude ha, when he produc of per period expeced frequency and jump sandard deviaion is held consan, he lower he expeced jump frequency per period, or he greaer he jump sandard deviaion, he more significan is he ail-faness in reurn disribuions. Namely, exclusion of rare bu exreme evens may fuel he underesimaion of VaR. 17 For example, p 11 and p 22 are 0.97 and 0.94, respecively, for TAIEX, and are 0.96 and 0.96, respecively, for TAIEX-ELEC.

12 Journal 72 of Financial Sudies Vol.7 No.3 December 1999 (61-94) reurn process is no reducible, in each and every phase in our ess he ransiion probabiliy (p 11 and p 22 ) is less han uniy. To remedy he general mixing normal models omiing he persisence of regimes, Venkaaraman (1997) incorporaed quasi-bayesian ML esimaion of Hamilon (1991) and used Mone Carlo simulaions o measure VaR 18. Hamilon (1991) firs proposed he use of quasi-bayesian ML esimaion, saing ha via inroducing prior consrains, analyss can miigae he singulariy problems in he converging process of ML funcion and hus eliminae overflows or underflows in ML esimaion. A singulariy arises whenever (1) he analyss over-reac o an ouliner and rea ha single observaion as a prevalen sae, or (2) one of he disribuion is impued o have a mean exacly equal o one of he observaions wih no sandard deviaion 0, say.the resul of our rolling esimaion in his sudy, neverheless, indicaes ha he prior consrains in Bayesian, insead of eliminaing overflows or underflows, serve a bes o lower he frequency of his unpleasanness. More imporanly, his alernaive is wih limied srengh for our sample of TAIEX socks, which are subjec o a price limi of 7%. Furhermore, our findings sugges ha incorporaing prior probabiliies would add lile o empirical sudies for TAIEX series. Specifically, we documen ha he difference in regime volailiy for TAIEX reurns significanly conrass wih ha in he seing in Hamilon (1991), who showed ha wih rivial difference beween he parameers in wo-regime models, addiional prior consrains help enhance forecas accuracy and eliminae he muliple local maxima wih EM-Algorihm. Neverheless, we documen ha he sandard deviaion for TAIEX s high -volailiy regime is almos 2.5 imes as grea as ha for he low-volailiy regime. Finally, he subjecive prior consrains are essenial in Bayesian models bu are difficul o be relayed by risk managers or regulaors for implemenaion wihin heir specific economy or marke. GARCH, he fourh alernaive, specifies he period volailiy as a funcion of prior period volailiy and reurn shock squares., he volailiy persisence measure, is equal o a i plus b i as described in Secion 2.2. Wih parameers a i and b i unable o adjus for any srucure changes in he esimaion periods, he GARCH models are a linear volailiy seing. Thus, conras wih Mixing-normal as well as MS, GARCH models are subjec o over-esimaing he persisence of reurn volailiy. 18 See Hamilon (1991).

13 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 73 To sum up, he MS model is he leas biased choice. Firs, he suden seing lacks he generalizaion for series such as TAIEX reurns. Second, MS models effecively adjus for srucural changes in he sample period and may hus ouperform GARCH specificaions. Third, MS models use Markov chain process o incorporae he persisence of volailiy regimes and may hus ouperform he mixure of normal models, which virually disregard he persisence Empirical Design and Tes Resuls This secion firs conrass boh kurosis and skewness coefficiens as well as he criical values of TAIEX reurn disribuions wih srucural changes being filered via MS wih he (unfilered) measures corresponding o he linear models. We hen compare he effeciveness of MS, mixing normal and GARCH in miigaing Lepo kuric and ail faness. Furhermore, we examine he violaion raes corresponding o linear and MS (non-linear) models for boh ailed regions in reurn disribuions. Consisen wih our prior, afer our filering srucural changes for he index reurns via MS, each originally non-normal error erm series appears o conform o sandard normal disribuions. The sample daa is provided by Taiwan Economic Journal (TEJ) and include Taiwan Sock Exchange marke (TAIEX) reurns as well as major indusrial index reurns. The laer es group includes TAIEX- CONS, TAIEX-FIN, TAIEX-EL&MACH, and TAIEX-ELEC. For TAIEX and he firs four indusrial indices, he sample period is beween January 3, 1991 and May 28, 1998, whereas he sample period for TAIEX-ELEC only daed back o he beginning of Accordingly, we have 2,382 observaions for he former indices and 1,236 observaions for he elecronics indusry Kurosis, Skewness and Criical Values of TAIEX Reurn Disribuions wih Srucural Changes Being Filered or Unfilered Table 1 presens saisics for reurn shocks including skewness and kurosis as well as he 1%, 2.5% and 5% criical values for boh ails in TAIEX reurn disribuions. Wihou regarding any srucural changes in mean and variance, as 19 Please refer o Li and Lin (1999) for furher discussions. 20 This sudy adops he Basle Commiee s proposal ha he learning window should a leas include 250 pre-var rading days. However, for TAIEX here are virually approximaely 281 daily observaions of close prices per calendar year.

14 Journal 74 of Financial Sudies Vol.7 No.3 December 1999 (61-94) opposed o he sandard normal disribuions, he disribuions for TAIEX-ELEC, TAIEX-FIN, TAIEX-CONS and TAIEX-EL&MACH reurn shocks are subsanially skewed 21. The kurosis coefficien for TAIEX-ELEC and TAIEX-EL&MACH are 3.81 and 4.51, respecively. Moreover, he kurosis coefficiens for TAIEX-FIN and TAIEX-CONS reurns are boh greaer han 5. We furher explore he significance of ail-faness in he disribuions of index reurn shocks. Table 1 shows ha he realized absolue 1% criical values on boh ails are significanly greaer han ha for a sandard normal disribuion. Moreover, he realized absolue 2.5% criical values for boh ails are greaer han ha for a sandard normal seing. The deviaion, neverheless, appear o decrease as we move owards he cener. Up o he 5% criical probabiliy, here appears no significan difference 22. This finding shows ha here exis small lumps on boh ails in marke reurn disribuions. Table 1 Skewness, Kurosis, and 1%, 2.5%, 5% Criical Values for TAIEX and Taiwan s Major Indusrial Index Reurn Shocks TAIEX TAIEX ELEC TAIEX- FIN TAIEX- CONS TAIEX- EL&MACH Skewness Coefficiens (N=0) Kurosis Coefficiens (N=3) % Lef-ailed Criical Value (N= -2.33) % Lef-ailed Criical Value (N= -1.96) % Lef-ailed Criical Value (N= -1.65) % Righ-ailed Criical Value (N=2.33) % Righ-ailed Criical Value (N=1.96) % Righ-ailed Criical Value (N=1.65) Number of Observaions Denoe R he -h period sock index reurn. Also le u and denoe mean and sandard deviaion for index reurns, respecively. Then reurn shock e = (R - u) /. 21 The TAIEX reurn disribuion is an excepion. For major indusrial group index reurns, he absolue values of he skewness coefficiens are greaer han or equal o For he lef-ailed region, which is he focus of Va R esimaion, boh TAIEX-ELEC and TAIEX-EL&MACH reurn shocks have realized absolue 5% criical values slighly greaer han hose for he sandard normal disribuions. Moreover, for TAIEX, TAIEX-FIN and TAIEX-CONS, he corresponding absolue values are slighly less han ha for he sandard normal disribuions.

15 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) N represens he measure for a sandard normal disribuion. 3. The reurn shocks for TAIEX slighly skew o he lef. In conras, TAIEX-ELEC (TAIEX-FIN and TAIEX-CONS) reurn shocks significanly skewed o he lef (righ). 4. The disribuion for TAIEX-ELEC, which is slighly Lepo kuric, is an excepion. All he oher indusrial index shocks are significanly Lepo kuric. 5. Surrounding he 1% and 2.5% regions, TAIEX and indusrial reurn shocks all appear o be significanly fa-ailed on boh sides as opposed o sandard normal disribuions. There exiss less significan ail faness, however, for he 5% regions on boh ails. On he ground ha he disribuions of unfilered index reurn shocks are subjec o non-normaliy, we use a wo-regime seing for means and variances, wih he Markov-chain process o capure regimes ranslaions. The wo-variance-regime seing serves o conrol Lepo kuric and fa ail in reurn disribuion. 23 Our modeling he wo regimes wih differenial means, on he oher hand, helps filering skewness for our sample disribuions. Table 2 Skewness, Kurosis, and 1%, 2.5%, 5% Criical Values for Taiwan s Index Reurn Shocks, wih he Srucural Change Componens Being Filered via MS Models TAIEX TAIEX- ELEC TAIEX- FIN TAIEX- CONS TAIEX- EL&MACH Skewness Coefficiens (N=0) Kurosis Coefficiens (N=3) % Lef-ailed Criical Value (N=-2.33) % Lef-ailed Criical Value (N=-1.96) % Lef-ailed Criical Value (N=-1.65) % Righ-ailed Criical Value (N=2.33) % Righ-ailed Criical Value (N=1.96) % Righ-ailed Criical Value (N=1.65) Number of Observaions N represens he measure corresponding o sandard normal disribuion. Also, le denoe regime 1 (regime 2) sample mean and sandard error, respecively. Then he filered sock index reurn shocks is: e = P(s =1 I T )(r -u 1 )/ 1 +P(s =2 I T )(r -u 2 )/ 2, 23 The noion is consisen wih Duffie and Pan (1997) and Venkaarman (1997).

16 Journal 76 of Financial Sudies Vol.7 No.3 December 1999 (61-94) where P(s I T ), s =1,2, denoes he esimaed smoohing probabiliy for he full sample. Table 2 presens he error erm saisics for Taiwan s index reurn shocks, wih he srucural change componens being filered by MS models 24. Comparing Tables 1 and 2, we may conclude ha, wih srucural changes being filered via MS models, he error erm non-normaliy drops significanly. For insance, wih srucural change being unfilered, he 1% lef-ailed criical value, skewness and kurosis for TAIEX-FIN error erm are 2.83, 0.29, and 5.17, respecively. Wih he srucural changes being filered, in conras, he corresponding measures are 2.30, 0.03, and 2.82, respecively. Moreover, he filering effec for ail-faness for he 2.5% (1%) regions appears o be less (more) significan. The filering effec for ail-faness for he 5% region appears o be he leas significan among he hree alernaives. Noe ha we adop a seing o which boh of he wo mean regimes are exougenous. Our seing, as compared wih Venkaarman (1997), who se he firs regime mean reurn o be zero, appears o be more generalized The Performance of he Compeing Models in Miigaing Non-Normaliy Problems There may be hree alernaives o miigae Lepo kuric and fa ails in reurn disribuion: he mixing normal, MS and GARCH models. In he mixure of normal disribuion models, regimes swiches are purely random, wih no persisence in regimes cross ime. In conras, MS models adop Markov chain process o incorporae he measure of regime persisence. The GARCH models differ from MS and mixing normal models in ha he former seing parameers do no adjus for srucural changes during he esimaion periods and hus may over-esimae he volailiy persisence. To compare he performance of MS, he mixing normal and GARCH for Lepo kuric and/or fa-ailed reurns, we use he linear models as he benchmark seings and apply he hree compeing models o he probabiliy densiy funcion 24 We use OPTIMUM, a package program from GAUSS, and he buil-in BFGS (Boyden, Flecher, Goldfarb, and Shanno) algebra o derive he negaive minimum likelihood (ML) funcion values of all he models. BFGS algebra is effecive for deriving he maximum value of he non-linear likelihood funcions. See Luenberger (1984). We randomly generae 100 ses of iniial values. We hen derive he ML funcion value for each of he 100 ses of iniial values. The mapped converged measure wih he greaes ML funcion value hen serves as he esimae. 25 The models we adop, neverheless, require an addiional parameer as opposed o ha in Venkaarman (1997).

17 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 77 of Taiwan Sock Exchange index reurns. Table 3 presens he relaive srenghs of he compeing models in handling non-normaliy problems in he TAIEX reurn shocks 26. Our firs horse race focuses on he Lepo kuric problem. Noe ha he linear models resul a large kurosis coefficien of In conras, MS, Bayesian mixing normal and GARCH models all generae lower kurosis coefficiens (2.93, 2.38 and 4.66, respecively). Specifically, he kurosis coefficien wih MS model is closes o ha of sandard normal disribuion. In conras, he coefficien wih respec o Bayesian mixing normal (GARCH) model appears o be lower (higher) as opposed o sandard normal disribuions. As o ail-faness, he 1% lef-ailed criical values for linear, MS, Bayesian mixing normal and GARCH models are -3.05, -2.34, and -2.84, respecively. The resuls sugges ha he laer hree seings are all effecive in filering srucural change componens for fa-ailed reurns. Neverheless, as compared wih sandard normal disribuions, for which he 1% lef-ailed criical value is -2.33, MS models perform he bes. Bayesian mixing normal (GARCH) models, relaively speaking, under- (over-) esimae he expeced loss. As o he 2.5% and 5% lef- and righ-ailed regions, all he hree remedial models are wih decreasing magniudes in reducing ail-faness. These findings are consisen wih our linear model resuls ha ail-faness is significan for he 1% region bu less significan for he 2.5% and 5% regions on boh sides. The relaive performance of he compeing models may be measured by he exen he saisics for filered reurn shocks resemble hose for sandard Gaussian, no by he absolue measures of he saisics. In oher words, he benchmark kurosis coefficien, skewness coefficien, and lef-ailed 1% criical value are 3, and 0, -2.33, respecively. Wih he corresponding esimaes for kurosis coefficien of 2.93, MS models appear o generae reurn shock saisics closer o he benchmark measure of 3.0 as opposed o Bayesian mixing-normal models (he corresponding esimaes is 2.38.) Likewise, he lef-ailed 1% criical value for MS is and lies much closer o he benchmark of 2.33 as compared wih Bayesian mixing-normal (he corresponding esimaes is ) Figure 1 presens he probabiliy densiy funcion of TAIEX reurn shocks wih he seing of no swiches in mean or variance. As opposed o sandard normal, here exis significan Lepo kuric and ail faness. Figures 2, 3 and 4 documen he probabiliy densiy funcions of TAIEX reurn shocks for MS, Bayesian mixing normal and GARCH, respecively. Taking Figure 1 as our 26 Table 2 shows ha he TAIEX reurns are no significanly skewed if we apply he linear model for he reurn shocks.

18 Journal 78 of Financial Sudies Vol.7 No.3 December 1999 (61-94) benchmark, Figures 2, 3 and 4 show ha he hree non-linear models help miigae Lepo kuric and ail-faness problems. Le us furher examine Figures 2 o 4. Bayesian mixing normal models appear o over-reac o Lepo kuric and ail faness. As shown in Figure 3, he ail areas differ significanly from ha in he sandard normal specificaion. GARCH models, on he oher hand, under-reac o Lepo kuric and fa ails. In conras, MS models, which generae less biased resuls, appear o be he mos efficien in filering srucural changes. The error erm wih MS, as compared wih hose wih he wo compeing models, closely conforms o sandard normal disribuions. The finding is consisen wih he noion ha Table 3 Effeciveness of he Compeing Models in Miigaing Lepo kuric, Tail-Faness and Skewness Problems Linear MS Bayesian Mixing-Normal GARCH (1,1) Kurosis Coefficiens (N=3) * Lef-ailed 1% Criical Value (N=-2.33) * Lef-ailed 2.5% Criical Value (N=-1.96) * Lef-ailed 5% Criical Value (N=-1.65) * Righ-ailed 1% Criical Value (N=2.33) * Righ-ailed 2.5% Criical Value (N=1.96) * Righ-ailed 5% Criical Value (N=1.65) * * Number of Observaions 2,382 2,382 2,382 2, This able shows ha wih he linear models, TAIEX reurn shocks slighly skew o he lef and are significanly Lepo kuric wih respec o criical probabiliies of 1% and 2.5% for boh ails. Also, here appears significan ail-faness for boh ends. However, ail faness phenomena become less significan when he criical probabiliy is se o 5% for boh lef- and righ-ailed regions. 2. * denoes he measure closes o he saisics for sandard normal disribuions. he general mixing normal models disregard he informaion for regimes persisence cross ime, whereas GARCH models over-esimae he persisence in volailiy. Namely, he prevalen normal specificaions for he error erm can benefi from incorporaing MS models o conrol he srucural change componens in he esimaion period probabiliy

19 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 79 Figure 1. PDF for TAIEX Reurn Shocks wih he Linear Models probabiliy reurn shock% Figure 2. PDF for TAIEX Reurn Shocks wih Markov-swiching Models probabiliy

20 Journal 80 of Financial Sudies Vol.7 No.3 December 1999 (61-94) Figure 3. PDF for TAIEX Reurn Shocks wih Mixure of Normal Models probabiliy reurn shock% Figure 4. PDF for TAIEX Reurn Shocks wih GARCH Models %

21 Journal of Financial Sudies Vol.7 No.3 December 1999 (61-94) 81 Figure 5. TAIEX Reurns during he Sample Period Probabiliy years Figure 6. Regime 1 Predicing Probabiliies for TAIEX Reurns via MS Models regmie 1 sandard error regime 2 sandard error

22 Journal 82 of Financial Sudies Vol.7 No.3 December 1999 (61-94) Figure 7. Esimaed Regime 1 and Regime 2 Sandard Errors for TAIEX reurns via MS Models

23 82 Journal of Financial Sudies Vol.7 No.3 December regime 1 mean regime 2 mean Figure 8. Esimaed Regime 1 and Regime 2 Means for TAIEX Reurns via MS Models TAIEX Reurns Series Predicing VaR for 95%

24 Journal of Financial Sudies Vol.7 No.3 December (a)the Predicing VaR for a Criical Inerval of 95% TAIEX Reurns Series Predicing VaR for 97.5% (b)the Predicing VaR for a Confidence Inerval of 97.5% TAIEX Reurns Series Predicing VaR for 99% (c)the Predicing VaR for a Confidence Inerval of 99%

25 84 Journal of Financial Sudies Vol.7 No.3 December 1999 Figure 9. The Predicing VaR wih 250-Prior-Trading-Day Windows via MS Model 3. Violaion Rae Examinaions (1) Lef-Tailed and Righ-Tailed Violaion Raes for Linear and MS Models wih 250- and 500-Prior-Trading-Day Windows We conduc horse races for boh linear and non-linear (MS) models on TAIEX, TAIEX-ELEC, TAIEX-FIN, TAIEX-CONS and TAIEX-EL & MACH. We alernaely use 250- and 500- prior-rading-day windows 27 in he rolling esimaion process and measure he viola ion raes on boh ails of he disribuions. (a) Esimaing he Criical Values for Inervals wih Negaive Reurns R Our research design may be illusraed via he ess wih 250 pre-var daily observaions. For each dae, we ge he esimaes from hisorical reurns series { } 250 i i=1. Then our Mone Carlo algorihm generaes 10,000 random numbers o simulae he disribuion of index reurn R. For 1%, 2.5% and 5% lef-ailed regions, we furher esimae VAR and make comparisons beween VAR and sock reurns R. If R < VAR (R > VAR ), he number of violaions would increase by one (would remain unchanged). The violaion rae measures hen serve as our indicaor of forecas accuracy. 28 For esimaion pracices, we adop he ex ane predicing probabiliies P(s y -1,y -2, ) o weigh he wo normal disribuions wih differen means and variances. 29 To illusrae wih our TAIEX reurns series, he sample period is beween January 3, 1991 and May 28, 1998, wih 2,382 observaions. For our ess wih 250 prior rading days as he learning sage, we have 2,132 violaion rae observaions. Figures 5 and 6 presen TAIEX reurns and he predicing probabiliies wih respec o regime 1, respecively. Moreover, Figure 7 presens he esimaes for daily sandard deviaion of reurns corresponding o regimes 1 and 2. Furhermore, Figure 8 presens he esimaed mean reurns for regimes 1 and In our rolling esimaion process, we move one sep furher and inroduce one more observaion for each ime poin. To faciliae convergence in our esimaion process, we use he esimaes for each period as he iniial values in our non-linear esimaion for he immediaely subsequen period. 28 The closer he percenage violaion o he criical probabiliy, he higher is he forecas accuracy. 29 For specific applicaions, condiional probabiliy p(s may be ailored for differen applicaions wih simply subsiuing a differen measure of q. Specifically, when q=0, i is a filering probabiliy. If q<0, i is an ex ane predicing probabiliy. Whereas when q>0, i is an ex pos smoohing probabiliy.

26 Journal of Financial Sudies Vol.7 No.3 December Since he sandard deviaion for Regime 1 (Regime 2) reurns is significanly greaer (less) 30, Regime 1 (Regime 2) is labeled as he high (low) volailiy regime 31. Figure 9 presens he VaR esimaes corresponding o MS models wih 250-prior-rading-day windows for confidence inervals 95%, 97.5% and 99%. The lef-ailed region is he focus of concurren VaR sudies. Panel A of Table 4 presens he violaion raes associaed wih TAIEX and Taiwan s indusrial reurns when he lef-ailed criical probabiliy is se o 1%. In erms of violaion rae, he wo MS models rank he firs and he second in forecas accuracy. The findings lend suppor o he superioriy of MS models. MS model wih a 500 prior-rading-day learning window excels wih respec o TAIEX, TAIEX-ELEC, TAIEX-CONS and TAIEX-FIN. I also ranks he second for TAIEX-EL&MACH. In conras, MS wih a 250 pre-var-day learning window performs he bes for TAIEX-FIN as well as TAIEX-EL&MACH and ranks he second for TAIEX, TAIEX-ELEC and TAIEX-CONS. Panel B of Table 4 presens he resuls for he seing wih criical probabiliy equal o 2.5%. The linear models wih a 500-day learning window perform he bes for TAIEX-CONS and rank he second for TAIEX-FIN. The linear models wih a 250-rading-day learning period rank he second for TAIEX-ELEC. MS models wih a 500-day learning period perform he bes for TAIEX, TAIEX-FIN as well as TAIEX-EL&MACH and rank he second for TAIEX-CONS. The MS models wih a 250-day window perform he bes for TAIEX-ELEC and rank he second for TAIEX-EL&MACH and TAIEX-ELEC. Panel C of Table 4 documens he horse race resuls regarding he 5% lef-ailed criical probabiliies. The linear models wih a 500 pre-var observaions perform he bes for TAIEX. The linear models wih a 250 rading-day windows perform he bes for esimaing TAIEX-ELEC volailiy and ranks he second for TAIEX and TAIEX-EL&MACH. MS models wih previous 500 daily reurn as he learning period perform he bes for TAIEX-FIN, TAIEX-CONS, TAIEX-EL&MACH. MS models wih a 250-day window rank he second for TAIEX-ELEC, TAIEX-FIN, and TAIEX-CONS. 30 Namely, 31 In conras, here exis mixed resuls as o he order of sample means wih respec o regimes 1 and 2.

27 86 Journal of Financial Sudies Vol.7 No.3 December 1999 Table 4 Lef-Tailed Violaion Raes for he MS and he Linear Models wih Differenial Learning Windows for TAIEX Marke and Indusrial Reurns Panel A Criical Probabiliy = 1% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 2.11% 2.07% 1.97% * 1.38% ** TAIEX-ELEC 2.64% 3.80% 1.93% * 1.77% ** TAIEX-FIN, 1.88% 1.59% 1.13% ** 1.22% * TAIEX-CONS 2.11% 2.13% 1.92% * 1.43% ** TAIEX- EL&MACH 2.53% 2.60% 1.36% ** 1.43% * Panel B Criical Probabiliy = 2.5% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 3.28% 3.51% 3.19% * 2.60% ** TAIEX-ELEC 3.65% * 5.84% 3.25% ** 4.89% TAIEX-FIN 2.77% 2.60% * 2.95% 2.44% ** TAIEX-CONS 2.95% 2.82% ** 3.24% 2.92% * TAIEX-EL&MACH 3.71% 4.04% 3.42% * 2.82% ** Panel C Criical Probabiliy = 5% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 4.74% * 4.89% ** 5.77% 5.31% TAIEX-ELEC 5.78% ** 7.07% 6.09% * 7.07% TAIEX-FIN 4.22% 3.83% 5.25% * 5.10% ** TAIEX-CONS 4.16% 4.41% 5.35% * 5.10% ** TAIEX-EL&MACH 5.39% * 5.90% 5.91% 5.37% ** 1. We adop boh 250- and 500-day learning windows of prior daily reurns. For he violaion rae ess wih a 250-day window, here are 986 (2,132) observaions for TAIEX-ELEC (he oher indices). In our ess wih a 500-day window, he number of observaions for TAIEX-ELEC (he oher indices) is 736 (1,882). 2. ** (*) denoes he mos accurae (he second mos accurae) measure in erms of violaion rae.

28 Journal of Financial Sudies Vol.7 No.3 December To sum up, for users wih criical probabiliy of 1%, MS wih a 500-day learning window dominaes he compeing models in erms of violaion rae. As o he 2.5% criical probabiliy, MS wih a 500 prior-rading-day window sill beas he compeing models bu wih smaller differences. As he criical probabiliy is se o be 5%, here are even smaller differences. As o he choice of he learning window, generally speaking, he seing wih 500 (250) pre-var observaions generaes more (less) accurae esimaes. (b) Esimaing he Criical Value for he Inerval wih Posiive Reurns To gain insighs as o he srenghs/weaknesses of he compeing models as well as he saisical properies of TAIEX index reurns, we also apply he VaR algorihm o sudy he righ-ailed region. Specifically, we examine he violaion rae, he measure of forecas accuracy, associaed wih each of our combinaions of compeing models and learning windows for TAIEX marke and indusrial reurns. Panels A, B, and C in Table 5 presen resuls corresponding o criical probabiliies of 1%, 2.5% and 5%, respecively. In general, MS models bea he alernaives for boh 1% and 2.5% righ-ailed regions in erms of violaion rae. When he criical probabiliy is se o 5%, however, he MS models no longer ouperform he linear models. Moreover, wih MS models, we find he shorer 250-day learning window alernaive ouperforms he 500-day seing. A poenial explanaion is ha incorporaing more hisorical daa helps gain he insighs for he complee cycle bu drives he model o be less sensiive o regime swiches and hus over-esimae he persisence. Le us summarize he findings in his secion. For boh ails, MS dominaes he popular linear models in depicing he ailed regions in reurn disribuions 32. The difference, however end o decrease as we move owards he cener of he disribuion. Moreover, our empirical resuls show ha, o depic he lef- as opposed o he righ-ailed region, one needs o incorporae more hisorical daa. For he buss, one would need a longer window of 500 prior daily reurns (approximaely wo-year daa) o capure he odd evens wih large negaive losses. On he oher hand, a learning window of 250 prior observaions (nearly one-year daa) would be sufficien o depic he booms. 32 When he righ-ailed probabiliy is se o 1% or 2.5%.

29 88 Journal of Financial Sudies Vol.7 No.3 December 1999 Table 5 Righ-Tailed Violaion Raes for he MS and he Linear Models wih Differenial Learning Windows for TAIEX Marke and Indusrial Reurns Panel A Criical Probabiliy = 1% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 2.02% 1.86% 1.17% ** 1.28% * TAIEX-ELEC 1.83% 2.85% 0.91% ** 1.49% * TAIEX-FIN 3.10% 3.19% 1.69% ** 1.86% * TAIEX-CONS 2.39% 2.60% 1.41% * 1.28% ** TAIEX- EL&MACH 1.88% 2.07% 0.94% * 1.01% ** Panel B Criical Probabiliy = 2.5% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 3.19% 3.19% 2.67% ** 2.92% * TAIEX-ELEC 3.55% 4.89% 2.64% ** 2.72% * TAIEX-FIN 4.41% 4.46% 3.61% ** 4.04% * TAIEX-CONS 3.94% 3.72% 2.91% ** 3.45% * TAIEX- EL&MACH 3.42% 3.72% 2.67% ** 2.82% * Panel C Criical Probabiliy = 5% Sock Index Reurns Linear Model wih 250 Daily Reurn Linear Model wih 500 Daily Reurn MS Model wih 250 Daily Reurn MS Model wih 500 Daily Reurn TAIEX 5.21% ** 5.21% ** 5.72% 5.42% * TAIEX-ELEC 6.19% 7.74% 5.38% ** 5.71% * TAIEX-FIN 6.14% 5.74% ** 6.14% 6.06% * TAIEX-CONS 5.91% 5.47% ** 5.82% * 6.11% TAIEX-EL&MACH 5.96% 5.90% 4.83% ** 5.37% * When he criical probabiliy is se o 1% or 2.5%, MS models is he mos accurae alernaive in erms of violaion rae. In conras, when he criical probabiliy is se o be 5%, MS models are no longer superior o he compeing seings.