Kobe University Repository : Thesis

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1 Kobe Universiy Reposiory : Thesis 学位論文題目 Tile 氏名 Auhor 専攻分野 Degree 学位授与の日付 Dae of Degree 公開日 Dae of Publicaion 資源タイプ Resource Type 報告番号 Repor Number 権利 Righs JaLCDOI URL Recen evoluion in Financial Economerics: Modeling high frequency daa, join dependence and esing forecasing performance( 金融計量経済学における進展 : 高頻度データと相互依存関係のモデリング 及び予測力のテスト ) 田, 帅如 博士 ( 経済学 ) Thesis or Disseraion / 学位論文 甲第 6828 号 hp:// 当コンテンツは神戸大学の学術成果です 無断複製 不正使用等を禁じます 著作権法で認められている範囲内で 適切にご利用ください PDF issue:

2 博士論文 Recen evoluion in financial economerics: modeling high frequency daa, join dependence and esing forecas performance 金融計量経済学における進展 : 高頻度データと相互依存関係のモデリング 及び予測力のテスト 平成 28 年 12 月神戸大学経済学研究科経済学専攻指導教員羽森茂之田帥如

3 Conens 1. Inroducion 2 2. Modeling ineres rae volailiy: A Realized GARCH approach Inroducion Mehodology Daa Empirical resuls ARMA-RGARCH model Conclusion.. 18 Appendix...19 Table and Figure Realizing Momens Inroducion Esimaion mehod of Realizing momens Daa Join behavior beween reurn and realizing momens Sudying reurn predicabiliy using realizing momens Sudying informaion ransmission mechanism using realizing momens Conclusion.. 44 Table and Figure Improving densiy forecasing: he role of mulivariae condiional dependence Inroducion Model specificaion Densiy forecasing and evaluaion Empirical resuls Conclusion.. 79 Table and Figure.. 80 Reference

4 Chaper 1. Inroducion. Over he pas hree decades, research ineres of modeling financial reurns has been growing rapidly. Many lieraures in financial economerics have been focused on volailiy modeling by fiing parameric models wih disribuional assumpions (such as GARCH and Sochasic Volailiy ype models). While as discussed in Anderson (2001), hese models suffer from model misspecificaion, and none is sricly correc. I is also widely recognized ha hese models suffer from curse of dimensionaliy, and canno be applied in large dimension cases. In addiion, informaion driven by inraday daa is compleely ignored in hese models. This paper conribues o wo key opics addressed in recen lieraures which concerned hese problems: (1) modeling high-frequency daa and (2) capure comprehensive feaures of he join disribuion of financial reurns. In chaper 2, we esimae he daily volailiy of shor erm ineres rae using a Realized GARCH approach. We find he proposed mehod fi he daa beer and provide more accurae volailiy forecas by exracing addiional informaion from realized measures. In addiion, we propose using he ARMA-Realized GARCH model o capure he volailiy clusering and he mean reversion effecs of ineres rae behavior. We find he ARMA-RGARCH model fis he daa beer han he simple RGARCH model dose, bu i does no provide superior volailiy forecass. In chaper3, we propose a new ype of momens which can describe curren feaures of daily reurn. The so called realizing momens can be esimaed using high-frequency inraday daa. We show some usages of his new ype of momens, such as invesigaing join behavior of reurn and is momens, reurn predicabiliy, and informaion ransmission mechanism. Many insighful resuls are found. Fuure research can use his new ype of momens o sudy many oher problems in financial economics. In chaper 4, we invesigae wheher he densiy forecas forecass of sock reurns can be improved by aking accoun of condiional dependence. The Regular Vine ype Copula-GARCH models are applied o accuraely capure comprehensive feaures of he join disribuion of sock reurns. Densiy forecass are evaluaed by adoping he correc specificaion es based on he probabiliy inegral ransform. Presened resuls show ha densiy forecas can be improved significanly afer incorporaing condiional dependence. 2

5 Chaper 2 Modeling ineres rae volailiy: A Realized GARCH approach 2.1 Inroducion Shor-erm ineres raes are widely recognized as key economic variables. They are used frequenly in financial economerics models because hey play an imporan role in evaluaing almos all securiies and macroeconomic variables. However, many popular models fail o capure he key feaures of ineres raes and do no fi he daa well. A milesone in erms of ineres rae models was he developmen of he generalized regime-swiching (GRS) model, proposed by Gray (1996). Convenional GARCH-ype and diffusion models failed o handle cerain ineres rae evens, such as explosive volailiy, which would cause serious problems in cerain applicaions. He believed his failure may be due o ime variaions in he parameer values. The GRS model ness many ineres rae models as special cases, and allows he parameer values o vary wih regime changes. Thus, he model should provide a soluion o he problem. He found ha he GRS model ouperforms convenional single-regime GARCH-ype models in ou-of-sample forecasing. Unforunaely, he GRS model sill does no fi he daa well, since almos all he repored parameers of he mos generalized version are nonsignifican. Anoher conceivable reason for he failure o model shor-erm ineres raes adequaely is ha he σ-field is no sufficienly informaive. When modeling ineres raes using convenional GARCH-ype models, he only daa used are he daily closing prices. All daa during rading hours are ignored. As high-frequency daa has become more available, recen lieraures have inroduced a number of more efficien nonparameric esimaors of inegraed volailiy (see Barndorff-Nielsen and Shephard, 2004; Barndorff-Nielsen e al., 2008; and Hansen and Horel, 2009). Examples of models ha incorporae hese realized measures include he muliplicaive error model (MEM) of Engle and Gallo (2006), and he HEAVY model of Shephard and Sheppard (2010). These models incorporae muliple laen variables of daily volailiy. Then, wihin he framework of sochasic volailiy models, Takahashi e al. (2009) propose a join model for reurn and realized measures. Shiroa e al. (2014) inroduce he realized sochasic volailiy (RSV) model, which incorporaes leverage and long memory. A Heson model sudies he join behavior of reurn and volailiy, and shows decisively differen exremal behavior o he convenional GARCH and SV models proposed by Ehler e al. (2015). 3

6 In his sudy, we model he shor-erm ineres rae in he euro-yen marke using he Realized GARCH (RGARCH) framework proposed by Hansen e al. (2012). Secion 2 inroduces he RGARCH framework, including he log-linear specificaion, he esimaion mehod, he condiional disribuions, he robus QLIKE loss funcion, and he value-a-risk-based loss funcion. Here, we also presen he model confidence se (MCS) procedure we use o evaluae he volailiy forecasing performance. Secion 3 provides a brief descripion of our daa and he realized measures. Secion 4 repors he in-sample empirical resuls and evaluaes he rollingwindow volailiy forecasing performance. Then, in Secion 5, we inroduce an ARMA-Realized GARCH (ARMA-RGARCH) framework. Here, we presen our empirical resuls, and compare hem o hose of he simple RGARCH model. Lasly, Secion 6 concludes he paper. 2.2.Mehodology Model specificaion We adop he RGARCH model wih is log-linear specificaion, as proposed by Hansen e al. (2012). In Secion 5, we propose an exension of he model o capure he well-known mean reversion and volailiy effecs in shor-erm ineres rae behavior. The RGARCH(p,q) model is specified as follows: r h z (2.1) p q i1 i i j1 j j (2.2) log h log h log x log x log h ( z ) u, (2.3) where ~.. 0,1 2 z I I D, ~. I. 0, u u I D, r is a zero-mean series, and h and x denoe he condiional variance and realized measure, respecively. Then, ( z ) is he leverage funcion consruced from Hermie polynomials. Here, we adop he simple quadraic form: 2 ( z ) 1z 2( z 1). This choice is proper for wo reasons. Firs, i saisfies E[ ( z )] 0 for any sandard disribuion wih Ez [ ] 0 and Var[ z ] 1. Second, i is proporional o he news impac curve discussed in Engle and Ng (1993). Thus, i can capure he asymmeric effec of 4

7 price shocks on daily volailiy. The news impac curve is defined as v( z) E(log h z z) E(log h ). 1 Equaions (1) and (2) consruc a GARCH-X model, wih he resricion ha he coefficien of he squared reurn is zero. Equaion (3) is called he measuremen equaion, because x is a measure of h. The measuremen equaion complees he model, using he leverage funcion ( z ) o provide a simple way o invesigae he join dependence of r and x. Numerous sudies on marke microsrucures argue ha reurns are dependen on rading inensiy and liquidiy indicaors, such as volume, order flow, and he bid ask spread (see Amihud (2002), Admai and Pfleiderer (1988), Brennan and Subrahmanyam (1996), and Hasbrouck and Seppi (2001)). Because he inraday volailiy is linked o he volume, order flow, and rading inensiy direcly, i should be dependen on he reurn. Therefore, he measuremen equaion is imporan from a heoreical poin of view. Unlike oher models ha incorporae realized measures in he variance equaion, such as he MEM and HEAVY models, he realized measure is no reaed as an exogenous variable. When he realized measure is a consisen esimaor of inegraed volailiy, i should be viewed as he condiional variance plus an innovaion erm. The condiional variance is adaped o a much richer -field, F ( r, r 1,, x, x 1, ). Convenional GARCH and SVype models only use daily closing prices. In comparison, he addiional informaion included in he realized measure on inraday volailiy is expeced o promoe he fi o he daa and he forecasing accuracy of condiional volailiy. Moreover, he logarihmic condiional variance can be shown o follow an ARMA process: pq log h ( )log h [ ( z ) u ] i i i j j j i1 j1 q. (2.4) The logarihmic condiional variance, h, is driven by boh he innovaion of reurn and he realized measure. The leverage effec is already indirecly embedded in he variance equaion. The persisence parameer, π, is given by ( i i) Esimaion mehod i 5

8 Following Hansen e al. (2012), we summarize he esimaion mehod in his secion. The model is esimaed using he quasi-maximum likelihood (QML) mehod. The likelihood funcion of he Gaussian specificaion is given by 1 r u l( r, x; ) [log log( ) ]. (2.5) 2 n h u 2 1 h u Since convenional GARCH-ype models do no model realized measures, i is meaningless o compare he join log likelihood o hose of he convenional GARCH models. However, we can derive he parial likelihood of he RGARCH model and compare ha wih he log likelihood of he GARCH models. The join condiional densiy of he reurn and realized measures is f r, x F f r F f ( x r, F ). (2.6) Then, he logarihmic form is log f r, x F log f r F log f ( x r, F ). (2.7) Thus, he join log likelihood of z and u under he Gaussian specificaion can be spli as follows: 1 r 1 u l( r, x; ) [log(2 ) log ] [log(2 ) log( ) ] 2 2 n 2 n 2 2 h u 2 1 h 1 u. (2.8) Then, he parial likelihood of he model is defined as n 2 1 r l( r; ) [log(2 ) log h ]. (2.9) 2 h 1 Hansen e al. (2012) obained ha n ˆ N I J I where expressions for I and 1 1 ( n ) (0, ) J are given in he appendix. An alernaive esimaion mehod for GARCH-ype models is proposed by Ossandon and Bahamonde (2011). I is possible o have a novel sae space represenaion and an efficien approach based on he Exended Kalman Filer (EKF). Since he srucure of he RGARCH model is quie complicaed, his is lef as a opic for furher research. 6

9 We also change he assumpion of a sandard normal disribuion on z o a more realisic disribuion. Here, we adop he disribuion proposed by Fernandez and Seel (1998) ha allows for skewness in any symmeric and coninuous disribuion by changing he scale of he densiy funcion: 2 f z f z H z f z H z 1 ( ) [ ( ) ( ) ( ) ( )] 1, (2.10) where is he shape parameer and H(.) is he Heaviside funcion. The disribuion is symmeric when is equal o 1. The mean and variance are defined as (2.11) 1 z M ( ) z ( M 2 M1 )( ) 2M1 M 2, (2.12) respecively, where M k k 2 z f ( z) dz. (2.13) 0 Here, M k is he k-h momen on he posiive real line. The disribuion is called a sandardized skewed disribuion if z 0 and he densiy funcion is given by 2 z 1. We adop he sandardized skewed disribuion. Here, 2 g[ ( sz ) ] 1 v if z f ( z, v), (2.14) 2 1 g[ ( sz ) ] 1 v if z where z follows a SKST ( 0,1,, ) disribuion, g(.) is he densiy funcion of he sandard disribuion, and and 2 are he mean and variance of a non-sandard skewed Suden disribuion, respecively. The disribuion is symmeric when Benchmark models We compare he RGARCH model o he benchmark GARCH and EGARCH models. A zeromean series can be specified by a GARCH model as 7

10 r h z (2.15) p q 2 i j j j i1 j1, (2.16) h r h and he EGARCH model of Nelson (1991) is defined as r h z (2.17) p q. (2.18) log h log h [ z ( z E z )] i i j j j j i1 j1 The log likelihood of he GARCH and EGARCH models under a Gaussian specificaion is given by n 2 1 r l( r; ) [log(2 ) log h ]. (2.19) 2 h 1 Noe ha he log likelihood of he benchmarks akes he same form as he parial likelihood of he RGARCH model. Therefore, we can compare he finess of hese models Forecasing performance evaluaion A reliable model should fi he daa well, bu should also generae an accurae forecas. We compare he forecasing performance by consrucing loss funcions. The firs loss funcion we consruc is he robus loss funcion QLIKE, as discussed in Paon (2011). This is called a direc evaluaion because i compares he volailiy forecass o heir realizaions direcly. As realizaions of volailiy are unobservable, a volailiy proxy is required o consruc he loss funcion. The QLIKE loss funcion is given by 2 2 ˆ QLIKE : L( ˆ, h) log h, (2.20) h where 2 ˆ is a proxy for he rue condiional variance and h is a volailiy forecas. The squared reurn is a widely used proxy, bu is also known as being raher noisy. Paon (2011) suggess using he realized volailiy and inra-daily range as unbiased esimaors. These are also more efficien han he squared reurn if he log price follows a Brownian moion. Alhough hese less 8

11 noisy proxies should lead o less disorion, he degree of disorion is sill large in many cases. Nagaa and Oya (2012) poin ou ha realized volailiy does no saisfy he unbiasedness condiion owing o marke microsrucure noise. They show ha bias in he volailiy proxy can cause misspecified rankings of compeing models. As he realized kernel is designed o deal wih microsrucure noise, i can clearly also be used as an unbiased esimaor of condiional variance. Since high frequency prices are only available during rading periods, he realized kernel and realized volailiy are direcly relaed o he open-o-close reurn. Thus, for he close-o-close reurn, he squared reurn is he only unbiased proxy. However, he squared open-o-close reurn is no an unbiased esimaor for daily (close-o-close) volailiy owing o he exisence of he overnigh reurn (see he appendix for he proof). In his sudy, we use he squared reurn, realized volailiy, and he realized kernel o consruc QLIKE loss funcion. The second loss funcion we consruc is a value-a-risk (VaR)-based loss funcion, as proposed by González-Rivera e al. (2004). The value-a-risk is esimaed from VaR 1 ( ), (2.21) where is he cumulaive disribuion funcion, and and variance, respecively. Then he asymmeric VaR loss funcion is given by are he condiional mean and L( y, VaR ) ( d )( y VaR ), (2.22) where d 1( y VaR ) is an indicaor variable. This is an asymmeric loss funcion since i penalizes VaR violaions more heavily, wih weigh (1 ). We call his an indirec evaluaion because of is risk measuremen poin of view. Smaller values are preferred for boh loss funcions. However, i is difficul o say wheher a model wih he smalles loss values significanly ouperforms alernaive models. Thus, we adop he model confidence se (MCS) procedure proposed by Hansen e al. (2011) o es he equal predicive abiliy (EPA) hypohesis and o obain he superior se of models (SSM) under each loss funcion. The esimaion and condiional volailiy evaluaion are all performed in R(R Core Team (2015)). 9

12 2.3. Daa Daa and he rolling-over mehod For our empirical applicaion, we use high-frequency daa of hree-monh euro-yen fuures from he Tokyo Financial Exchange (TFX). Since derivaive conracs have a finie life, we consruc a coninuous ime series using he rolling-over mehod and volume crieria, as discussed in Carchano and Pardo (2009). Our full sample comprises 980 observaions from January 2006 o December We esimae boh close-o-close reurn and open-o-close reurn series. To evaluae he forecasing performance, we use rolling-window mehod, wih a window size of 200 observaions, and obain one-sep-ahead ou-of-sample forecass Realized measures Recen lieraure has inroduced a number of realized measures of volailiy, such as realized variance, realized kernel, bipower variaion, inraday range, and oher nonparameric esimaors. In his sudy, we approximae he inegraed volailiy using wo esimaors. The firs is he fiveminue realized variance esimaor of Andersen e al. (2001), defined as he summaion of he squared inraday reurns: RV n 2 ri, i1. (2.23) The second esimaor is he realized kernel of Barndorff-Nielsen e al. (2008), given by H h K( X ) k( ) h, (2.24) H 1 hh n xx, (2.25) h j j h j h 1 where k(x) is a kernel weigh funcion and H is he bandwidh. Here, we adop he Parzen kernel, which guaranees non-negaiviy and saisfies he smoohness condiions, k'(0) k'(1) 0. The Parzen kernel funcion is defined as follows: 10

13 x 6x 0 x 1/ 2 3 k( x) 2(1 x) 1/ 2 x 1 0 x 1. (2.26) Figure 1 represens he series of reurns and realized measures Empirical resuls In-sample empirical resuls wih a normal disribuion In his secion, we presen our empirical resuls for he close-o-close reurn and he open-oclose reurn wih a sandard normal disribuion, and compare hem o hose of he benchmarks. For simpliciy, RG, G, and EG denoe he RGARCH, GARCH, and EGARCH models, respecively. 1 Table 1 repors he main empirical resuls for he close-o-close reurn (he complee ables are available online). Several poins are worh noing. Firs, he join likelihood of he reurn and volailiy indicae ha he simpliciy of an RG(1,1) model is proper. The AIC and SBIC also sugges ha he RG(1,1) model should be adoped. As a resul, we esimae he G(1,1) and EG(1,1) models for comparison. In erms of he parial log likelihood, he RG models clearly ouperform he benchmarks, alhough he RGARCH models and he QML esimaion mehods do no maximize he parial log likelihood. The addiional informaion conained in he inraday volailiy improves he fi o he daily reurn. Second, he RG persisence parameer, π, is abou This indicaes srong persisence, bu is also he covariance saionariy of he shor-erm ineres rae. Then, he value of γ2 is almos 0 and is nonsignifican, which indicaes ha he curren volailiy is srongly affeced only by he laes realized measures. In he case of he GARCH and EGARCH models, he condiional volailiy has exremely long memory owing o he high value of β. Furhermore, hese models are almos unaffeced by recen news, as he values of α in each case are very small and nonsignifican. In conras, he value of β in he RGARCH models is much smaller, indicaing a much shorer memory in he model s volailiy process. On he oher hand, a larger proporion of he condiional volailiy is due o recen news, since γ1 in he RG models is abou 0.29 and is 1 Following packages are used o obain he empirical resuls: rugarch (Ghalanos (2014)), copula (Hofer e al.(2015), exmex (Souhworh and Heffernan(2013)), MCS (Caania and Bernardi(2015)). 11

14 highly significan. The posiive values of parameers η1 and η2 of he leverage funcion confirm he leverage effec of he ineres rae. Furhermore, he significance of hese parameers implies he exisence of join reurn and inraday volailiy behavior, because he innovaion erm of he reurn is embedded in he measuremen equaion. Therefore, i is proper o consider ha rading inensiy and liquidiy indicaors impac he reurn. Then, is he exremal index, which describes he exremal behavior of a sochasic process, as discussed in De Haan e al. (1989), Bhaacharya (2008), and Laurini e al. (2012). The exremal index is esimaed by 5000 simulaions, wih fixed parameers, using he esimaion mehod proposed by Ferro and Segers (2003). Here, (0,1), and lower values indicae higher clusers of exreme values for a given confidence level. In his sudy, he confidence level is se o 95%. The value of is approximaely 0.91 for he GARCH and EGARCH models, and approximaely 0.72 for he RGARCH models. This means he GARCH and EGARCH processes assume almos no clusering of exreme values, while he RGARCH processes assume some clusering. Figure 2 shows he news impac curve defined in Secion 2. I is clear ha posiive price shocks have a larger impac on volailiy han do negaive shocks. As a resul, ineres raes rise slowly and fall quickly. Table 2 repors he empirical resuls for he open-o-close reurn. As in he case of he close-oclose reurn, he values of γ2 sugges ha he condiional variance is ied only o he laes realized volailiy. The persisence parameer, π, sill suggess srong persisence and covariance saionariy. Then, he exremal index shows ha he EGARCH process capures almos he same level of volailiy clusering as he RGARCH models do. On he oher hand, exreme open-oclose reurn values are a bi more clusered han are exreme close-o-close reurn values. Oher parameers sugges almos he same resuls. Alhough he parial log likelihood of he RGARCH models is smaller han ha of he EGARCH model, his is no a problem, because he RGARCH models do no maximize he parial log likelihood. However, in erms of he join log likelihoods and informaion crieria, i is no proper o adop he simpliciy of he RG(1,1) model. This is discussed in he nex secion In-sample empirical resuls wih a sandard skewed Suden disribuion 12

15 In his secion, we change he assumpion of a sandard normal disribuion o a more generalized disribuion, called a sandard skewed Suden disribuion, or SSKT(0,1). Table 3 presens he esimaes for he close-o-close reurn under he new assumpion. The coefficiens repor almos he same resuls as in he case of he normal disribuion assumpion. The skew parameer is approximaely 1.03, which means he condiional disribuion of he reurn is almos symmeric. Noe ha since a posiive reurn means a fall in he ineres rae, we refer o his as negaive. Thus, negaive news has a slighly larger impac on volailiy han does posiive news. The exremal indices are around 0.85, indicaing ha under he assumpion of a skewed disribuion, exreme reurns do no cluser as frequenly as hey do under a normal disribuion. As γ2 is again 0 and nonsignifican, i is proper o selec he bes model beween RG(1,1) and RG(2,1). The likelihood raio saisics beween hese wo models for he realized kernel and realized volailiy 2 are 8.18 and 8.22, respecively, which are boh greaer han 6.63 ( (1) ). The AIC and SBIC values indicae ha he RG(2,1) model fis he daa beer han does he RG(1,1) model. Therefore, an RG(2,1) model should be proper. Table 4 repors he empirical resuls for he open-o-close reurn under he assumpion of a skewed disribuion. Three poins deserve o be menioned. Firs, he skew parameers are equal o 1, indicaing he same impac of posiive and negaive news on volailiy. Second, he exremal indices are larger han hose of he normal disribuion, which indicaes relaively sligh volailiy clusering. Finally, he likelihood raio saisics for he RG(1,1) and RG(2,1) models for he 2 realized measures are 4.76 and 4.24, respecively, which are greaer han 3.84 ( (1) ), bu less han 6.63 selecion. 2 ( (1) 0.01). The AIC and SBIC values sugges differen resuls for model In a correcly specified model, robus and non-robus sandard errors should be in agreemen, see Hansen e al.(2012). The non-robus and robus sandard errors are diagonal elemens of he inverse Fisher informaion marix 1 I and I 1 J I 1. Tables 5 o 8 presen hese sandard errors for boh reurn series and he disribuion assumpions. The resuls are as follows. For he close-oclose reurn, he disances beween 1 I and I J I for he GARCH and EGARCH models are 1 1 quie large, indicaing serious misspecificaions. However, he RGARCH models are far less misspecified. On he oher hand, he RGARCH models wih he skewed disribuion show a 13

16 much beer fi han hey do wih he normal disribuion. The RG(2,1) model wih a skewed disribuion clearly performs beer han he RG(1,1) model, suggesing he same resul as he likelihood raio es and informaion crieria. For he open-o-close reurn, he EGARCH model does no perform worse han he GARCH model, bu sill canno compee wih he RGARCH models. However, i is quie difficul o disinguish beween he normal and skewed disribuion, as well as beween he RG(1,1) and RG(2,1) models. Thus, we evaluae he forecasing performance for all hese models Forecasing performance evaluaion As menioned in Secions 2 and 3, we evaluae he forecasing performance by consrucing QLIKE and asymmeric VaR loss funcions. Then, we es he equal predicive abiliy (EPA) hypohesis using he model confidence se (MCS) procedure. The ou-of-sample condiional volailiy is esimaed by he rolling-window mehod, wih a window size of 200 observaions. Each poin in rolling-window esimaion is checked o avoid sub opimizaion. The random seed of MCS sampling would affec he p-values o a small exen, bu no change he rankings. Following he resuls in Secion 4.1 and 4.2, for he close-o-close reurn, we evaluae six models, including GARCH, EGARCH, and RG(1,1) wih a normal disribuion and RG(2,1) wih a skewed disribuion, using boh realized measures. For he open-o-close reurn, we evaluae 10 models, including he benchmark models, and RG(1,1) and RG(2,1) wih boh he normal and skewed disribuions, and using boh realized measures. To evaluae he QLIKE loss funcion, he squared reurn, realized volailiy, and realized kernel are all used as volailiy proxies QLIKE evaluaion Table 9 repors he MCS evaluaion for he QLIKE loss funcion. The confidence level of he EPA hypohesis is 10%. Owing o microsrucure noise, he realized volailiy is no an unbiased esimaor of he open-o-close volailiy (Nagaa and Oya, 2012), bu he squared reurn and realized kernel are. In addiion, he squared reurn is quie noisy (Paon, 2011), while he realized kernel and realized volailiy are more efficien. Alhough biased proxies may lead o misspecified rankings, we sill repor hese resuls. Models wih ELI in he column of MCS p- values were eliminaed under he given confidence level using he eliminaion rule of Hansen e al. (2011). 14

17 For he close-o-close reurns, using he squared reurn as a volailiy proxy, he RG(1,1) model wih a normal disribuion and he realized kernel as he realized measure performs bes. However, he null hypohesis ha he GARCH and RG(1,1) models, using realized volailiy, have equal predicive abiliy canno be rejeced. Since he MCS p-values are close o 1, i is difficul o disinguish beween he predicive abiliy of he wo RG models, bu hey clearly ouperform he GARCH model. Using he realized kernel and realized volailiy as volailiy proxies, no oher model has a predicive abiliy equal o ha of he bes performing model. The RG(1,1) model wih a normal disribuion and he realized kernel performs bes under he evaluaion of all volailiy proxies. Therefore, i is proper o consider ha using he realized kernel as a realized measure provides beer volailiy forecass han when using he realized volailiy. For he open-o-close reurn, i is quie difficul o obain agreemen. The resuls depend on he volailiy proxy used. However, as menioned above, he realized kernel is unbiased and he mos efficien volailiy proxy, we adop he resul ha he RG(2,1) model wih he skewed disribuion and he realized kernel as he realized measure has he bes predicive abiliy Value-a-Risk evaluaion The VaR evaluaion is a widely used for condiional volailiy assessmen. A VaR esimae is said o be valid if i saisfies he uncondiional convergence condiion of Kupiec (1995) and he independence and condiional convergence of Chrisoffersen (1998). We also repor he resuls of he VaR independence duraion es proposed by Chrisoffersen and Pelleier (2004). Table 10 presens he VaR backesing resuls. For he close-o-close reurn, all models generae valid VaR forecass, excep he EGARCH model. For he open-o-close reurn, he RG model wih he normal disribuion and he realized kernel performs bes. Oher models fail o rejec several hypoheses. Since he exceedance raio of he RG model wih he skewed disribuion is 0 for 0.5% VaR, we canno calculae he likelihood raio saisics for he VaR backesing. Overall, he VaR forecass generaed by he RG models are reliable. However, insead of validiy, i is of more ineres o invesigae he predicive accuracy of he VaR esimae. Thus, we consruc he asymmeric VaR loss funcion menioned in Secion 2 and es he EPA hypohesis. Table 11 repors he MCS evaluaion for he VaR loss funcion. For he close-o-close reurn, i is clear ha he RG(1,1) model wih he normal disribuion dominaes 15

18 he alernaives for boh he 0.5% and 1% VaR esimaes. The predicive accuracy of esimaion using he realized kernel is a lile beer han ha of he realized volailiy. For he open-o-close reurn, he RG models wih he normal disribuion have almos equal predicive abiliy, and dominae he oher models. Considering all our in-sample empirical resuls, he QLIKE loss funcion, and he VaR esimae, he RGARCH models clearly dominae he benchmarks. Alhough he RGARCH models wih he skewed disribuion fi he daa beer for he in-sample close-o-close reurn, hey canno provide superior volailiy and VaR forecass. Clearly, he forecas performance is affeced more by he specificaions of he models han by he chosen realized measures. The RG model wih a normal disribuion and he realized kernel as a realized measure performs bes in he QLIKE evaluaion, using all volailiy proxies. The model also performs slighly beer han he model ha uses he realized volailiy in he VaR evaluaion. Thus, i is proper o consider ha he realized kernel is a beer realized measure han is he realized volailiy. On he oher hand, i is quie difficul o selec he bes performing model for he open-o-close reurn. Consequenly, we exend he mean process of RG(1,1) wih he normal disribuion and he realized kernel for he close-o-close reurn in Secion ARMA-RGARCH model Gray (1996) suggess ha models of shor-erm ineres raes should capure wo well-known empirical aribues, namely mean reversion and lepokurosis. Engle (1982) shows ha lepokurosis in an uncondiional disribuion may be caused by condiional heeroskedasiciy. Condiional heeroskedasiciy is already modeled by he volailiy process and leverage funcion. The RGARCH model assumes he condiional mean E[ r F 1] 0. To capure he level behavior of shor-erm ineres raes, a more general specificaion of he mean process is proper. A simple way o model he mean reversion effec is o impose an ARMA process on he mean process. In his secion, we generalize he mean process o an ARMA process o model boh he mean reversion and he volailiy clusering effecs simulaneously. The ARMA-RGARCH model is specified as follows: r p r i i j j i1 j1 q 16

19 hz r log h log h log x m m n n m1 n1 s log x log h ( z ) u. The RGARCH model is nesed wihin he ARMA-RGARCH model by imposing i j 0. Here, 1 should be posiive o model volailiy clusering, and 1 should be negaive if he mean reversion effec exiss. For simpliciy, we esimae an ARMA(2,2)- RGARCH(1,1) model wih a sandard normal disribuion and using he realized kernel as he realized measure, as we have already discussed he volailiy process in Secion 4. The resuls are as follows: r r 0.995r (6e 6) (0.003) (0.004) (3e 5) (3e 6) log h log h log x 1 1 (0.273) (0.012) (0.008) 2 log x log h z 0.02( z 1) u (0.964) (0.061) (0.038) (0.003) l( r, x ) lr ( ) Here, he numbers in parenheses are robus sandard errors. Noe he following poins. Firs, he likelihood raio saisics of he ARMA-RGARCH model and of is benchmark RG(1,1) model are , which is larger han 2 (5) This suggess a significan increase in fiing 0.01 he daa. This finding is also eviden in he increase of he parial likelihood, afer imposing an ARMA process. Second, he coefficiens and 1 1 are posiive and negaive, respecively, and also highly significan, showing he exisence of srong volailiy clusering and mean reversion effecs. Oher properies are similar o he simple RGARCH model. Moreover, since we assumed independence beween z and u, { zˆ, u ˆ } should be independenly disribued if he model is well specified. Genes and Rémillard (2004) sudied ess of independence and randomness based on a decomposiion of an empirical copula process. Kojadinovic and Yan (2011) proposed a 17

20 generalizaion of he decomposiion o es serial independence in a coninuous mulivariae ime series. Since he saisics of serial dependence are no disribuion free, hey sudied he consisency using he boosrap mehodology. We invesigae he serial independence beween z and u of he RGARCH and ARMA-RGARCH models, assuming a realized kernel and sandard normal disribuion on he innovaion process, using he empirical copula mehod. Table 12 repors he p-values of he global Cramer Von Mises es, he combined ess from he Mobius decomposiion wih Fisher s rule, and Tippe s rule. The able also evaluaes he forecasing performance of he ARMA-RGARCH and simple RGARCH models. The resuls are as follows. The p-values for he mulivariae serial independence es srongly sugges serial independence beween z and u for boh models. As a resul, neiher he ARMA-RGARCH model nor he simple RGARCH model are misspecified. The simple RGARCH model performs beer in he VaR esimae. However, he evaluaion of he QLIKE loss funcion does no sugges he same resul. Using a squared reurn as a volailiy proxy, he MCS p-values canno rejec he EPA null hypohesis. Then, using he realized kernel and realized volailiy as proxies, he ARMA- RGARCH model performs beer han he benchmarks. Figure 3 represens he series of parameers and he condiional sandard errors using he rolling-window mehod for he wo models. The scales of y-axis are 70 imes of sandard errors for bea and gamma and 10 imes for ea1 and ea2. I is clear ha he paerns of parameers for he wo models are almos he same. The variaion of parameers is quie small over ime. For he ARMA-RGARCH model, parameers are more volaile han simple RGARCH model. Overall, alhough modeling he volailiy clusering and mean reversion effecs improve he fi o he daa, we canno say ha he forecasing accuracy has improved Conclusion This sudy proposes an RGARCH approach o model shor-erm ineres raes. The imporan empirical resuls include he following. Firs, he RGARCH model is an effecive ool o model and forecas he volailiy of an ineres rae. The more informaive σ-field improves he fi o he daa and generaes more accurae volailiy forecass and value-a-risk esimaes. Second, generalizing he assumpion of he condiional disribuion does no improve he forecasing accuracy. The asympoic properies and he QML esimaion mehod perform well enough ha he RGARCH model wih a sandard normal disribuion can ouperform convenional GARCH- 18

21 ype models. Finally, he proposed ARMA-RGARCH model can capure he volailiy clusering and mean reversion effecs. The generalizaion of he mean process fis he daa beer han he simple RGARCH model does, bu does no improve he forecasing accuracy of daily volailiy. Neiher he RGARCH nor he ARMA-RGARCH models are misspecified. Finally, alhough we only invesigae one marke, our resuls should be a general indicaor for oher markes. Appendix A: Asympoic disribuion Following Hansen e al (2012), we wrie he leverage funcion as ( z ) 1a1 ( z1) a ( z ), ' ' 2 ' ' and denoe he parameers in he model by (,, u ), where (, 1, p, 1, q) and ' ' (,, ). To simplify he noaion we denoe ' (1,log h 1,,log h p,log x 1,log x q) and q ' ' (1,log h, a ). Thus, equaion (2) and (3) can be expressed as log h and ' l, respecively. The score, ' log x q u n 1 l, is given by 2 2 l 1 2 2u 2u u u 1 z u,, 2 h q u u u ' where h log h, u 1 u za ' log h 2 wih a az ( ). The second derivaive, z 2 l ' n 1 2 l, is given by ' u 2 uu ' 1 2 2uu 2 1 z h h z h 2 2 u 2 u 2 l u ' u ' 1 ' q ' 2 h v 2 h q 2 q u u u 2 2 u 1 2 u ' u ' u u h 4 q u u u 19

22 where h 2 log h, ' v 1 2 ' ' (0,1, za ) 1 wih 4 ' 2, u za z a a 2 az ( ). Suppose z 2 ( r, x,log h ) is saionary and ergodic, hen 1 n n 1 d 2 l 1 N(0, J ) and n l p I ' wih n u (1 2 ' E z u 2 ) E ( h h ) 4 u 1 ' 1 ' J E( u ) ( ) 2 qh E q 2 q u u E( u ) E( u ) ' E( u ) ' Eu ( / 1) E( h ) ( ) u 6 E q 6 4 2u 2u 4 u and 2 1 Eu ( ) ' ( ) 2 E hh 0 2 u 1 ' 1 ' I E 2 ( uq uv ) h E( q ) 0 2 q u u u are finie. Appendix B: Proof Here, we prove ha he squared open-o-close reurn is no an unbiased esimaor of he rue daily condiional variance. Denoe r, cc r, and oc r as he close-o-close, open-o-close, and ov overnigh reurn, respecively. I is clear ha rcc, roc, r, ov, 2 2 E[ r F ] E[ r F ] E[ r F ] 0, and Var[ r F ] E[ r F ], where 2 is cc, 1 oc, 1 ov, 1 he rue condiional variance. Thus, cc, 1 cc, 1 Var[ r F ] Var[ r r F ] oc, 1 cc, ov, 1 20

23 E[( r r ) F ] E[ r r F ] 2 2 cc, ov, 1 cc, ov, 1 E[ r 2 r r r F ] { E[ r F ] E[ r F ]} cc, cc, ov, ov, 1 cc, 1 ov, 1 2cov[ ] [ ]. 2 rcc, rov, F 1 Var rov, F 1 The squared open-o-close reurn is he unbiased esimaor of he rue condiional variance if and only if Var[ rov, F 1 ] 2cov[ rcc, rov, F 1], which is no an ordinary condiion. For he same reason, he realized volailiy and realized kernel are also no unbiased esimaors of he daily volailiy of he close-o-close reurn. 21

24 Table 1 Empirical resuls for close-o-close reurns wih sandard normal disribuion Realized Kernel Realized Volailiy G(1,1) EG(1,1) RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) α β1 (0.99) (1.17) β2 (21.55) (40.97) (43.41) (16.14) (59.82) (26.82) (33.02) (13.09) (38.75) (18.60) γ1 (2.48) (1.90) (2.35) (1.91) γ2 (1.38) (18.84) (5.28) (57.63) (4.71) (13.07) (3.89) (21.88) (3.84) 0 1e-6 1e-6 0 η1 (5e-6) (1.5e-5) (1.4e-5) (0) η2 (1.85) (1.83) (1.85) (1.83) (2.17) (2.17) (2.17) (2.17) l(r,x) l(r) π AIC SBIC (3.79) (3.77) (3.76) (3.74) (3.89) (3.91) (3.88) (3.88) Noe: G, EG, and RG represen he GARCH, EGARCH, and RGARCH models, respecively. The realized kernel and realized volailiy are he realized measures used in he esimaions. The 22

25 values in parenheses are values calculaed using a robus sandard error, and l(r, x) and l(r) denoe he join log likelihood and parial likelihood, respecively. Table 2 Empirical resuls for he open-o-close reurn wih sandard normal disribuion Realized Kernel Realized Volailiy G(1,1) EG(1,1) RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) α e-4 β1 (3.9e-3) (0.02) β2 (0.12) ( ) ( ) (280.29) ( ) (441.43) ( ) (126.37) (550.56) (120.65) γ1 (75.44) (81.29) (8.42) (8.21) γ2 (17.18) (179.97) (123.63) (58.25) (105.55) (443.75) (130.23) (534.07) (124.90) η1 (2e-6) (0) (0) (0) η2 (1.98) (1.97) (1.98) (1.97) (1.58) (1.59) (1.59) (1.59) l(r,x) l(r) π AIC (5.05) (4.93) (5.05) (4.97) (5.65) (5.53) (5.62) (5.51)

26 SBIC Noe: G, EG, and RG represen he GARCH, EGARCH, and RGARCH models, respecively. The realized kernel and realized volailiy are he realized measures used in he esimaions. The values in parenheses are values calculaed using a robus sandard error, and l(r, x) and l(r) denoe he join log likelihood and parial likelihood, respecively. Table 3 Empirical resuls for he close-o-close reurn wih skewed suden disribuion Realized Kernel Realized Volailiy RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) β1 β2 γ1 γ2 η1 η2 skew l(r,x) AIC (233.62) (32.51) (363.12) (35.94) (289.92) (20.18) (356.39) (24.32) (6.69) (6.88) (3.95) (4.37) (105.11) (7.80) (118.74) (9.29) (101.41) (6.64) (173.63) (8.52) (0) (0) (0) (0) (1.03) (1.53) (1.41) (1.53) (0.86) (1.96) (1.86) (1.96) (0.68) (1.42) (1.14) (1.42) (0.49) (2.03) (1.77) (2.04)

27 SBIC Noe: The realized kernel and realized volailiy are he realized measures used in he esimaions. The values in parenheses are values calculaed using a robus sandard error, and l(r, x) denoes he join log likelihood. Table 4 Empirical resuls for he open-o-close reurn wih skewed suden disribuion Realized Kernel Realized Volailiy RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) β1 β2 γ1 γ2 η1 η2 skew l(r,x) AIC SBIC ( ) (301.56) ( ) (485.13) ( ) (140.40) (342.98) (161.54) (116.61) (133.06) (24.86) (30.86) (38.65) (163.59) (9.19) (172.01) (24.39) (228.15) (8.81) (257.88) 0 1e-6 0 2e-4 (0) (2.8e-5) (0) (5.6e-5) (1.96) (1.93) (1.94) (1.93) (1.55) (1.54) (1.31) (1.54) (4.52) (3.69) (4.01) (3.67) (3.91) (3.44) (1.72) (3.40)

28 Noe: The realized kernel and realized volailiy are he realized measures used in he esimaions. The values in parenheses are values calculaed using a robus sandard error, and l(r, x) denoes he join log likelihood. Table 5 Robus and non-robus sandard errors for he close-o-close reurn wih normal disribuion Realized Kernel Realized Volailiy G(1,1) EG(1,1) RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) I 1 α β1 β2 γ1 γ2 η1 2.5e e e-3 7.2e-3 6e e e-3 9e e-3 6e e η2 4e-3 4.1e-3 4.1e-3 4.1e-3 3.6e-3 3.7e-3 3.7e-3 3.7e-3 I J I 1 1 α β e β2 γ1 γ2 η1 η2 Noe: I 1 and e e e-3 5.4e-3 5.5e-3 5.5e-3 5.2e-3 5.2e-3 5.2e-3 5.2e-3 I J I denoe he non-robus and robus sandard errors

29 Table 6 Robus and non-robus sandard errors for he close-o-close reurn wih skewed disribuion Realized Kernel Realized Volailiy RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) I 1 β1 2e e e e β2 γ1 γ2 η e e e e η I J I 1 1 β1 β2 γ1 γ2 η1 η2 Noe: I 1 and e e e e e e e e I J I denoe he non-robus and robus sandard errors. 1 1 Table 7 Robus and non-robus sandard errors for he open-o-close reurn wih normal disribuion Realized Kernel Realized Volailiy G(1,1) EG(1,1) RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) I 1 27

30 α β1 β2 γ1 γ2 η1 4.6e e-4 1.3e-3 3.3e-4 1.3e-3 3.3e-4 2.5e-3 1e-3 3e-3 4.5e-3 4.5e e-3 2e e e-3 3.7e-3 2e-3 4e η I J I 1 1 α β e-4 3e-3 4e-4 1.3e-3 3e-4 8e-4 5.6e-4 2.9e-3 1e-3 β2 γ γ2 η1 η2 Noe: I 1 and e-3 3.5e e-3 2.8e-3 5.5e-3 3.3e-3 1e-3 3.2e-3 7.5e-4 3.4e I J I denoe he non-robus and robus sandard errors. Table 8 Robus and non-robus sandard errors for he open-o-close reurn wih skewed disribuion Realized Kernel Realized Volailiy RG(1,1) RG(2,1) RG(1,2) RG(2,2) RG(1,1) RG(2,1) RG(1,2) RG(2,2) I 1 β1 5e-4 2e-3 3e-4 2e-3 5e-4 3.5e-3 1e-3 4e-3 β

31 γ1 γ2 η e e η I J I 1 1 β1 β2 γ1 γ2 η1 η2 Noe: I 1 and 2e-3 5e-4 1e-3 3e-4 6.5e-4 4e-4 2e-3 2e-3 3e-3 2.3e e e e e I J I denoe he non-robus and robus sandard errors. 1 1 Table 9 MCS evaluaion for he QLIKE loss funcion Volailiy proxy: 2 r Close-close reurn Open-close reurn QLIKE p-mcs Rank QLIKE p-mcs Rank G G ELI 10 EG ELI 4 EG RG11normRK RG11normRK RG11normRV RG21normRK RG21sksRK ELI 6 RG11normRV RG21sksRV ELI 5 RG21normRV Volailiy proxy: Realized Kernel RG11sksRK ELI 7 RG21sksRK ELI 8 RG11sksRV ELI 6 RG21sksRV ELI 9 G ELI 3 G ELI 10 29

32 EG ELI 6 EG ELI 9 RG11normRK RG11normRK ELI 8 RG11normRV ELI 2 RG21normRK ELI 6 RG21sksRK ELI 5 RG11normRV ELI 7 RG21sksRV ELI 4 RG21normRV ELI 5 Volailiy proxy: Realized Volailiy RG11sksRK ELI 3 RG21sksRK RG11sksRV ELI 4 RG21sksRV G ELI 3 G ELI 10 EG ELI 6 EG ELI 9 RG11normRK RG11normRK ELI 8 RG11normRV ELI 2 RG21normRK ELI 6 RG21sksRK ELI 5 RG11normRV ELI 7 RG21sksRV ELI 4 RG21normRV ELI 5 RG11sksRK ELI 3 RG21sksRK RG11sksRV ELI 4 RG21sksRV Noe: 2 r, realized kernel and realized volailiy are he volailiy proxies used in consrucing QLIKE loss funcions. The values of p-mcs are he p-values of he model confidence se procedure. Models wih ELI in he column of p-mcs were eliminaed using he MCS eliminaion rule under he given 10% confidence level. Table10 VaR backesing VaR0.5% VaR1% Raio UC CC INDDR Raio UC CC INDDR close-close reurn G EG RG11normRK RG11normRV RG21sksRK RG21sksRV

33 open-close reurn G EG RG11normRK RG21normRK RG11normRV RG21normRV RG11sksRK RG21sksRK RG11sksRV RG21sksRV Noe: Raio, UC, CC, and INDDR represen he value-a-risk exceedance raio, p-values for uncondiional convergence, condiional convergence, and duraion-based independence hypohesis, respecively. Table 11 MCS evaluaion for he VaR loss funcion Close-close reurn Open-close reurn VaRLoss p-mcs Rank VaRLoss p-mcs Rank VaR 0.5% G G ELI 10 EG ELI 6 EG RG11normRK RG11normRK RG11normRV RG21normRK RG21sksRK RG11normRV RG21sksRV RG21normRV RG11sksRK ELI 7 RG21sksRK ELI 8 RG11sksRV ELI 6 RG21sksRV ELI 9 VaR 1% G G ELI 10 EG ELI 6 EG RG11normRK RG11normRK RG11normRV RG21normRK RG21sksRK ELI 4 RG11normRV

34 RG21sksRV ELI 5 RG21normRV RG11sksRK ELI 7 RG21sksRK ELI 8 RG11sksRV RG21sksRV ELI 9 Noe: VaRLoss and p-mcs represen he values of he asymmeric VaR-based loss funcion 10 6 and p-values for MCS procedure, respecively. Models wih ELI in he column of p-mcs were eliminaed using he MCS eliminaion rule under he given 10% confidence level. Table 12 Comparison of ARMA-RGARCH and RGARCH VaRLoss*10^6 QLIKE Serial independence 0.5% 1% R2 RK RV CVM Fisher Tippe RG (1) (1) (1) (5e-3) (0.03) ARMA RG (7e-4) (0) (0.35) (1) (1) Noe: he values in parenheses are p-values for he MCS procedure; CVM, Fisher, Tippe represen he Cramer Von Mises es, Fisher s rule, and Tippe s rule, respecively. 32

35 Figure 1 Time series of reurns and realized measures Close-o-close reurn Jan Jan Jan Jan Jan Open-o-close reurn Jan Jan Jan Jan Jan

36 Realized kernel 0 2.0e e e e e-06 Jan Jan Jan Jan Jan Realized volailiy 0 5.0e e e-06 Jan Jan Jan Jan Jan

37 Figure 2 News impac curve Figure 3 Parameers and condiional sandard errors Gamma Jan Jan Jan Jan RG ARMARG 35

38 Bea Jan Jan Jan Jan RG ARMARG Ea Jan Jan Jan Jan RG ARMARG 36

39 Ea Jan Jan Jan Jan RG ARMARG Condiional sandard error Jan Jan Jan Jan RG ARMARG 37

40 Chaper 3 Realizing momens 3.1. Inroducion Many financial problems such as asse pricing, risk managemen and porfolio allocaion require disribuional characerisics of asse reurns. The mos criical feaure of reurn disribuion is is second momen, which has been sudied by using various mehods over he pas hree decades. Such approaches include fiing a parameric model (such as GARCH and SV ype models) wih a disribuional assumpion, exracing implied volailiy from opion price using specific opion pricing models, and calculaing realized volailiy which is he sum of inraday squared reurns. As discussed in Andersen e al. (2001), parameric volailiy models and opion pricing models suffer from model misspecificaion. Regardless of model misspecificaion, volailiy esimaes obained from he parameric volailiy models and opion pricing models are condiioned on he pas informaion, bu financial praciioners prefer indicaors which conain informaion up-odae. On he oher hand, Andersen e al. (2001, 2003) argue ha under suiable condiion, realized volailiy is he unbiased and consisen esimaor of quadraic variaion. This ype of volailiy can be considered as he volailiy during a cerain period, however, volailiy a he momen is of more ineres. In addiion, invesigaing higher momens such as skewness and kurosis arac increasing aenion, and such sudies (Jondeau and Rickinger (2003), Conrad e al. (2013), Kang and Lee (2016)) find srong relaionship beween higher momens and reurns. These sudies also belongs o he framework of condiional, implied and realized momens. In his paper, we propose a new ype of momens which can be esimaed from inraday daa. The so called realizing momens differ from condiional, implied and realized momens, since hese momens conain informaion up-o-dae and describes he curren reurn disribuion a daily frequency. Generally, we only have he poin observaion of reurn a daily, weekly, monhly and longer frequencies, bu we canno observe he probabiliy densiy of reurn. Finance lieraures consider ha all informaion would be absorbed and refleced by marke price. However, i is hardly o consider ha a poin observaion of reurn can describe hese informaion well. Financial reurns 38

41 are raher noisy, especially a daily frequency. I is more accepable o consider ha informaion would be well described by he probabiliy densiy of reurn. In addiion, he characerisics of a reurn disribuion can be described well by is momens. Therefore, we exrac he momens of he reurn disribuion and use hese momen o sudy some problems widely concerned in finance. In his paper, we invesigae he join behavior of reurn and is momens, he predicabiliy of reurn, and informaion ransmission mechanism using realizing momens, and provide many insighful resuls. This new ype of momens can be used for many purpose in furher finance research. This paper is organized as follows: Secion 2 describe he esimaion mehod of realizing momens. Secion 3 briefly describes our daa. Secion 4 provides insighful resuls of join behavior beween reurn and realizing momens. Secion 5 invesigaes he predicabiliy of reurn using realizing momens. Secion 6 sudies he informaion ransmission mechanism using his new ype of momens based on he framework of spillover index proposed by Diebold and Yilmaz (2012). Secion 7 concludes Esimaion mehod of Realizing momens To esimae he momens of daily reurn from inraday daa, we build our framework on he Hallam and Olmo (2014) o firs approximae he probabiliy densiy of curren reurn a daily frequency. The esimaion mehod relies on he heory of self-affine process Self-Affine process A self-affine process performs disribuional scaling behavior, which suggess ha he disribuion of he process a differen ime scales are idenical afer an appropriae ransformaion. Definiion 1: A sochasic process { X( )} saisfies d H 1 k 1 H { X ( c ),, X ( c )} { c X ( ),, c X ( )} (3.1) k for some H 0 and c, k, 1,, k 0, is called self-affine. H is he self-affiniy index and describes he relaionship beween disribuions of{ X( )} a differen ime scales Esimaion of Realizing momens 39

42 Many empirical sudies have confirmed he exisence of disribuional scaling behavior in a wide range of asses (see Calve and Fisher (2002), Maia e al. (2003), Calve and Fisher (2004), Di Maeo e al. (2005), Di Maeo (2007), Onali and Goddard (2009)). To approximae he probabiliy densiy of reurn using inraday daa and esimae he corresponding realizing momens, we assume Assumpion 1: The sochasic logarihmic price process{ X( )} is self-affine and has saionary incremens X ( ) X ( ) X ( ), where X() is he reurn process. Le r D denoes daily reurn and r i denoes inraday reurn. Under he assumpion of self-affiniy, he reurn process saisfies he disribuion scaling behavior and we have he relaionship H f ( r ) f ( c r ) (3.2) D i Equaion 2.2 says he probabiliy densiy of daily reurn r D is idenical o he probabiliy densiy of inraday reurn r i rescaled by he facor c H, where c is called he prefacor and equals o he relaive lengh of wo sampling inervals. For example, for a marke wih 9 hours rading, if we collec 5 minue inraday reurn, hen c would be 108. H is he self-affiniy index and can be esimaed using various. Here, we apply he Derended Moving Average (DMA) mehod. The DMA esimaor can be obained in following way. For a discree ime series x( ), 1,, T, selec a range of window sizes n, nmin n nmax, and filer he original series x () using a sandard moving average wih each n. n1 MA 1 xn ( ) x( k) (3.3) n k 0 MA 2 For each MA filered series{ xn ( )}, we calculae he value of n, where 1 (3.4) T 2 MA 2 n [ x( i) xn ( i)] T n in 40

43 Under he assumpion of self-affiniy, we have he relaionship n n H, and we can obain he 2 esimaes of he self-affiniy index H by running a linear regression of he logarihm of n on he logarihm of n, logn Hlog n (3.5) Then we can rescale inraday reurns by c Ĥ. The disribuion of hese rescaled inraday reurns is idenical o he daily reurn disribuion. Consequenly, we can obain he momens of daily reurn by calculaing sample momens of rescaled inraday reurns. 1 E[ r ] D T T i 1 r R T 1 V[ r ] ( r r ) D R R T 1 i1 2 1 T Skew[ r ] D T i1 ( r r ) R Vr [ ] D 3/2 R 3 T 1 4 ( rr rr) T i1 Kur[ rd ] T 1 [ ( rr rr)] T i Daa Our daa consiss of 7 frequenly raded commodiies in Tokyo commodiy exchange, including gold, silver, plainum, gasoline, crude oil, kerosene and rubber. Since derivaive conracs have a finie life, we consruc our ime series using he rolling-over mehod and volume crieria, as discussed in Carchano and Pardo (2009). The sample period is from Jan 2005 o Dec 2011, including 1707 observaions. The consruced price series are represened in figure Join behavior beween reurn and realizing momens 41

44 Figure 2 represen he ime series of reurns and esimaes of realizing momens in each marke. I is obvious ha reurn, (realizing) mean and skewness flucuae around zero, and reurn has much larger flucuaion han is mean. We also sudy he join behavior of reurns and heir momens. Figure 3 shows he join hisogram beween reurn and realizing momens in crude oil marke. For oher markes, hese figures of join hisogram are provided in appendix and hey sugges similar resuls. Figure 3a represens he relaionship beween reurn and mean. If marke is perfecly efficien, reurn should be coincide wih is mean, bu his is no he case in crude oil (and oher commodiy) marke, suggesing daily reurn is raher noisy. In general, reurn is posiively dependen wih he mean, and has a relaive larger probabiliy o coincide wih he mean in he ail and cener par. Tha is o say, even under he condiion of imperfecly efficien marke, reurn would coincide wih he mean of is probabiliy densiy in wo condiions: (1) an exreme even occurred and (2) nohing happened. From figure 3b, we can find he reurn is almos independen wih is variance, suggesing ha marke paricipans are risk neural in crude oil marke. Figure 3c shows ha reurn are posiively correlaed wih skewness, wih explici ail dependence. This is very inuiive: if he reurn disribuion is righ skewed, a posiive reurn would be archived and vice versa; on he oher hand, an exreme large/small reurn would happen if is disribuion is exremely righ/lef skewed. Figure 3d indicaes ha he cluser level of probabiliy densiy of reurn rise when he absolue reurn is large, and he kurosis is relaive low when absolue reurn is small. Figure 3e invesigaes he relaionship beween mean and variance of daily reurn densiy. We can see a clear V shape join hisogram. Generally, he variance and absolue mean of reurn densiy are almos posiively correlaed. The larger he absolue mean, he larger he price variaion. While he absolue mean is small (around 0.5), he variance is always small. Tha is o say, when here is nohing large happens in he marke, he price flucuae less. If he mean (and corresponding reurn) is exremely large or small, he price variaion is also very large. Figure 3f represen he relaionship of realizing mean and skewness. Basically, we can see ha mean and skewness are posiively dependen. From figure 3g, 3h and 3i, we can say ha mean is independen of kurosis, and variance is independen of skewness and kurosis. 42

45 Finally, figure 3j sugges ha here is a V shape relaionship beween skewness and kurosis. If he reurn disribuion is highly skewed (which means he absolue reurn is exremely large), i is also highly clusered. If he reurn disribuion is almos no skewed, i is also no clusered Sudying reurn predicabiliy using realizing momens The predicabiliy of asse reurn is a key issue in financial economics. In his paper, we focus on invesigaing wheher daily reurn is predicable based on is pas informaion. All informaion should be described by he probabiliy densiy of reurn. Therefore, we sudy he reurn predicabiliy by invesigaing wheher reurn is dependen of is pas realizing momens. In his paper, we apply he independence es using he empirical copula mehod proposed by Genes and Remillard (2004). The null hypohesis is ha wo series are independen. We es he lags up o 50. Figure 4 represens p-values of he independence es of reurn and is momens in each marke. Generally, he null hypohesis of independence is rarely rejeced in he long erm. Bu in shor erm, someimes he null hypohesis is rejeced. These resuls sugges ha pas informaion of reurn would help o predic only in shor erm (generally 2 or 3 days, and no more han one week) Sudying informaion ransmission mechanism using realizing momens. Sudying cross marke informaion ransmission mechanism is a ypical issue in financial economics. As he realizing momens describe informaion conained in reurn densiy, we can use his new ype of momens o sudy cross marke informaion ransmission mechanism. We build our framework on he spillover index proposed by Diebold and Yilmaz (2012), and sudy volailiy, skewness and kurosis spillover cross gold, silver, plainum, gasoline, crude oil, kerosene and rubber markes. Table 1, 2 and 3 represens he oal volailiy, skewness and kurosis spillover index of 1-, 3-, and 10- seps. I is obvious ha volailiy shocks almos ransmi nohing o oher markes, however, a considerable proporion of informaion driven by skewness and kurosis ransmi o oher markes and help o forecas he skewness and kurosis in oher markes. 43

46 In addiion, we can observe ha he gold, silver and plainum markes ransmi much more informaion o each oher han oher markes. Same phenomenon happens in gasoline, crude oil and kerosene markes. Therefore, we can conclude ha informaion mosly ransmi beween markes belong o same class. Figure 5, 6 and 7 represen he cross marke dynamic oal volailiy, skewness and kurosis spillover. I is clear ha during he global financial crisis and European deb crisis, here is a significan increase in ransmission of volailiy shock. Moreover, effec of volailiy shock spillover generally reach he peak in 3 o 5 days, and decay fas. For skewness shock and kurosis shock, he spillover level reach he peak level immediaely, and almos do no decay in 10 days Conclusion We propose a new ype of momens, which can describe he curren probabiliy densiy of daily reurn. The so called realizing momens conain informaion up-o-dae. This new ype of momens can be esimaed using high-frequency inraday daa by assuming he self-affiniy of reurn process. We also show various use of realizing momens, including sudying he join behavior of reurn and is momens, predicabiliy of reurn, and cross marke informaion ransmission mechanism. Many insighful resuls are provided. Fuure research can invesigae many financial problems using his new ype of momens. 44

47 Tabel 1 Volailiy spillover index Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional from ohers 1-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional % including own 3-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional % including own 10-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber

48 Direcional ohers Direcional including own o % Tabel 2 Skewness spillover index Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional from ohers 1-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional % including own 3-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional % including own 10-sep ahead 46

49 Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional including own % Table 3 Kurosis spillover index Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional from ohers 1-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional % including own 3-sep ahead Gold Silver Plainum Gasoline CrudeOil

50 Kerosene Rubber Direcional o ohers Direcional % including own 10-sep ahead Gold Silver Plainum Gasoline CrudeOil Kerosene Rubber Direcional o ohers Direcional including own % 48

51 Figure 1 Price series 49

52 50

53 51

54 Figure 2 reurn and esimaes of realizing momens 52

55 (a) Join hisogram of reurn and is mean (b) Join hisogram of reurn and is variance 53

56 (c) Join hisogram of reurn and skewness (d) Join hisogram of reurn and kurosis 54

57 (e) Join hisogram of mean and variance (f) Join hisogram of mean and skewness 55

58 (g) Join hisogram of mean and kurosis (h) Join hisogram of variance and skewness 56

59 (i) Join hisogram of variance and kurosis (j) Join hisogram of skewness and kurosis Figure 3 Join hisogram of reurn and realizing momens in crude oil marke 57

60 58

61 59

62 60

63 Figure 4 P-values for independence es beween reurn and realizing momens in each marke 61

64 Figure 5 Dynamic oal volailiy spillover Figure 6 Dynamic oal skewness spillover 62

65 Figure 7 Dynamic oal kurosis spillover 63

66 64