The Influence of Posiive Feedback Trading on Reurn Auocorrelaion: Evidence for he German Sock Marke Absrac: In his paper we provide empirical findings on he significance of posiive feedback rading for he reurn behavior in he German sock marke. Relying on he Shiller- Senana-Wadhwani model, we use he link beween index reurn auo-correlaion and volailiy o obain a beer undersanding ino he reurn characerisics generaed by raders adhering o posiive feedback rading sraegies. Our empirical evidence shows ha in he German sock marke a significan proporion of invesors are posiive feedback raders and ha his posiive feedback rading seems o be responsible for he observed negaive reurn auocorrelaion during periods of high volailiy. JEL Classificaion: G14, C Keywords: Reurn Auocorrelaion, Posiive and Negaive Feedback Trading, German Sock Marke
1. Inroducion There can be no doub ha some invesors ry o discover rends in pas sock prices and base heir porfolio decisions on he expecaion ha hese rends will persis. In he behavioral finance lieraure his ype of invesors is usually called a feedback rader. Posiive feedback raders buy socks in a rising marke and sell socks in a falling marke, while negaive feedback raders adhere o a buy low, sell high invesmen sraegy. One of he consequences of he exisence of a sufficienly large number of feedback raders in he sock marke is he auocorrelaion of reurns and, hence, he parial predicabiliy of aggregae sock reurns. On he one hand, he behavioral finance lieraure provides a fair amoun of heoreical models of feedback rading, and he experimenal findings, as well as, he survey evidence overwhelmingly suppor he exisence of posiive feedback raders. 1 On he oher hand, he empirical evidence is mixed wih respec o he presence of feedback raders in sock markes and he resuling consequences for reurn behavior. For example, Shefrin and Saman (1985) and Odean (1998) provide evidence in favor of he disposiion effec, i. e. invesors are relucan o realize losses and hey sell winners oo early, which conradics he posiive feedback hypohesis. Lakonishok, Shleifer, and Vishny (199) invesigae posiive feedback sraegies aken by insiuional invesors and find, wih he excepion of small socks, no evidence of 1 Theoreical models on feedback rading can be found in Shiller (1984), DeLong, Shleifer, Summers, and Waldmann (1990), Culer, Poerba, and Summers (1990), Kirman (1993), Campbell and Kyle (1993), and Shleifer (000). Kroll, Levy, and Rapopor (1988), Shiller (1988), De Bond (1993), and Bange (000) among ohers provide experimenal and survey evidence.
3 posiive feedback rading in pension funds. Whereas, he ime series evidence conained in Senana and Wadhwani (199), Campbell and Kyle (1993), Koumos (1997), and Koumos and Said (001) suppors o a large exen he noion of posiive feedback rading in developed, as well as, emerging sock markes. The aspecs oulined above shows ha empirical sudies analyzing feedback rading provide inconclusive evidence and ha here are only a few empirical sudies in he exising lieraure. Lack of daa, as well as, he difficuly o discriminae empirically beween feedback rading and oher heoreical explanaions for reurn auocorrelaion mos prominenly non-synchronous rading (Lo and MacKinlay, 1990), ime-varying expeced reurns (Conrad and Kaul, 1988, 1989) and ransacion coss (Mech, 1993) are responsible for he gap in he lieraure o find sufficien evidence on he conribuion of feedback rading for auocorrelaed reurns. In his paper we use he link beween reurn auocorrelaion and volailiy o beer undersand he significance of posiive feedback rading in Germany s sock marke by analyzing daily daa of he C-Dax, he Dax, and he Nemax50 index over he 1998 001 period. The small number of empirical sudies on he impac of feedback rading on reurn auocorrelaion and he concenraion in he empirical finance lieraure on he US sock marke moivaes our selecion of he German sock marke. Providing empirical evidence for German sock price indices reduces he daa snooping bias and allows o compare our findings wih he previous lieraure. The heoreical poin of deparure is he feedback rader model pu forward by Shiller (1984) and Senana and Wadhwani (199). Nelson s (1991) exponenial GARCH model and an even sudy focusing on he Sepember 11, 001, crash provide he mehodological basis. There has been no empirical sudy on he presence of
4 feedback rading as one of he possible forces deermining he properies of reurns in Germany s sock marke. We are ineresed in he quesion of wheher posiive feedback raders are presen in Germany s sock marke and, if so, wha i implies for reurn behavior. The res of he paper is organized as follows: Secion oulines he feedback rader model. The discussion of he esing sraegies and he empirical findings are presened in Secion 3. Secion 4 provides he conclusion.. Feedback Trading and Auocorrelaed Reurns The Shiller-Senana-Wadhwani model (Shiller, 1984; Senana and Wadhwani, 199) capures he behavior of wo disinc ypes of invesors in he sock marke. Feedback raders or rend chasers as a group do no base heir asse decisions on fundamenal value and insead reac o price changes. Their demand for socks is based on he hisory of pas reurns raher han expeced fundamenals. The second group, smar money invesors, responds raionally o expeced reurns subjec o heir wealh limiaion. The presence of boh groups in he sock marke and heir specific behavior provides he heoreical raional for serially correlaed sock reurns and he imporance of volailiy for he reurn auocorrelaion characerisics. The relaive demand for socks by feedback raders, F, is modelled as: F γ R, (1) = 1 where R 1 denoes he reurn in he previous period. The value of he parameer γ permis he differenaion beween he wo ypes of feedback raders. γ > 0 refers o he case of posiive feedback raders, who buy socks afer a price rise and sell socks
5 afer a price fall. Buying in a rising marke and selling in a falling marke can resul from exrapolaing expecaions abou sock prices or rend chasing. Furhermore, porfolio insurance is an example of a posiive feedback rading sraegy. This sraegy implies ha in a rising marke a higher proporion of wealh is invesigaed in socks, which generaes sock price increases. In a falling marke, a lower proporion of wealh is invesigaed in socks by he porfolio insurance sraegy, which resuls in sock sales and sock price decreases. Anoher form of posiive feedback rading is he use of sop loss orders, which prescribe selling afer a cerain level of losses regardless of fuure prospecs. Moreover, he effecs of he liquidaion of invesors posiions who are unable o mee margin calls are comparable o he impacs of a posiive feedback rading sraegy. γ < 0 indicaes he case of negaive feedback rading. Unlike a posiive feedback rader, he negaive feedback rader exhibis a buy low, sell high sraegy, i. e. selling socks afer price increases and buying socks afer price declines. Negaive feedback rading can resul from profi aking as markes rise or from invesmen sraegies ha arge a consan share of wealh in differen asses. The proporionae demand for socks by smar money raders, S, is deermined by a mean-variance model: S = ( E 1R α )/ µ, () where E 1 denoes he expecaion operaor and α he reurn on a risk free asse. In his model smar money raders hold a higher proporion of socks, he higher he expeced excess reurn, E 1 R α, and he smaller he riskiness of socks, µ. The risk measure is modelled as a posiive funcion of he condiional variance, σ, of
6 σ sock prices µ = µ ( ), where he firs derivaion is posiive reflecing risk averse invesing behavior. Equilibrium in he sock marke requires ha all socks are held: S F =1. (3) + If all invesors are smar money raders, F = 0, hen marke equilibrium, S = 1, yields Meron s (1973) capial asse pricing model: 1R E α = µ ( σ ). (4) Allowing he exisence of boh groups in he sock marke and subsiuing (1) and () in (3) yields, afer rearranging and under he assumpion of raional expecaions, R = E 1 R + ε : R = + µ σ ) γµ ( σ ) R 1 α ( + ε. (5) As can be seen from equaion (5) in a marke wih smar money invesors, as well as, feedback raders, he resuling reurn equaion conains he addiional erm R 1 so ha sock reurns exhibi auocorrelaion. The paern of auocorrelaion in reurns depends on he ype of feedback rader capured by he parameer γ, where posiive (negaive) feedback rading, γ > 0 ( γ < 0), implies negaively (posiively) auocorrelaed reurns. Furhermore, he exen o which reurns exhibi auocorrelaion varies wih he σ level of reurn volailiy, µ ( ). For example, if here is an increase in volailiy, smar money readers reduce he demand for socks (see equaion ()), which allows feedback raders o have a greaer impac on he sock price. Consequenly, a larger discrepancy beween he curren sock price and is fundamenal value resuls. This is
7 due o he larger proporion of socks demanded by feedback raders so ha sock reurns exhibi sronger auocorrelaion. The paern of auocorrelaion is deermined by he ype of feedback rader and he exen of volailiy, which becomes obvious when relying on a linear form for µ ( ) in equaion (5): σ R = + µ σ ) ( γ 0 + γ 1σ ) R 1 α ( + ε. (6) Equaion (6) is crucial for our empirical invesigaion. Firs of all, a a consan risk level, σ, he direc impac of feedback raders is given by he sign of he parameer γ, where negaive (posiive) feedback rading, γ 0 ( γ 0 > 0), resuls in posiively 0 0 < (negaively) auocorrelaed reurns. Suppose γ 0 is negaive and γ 1 is posiive. A low volailiy levels Senana and Wadhwani hypohesize ha negaive feedback rading dominaes, which induces posiive serial correlaion in reurns due o he relaive srengh of γ 0 compared o γ 1σ. As risk increases, he larger influence of γ 1σ compared o γ 0 induces negaively auocorrelaed sock reurns due o he dominance of posiive feedback raders. Negaive feedback rading is only one hypohesis ha explains posiive auocorrelaion in daily sock reurns. Oher poenial explanaions ofen proposed in he finance lieraure are non-synchronous rading, ime-varying expeced reurns, and ransacion coss. The firs, and mos prominen, explanaion saes ha index reurn auocorrelaion resuls due o non-synchronous rade price observaions of he socks in an index. Sock prices are compued a fixed poins in ime, for example, a he close of each rading day. Generally, he las price observed for each share prior o poin is used o compile he index a ime. Since rading occurs a discree poins in ime for
8 some socks he las rade may have occurred a an earlier poin in ime, while for oher socks he las rade may have occurred jus prior o ime. Consequenly, he value of he index reflecs a mixure of sale, as well as, conemporaneous rade prices. The posiive auocorrelaion in index reurns is induced because raded and non-raded shares are grouped ino an index and, hence, some of he reurns for he inerval 1 o reflec informaion arriving in he previous inerval o 1 (Lo and MacKinlay, 1990). The second explanaion posulaes ha he expeced reurns on socks share a common, posiively auocorrelaed process. Auocorrelaion in expeced reurns is driven by serially correlaed risk premiums ha in urn induces auocorrelaion in raw reurns of he individual and index reurns. Time-varying risk premiums can be explained by ineremporal asse pricing models, such as condiional versions of he arbirage pricing heory or he consumpion based asse pricing model. Variaion in risk facors induce variaion in shor-horizon risk premiums (Conrad and Kaul, 1988, 1989). According o he hird explanaion invesors do no rade on new informaion if gains due rading are lower han informaion and ransacion coss. Coss of processing informaion and direc rading coss may inhibi rading and herefore, delay he ransmission of new informaion ino sock prices. If he index conains socks ha immediaely reflec new informaion, as well as, socks ha do no, hen index reurns exhibi posiive auocorrelaion (Mech, 1993). Available empirical evidence demonsraes ha he degree of daily aggregae reurn auocorrelaion is oo large o be explainable by he argumens menioned above. For example, Mech (1993), Ogden (1997), McQueen, Pinegar, and Thorley (1996) provide lile empirical suppor ha reurns are serially correlaed due o ime-
9 varying risk premiums. Similarly, Mech s (1993) ransacion coss argumen and Lo and MacKinlay s (1990) non-synchronous rading hypohesis canno compleely accoun for he observed auocorrelaions (see also, Boudoukh, Richardson, and Whielaw, 1994). Neverheless, we canno enirely ignore hese hypoheses as empirically valid heoreical explanaions for posiively auocorrelaed index reurns alhough none of hese approaches explicily relies on he relaionship beween reurn auocorrelaion and volailiy. Our proposed mehod o answer he quesion of wheher posiive feedback raders ac in Germany s sock marke is o idenify periods of high volailiy and invesigae he specific reurn characerisics for hese periods. Are here enough posiive feedback raders during periods of high volailiy o generae negaive reurn auocorrelaion and o overcompensae he posiive auocorrelaion in reurns due o negaive feedback rading and/or due o he oher possible explanaions? We answer his quesion in he nex secion. 3. Daa, Mehodology, and Empirical Findings The ime series used for our empirical invesigaion consis of daily daa of he C- Dax, he Dax, and he Nemax50 index for he period from January 1, 1998 o November 1, 001, which amouns o abou 1000 observaions. The C-Dax covers approximaely 675 shares and is herefore a very broad index. The Dax conains 30 German blue chips and reflecs sock price developmen in he marke segmen belonging o he more radiional firms. The Nemax50 conains he larges high-ech companies in he Neuer Mark. The uilizaion of hese indices enables us o provide a broad picure of he quesion under scruiny and he unique sample lengh allows a
10 direc comparison of he empirical findings. From he daily close prices, we calculae he index reurn as he percenage of he logarihmic difference, i. e. R (ln P ln P 1) 100 =, where P is he index a ime. To provide preliminary evidence on he link beween volailiy and auocorrelaion of index reurns, we underake he following experimen. Few economiss would disagree ha sock marke volailiy dramaically increased during he days afer he erroris acs in he U.S. on Sepember 11, 001. This sock marke crash enables us o assess he effecs of volailiy on he auocorrelaion properies of sock reurns wihou having o model a measure of volailiy. Therefore, we esimae he following auoregression: R α ( γ γ Crash R + ε, (7) = + 0 + 1 ) 1 where he dummy variable Crash is equal o one during he crash week (Sepember 11 o 14), during he five rading days afer he crash (Sepember 11 o 18), during he Sepember 19 o 5 period, and equal o zero oherwise. According o he heoreical discussion in Secion, we expec a saisically significan negaive parameer γ 1 a leas for he wo periods direcly afer he Sepember 11 crash due o posiive feedback rading sraegies. Wih a reducion in volailiy he negaive auocorrelaion in sock reurns possibly vanishes during he hird period resuling in no, or posiive auocorrelaed reurns. Table 1 abou here
11 Table 1 conains he resuls of our experimen. Wih only one excepion, he esimaed parameers of he crash dummies are for he firs wo periods (Sepember 11 o 14 and Sepember 11 o 18) saisically significan negaive (a leas) a he 5 % level. In conras, all esimaed parameers γ 1 for he Sepember 19 o 5 period are saisically insignifican from zero. These resuls sugges ha here are enough posiive feedback raders in he German sock marke generaing negaive serially correlaed reurns during periods of high volailiy. During he period of lower volailiy, afer mos of he impac of he erroris aacks vanished, C-Dax, Dax, and Nemax50 reurns do no show saisically significan negaive firs order reurn auocorrelaion. Clearly, he simple dummy analysis canno be fully convincing. The repored negaive auocorrelaion is based only on a few observaions, he selecion of he dummy periods is arbirary, and here is no explici measure of volailiy. These hree argumens indicae a more rigorous analysis. Table provides an overview of he ime series characerisics of he C-Dax, Dax, and Nemax50 indices by reporing mean, variance, skewness, and kurosis for he daily reurns. The imes series of all index reurns are drifless and he uncondiional variance for he Nemax50 index reurns is significanly higher han he variances for he C-Dax and he Dax. Like almos all high frequency financial daa, normaliy of he reurn disribuion is rejeced by he measures of skewness and kurosis. An inspecion of Table suggess ha he C-Dax, Dax, and Nemax50 reurns have o be modelled as heeroskedasic and/or fa-ailed. Table abou here
1 As is shown in Secion of he paper, he index reurn auocorrelaion may vary over ime wih he dominance of posiive or negaive feedback raders, which in urn should be a funcion of reurn volailiy. To inroduce a volailiy erm ino he mean equaion, we use he exponenial GARCH (EGARCH) mehodology proposed by Nelson (1991) where equaion (6) is joinly esimaed wih: lnσ g = β0 + β1g 1 + β lnσ 1 ( z E z ) = ψ z + δ. (9) (8) In equaion (9), z = ε / σ denoes he sandardized innovaion. The consrucion of g allows he condiional variance process and decreases in index reurns. If z > 0 if z 0, hen g is linear in z wih slope σ o respond asymmerically o increases, hen g is linear in z wih slope ψ + δ, and ψ δ. This allows us o provide empirical evidence on he leverage effec (Black, 1976) as a heoreical jusificaion of asymmeric sock reurn volailiy. According o he leverage effec, sock price declines increase he deb o equiy raio, which in urn increases sock reurn volailiy relaive o sock price increases. Many sudies dealing wih index reurns employ he normal densiy funcion. However, in his case he parameer esimaes are no asympoically efficien because he sandardized residuals appear o be lepokuric. To preven parameer esimaes from being influenced by ouliers wih low probabiliy we use he generalized error disribuion.
13 The esimaion resuls are summarized in Table 3. The coefficiens describing he condiional variance process are saisically significan in all cases. When looking a he esimaes for ψ and δ here is evidence of asymmery in he dependence of he volailiy from negaive and posiive innovaions. The impac of negaive innovaions is a leas wice as large as he impac of posiive innovaions. This implies ha in he index reurns under consideraion he volailiy is higher in periods of marke decline han in marke upurns, which can be heoreically jusified by he leverage effec. The esimaes of he β coefficiens reveal a high degree of shock persisence in volailiy. Furhermore, he esimaed model generaes hick ails wih boh a randomly changing condiional variance and a hick ailed condiional disribuion for he sandardized errors. According o he values of υˆ he disribuion of he εˆ is significanly hickerailed han he normal disribuion. Table 3 abou here We now urn o he crucial findings of he parameer esimaes γ ˆ 0 and γ ˆ 1 o answer he quesion abou he exisence of posiive feedback raders in he hree German sock marke segmens. The resuls are consisen wih our heoreical suggesions because all γ ˆ 0 coefficiens are saisically significan negaive and he γˆ 1 In addiion o he EGARCH(1, 1) specificaion, we experimened wih processes of higher order. The coefficiens of higher order processes are saisically insignifican (resuls are no shown bu available on reques), which jusifies he use of he parsimonious EGARCH(1, 1) model.
14 parameers are significan posiive. During periods of high volailiy here is enough posiive feedback rading in he German sock marke o produce negaive firs order auocorrelaed reurns, even hough oher facors end o generae posiive auocorrelaion. These findings are broadly consisen wih he empirical evidence in Senana and Wadhwani (199), Koumos (1997), and Koumos and Said (001) for oher developed, as well as, emerging sock markes. So far, he empirical resuls mee he necessary condiion ha he esimaes for γ 0 and γ 1 have he expeced signs. Bu according o equaion (6), sock index reurns only exhibi negaive auocorrelaion if he magniude of a negaive γ 1 is sufficienly high o compensae for a posiive γ 0, given condiional reurn volailiies. Therefore, we assess he empirical relevance of posiive feedback rading by calculaing he auocorrelaion coefficien, 0 1 ˆ ρ = γˆ + γˆ σˆ, for he esimaed minimum, mean, and maximum condiional volailiy. The resuls are repored in Table 4. Table 4 abou here The calculaed values ˆρ min, ˆρ mean, and ˆρ max indicae ha posiive feedback rading is no a phenomenon of a few rading days wih peaking volailiy, bu can be found a (fairly low) mean volailiy levels. Wih increasing volailiy posiive feedback raders have an even greaer influence on he index reurns inducing negaive reurn auocorrelaion which confirms he heory suggesed above.
15 4. Conclusion In his paper we provide empirical evidence on he imporance of posiive feedback rading for he reurn behavior in differen German sock marke segmens. Relying on he heoreical models pu forward by Shiller (1984) and Senana and Wadhwani (199) we use he link beween index reurn auocorrelaion and volailiy o beer undersand he reurn characerisics generaed by raders adhering o posiive feedback rading sraegies. Germany s C-Dax, Dax, and Nemax50 indices for he period from January 1, 1998 o November 1, 001, represen differen sock marke segmens, hereby providing an ineresing and broad plaform for an analysis of feedback rading sraegies. Firs, we provide empirical evidence relying on he sock marke crash due o he erroris acs in he U.S. on Sepember 11, 001. Few economiss will disagree ha volailiy had enormously increased in he days afer he sock price crash, which lead direcly o he quesion of he auocorrelaion properies in reurns during his urbulen period. Our simple dummy variable approach exhibis empirical resuls ha are consisen wih he heory regarding he relaionship beween volailiy and auocorrelaion in index reurns. Whereas index reurns show shorly afer he crash srong negaive auocorrelaion indicaing he exisence of posiive feedback raders, he negaive serial correlaion in reurns vanishes he week afer he crash when volailiy has decreased. The applicaion of Nelson s (1991) exponenial GARCH model as a more sophisicaed approach relies on an explici volailiy measure and allows he condiional variance o respond asymmerically o posiive and negaive innovaions. Our findings provide srong suppor for he exisence of a leverage effec. This implies
16 ha in Germany s C-Dax, Dax, and Nemax50 index volailiy is higher in bearish periods compared o bullish periods. More imporanly and consisen wih he empirical resuls of he even sudy of he Sepember 11 crash, our empirical evidence shows ha posiive feedback raders are presen in hese sock marke segmens and generae negaive reurn auocorrelaion even a mean levels of reurn volailiy.
17 References Bange, M. M. (000), Do he Porfolios of Small Invesors Reflec Posiive Feedback Trading?, Journal of Financial and Quaniaive Analysis 35, 39 55. Black, F. (1976), Sudies of Sock Volailiy Changes, Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, 177 81. Boudoukh, J., Richardson, M. P. and M. P. Whielaw (1994), A Tale of Three Schools: Insighs on Auocorrelaions of Shor-Horizon Sock Reurns, Review of Financial Sudies 7, 539 73. Campbell, J. Y. and A. Kyle (1993), Smar Money, Noise Trading, and Sock Price Behavior, Review of Economic Sudies 60, 1 34. Conrad, J. and G. Kaul (1988), Time-Variaion in Expeced Reurns, Journal of Business 61, 409 5. Culer, D. M, J. M. Poerba and L. H. Summers (1990), Speculaive Dynamics and he Role of Feedback Traders, American Economic Review, Papers and Proceedings 80, 63 8. DeBond, W. E. M. (1993), Being on Trends: Inuiive Forecass of Financial Risk and Reurn, Inernaional Journal of Forecasing 9, 355 71.
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1 Table 1: Sepember 11 Crash and Auocorrelaion in Sock Reurns Index Dummy Period αˆ ˆ 0 C-Dax Sepember 11 o 14 0.00 (0.05) Sepember 11 o 18 0.00 (0.04) Sepember 19 o 5 0.001 (0.0) γ γ 1 0.03 (0.06) 0.0 (0.06) 0.01 (0.5) ˆ 0.8* (.8) 0.5* (.69) 0.14 (0.51) R 0.003 0.003 0.001 Dax Sepember 11 o 14 0.00 (0.04) 0.003 (0.06) 0.31* (.38) 0.004 Sepember 11 o 18 0.003 (0.05) 0.00 (0.06) 0.8* (.37) 0.004 Sepember 19 o 5 0.01 (0.13) 0.0 (0.4) 0.19 (0.7) 0.001 Nemax50 Sepember 11 o 14 0.001 (0.01) Sepember 11 o 18 0.001 (0.01) Sepember 19 o 5 0.003 (0.03) 0.14* (3.41) 0.14* (3.39) 0.13* (3.4) 0.8* (.73) 0.3 (1.89) 0.07 (0.16) 0.0 0.0 0.0 The esimaed parameers rely on he model R = α + ( γ 0 + γ 1Crash ) R 1 + ε. R denoes he adjused coefficien of deerminaion. -saisics in parenheses are based on heeroskedasic-consisen sandard errors. * denoes saisical significance (a leas) a he 5 % level. Daily daa from 1998:1: o 001:11:1 (1000 observaions) are used.
Table : Time Series Characerisics of Index Reurns Mean 0.00 (0.96) C-Dax Dax Nemax50 0.008 (0.88) 0.004 (0.96) Variance.09.81 7.48 Skewness 0.51 (0.00) Kurosis 6.90 (0.00) 0.50 (0.00) 0.06 (0.00) 5.10 5.3 (0.00) (0.00) Index reurns are calculaed as R = (ln P ln P 1) 100, where P is he index a ime. P-values are in paranheses. Daily daa from 1998:1: o 001:11:1 (1000 observaions) are used.
3 Table 3: EGARCH(1,1) Parameer Esimaes αˆ 0.00 (0.0) µˆ 0.0 (0.15) γˆ0 0.15 (4.35)* γ ˆ1 0.09 (4.43)* ˆβ 0.03 0 (.63)* ˆβ 0.11 1 (13.94)* ˆβ 0.94 (53.43)* ψˆ 1.00 (4.56)* δˆ 1.51 (9.31)* C-Dax Dax Nemax50 0.05 (0.83) 0.06 (0.70) 0.1 (5.7)* 0.08 (5.80)* 0.04 (.9)* 0.10 (13.7)* 0.95 (63.35)* 0.93 (4.37)* 1.46 (9.65)* 0.3 (0.38) 0.10 (0.40) 0.45 (4.00)* 0.10 (.74)* 0.18 (3.03)* 0.14 (14.84)* 0.90 (8.50)* 0.78 (3.86)*.30 (1.48)* υˆ 1.59 1.55 1.5 (14.13)* (15.73)* (16.34)* The esimaed parameers rely on he equaions (6), (8), and (9), ha are joinly esimaed via maximum likelihood. - saisics are in parenheses and * denoes saisical significance (a leas) a he 5 % level. Daily daa from 1998:1: o 001:11:1 (1000 observaions) are used.
4 σˆ min ˆρ min σˆ mean ˆρ mean Table 4: Volailiy and Reurn Auocorrelaion C-Dax Dax Nemax50 0.65 0.09.04 0.03 0.86 0.05.73 0.10 0.93 0.36 7.37 0.9 σˆ max ˆρ max 14.14 1.1 17.41 1.7 30.36.59 The auocorrelaion coefficiens are calculaed as ˆ ρ = γˆ + γˆ σˆ. 0 1