A mixed data sampling model for the return-liquidity dependence in stock index futures markets

Transcription

1 A mixed daa sampling model for he reurn-liquidiy dependence in sock index fuures markes Yuing GONG (SHU-UTS SILC Business School, Shanghai Universiy) Absrac Undersanding and quanifying he dependence of reurns and liquidiy is criical for liquidiy risk managemen. In his paper we exend he idea of mixed daa sampling (MIDAS) from linear correlaion in Colacio e al. (212) o he more general dependence measure: copula, and propose a copula-midas model o describe he asymmeric and periodic reurn-liquidiy dependence of CSI3 fuures. Based on he skewed copula-midas model, i is found ha exreme decreases in reurns end o be accompanied by exreme increases in bid-ask spreads, bu exreme increases in reurns may no coincide wih exreme reducions in bid-ask spreads. Furhermore, he reurnspread dependence consiss of boh shor-run and long-run componens, and he longrun componen will influence he reurn-spread dependence in he nex wo weeks. Las, he ou-of-sample forecas of liquidiy risk sresses he imporance of considering asymmery and periodiciy in reurn-spread dependence as i enables invesors o well predic liquidiy risk in imes of marke crashes. Our resuls imply ha high frequency rading invesors of CSI3 fuures should pay more aenions o preven he poenial liquidiy risk when he bid-ask spreads are widened. And invesors are suggesed o use he pas wo-week high frequency daa o forecas he curren reurn-spread dependence in liquidiy risk managemen. Keywords Mixed daa sampling, copula, CSI3 fuures, liquidiy, reurn 1. Inroducion Undersanding and quanifying he dependence of reurns and liquidiy is criical for liquidiy risk managemen. Liquidiy risk has been of major concern o boh Chinese regulaors and invesors, since he sock marke collapses in 215. In he firs half year of 215, he sock indices and fuures are rising rapidly wih excessive liquidiy, which is fueled by he inroducion of sock wih-funding. However, from June he sock marke sars o crash and liquidiy is dried up due o panic invesor senimen and irraional selling. The unprecedened urmoil brings grea challenges o regulaors and incurs unexpeced losses o mos invesors. Hence, i is necessary o ake he reurnliquidiy dependence ino accoun when we measure and forecas he liquidiy-adjused risk by means of a join disribuional model. Corresponding Auhor: Yuing Gong, Deparmen of Economics, SHU-UTS SILC Business School, Shanghai Universiy, Shanghai 218, China. Address: Room 511, Wenhui Building, 2 Chengzhong Road, Jiading Disric, Shanghai 218, China. Tel: yuinggong1985@163.com. 1

2 The reurn-liquidiy dependence is shown o exhibi wo feaures: asymmery and periodiciy. Firs, reurns and liquidiy measures have asymmeric dependence paerns in normal or exreme periods. Take he CSI3 fuures as an example. In 214 when he marke is relaively ranquil, he 5-minue reurns and bid-ask spreads (a proxy for liquidiy) are weakly correlaed wih average correlaion -.2. However, in June 215 wih exreme sock crash and liquidiy dry-ups, he reurn-spread correlaion drops sharply o -.23, en imes sronger han in 214. The asymmeric behaviors of reurnspread ail dependence have also been discussed in Weißand Supper (213), Hameed e al. (21). Second, he high frequency reurn-liquidiy dependence display a longrun rend as i is also affeced marke-wide facors sampled a low frequency, such as marke urmoil, invesor senimen, funding ighness, ec. For example, in December 214 during he imes of ighness in funding marke, he reurns and bid-ask spreads have pronounced negaive correlaion a While in Augus 215 when he cenral bank reduces ineres raes and he funding condiion is improved, he reurn-spread correlaion goes up o -.3. Curren economeric models are able o model eiher of he wo feaures individually, bu i is sill difficul o capure boh of hem simulaneously. Firs, he asymmeric reurn-liquidiy dependence can be well characerized by copula models, which are believed o be a more general dependence measure han linear correlaions. Bu exising copulas are unable o disinguish he shor-run and long-run componens in he evoluion of nonlinear dependence. Second, he periodic reurn-liquidiy dependence can be decomposed ino shor-run and long-run componens via he mixed daa sampling (MIDAS) scheme in Colaicio e al. (212). Bu heir componen model is buil upon Engle s dynamic condiional correlaion (DCC) and hence is no flexible enough o specify he shor-run and long-run componens in nonlinear dependence paerns. To overcome he problem, we propose he copula-midas model by exending he idea of mixed daa sampling from correlaion o copula. The newly proposed model enables us no only o describe he nonlinear dependence srucures unaffeced by he marginal disribuions, bu also o exrac he shor-run flucuaions and he long-run rend from he dynamics of nonlinear dependence. In line wih earlier MIDAS models, our copula-midas model exends he idea of mixed daa sampling from modelling linear correlaion o modelling he more general dependence srucure. So far he developmen of MIDAS models has experienced hree periods: MIDAS on mean, MIDAS on volailiy and MIDAS on correlaion. (1) MIDAS on mean. Ghysels e al. (24) propose he firs basic MIDAS model, which accommodaes he low frequency explained variable and he high frequency explanaory variables using highly parsimonious disribued lag polynomials. Ghysels e al. (27) hen include he auoregressive explained variable ino he regression (AR MIDAS). These models are widely used in nowcasing and shor-erm forecasing 2

3 macroeconomic indicaors, such as Clemens and Galvao (28), Andreou e al. (211), Ferrara and Marsilli (213), Monefore and Morei (213), Foroni e al. (215). (2) MIDAS on volailiy. Ghysels e al. (25) incorporae he mixed daa sampling scheme ino modelling volailiy dynamics and invesigae he risk-reurn rade-off based on monhly and daily sock daa. Engle e al. (213) develop he GARCH-MIDAS model o revisi he relaion beween US sock marke volailiy and macroeconomic aciviy (inflaion and indusrial produc growh). Kong e al. (28), Girardin and Joyeux (213) apply he above models o sudy Chinese A-share and B-share sock markes. (3) MIDAS on correlaion. Colacio e al. (22) propose he DCC-MIDAS model of dynamic correlaions wih shor-run and long-run componen specificaions and explore he correlaion dynamics of indusry porfolios and he1-year bond. Conrad e al. (214) endogenize he variaion of key macroeconomic figures ino DCC-MIDAS and examine he effec of US macroeconomic environmen on he long-erm sock-oil correlaion. Therefore, curren MIDAS models only specify he shor-run and long-run componens in correlaion, which is inadequae o capure he dynamics of nonlinear dependence, in paricular, he dynamics of ail dependence. To his end, our copula- MIDAS model fulfills his gap by exending MIDAS mehod ino dynamic copula models. Also, our empirical findings abou he reurn-spread dependence complemen he lieraure on he conemporaneous dependence of reurn and liquidiy. The dependence of reurn and bid-ask spread is shown o be negaive, consisen wih Amihud (22), Bekaer e al. (27), Hameed e al. (21) and Bali e al. (214). Unlike hem, our analysis using copula-midas model makes one-sep furher by saing ha he dependence srucure of reurns and bid-ask spreads are no only asymmeric bu also periodic. The negaive dependence is more pronounced in he case of exreme reurn decrease and large bid-ask spread han he case of exreme reurn increase and small bid-ask spread. Furhermore, he reurn-spread dependence is periodic wih a long-run rend ha is less flucuaed wihin wo weeks. I implies o invesors o use high frequency reurns and bid-ask spreads daa in he pas wo weeks o forecas he curren reurn-liquidiy dependence in liquidiy risk managemen. The paper is organized as follows. Secion 2 presens he copula-midas model and is esimaion. Secion 3 provides a preliminary analysis of he 5-minue reurns and bid-ask spreads daa. Secion 4 invesigaes he shor-run and long-run componens in he ail dependence of reurns and bid-ask spreads and hen forecass he liquidiy risk of CSI3 fuures. Secion 5concludes. 2. Economeric mehodology 2.1 Copula-MIDAS model The copula-midas model is a naural exension of he DCC-MIDAS model by Colacio e al. (211) o he copula framework. Consider he bivariae ime series 3

4 process y1, y 2, 1,, T. Le F 1 and F 2 represen he marginal disribuion funcions (CDF) of i i i y 1 and y 2 a ime. By he probabiliy inegral ransformaion, u F y for i 1, 2. The copula-midas model is wrien as follows: u u C u u,, ;,,, (1) g u, u, (2) K 1 k k. (3) k1 Equaion (1) shows ha he copula-midas model belongs o he ime-varying copula family. parsimonious parameer vecor. sochasic process ransformaion o ensure ha is he ime-varying parameer and 1,, Q is he is assumed o be driven by an unobserved such ha, where is a non-decreasing remains in is domain as in Paon (26). Equaion (2) differeniaes our copula-midas model from he exan dynamic copula models. The laen variable has wo sampling schemes: one is he high frequency wih subscrip, 1,, T, he same sampling frequency as y1, y 2 ; he oher is he low frequency afer wih subscrip, T N 1,, c, changes is value N c periods of. Then he dynamics of dependence srucure can be decomposed ino wo componens: a shor-run componen and a long-run componen (discussed in (3)). The specificaion of he shor-run componen follows Paon (26) and he shor-lived effecs are capured by an auoregressive lag 1 and a daa-driven erm g u, u whose funcional form depends on he copula ypes. and are he coefficiens and 1,1. Equaion (3) gives he long-run componen of dependence srucure reflecs he fundamenal or secular causes of ime variaion in dependence. ha is he ransformed realized correlaions of he high frequency observaions for 1 Nc 1,, Nc, and is a weighed average of in he pas K periods. Like Colacio e al. (211), he weigh scheme is Bea funcion such ha x1 K x1 k x 1 k / K 1 k / K. is he weigh coefficien and K is he maximum lag of. k1 Noe ha he copula-midas model has he desirable feaures of boh copulas and MIDAS models. Firs, he copula-midas model is able o describe nonlinear dependence srucures unaffeced by he marginal disribuions. The DCC-MIDAS 4

5 model by Colacio e al. (211) only invesigaes he shor-run and long-run componens of linear correlaion, and o a cerain degree i can be regarded as a special case of copula-midas (Gaussian copula-midas). Since linear correlaion is no a good measure of dependence for non-normal financial variables (Embrechs e al., 23; Okimoo, 28), our copula-midas model provides more flexibiliy in modelling he ail dependence of financial variables. Second, he copula-midas model capures he periodic evoluion of dependence relaed o he fundamenal or secular causes. I exracs he shor-run flucuaions and he long-run rend from he dynamics of dependence. While he exan ime-varying copulas, such as he dynamic copula in Paon (26), Hafner and Manner (212), only describes he shor-run dynamics in he dependence srucure. 2.2 Esimaion We follow he wo-sep procedure in Joe (1997) o esimae he join disribuion y, y. Bu one addiional sep is required: o selec he opimal MIDAS lag K of 1 2 in he formaion of he long-run componen. To be specific, he join disribuion of bivariae ime series y1, y 2 is esimaed by he following wo seps: Firs, we esimae each univariae marginal model and calculae he marginal CDF for y1, y 2, 1,, T, denoed by ˆ, ˆ u u. 1 2 Second, we plug uˆ, u ˆ ino he copula log-likelihood funcion and esimae 1 2 he copula-midas parameer,, by maximum likelihood esimaion (MLE). Since he value of log-likelihood funcion depends on he MIDAS lag K, i is necessary o choose he opimal lag via profiling of likelihood funcion. The approach is o selec he smalles number of MIDAS lags afer which he log-likelihood values seem o reach he plaeau, which is also used in Engle e al. (213), Colacio e al. (211). Under cerain regulariy condiions, he asympoic properies of he wo-sep esimaor are normally disribued. And he variance-covariance marix is esimaed by he block boosrapping procedure wih block lengh T. Besides, he specificaion of copula-midas model varies wih copula ypes. For Gaussian copula, 1 x x x e / 1 e o hold he correlaion parameer in he inerval 1,1, Q indicaing he absence of parsimonious parameers. g u, u 1, F u is he inverse of univariae sandard normal i1 i i1 CDF, i 1, 2. The realized correlaion is compued as Nc Nc Nc s2s 1s 2s s 1 Nc 1 s 1 Nc 1 s 1 Nc 1. 5

6 For copula, x and g u, u are he same, excep ha Q 1 wih represening he degree of freedom v and suden s CDF wih degree of freedom v. 1 F i For he skewed copula in Chrisoffersen e al. (212), are he same, excep ha 2 v and skewness parameer, Q wih v, 1 F i CDF wih degree of freedom v and skewness parameer. is he inverse of univariae x and g u, u represening he degree of freedom is he inverse of univariae skewed suden s 3. Daa and descripive saisics Our empirical sudy analyzes he dependence of high-frequency reurns and bidask spreads in Chinese CSI3 fuures marke. The opic is ineresing as he join disribuion of reurns and bid-ask spreads is useful in measuring and forecasing he liquidiy risk of sock index fuures marke, which is of major concern o boh invesors and regulaors, especially afer he sock marke crash in he summer of 215. We collec he inraday bid and ask prices of CSI3 fuures conracs from January 27 h, 214 o Sepember 2 nd, 215. The daa is obained from CSMAR Soluion provided by GTA Finance and Educaion Group. To consruc he coninuous nearby fuures price series, we use he prices for he nearby fuures conrac unil he conrac reaches he firs day of he delivery monh. Then prices for he nex nearby conrac are used. Furhermore, we exclude he pre-marke and afer-marke rading hours as well as he firs and las hree 5-minue inervals of he regular marke hours. Hence, we consider he period from 9:3 a.m. o 3: p.m. in each rading day. To provide a quaniaive descripion of reurns and bid-ask spreads, we sar wih some noaions. We assume he disance beween wo observaion poins o equal 5 minues. Le P denoe he average mid-prices (average of bid and ask prices) of CSI3 fuures in he h 5-minue inerval, and le 6 S denoe he average bid-ask spread (bid price minus ask price) of CSI3 fuures in he h 5-minue inerval, 1,, T, T M Nc, where M is he number of rading days and N c is he number of observaions per day. The 5-minue inraday reurn is compued as R log regardless of he firs observaion in each day ( P P 1 mod, Nc 1). Hence, we have Nc 47 observaions per day and observaions for he full sample. Figure 1 plos he reurns and bid-ask spreads of CSI3 fuures. Reurns and bidask spreads co-move hroughou he sample period. Furhermore, large negaive reurns coincide wih sudden increase in bid-ask spreads, especially in December 214 and from June o Sepember in 215.

7 Panel (a) of Table 1 repors he summary saisics of he sample daa. The 5- minue reurns of CSI3 fuures are non-normally disribued wih negaive skewness and excess kurosis, indicaing he downside risk in sock index fuures. The reurns also exhibi self-persisence and volailiy clusering feaures. The 5-minue bid-ask spreads of CSI3 fuures have a righ-skewed empirical disribuion and vary considerably from.2 o 6.22 poins. The spreads are also auo-correlaed, which means he curren bid-ask spread will influence he spread in he near fuure. Panel (b) of Table 1 is designed o es for ime-variaion in he reurn-spread dependence. We follow Paon (212) and perform a es based on auocorrelaions in rank correlaions wih lags up o p 1 (5 minues), p 12 (one hour), p 24 (half day) and p 47 (one day), respecively. The rejecion of he null hypohesis indicaes he dynamically evolving reurn-spread dependence hroughou he sample period. Panel (c) of Table 1 illusraes he inraday reurn-spread dependence may vary from quarer o quarer. To sar wih, we compue he correlaion of reurns bid-ask spreads R and S in each quarer and find ha hey are mosly negaively correlaed excep for he second quarer of 214. Then, o examine he asymmery in he negaive reurn-spread dependence, we calculae he exceedance correlaions a 5% and 95% perceniles for he pair R, S raher han R, S..5 measures he conemporaneous correlaion of he lowes 5% reurns and he highes 5% bid-ask spreads, and.95 measures he conemporaneous correlaion of he highes 5% reurns and he lowes 5% bid-ask spreads. In mos cases R S.95 R, is larger han.5, S in absolue values. I implies exreme decreases in CSI3 fuures reurns end o be followed by exreme rise in bid-ask spreads, bu he endency for exreme increase in reurns and exreme reducion in bid-ask spreads is no so srong. In paricular, his asymmeric dependence paern is ousanding especially in he fourh quarer of 214 and in he second quarer of 215, when he sock marke and sock index fuures marke suffer from grea urmoil. Las, we es for he pairwise exceedance correlaions symmery in each quarer based on he ess of Hong e al. (27). The rejecion of he ess reveals he symmeric copulas (Gaussian or ) are no good choices in modelling he reurn-spread dependence, hence some copulas wih asymmeric dependence is suggesed. Overall, he preliminary analysis ells us ha he dependence of CSI3 reurns and bid-ask spread is asymmeric and periodic from quarer o quarer. The periodic dynamics implies an unknown long-run rend underlying in he high frequency reurnspread dependence. I hen provides some hins for us o use he copula-midas model in he following empirical analysis. 4. Empirical analysis 7

8 The empirical analysis secion includes wo main pars: he in-sample esimaion of he reurn-spread join disribuion and he ou-of-sample forecas of liquidiyadjused inraday value-a-risk (LIVaR). Correspondingly, we divide he sample ino wo sub-periods, such ha he observaions in 214 are used for he in-sample esimaion and he remaining observaions in abou 8 monhs of 215 are reserved for he ou-of-sample LIVaR forecas. 4.1 In-sample esimaes of marginal disribuions Before applying copula models o sudy he reurn-spread dependence, we inroduce he marginal models of reurns and bid-ask spreads. The reurns R are firs deseasonalized and hen modelled by he auoregressive and GARCH models wih Demara and McNeil (25) s skewed errors, as in (1), (2) and (3), in order o capure he non-normal feaures illusraed in Table 1. R r IVn, (1) r r h, i. i. d. skew s, d, (2) k f h h h (3) IV n is he deerminisic paern of he inraday volailiy for he n -h 5-minue inerval, 2 n 1,, Nc, Nc 48 in one rading day. Like Gio (25), IVn 1 M Rs, where,,2,, 1 B n N n N n M N n, M is he number of rading days. n c c c The auoregressive and GARCH coefficiens saisfy 1 1 1, 1, 2 1, The bid-ask spreads 8 sbn S are modelled by he auoregressive condiional duraion (ACD) by Engle and Russell (1998) in (4) and (5). Noe ha coninuous raher han discree. This is because we expec boh S is assumed o be R and S o reflec he average levels of gains (or losses) and liquidiy for he 5-minue long inerval. Bekaer e al. (27), Wei and Frino (213) also use he aggregaed bid-ask spread as he liquidiy measure of CSI3 fuures S, i. i. d. Weibull 1,, (4), S (5) Table 2 presens he in-sample resuls of he CSI3 reurns and bid-ask spreads marginal disribuions. Three conclusions can be wihdrawn. Firs, he marginal models are shown o be correcly specified, which ensures he consisency of he copula esimaes in he nex subsecion. We fail o rejec he Kolmogorov-Smirnov es of model specificaion in each marginal model and deec no auocorrelaion in he residuals by he Ljung-Box es. Second, he esimaes in he marginal model of R confirm he non-normal

9 feaures of reurns in Table 1, including heavy ails, serial correlaion and volailiy clusering. Similar characerisics of he high-frequency CSI3 fuures reurns can also been found in Yang e al. (212), Hou and Li (214) and Suo e al. (215). Third, he CSI3 fuures bid-ask spreads 9 S are skewed o he righ and auocorrelaed over he ime, which is also consisen wih Table 1. The shape parameer in Weibull disribuion is.59, indicaing in some cases he CSI3 fuures marke may be illiquid and invesors have o suffer from high ransacion cos. Bu Xu and Wan (215) find ha he disribuion of bid-ask spreads is symmeric from 21 o 212. The righ skewness of CSI3 fuures spreads from 214 o 215 is more significan han he period 21 o 212, implying ha he marke liquidiy condiion is worsening in recen years, possibly due o he 215 sock marke crash. The auoregressive coefficien 2 is.56, implying he bid-ask spread is predicable over cerain ime horizon. The auocorrelaed effecs in sock index fuures bid-ask spreads has also been found in Bekaer e al. (27), Weißand Supper (213), Groß-KlußMann and Hausch (213), no maer wheher discree (end-poin) or coninue (average) bid-ask spreads are considered. 4.2 In-sample esimaes of copula models Table 3 provides he in-sample esimaes of five copula models. Among hem, he firs hree are copula-midas models wih ellipical copulas (Gaussian, and skewed ). We choose ellipical copulas insead of Archimedean copulas because mos Archimedean copulas feaured by posiive dependence paern are unable o describe he negaive reurn-spread correlaion and ail dependence shown in Table 1. The maximum lag K is chosen o be 1 as i is he minimum lag ha maximizes he loglikelihood values of copula-midas models. Besides, o undersand he imporance of disinguishing shor-run and long-run componens in dependence, we include wo copulas wihou mixed daa sampling schemes: he dynamic skewed copula evolved like Paon (26) and he consan skewed copula. Dynamic skewed copula: Consan skewed copula:,, ;,,,,, sk u1 u2 C u1 u2 v c g u, u sk u u C u u v,, ;,,. (7) From Table 3, i is concluded ha he CSI3 fuures reurns and bid-ask spreads no only exhibi asymmeric ail dependence, bu also display shor-run flucuaions and a long-run rend in he dynamic dependence. Among he five copulas, he skewed copula-midas model has he bes in-sample goodness-of-fi performance wih he highes log-likelihoods and he lowes AIC and BIC. The log-likelihoods almos double when we swich from he dynamic or consan copulas o he copula-midas models, (6)

10 implying he necessiy of considering he periodic feaure in reurn-spread dependence. Nex, he log-likelihoods increase furher bu a a lower speed when we swich from he symmeric Gaussian or copula-midas models o skewed copula-midas. I suggess ha he reurn-spread dependence is more likely o be asymmeric a boh ails. Figure 2 plos he reurn-spread exceedance correlaions based on he skewed copula-midas model: he black lines represen he oal exceedance correlaions and he red lines denoe he long-run rends (discussed laer). Remember ha 1.5 in Panel (a) measures he conemporaneous correlaion of he lowes 5% reurns and he highes 5% bid-ask spreads, and.95 in Panel (b) measures he conemporaneous correlaion of he highes 5% reurns and he lowes 5% bid-ask spreads. The reurn-spread dependence is asymmeric as skewness parameer.5 is always lower han.95 due o he negaive.3 in he skewed copula-midas model. In oher words, exreme decreases in reurns end o be accompanied by exreme increases in bid-ask spreads (marke liquidiy dry-ups). Exreme increases in reurns, on he oher hand, don coincide wih exreme reducions in bid-ask spread (marke liquidiy abundance). In paricular, in he afernoon of December 9 h 214, he CSI3 fuures mid-price slumps by % (from poins o poins). A he same ime, he marke liquidiy condiion worsens wih ripled bid-ask spreads (from.4 poins o 1.2 poins). As a resul,.5 falls dramaically o below -.41, while.95 says close o zero. Our sudy of he reurn-spread dependence based on copulas is direcly in line wih earlier work in Weißand Supper (213), bu we focus on differen ail dependence. Weißand Supper (213) examine he reurn-spread co-movemen when boh of hem increase exremely or boh of hem decrease exremely, as he vine copula is consruced by Archimedean copulas feaured wih posiive ail dependence. I is found in heir analysis ha, exreme increases in bid-ask spreads are associaed wih exreme rise in conemporaneous reurns as invesors ask for high risk premium, bu exreme decreases in bid-spreads may no happen wih sock price crashes. Their analysis of ail dependence in he firs and hird quadrans is applicable o US sock marke, bu may no be appropriae for Chinese CSI3 fuures markes since reurns and bid-ask spreads are generally negaively dependen. In his case, we focus on he ail dependence in he second and fourh quadrans and find ha he dependence of low reurns and large bid-ask spreads is sronger han he dependence of high reurns and small bid-ask spreads. Therefore, i is he sign of he reurn-spread dependence ha deermines which ail dependence we are going o invesigae and which copula we are going o use. Figure 3 depics he esimaed correlaions based on he five copulas. Again, he black lines represen he oal correlaions and he red lines are he long-run rends. The

11 periodic characerisic in reurn-spread dependence is refleced in Panel (c) of Figure 3 in conjuncion wih Figure 2. Two poins should be addressed. Firs and mos imporan, he reurn-spread dependence can be decomposed ino he shor-run flucuaions and he long-run rend. The shor-run componen capures he shor-lived reurn-spread dependence coming from high frequency microeconomic srucures and i moves around he long-run rend. The long-run componen measures he weighed average of realized reurn-spread correlaions over he pas 2 weeks ( K 1 rading days), and is expeced o be relaed wih he marke-wide fundamenal or secular causes, as noed in Colacio e al. (211). The slowly-moving long-run componen in he negaive reurn-spread dependence is possibly due o biding capial consrains of firms and marke urmoil in Hameed e al. (21). Comparing Panel (c) and he dynamic skewed copula in Panel (d), we can see he cres-o-ough waves from.5 o -.3 in he copula-midas reurn-spread correlaions, while he correlaions from he dynamic skewed copula are mean-revering and move much more frequenly. Second, he reurn-spread dependence, eiher correlaion or exceedence correlaion, is ime-varying raher han consan. The negaive dependence is more pronounced in imes of exreme downward movemen in CSI3 fuures prices. Look a Panel (c) and he consan skewed copula in Panel (e). If he ime-variaion in dependence was ignored, he reurn-spread dependence would be misakenly underesimaed wih average correlaion only a -.6 (.6 for he consan skewed copula in Table 3). Then urn o Panel (c) and he oher wo copula-midas models in Panel (a) and (b). The skewed copula-midas model has he mos volaile correlaions, which is followed by copula-midas and Gaussian copula-midas. For example, in he afernoon of December 9 h 214, he reurn-spread correlaion is as low as -.26 for skewed copula-midas, -.22 for copula- MIDAS and -.18 for Gaussian copula- MIDAS. I is concluded ha, accouning for asymmeric dependence avoids underesimaing he exreme reurn-spread dependence and enables invesors o conrol he liquidiy risk in imes of marke crashes. Figure 4 visualizes he decaying impacs of pas realized correlaions on curren correlaions for he hree copula-midas models. The impacs are measured by weighs a differen lags, which are calculaed according o he weigh funcion k, k 1,, K in (3). For skewed copula-midas, he pas daily realized correlaions will influence he fuure correlaions in he nex 9 days (abou 2 weeks), much longer he oher wo copula-midas models which only ake 5 days (1 week) for he weighs o decay o zero. The implicaion is ha he marke needs longer ime o absorb he exreme negaive informaion shocks capured by he skewed copula han he normal informaion shocks capured by symmeric ellipical copulas. The resul is also discussed by Gong and Zheng (216), alhough hey invesigae he persisence of dependence by esing he long-memory effec in he evoluion of dependence. 11

12 Our findings of negaive conemporaneous reurn-spread dependence are suppored by he hypohesis ha liquidiy risk is priced, which has been saed in Amihud (22), Bekaer e al. (27) and Bali e al. (214). However, we push heir resuls furher by poining ou ha he dependence is asymmeric and periodic. On one hand, he negaive dependence is found o be much sronger in marke downurn (low reurns) wih exreme illiquidiy (large bid-ask spreads) han in marke upurn (high reurns) wih sufficien liquidiy (small bid-ask spreads). Hameed e al. (21) have documened he asymmeric responses of liquidiy o changes in sock prices, bu hey concenrae on he lead-and-lag effecs of curren marke declines on fuure liquidiy dry-ups in he nex few weeks. On he oher hand, he reurn-spread dependence is shown o be periodic wih a long-run rend ha is less flucuaed wihin wo weeks. The mixed daa sampling scheme excludes he shor-lived effecs in dependence relaed wih microsrucure noise from he oal reurn-spread dependence, hence he long-run componen in dependence is expeced o be linked wih some marke-wide facors like funding ighness, invesor senimen or marke uncerainy. Alhough few scholars have discussed he shor-run and long-run componens in reurn-spread dependence, his periodic feaure can sill be observed in Weißand Supper (213) when hey plo he ail dependence evoluion of reurns and bid-ask spreads in a rolling-window way. The implicaions of sudying he asymmeric and periodic reurn-spread dependence are wofold. Firs, he liquidiy risk of CSI3 fuures markes is priced under illiquid marke condiions, bu is no well priced under liquidiy marke condiions. The high frequency rading invesors of CSI3 fuures should pay more aenions o cope wih he poenial liquidiy risk when bid-ask spreads are widened. Second, he invesors engaging in high frequency rading are suggesed o use reurns and bid-ask spreads daa in he pas wo weeks o forecas he curren reurn-liquidiy dependence in liquidiy risk managemen. The sample is no necessarily he larger he beer, as he upwards and downwards in he long-run rend may offse each oher. Meanwhile, he high frequency sample should no be oo small such ha we may no have enough informaion o predic he ail dependence of exreme decrease in reurns and exreme increase in bid-ask spreads. 4.3 Ou-of-sample forecass of liquidiy adjused inraday value-a-risk To explore he economic value of modelling he asymmeric and periodic reurnspread dependence, we hen forecas he liquidiy adjused inraday value-a-risk (LIVaR) of CSI3 index fuures according o he five copulas presened above. The in-sample resuls have old us ha he impacs of realized correlaions (or exceedance correlaions) will las less han 1 rading days. Hence, we employ he in-samples wih sizes of 47 observaions (1 rading days) a 5-minue inervals and forecas he following 23 observaions. The in-sample window is hen shifed forward and he nex 24 observaions are forecased using he re-esimaed models. Hence, each day wih 12

13 N 47 observaions is spli up ino wo forecasing periods of lengh 23 and 24, c respecively. The LIVaR is calculaed from he M 1 simulaed pairs of reurns and bidask spreads m m R, S based on he copula and marginal models, m 1,, M. Here we follow he LIVaR measure proposed by Weißand Supper (213) as i is direcly applicable o copula-based models. The forecased LIVaR a ime a he h quanile is where LIVaR P 1 exp IVn q h IV n, P is he acual mid-price a ime, inraday volailiy defined in (1), n N mod, c., 13 (8) IV n is he deerminisic paern of he h and q characerize he m mean, volailiy and he h quanile of he simulaed liquidiy adjused reurns r based on m m m m R, S. The adjused reurn is compued as 1 exp P P R, P 1 is he acual mid-price a ime 1. m m m 1 S r R IV 2 m P n The ou-of-sample performance of hese models is assessed by comparing he simulaed LIVaR forecass and he realized liquidiy-adjused profis and losses ( PL ). Focusing on he downward risk, bid price B. PL a ime is based on he mid price P 1 and PL. P 1 P P B B P 1 (9) Figure 5 plos he simulaed LIVaR forecass based on five copulas and he realized liquidiy-adjused profis and losses PL. Figure 6 presens he corresponding LIVaR exceedances of he five copulas. The wo figures provide ample evidence of he skewed copula-midas model s abiliy o predic he CSI3 fuures liquidiy risk. The LIVaR values prediced by skewed copula-midas in Panel (c) of Figure 5, say relaively close o he realized losses, bu he LIVaR forecass are only exceeded in 33 cases in Panel (c) of Figure 6. In paricular, he model wih asymmeric and periodic dependence well predics he liquidiy risk increases from June o Augus in 215 in imes of sock marke crashes. The maximum underesimaion of LIVaR is 127 poins, while oher four copulas underesimae LIVaR by above 2 poins. Meanwhile, our forecass are no oo conservaive o preven invesors from reserving oo much risk capial. Now le s look a he plos of dynamic skewed copula and consan skewed copula in panel (d) and (e). These wo models disregard he long-run componen in reurn-spread dependence and heir ou-of-sample forecas performances of LIVaR are no as saisfacory as he skewed copula-midas model. The number of exceedance,

14 is 38 for he dynamic skewed copula and is 42 for he consan skewed copula. When he marke liquidiy risk is driven up in he summer of 215, he dynamic skewed copula based sraegy underesimaes LIVaR by 293 poins in June and by 546 poins in Augus, in depiced in Panel (d) of Figure 6. The consan skewed copula performs even worse, underesimaing LIVaR by 298 poins in June and by 549 poins in Augus, as shown in Panel (e) of Figure 6. The underperformance of hese wo models is possibly due o he ignorance of he long-run componen in dependence from heir model specificaions. Since he models only capure he shor-run dynamics or consancy in reurn-spread dependence using he nearby hisorical daa, he forecased dependence will say relaively sable and exhibi no long-run upward and downward rends. The model specificaions are likely o underesimae of he reurn-spread dependence, hus unable o predic LIVaR as exacly as he skewed copula-midas model. Overall, he ouperformance of he skewed copula-midas model over he dynamic and consan skewed copulas in LIVaR forecass sresses he imporance of mixed daa sampling in modelling he shor-run and long-run componens of dependence. Then urn o he LIVaR plos of Gaussian and copula-midas models in panel (a) and (b). During 215 he sock index fuures marke encouners wo rounds of crashes in June and in Augus. Alhough he wo models ha assumes symmeric reurn-spread dependence are able o predic LIVaR in he second round of marke crash, hey fail o predic LIVaR in he firs round of marke crash. In Augus, he simulaed LIVaR of Gaussian copula-midas only exceeds PL in 4 cases wih maximum underesimaion of 183 poins, and he simulaed LIVaR of copula-midas also exceeds 14 PL in 4 cases wih maximum underesimaion of 15 poins. A ha ime he performance of hese wo copulas are comparable wih he skewed copula-midas model. However, in June, eiher of he wo models can forecas he sudden increase in liquidiy risk. The reason behind his is ha, he hisorical daa for LIVaR forecass don conain enough informaion abou marke crash and liquidiy dry-ups, as he sock index marke is booming in he firs half year of 215. If symmeric dependence paern is specified, he forecased reurn-spread dependence in June will be much lower (in absolue values) han hey ruly are, hus leading o he underesimaion of LIVaR. In shor, he comparison of he skewed copula-midas model wih he wo symmeric copula- MIDAS models highlighs he significance of characerizing asymmeric dependence paerns. To formally es he adequacy of hese ou-of-sample LIVaR forecass we employ he condiional coverage es proposed by Chrisoffersen and Pelleier (24). The null hypohesis is ha he forecased LIVaR values have correc number of exceedances and serially independen. Among he five copulas, we rejec he dynamic skewed copula and he consan skewed copula a he 1% significance level, as heir es saisics

15 are 5.79 (p value.6) and 5.6 (p value.8). On he conrary, he Gaussian, and skewed copula-midas models can be rejeced a 1% significance level, and heir es saisics are 3.14 (p value.21), 2.13 (p value.34), 1.84 (p value.4), respecively. I is inferred ha he copula-midas models can well predic LIVaR wih correc number of exceedances, which confirms he plos of simulaed LIVaR and in Figure 5. PL In summary, in his secion we apply he copula-midas model o analyze he dependence srucure of 5-minue CSI3 fuures reurns and bid-ask spreads during 214 and 215. I is found ha he reurn-spread dependence is asymmeric: he negaive dependence of low reurns and large bid-ask spreads is sronger han he dependence of high reurns and small bid-ask spreads. Furhermore, he reurn-spread dependence display shor-run flucuaions and a long-run rend relaed wih secular or fundamenal facors. In addiion, he ou-of-sample forecas of LIVaR illusraes he economic value of modelling he asymmeric and periodic reurn-spread dependence srucure as i helps invesors o predic he sudden increase in liquidiy risk in imes of marke slumps. 5. Conclusion Liquidiy risk has received much more aenion ever since he sock marke crash in he summer of 215 and he launch of circui-breaker mechanism in 216. Modelling liquidiy risk should ake ino accoun he dependence srucure of reurns and bid-ask spreads. In his paper, we propose a copula-midas model o capure he asymmeric and periodic feaures in he evoluion of reurn-spread dependence. The new model, which is buil upon Colacio e al. (211), exends he approach of mixed daa sampling from correlaion o he more general dependence measure: copula. I is no only flexible enough o describe he reurn-spread dependence regardless of heir marginal disribuions, bu also able o disinguish he shor-lived effecs and he long-run rend underlying in he dynamic dependence. Applying he copula-midas model o he 5-minue CSI3 fuures daa, we find ha he skewed copula-midas model has he beer in-sample goodness-of-fi and ou-of-sample LIVaR forecas performance han oher copulas. The reurns and bid-ask spreads are negaively dependen and exhibi asymmeric ail dependence. Equivalenly, he crash in CSI3 fuures reurns is likely o be accompanied by exreme increase in bid-ask spreads, bu i is no he case he oher way round. Furhermore, he reurnspread dependence consiss of boh shor-run and long-run componens, and he longrun componen relaed wih marke-wide fundamenals or secular causes will influence he reurn-spread dependence in he nex wo weeks. The resuls sugges ha high frequency rading invesors of CSI3 fuures should pay more aenions o preven 15

16 he poenial liquidiy risk when he bid-ask spreads are widened. I also advises invesors o use wo-week high frequency hisorical daa of reurns and bid-ask spreads o forecas he nearby reurn-liquidiy dependence in liquidiy risk managemen. The paper leaves several opics for furher research. One exension is o add exogenous explanaory variables ino he dynamic process of dependence srucure, such as he marke-wide variables menioned in Hameed e al. (21), Bali e al. (214). Anoher exension is o invesigae he liquidiy risk problem in oher financial markes, including socks, bonds and commodiy fuures, if he high frequency rading daa is available. The resuls abou liquidiy risk managemen in hese markes may be of ineres o boh regulaors and invesors. 16

17 References [1] Amihud, Y. Illiquidiy and sock reurns: cross-secion and ime-series effecs [J]. Journal of Financial Markes. 22, 5(1): [2] Andreou, E., Ghysels, E., Kourellos, A. Forecasing wih mixed-frequency daa [J]. Oxford Handbook of Economic Forecasing. 211: [3] Bali, T.G., Peng, L., Shen, Y., Tang, Y. Liquidiy shocks and sock marke reacions [J]. Review of Financial Sudies. 214, 27(5): [4] Bekaer, G., Harvey, C.R., Lundblad, C. Liquidiy and expeced reurns: Lessons from emerging markes [J]. Review of Financial Sudies. 27, 2(6): [5] Chrisoffersen, P., Errunza, V., Jacobs, K., Langlois, H. Is he poenial for inernaional diversificaion disappearing? A dynamic copula approach [J]. Review of Financial Sudies. 212, 25(12): [6] Chrisoffersen, P., Pelleier, D. Backesing value-a-risk: A duraion-based approach [J]. Journal of Financial Economerics. 24, 2(1): [7] Clemens, M.P., Galvão, A.B. Macroeconomic forecasing wih mixed-frequency daa: Forecasing oupu growh in he Unied Saes [J]. Journal of Business & Economic Saisics. 28, 26(4): [8] Colacio, R., Engle, R.F., Ghysels, E. A componen model for dynamic correlaions [J]. Journal of Economerics. 211, 164(1): [9] Conrad, C., Loch, K., Riler, D. On he macroeconomic deerminans of long-erm volailiies and correlaions in US sock and crude oil markes [J]. Journal of Empirical Finance. 214, 29: [1] Demara, S., McNeil, A.J. The copula and relaed copulas [J]. Inernaional Saisical Review. 25, 73(1): [11] Embrechs, P., Lindskog, F., McNeil, A. Modelling dependence wih copulas and applicaions o risk managemen [J]. Handbook of Heavy Tailed Disribuions in Finance. 23, 8(1): [12] Engle, R.F., Ghysels, E., Sohn, B. Sock marke volailiy and macroeconomic fundamenals [J]. Review of Economics and Saisics. 213, 95(3): [13] Engle, R.F., Russell, J.R. Auoregressive condiional duraion: A new model for irregularly spaced ransacion daa [J]. Economerica. 1998: [14] Ferrara, L., Marsilli, C. Financial variables as leading indicaors of GDP growh: Evidence from a MIDAS approach during he Grea Recession [J]. Applied Economics Leers. 213, 2(3): [15] Foroni, C., Marcellino, M., Schumacher, C. Unresriced mixed daa sampling 17

18 (MIDAS): MIDAS regressions wih unresriced lag polynomials [J]. Journal of he Royal Saisical Sociey: Series A (Saisics in Sociey). 215, 178(1): [16] Ghysels, E., Sana-Clara, P., Valkanov, R. The MIDAS ouch: Mixed daa sampling regression models [R]. Working paper, Anderson Graduae School of Managemen, Universiy of California, Los Angeles, 24. [17] Ghysels, E., Sana-Clara, P., Valkanov, R. There is a risk-reurn rade-off afer all [J]. Journal of Financial Economics. 25, 76(3): [18] Ghysels, E., Sinko, A., Valkanov, R. MIDAS regressions: Furher resuls and new direcions [J]. Economeric Reviews. 27, 26(1): [19] Gio, P. Marke risk models for inraday daa [J]. The European Journal of Finance. 25, 11(4): [2] Girardin, E., Joyeux, R. Macro fundamenals as a source of sock marke volailiy in China: A GARCH-MIDAS approach [J]. Economic Modelling. 213, 34: [21] Gong, Y., Zheng, X. Long memory in asymmeric dependence beween LME and Chinese aluminum fuures [J]. Journal of Fuures Markes. 216, 36(3): [22] Groß-KlußMann, A., Hausch, N. Predicing bid-ask spreads using long-memory auoregressive condiional Poisson models [J]. Journal of Forecasing. 213, 32(8): [23] Hafner, C.M., Manner, H. Dynamic sochasic copula models: Esimaion, inference and applicaions [J]. Journal of Applied Economerics. 212, 27(2): [24] Hameed, A., Kang, W., Viswanahan, S. Sock marke declines and liquidiy [J]. The Journal of Finance. 21, 65(1): [25] Hong, Y., Tu, J., Zhou, G. Asymmeries in sock reurns: Saisical ess and economic evaluaion [J]. Review of Financial Sudies. 27, 2(5): [26] Hou, Y., Li, S. The impac of he CSI 3 sock index fuures: Posiive feedback rading and auocorrelaion of sock reurns [J]. Inernaional Review of Economics & Finance. 214, 33: [27] Joe, H. Mulivariae models and mulivariae dependence conceps [M]. Chapman & Hall/CRC, [28] Kong, D., Liu, H., Wang, L. Is here a risk-reurn rade-off? Evidences from Chinese sock markes [J]. Froniers of Economics in China. 28, 3(1): [29] Monefore, L., Morei, G. Real-ime forecass of inflaion: The role of financial variables [J]. Journal of Forecasing. 213, 32(1): [3] Okimoo, T. New evidence of asymmeric dependence srucures in inernaional 18

19 equiy markes [J]. Journal of Financial and Quaniaive Analysis. 28, 43(3): [31] Paon, A. Copula mehods for forecasing mulivariae ime series [J]. Handbook of Economic Forecasing. 212, 2: [32] Paon, A.J. Modelling asymmeric exchange rae dependence [J]. Inernaional Economic Review. 26, 47(2): [33] Suo, Y. Y., Wang, D. H., Li, S. P. Risk esimaion of CSI 3 index spo and fuures in China from a new perspecive [J]. Economic Modelling. 215, 49: [34] Wei, W.C., Frino, A. The impac of underlying marke closure on fuures marke: Evidence from China [J]. Inernaional Journal of Banking and Finance. 213, 9(2): 2. [35] Weiß, G.N., Supper, H. Forecasing liquidiy-adjused inraday Value-a-Risk wih vine copulas [J]. Journal of Banking & Finance. 213, 37(9): [36] Xu, F., Wan, D. The impacs of insiuional and individual invesors on he price discovery in sock index fuures marke: Evidence from China [J]. Finance Research Leers. 215, 15: [37] Yang, J., Yang, Z., Zhou, Y. Inraday price discovery and volailiy ransmission in sock index and sock index fuures markes: Evidence from China [J]. Journal of Fuures Markes. 212, 32(2):

20 Tables Table 1. Descripive saisics and preliminary analysis of reurn-spread dependence Reurn Panel (a) Summary saisics 2 Bid-ask spread Mean( 1 5 ) Mean.3481 Sandard deviaion.31 Sandard deviaion.199 Skewness Skewness Excess kurosis Excess kurosis Minimum Minimum.2 Maximum.481 Maximum J-B( 1 ) 3.38 J-B( 1 ) Ljung-Box(12) Ljung-Box(12)( 1 ) ARCH(12) Panel (b) Tess for ime-varying rank correlaions 7 7 p 1 ( 1 ) p 24 ( 1 ) p 12 ( 1 ) p 47 ( 1 ) Panel (c) Tess for exceedance correlaions symmery Quarer R, S R S R S.5, Saisics.95, 214/1-214/ /4-214/ /7-214/ /1-214/ /1-215/ /4-215/ /7-215/ Panel (a) repors he summary of descripive saisics for he 5-minue reurns and bid-ask spreads of CSI3 fuures from January 27 h, 214 o Sepember 2 nd, 215. J-B is he Jarque-Bera es for normaliy. Ljung-Box(12) and ARCH(12) are he Ljung-Box es for serial correlaion and for GARCH effec wih 12 lags., and denoe significance a he 1%, 5% and 1% levels, respecively. Panel (b) repors he saisics from Paon (212) s ess of auocorrelaions in uu 1 2 wih lags P 1,12, 24, 47, and he rejecion of he hypohesis suppors he presence of dynamically evolving dependence. Panel (c) repors he reurn-spread correlaion R, S exceedance correlaions of reurns and negaive bid-ask R S and R S.5,.95,, he as well as he saisics in Hong e al. (27) o es for exceedance correlaions symmery in each quarer of our sample. The rejecion of he hypohesis indicaes asymmeric reurn-spread dependence.

21 Table 2. Marginal model esimaes of reurns and bid-ask spreads over he in-sample period Reurn Bid-ask spread.792 (.329).2133 (.77) (.49) (.118).3713 (.271) (.1535) (.63).5946 (.3) (.134) d f (.268) s k (.45) 4 logl( 1 ) logl( 1 ) Ljung-Box Ljung-Box K-S.219 K-S.453 The able repors he resuls of marginal models for CSI3 fuures reurns and bid-ask spreads for he in-sample period from January 27 h, 214 o December 31 s, 214. The condiional mean and variance models of reurns are given in (1), (2) and (3), and he auoregressive duraion model of bid-ask spreads is given in (4) and (5). Sandard errors are given in he brackes on he righ of he parameers., and denoe significance a he 1%, 5% and 1% levels, respecively. logl is he marginal model s log likelihood value. Ljung-Box es examines he one-order serial correlaions in sandardized residuals of marginal models and he null hypohesis is no serial correlaion is deeced. K-S is he Kolmogorov-Smirnov es saisic wih he null hypohesis ha he model is correcly specified. 21

22 Table 3. Copula model esimaes over he in-sample period Gaussian copula-midas copula-midas skewed copula-midas dynamic skewed copula consan skewed copula c (.3472) (1.3382) (.9568) (.216) (.64) (.267) (.64) (.741) (.1183) (.34) (.225) (.123) (.559) v v v (1.311) (.8292) (.616) (.639) (.391) (.199) (.249) K logl AIC BIC The able repors he esimaes of five copulas models for he CSI3 fuures reurn-spread dependence: hree copula-midas models in equaion (1), (2) and (3), he dynamic skewed copula in equaion (6), and he consan skewed copula in equaion (7). The in-sample period is from January 27 h, 214 o December 31 s, 214. Sandard errors are given in he brackes below he parameers., and denoe significance a he 1%, 5% and 1% levels, respecively. logl includes he log likelihoods of boh copula densiy and marginal models. AIC and BIC are he Akaike and Bayesian informaion crieria. 22

23 Figures.5 Panel (a): 5-minue reurns /3 214/5 214/7 214/9 214/11 215/1 215/3 215/5 215/7 215/9 Panel (b): 5-minue bid-ask spreads /3 214/5 214/7 214/9 214/11 215/1 215/3 215/5 215/7 215/9 Noe: The figure plos he 5-minue reurns and bid-ask spreads of CSI3 fuures. The sample period is from January 27 h, 214 o Sepember 2 nd, 215, and we consider he period from 9:3 a.m. o 3: p.m. in each rading day. The daa is aken from CSMAR Soluion. Figure 1. CSI3 fuures mid-price reurns and bid-ask spreads 23

24 .4 Panel (a): skewed copula-midas: oal long-run /2 214/3 214/4 214/5 214/6 214/7 214/8 214/9 214/1 214/11 214/12.2 Panel (b): skewed copula-midas: oal long-run /2 214/3 214/4 214/5 214/6 214/7 214/8 214/9 214/1 214/11 214/12 Noe: The figure plos he exceedance correlaions of CSI3 fuures reurns and bid-ask spreads from January 27 h, 214 o December 31 s, 214 based on he skewed copula-midas model..5 in Panel (a) measures he conemporaneous correlaion of he lowes 5% reurns and he highes 5% bid-ask spreads, and in Panel (b) measures he conemporaneous correlaion of.95 he highes 5% reurns and he lowes 5% bid-ask spreads. The black lines represen he oal exceedance correlaions and he red lines denoe he long-run rends. Figure 2. Exceedance correlaions of CSI3 fuures reurns and bid-ask spreads 24