Testing Stationarity of Futures Hedge Ratios

Transcription

1 Tesing Saionariy of Fuures Hedge Raios Chrisos Floros * Deparmen of Accouning and Finance, Technological Educaional Insiue of Cree, Esavromenos, GR 71004, Heraklion, Cree, Greece and Hellenic Open Universiy, School of Social Sciences, Paras, Greece cfloros@saff.eicree.gr Savros Degiannakis Deparmen of Economics and Regional Developmen, Paneion Universiy of Social and Poliical Sciences, 136 Andrea Syngrou Avenue, GR Ahens, Greece Enrique Salvador Financial Mahemaics and Compuaion Cluser (FMC2), Smurfi School of Business, Universiy College Dublin, Blackrock, Co. Dublin, Ireland Telephone number: / enrique.salvador@ucd.ie; and Universia Jaume I, Finance and Accouning Deparmen Avda. Sos Byna s/n, E Casellon de la Plana, Spain Absrac This paper re-invesigaes he saionariy of sock index fuures hedge raios. We show ha he dynamic hedge raios, calculaed from ime-varying variancecovariance marices, are saionary over ime. However, he examinaion of he evoluion of spo and fuures dynamics, provides evidence ha he hedge raios are beer described as a combinaion of wo differen mean-revering saionary processes which depend on he sae of he marke. Addiionally, we analyse he dynamics of hedge raios a inraday level, which display a complex picure, suggesing ha inraday movemens in he spread beween spo asse and fuures posiion are driven mainly by marke paricipans wih differen perspecives of invesmen horizon. Keywords: Diag-BEKK, Fuures, Hedge Raios, Inra-day Daa, Mulivariae Volailiy Modelling, Regime-Swiching, Saionariy, Uni Roos. JEL: G13; G15; C32; C58. * Corresponding auhor 1

2 1. Inroducion Are fuures hedge raios saionary? This is a highly imporan quesion in he financial lieraure. The saionariy of he hedge raios indicaes a sable relaionship beween spo and fuures prices. Since hedgers seek for reducing he risk of heir invesmens, reliable dynamics of hedge raios are expeced. If no, fuures markes may lose is usefulness o hedgers since he risk diversificaion can be hard o achieve. The propery of saionariy moivaes invesors o use hese sraegies and can be uilised by policy makers o sabilize financial markes. This imporan quesion is also relaed o he performance of variances of spo and fuures reurns and heir covariance. In his paper we pursue he answer o his quesion by analysing he properies of dynamic hedge raios (HRs). There are several echniques available for managing financial risk. One of he mos widely used is hedging wih fuures conracs. A hedge is a spread beween a spo asse and a fuures posiion ha reduces risk 1. A considerable amoun of research has focused on modelling he disribuion of spo and fuures prices and applies he resuls o esimae he opimal hedge raio using various ype of models such as OLS, GARCH, ECM and VECM models (see Chen e al., 2003; Floros and Vougas, 2004; Salvador and Arago, 2014). The hedge raio is defined as he number of fuures conracs bough or sold divided by he number of spo conracs whose risk is being hedged. Several sudies have invesigaed he opimal hedge raio using sock index fuures under boh a consan (saic) and a ime-varying (dynamic) seing. To esimae he opimal hedge raios, early works used he slope of an OLS regression of he spo on he fuures reurns, while an improvemen has been made by adoping a bivariae GARCH framework (see Park and Swizer, 1995). This las approach has become one of he mos popular echniques since i allows he modelling of he empirical characerisics of he spo-fuures disribuion. Alhough mos of he previous sudies are successful in capuring he imevarying covariance-variances, almos all of hem focus only on he esimaion of he 1 In his paper we follow he radiional view of hedging, i.e. risk minimizaion. There are oher alernaive HRs, e.g. auhors use oher objecives such as (i) HR based on raes of reurns siuaions where spo is fixed, (ii) HR for he case when rader wishes o maximize he raio of he expeced reurn on he hedged porfolio o is variance, or (iii) when here is marking o marke and sochasic ineres raes ec. These alernaives HRs involve boh risk and reurn, bu hey are generally more complicaed han he radiional minimisaion of risk, and hence hey are no considered in mos empirical sudies. 2

3 hedge raios. The main purpose of his paper is o furher examine and undersand he saionariy of hedge raios over ime, as he lieraure provides limied informaion abou i 2. Previous sudies such as Ederingon (1979) and Anderson and Danhine (1981) assume ha he opimal hedge raio is consan when i can be obained as a slope coefficien of an OLS regression. When he opimal hedge raios depend on he condiional disribuions of spo and fuures price movemens, hen he hedge raios vary over ime as his disribuion changes. Previous sudies show he variabiliy of he hedge raios over ime, and suppor he hypohesis ha he opimal hedge raios of commodiies are ime-varying and non-saionary (see Baillie and Myers, 1991). They repor ha he hedge raios conain a uni roo and herefore behave much like a random walk. Grammaikos and Saunders (1983) were he firs o examine he sabiliy of hedge raios. They concluded ha he hedge raio sabiliy (saionariy) in currencies could no be rejeced. Furhermore, Malliaris and Urruia (1991) examined he random walk hypohesis and concluded ha he hedge raios of he seleced indices and currencies follow a random walk. However, Ferguson and Leisikow (1998) repor ha fuures hedge raios are saionary using a simple OLS regression approach. They argue ha he hedge raios in previous sudies follow a random walk due o a small sample size of daa and hedge raio calculaion overlap. Furhermore, Lien e al. (2002) rejec he null hypohesis ha he opimal GARCH hedge raios have a uni roo. Recenly, Lai and Sheu (2010) propose a new class of mulivariae volailiy models encompassing realized volailiy (RV) esimaes o obain he riskminimizing hedge raios. Their resuls show ha hedging improvemen is subsanial when swiching from daily o inraday frequencies. They also repor ha he ADF es on he RV-based hedge raios (inraday) is rejeced, excep for he resuls based on he (daily) OLS and he ECT-GARCH-CCC models for pos-crisis period of The conribuion of his aricle is o examine wheher he ime-varying hedge raios calculaed from a se of European sock indices (German DAX30, Briish FTSE100, French CAC40 and Spanish IBEX35) are saionary over ime. Tesing for a uni roo in fuures and spo prices is ricky due o he propensiy of such prices o jumps (Alexander, 2008). In order o overcome he predicamen due o jumps, we 2 There is o dae no definie conclusion concerning he saionariy of he dynamic HRs, which may be used o improve hedging performance. 3

4 invesigae he hedge raios saionariy in low volaile periods and highly volaile periods based on he Regime-Swiching Augmened Dickey Fuller (RS-ADF) uni roo es inroduced by Kanas and Genius (2005). The paper provides empirical evidence ha he ime-varying hedge raios are saionary over ime. Thus, we confirm he sable relaionship beween fuures and spo reurns across ime. However, if we ake a closer look a he evoluion of spo and fuures dynamics, we find ou ha he hedge raios are beer described as a combinaion of wo differen mean-revering saionary processes which depend on he sae of he marke. Alhough correlaions follow a sable saionary process in boh saes, during periods of financial urmoil he correlaions beween spo and fuures are differen han during calm periods. This resul sheds ligh on he conroversy caused by he evidence of greaer hedging effeciveness using saic hedge raios han using simple dynamics ones, and why here have been several recen papers which boh heoreically (Lien, 2010) and empirically (Alizadeh and Nomikos, 2008; Salvador and Arago, 2014) showed a greaer effeciveness of regime-swiching models. The inuiion is ha omiing he regime-swiching specificaion leads o inefficien hedges compared no only o he ones considering his sae-dependence bu also o he saic ones. Moving one sep beyond, we furher analyse he dynamics of opimal hedge raios a inraday level (we exend he sudy by Lai and Sheu, 2010). Since execuing an inraday hedging sraegy would be very expensive, we focus in providing new insighs abou he dynamics of he spo and fuures markes a ulra-high frequency. The resuls display a complex picure on he dynamics a ulra-high frequency, suggesing ha inraday movemens in he spread beween spo asse and fuures posiion are driven mainly by marke paricipans wih differen perspecives of invesmen horizon. This is because here is very lile hedging a shor horizons (speculaive acions are more han he invesmen acions), bu a long horizons (speculaive acions are less han he invesmen acions) here is more presence of hedging sraegies. The res of he paper is organised as follows. Secion 2 provides a descripion of he daabase. Secion 3 develops he model used o obain he dynamic hedge raios. Secion 4 analyses he ime-series saionariy of he esimaed hedge raios no only from a sandard perspecive bu also from he regime-swiching framework. In 4

5 secion 5 we ake a closer look a he saionariy properies of hedge raios using inraday daa and secion 6 concludes. 2. Daa Descripion The daase is comprised by daily daa (spo and fuures closing prices) from he main sock indices and is corresponding fuures conracs in Germany (DAX30), France (CAC40), Unied Kingdom (FTSE100) and Spain (IBEX35). The ime horizon includes observaions from May o November Wihin his sample period we have wo differen conexs, i.e. before (under several years of sabiliy and susained growh) and afer he global financial crisis and he Eurozone deb problems sared in The sock markes analysed are he mos raded European financial markes and all of hem are raded on an elecronic rading sysem. The ime-series for he indices and heir near-ime delivery (nearby) fuures conrac 4 are provided by Daasream. [Inser Table 1 abou here] Table 1 presens he saisical properies of he price and reurns series. The reurns of spo and fuures prices follow all sylized facs of financial ime series such as lepokurosis, volailiy clusering, leverage effecs, ec. (see Bollerslev e al., 1994). Furher, he log of prices are found o be I(1), i.e. he series are non-saionary, hus we model he log-reurns for he ime-series analysis 5. We esimae ime-varying hedge raios using GARCH models which are very popular in he lieraure o capure he sylized facs of financial ime series (see, for example, Degiannakis and Floros, 2010). In he nex secion we develop he empirical models o obain dynamic hedge raios and we describe heir paerns. 3. Esimaing Time-Varying Hedge Raios 3.1. Mehodology 3 Since May of 2000 daa is available for all he examined indices. 4 Carchano and Pardo (2008) show ha rolling over he fuures series has no significan impac on he resulan series. Therefore, he leas complex mehod can be used for he consrucion of he series o reach he same conclusions. 5 These resuls are available upon reques. 5

6 According o Lee (1999), given he ime-varying naure of he covariance in financial markes, he OLS assumpion is inappropriae when esimaing opimal hedge raios. In he GARCH model 6 he condiional variance of a ime series depends upon he squared residuals of he process (Bollerslev, 1986). I also capures he endency for volailiy clusering in financial daa, and uilises he informaion in one marke own hisory (univariae GARCH) or uses informaion from more han one marke hisory (mulivariae GARCH). According o Conrad e al. (1991), mulivariae GARCH models provide more precise esimaes of he parameers because hey uilise informaion in he enire variance-covariance marix of he errors and allow he variance and covariance depend on he informaion se in a vecor of he ARMA manner (Engle and Kroner, 1995). In he more radiional hedge raio esimaion mehodology, he covariance marix of spo and fuures prices (and herefore he hedge raio) is consan hrough ime. However, a large body of research has applied he GARCH framework o infer ime-varying hedge raios (Cecchei e al., 1988; Kroner and Sulan, 1993; Park and Swizer, 1995). Alhough GARCH models are useful for esimaing ime-varying opimal hedge raios, a ime-varying covariance marix of spo and fuures prices is no sufficien o esablish ha he opimal hedge raio is ime-varying 7. In his sudy we use a bivariae model wih GARCH errors, he Diag- BEKK(p,q) model, o esimae he dynamic variance-covariance marix of spo and fuures log-reurns. The Diag-BEKK(p,q) framework of log-spo (s) and log-fuures (f) is esimaed in he form 2 σ s, = σ sf, y σ = ε ( 1 L) log( s ) ( 1 L) log( f ) ε ε s, f, = C0C + sf, H 2 0 σ f, Ψ 1 a = b ~ N ε ε ( 0, H ) s, f, q p ( Aε ) + ( i iε iai B jh jb j ) i= 1 where Ψ 1 is he informaion a ime 1 and he variance-covariance marix specificaion, H, is he BEKK model of Baba e al. (1990). The marices j= 1, (1) A i and B j 6 The advanage of he GARCH specificaion is ha i is a model ha allows for lepokurosis in he disribuions of price changes. 7 Consancy of he HR refers o he raio of he covariance (beween he spo and fuures price) o he variance of he fuures price, which is consan (Moschini and Myers, 2002). 6

7 are resriced o be diagonal. The Diag-BEKK(p,q) model is guaraneed o be posiive definie and requires he esimaion of fewer parameers compared o oher mulivariae models; i.e. Diag-VECH, BEKK, ec. This mulivariae specificaion allows us obain ime-varying hedge raios hrough he condiional covariance marix HR σ =, (2) σ sf, 2 f, where he dynamic hedge raios are compued as he quoien beween he condiional spo-fuures covariance and he fuures variance Empirical Resuls The esimaion of he model is carried ou using condiional quasi maximum likelihood esimaion 8. The p and q lag orders have been seleced according o he Schwarz s (1978) Bayesian crierion. The resuls from he Diag-BEKK(1,1) model (eq.1) are presened in Table 2. The coefficiens are all saisically significan and imply volailiy clusering. Boh spo and fuures log-reurns exhibi srong persisence in volailiy bu i is he fuures marke which shows he sronges persisence. [Inser Table 2 abou here] Figure 1 shows he esimaed variances over ime for he DAX30, FTSE100, CAC40 and IBEX35 spo and fuures indices. We observe several peaks in he volailiy measures common o all markes; e.g. around 2003, in laes 2008 coinciding wih global financial crisis, and one covering end beginnings 2012 wih he wors par of he Eurozone deb problems which refleced in he sock markes. Also in Spain here is a peak during he beginnings of 2013 showing furher problems wih he sabiliy of ha marke. [Inser Figure 1 abou here] Figure 2 shows he plo of ime-varying hedge raios obained using eq.2. The DAX hedge raios are quie volaile during he firs par of he sample bu hey seem o sabilise afer Despie he eviden peaks in volailiies in all counries, he hedge raios follow a smooh paern along he sample period where hey seem o 8 The condiional log-likelihood funcion for a single observaion can be wrien as L ( θ ) = ( n / 2) log(2π ) ( 1/ 2) log( H ( ) ) (1/ 2) ( )' 1 θ ε θ H ( θ ) ε ( θ ), where θ represens a vecor of parameers and n is he sample size (for more deails see Xekalaki and Degiannakis, 2010). 7

8 reurn always o a predeermined value. As, from visual descripion of he hedge raios we canno infer abou heir saionariy, nex secion provides a formal sudy of he hedge raio saionariy and he implicaions for opimal hedging. [Inser Figure 2 abou here] 4. Analysing he (Non) Saionariy of he Hedge Raios 4.1. Uni Roo Theory The Augmened Dickey Fuller (ADF) es assumes ha he an AR(p) process where y y = ay + x δ + β y β y + u y series follows ' p p, (3) 2 defines he firs difference of hedge raios, and N( 0, ) H : a 0 and H : a 0. 0 = 1 < u σ, wih ~ u Phillips and Perron (1998) propose a nonparameric mehod o conrol for serial correlaion when esing for a uni roo (his es is popular in he analysis of financial ime series). The PP es esimaes he es equaion y = ay δ + + x ' 1 u, and modifies he -raio of he a coefficien; hence, he serial correlaion does no affec he asympoic diribuion of he es saisic Regime-Swiching ADF Tes Recen lieraure has quesioned he asympoic power and saisical properies of radiional ADF ess; e.g. Chorareas e al. (2002) and Solis e al. (2002). In his paper we are ineresed in he saionariy properies of hedge raios condiioned o volailiy levels in he markes (low and high volailiy), i.e. if he hedge raios are (non)saionary wihin high and low volailiy periods independenly of which is is saionariy in he long-run (assuming a single regime in he long-run). This can be done by applying he mehodology developed by Kanas and Genius (2005). They exend he ADF regression by allowing boh he auoregressive parameers and he volailiy of he hedge raios o change over ime following a firs order Markov process. Hence, he regime-swiching ADF, or RS-ADF, specificaion 9 The es correcs for any serial correlaion and heeroskedasiciy in he errors u of he es regression. 8

9 es for he (non)saionariy of hedge raios under differen saes of volailiy is defined as: y = a p 2 0, s + ak s y k + bs y + u, 1, ~ N( 0, s ) k= 1 u σ, (4) where a,..., a,, b 0, s k s s are regime-swiching parameers, s is he unobservable regime, and u are normal innovaions wih sae-dependen variances Empirical Resuls Table 3 shows he resuls from he ADF and PP ess applied o he esimaed hedge raios under hree cases: i) a simple AR(p) process, ii) a consan rend and iii) a ime saionary rend. Tess when considering a specific rend show ha he hedge raio esimaed from he Diag-BEKK model considering he reurns of spo and fuures prices are saionary, or I(0). This does no hold, however, when we do no specify a rend in he daa which shows is imporance when esing for saionariy in hedge raios. Our resuls are in line wih previous papers such as Ferguson and Leisikow, 1998 and Lien e al who also found ha ime-varying hedge raios are saionary over ime. [Inser Table 3 abou here] The implicaion of his resul is ha opimal hedges on sock indices end o flucuae around a mean-revering value. This sable relaionship beween he correlaions of spo and fuures markes can be exploied by hedgers o reduce he risk of heir invesmens. However, his adds furher conroversy on he debae abou he superioriy or no of dynamic hedge raios agains saic sraegies for minimising he risk of a hedged porfolio. Several auhors found ha more complex models do no provide a beer performance han simple saic ones (Lien e al., 2002; Coer and Hanley, 2012). The variabiliy of he ime-varying hedge raios around his mean is wha may cause a worse performance of his kind of dynamic models compared o he saic ones. Neverheless, his resul of saionariy in he hedge raios can be viewed as good news, since i implies a reliable relaionship beween he spo and fuures prices and a confirmaion ha fuures markes are useful for hedgers. 10 The model is esimaed by he maximum likelihood mehod using an algorihm where ex-ane and filered probabiliies are inferred in firs place and hen based on hem sandard maximisaion of he likelihood funcion is performed (see Hamilon, 1994; Floros and Salvador, 2014). 9

10 Besides his firs analysis, we also examine he saionariy of hedge raios by looking a low and high volaile periods. The advanage of our approach is ha we do no need o assume which periods correspond o low/high volailiy saes bu i is he esimaion procedure iself which makes his classificaion. Table 4 shows he esimaions of he RS-ADF model presened in eq.4. We observe ha mos of he coefficiens represening a consan drif in he ime-series are saisically significan, bu if we look a he auoregressive coefficien jus a few of hem presen significance. The mos relevan coefficien in Table 4 is b s which represens he exisence or no of a uni roo in he sae-dependen process. Some resuls are noeworhy. [Inser Table 4 abou here] Firs, in boh saes he coefficiens b s are negaive and significan which implies saionariy wihin each sae-dependen process. This confirms he resuls of saionariy (Francq and Zakoian, 2001; Timmerman, 2000; Yang, 2000) on imevarying hedge raios previously obained, bu is inerpreaion is differen. Here we have wo differen mean-revering processes, one when he process is in low-volailiy periods, and anoher one when he process is in high-volailiy periods. Wihin each sae he hedge raios end o flucuae around differen values insead of jus one common value independen of he sae. Figure 3 shows he probabiliy of being in a sae of low volailiy and complemens Figure 2 which shows in shaded areas he observaions ha correspond o high volailiy periods when compared o he esimaed hedge raios. [Inser Figure 3 abou here] The hedge raios process changes among regimes. The hedge raios wihin each regime are saionary bu he dynamics of he correlaion in he differen regimes are no he same. Thus, if we are ineresed in shorer horizons hedges he omission in considering differen saes can be a cause of a worse hedging performance. In fac, his resul sheds ligh o very recen evidence which shows boh heoreically and empirically ha hedge raios obained from regime swiching models ouperform he res of sraegies (boh saic and dynamic). Lien (2010) characerizes condiions under which he regime-swiching hedge sraegy performs beer han he OLS hedge sraegy and where he GARCH effecs prevail. These condiions would 10

11 allow he RS-GARCH hedge sraegy o dominae boh OLS and GARCH hedge sraegies. Recenly, Alizadeh and Nomikos (2008) for commodiies and Salvador and Arago (2014) for sock indices repor a greaer performance of regime-swiching sraegies han hose obained hrough single-regime models. Our resuls abou his sae-dependen saionariy of hedge raios suppor his previous evidence. When analysing he performance of hedging sraegies we usually look a shorer horizons and we end o follow he false dynamics. So, no considering he swiching of HRs' regimes causes a worse hedging effeciveness. [Inser Table 5 abou here] In Table 5 we repea he esimaions of he RS-ADF model, bu in his case we do no consider a drif in he model. Here we obain a surprising resul. The coefficien b s in he low volailiy sae is negaive and significan providing evidence of saionariy of hedge raios during his low volailiy sae. However, if we look a high volailiy saes i seems ha he process followed by opimal hedge raios is nonsaionary. This resul highlighs he imporance of modelling properly he rend of he ime-series (similar resuls when using sandard uni-roo ess) since is wrongspecificaion could lead o wrong conclusions abou he saionariy of hedge raios. 5. Hedge Raio Saionariy for Inraday Daa Dynamic hedging is usually expensive o implemen since i involves ransacion coss any ime he hedged porfolio is re-balanced. Therefore, hedging is more raional a low frequencies. However, if he hedging is conduced only by invesors, he hedge dynamics will no differ across differen sampling frequencies. On he oher hand, if he hedging is conduced by invesors and raders (i.e. swap rading beween fuures and spo for speculaion), hen he hedge dynamics will differ across differen sampling frequencies. In his secion, we ry o unmask his hypohesis by looking a he saionariy paerns of inraday hedge raios boh from sandard and regime-swiching echniques. Given he coss associaed wih a hedging sraegy a inraday level we do no associae any of hese resuls wih hedging effeciveness. The daase is comprised by hourly observaions of he DAX index and is corresponding fuure conrac from 3 rd of January, 2000 o 30 h of December,

12 (25138 observaions) 11. As in he previous daases, we firs compue he dynamic hedge raios based on eqs.1 and 2. A plo of he esimaed inraday hedge raios is displayed in Figure 4. The hedge raios seem o follow a smooh paern alhough i is no possible o draw any conclusion abou is saionariy from his figure. Therefore, we run he sandard and regime-swiching saionariy ess o provide new insighs. [Inser Figure 4 abou here] Panel A in Table 6 displays he sandard uni-roo ess for he German inraday hedge raios. Similar o he resuls above, we rejec he null hypohesis of a uni roo in he inraday hedge raio series. However, if we consider he regimeswiching approach and disinguish beween high and low volailiy regimes (panel B Table 6) we canno rejec he uni roo in any of he regimes. [Inser Table 6 abou here] This resul draws a complex picure for he disribuions of spo and fuures reurns a ulra-high frequency. Alhough when looking a longer horizons he spofuure correlaions seem o follow a saionary process, when looking a inra-day horizons he dynamics of he spreads beween hese wo markes follow unpredicable dynamics. Taken his resul ogeher wih he ones repored in previous secions, we conclude ha he dynamics of hedge raios vary across differen sampling frequencies. Given hese resuls and according o our hypohesis, he agens driving he spread of hese markes a inraday level are mainly speculaors. In oher words, our resuls suppor ha, a he ulra-high frequency, invesors who hedge heir sraegies are dominaed by speculaors. This is due o he fac ha marke paricipans have differen perspecives of heir invesmen horizon. On he one hand, we have he invesors who prevail a he daily frequency, and on he oher hand, we have he speculaors who prevail a he inra-day frequency. The implied ransacion cos when rebalancing he opimal hedge posiion can be he reason o discourage he hedgers o operae a his ulra-high frequency. Also, he unsable dynamics followed by he correlaions of spo and fuures markes a his ulra-high frequency can make difficul for hedgers o achieve he desired risk reducion in heir invesmens. On he oher hand, day rading or speculaion in securiies is conduced no only by financial firms and professional speculaors (i.e. equiy invesmen and fund 11 The hourly sampling frequency has been seleced in order o minimize he effec of microsrucure noise, see Degiannakis and Floros (2013). 12

13 managemen specialiss) bu, hanks o elecronic rading and margin rading, i has become increasingly popular among a-home raders as well 12. This is increasingly giving o his kind of marke paricipans a very imporan role when defining he dynamics of he spo-fuures markes a his ulra-high frequency. Our resuls are in line wih Tse and Williams (2013) who suppor ha any fuure effors sudying speculaion in he fuures markes mus be done using high frequency inraday daa. 6. Conclusion Saic and dynamic models of various forms have been well acceped o calculae hedge raios. However, here is o dae no definie conclusion concerning he saionariy of he dynamic hedge raios. We focus on he characerisics of opimal hedge raios for he DAX30 (Germany), FTSE100 (UK), CAC40 (France), and IBEX35 (Spain) indices over he period We examine he saionariy of hedge raios by employing sandard economeric mehods of uni roo ess and a new sae-dependen approach following he RS-ADF es. Dynamic hedge raios are esimaed by a bivariae GARCH-ype model. We find ha dynamic hedge raios are saionary over ime when he enire sample is considered. This resul implies a sable relaionship in he spo-fuures correlaions ha can be used for hedgers o reduce he risk in heir invesmens. However, when we consider shorer horizons and disinguish beween volailiy saes (i.e. high and low volaile periods), we show ha he dynamic hedge raios follow differen saionary processes during periods of calm and periods of financial urmoil. These resuls suppor evidence from previous sudies which repor a greaer hedging performance of dynamic sraegies using regime-swiching models. The differen processes followed by he hedge raios for volaile periods are associaed wih changes in he variances and he covariance beween spo and fuure reurns. This has imporan implicaions for hedgers. Firs, financial analyss and hedgers mus deermine he effec of his unexpeced change in he risk on heir posiion. Second, hey should deermine he facors causing his shifed saionariy. 12 Speculaors are more acive a inraday level since hey profi from heir inraday invesmens in informaion. Moreover, a speculaive aciviy is imporan for inraday markes. Speculaors make markes more liquid and efficien, while hey benefi from he high price volailiy. We argue ha wihou speculaion a inraday level, markes would be less complee in ha here would be fewer opporuniies for oher marke paricipans. 13

14 The resuls for he dynamic hedge raios a inraday level draw a complex picure suggesing ha he spreads are mainly driven by shor-erm marke paricipans. We argue ha we have invesors who prevail a he daily frequency and speculaors who prevail a he inra-day frequency in he spo-fuures sock markes. References Alexander, C Pracical Financial Economerics. John Wiley & Sons Ld. Anderson, R. W., & Danhine, J. P Cross hedging. Journal of Poliical Economy, 89, Baillie, R. T., & Myers, R. J Bivariae GARCH esimaion of he opimal commodiy fuures hedge. Journal of Applied Economerics, 6, Bollerslev, T Generalised Auoregressive Condiional Heeroscedasicy. Journal of Economerics, 33, Bollerslev, T., Engle, R.F. & Nelson, D.B ARCH models. in R.F. Engle, D. McFadden, Handbook of economerics, Vol. 4 Elsevier Science B.V, Amserdam. Cecchei, S. G., Cumby, R. E., & Figlewski, S Esimaion of opimal fuures hedge. Review of Economics and Saisics, 70, Chen, S.-S., Lee, C.-F., & Shresha, K Fuures hedge raios: A review. The Quarerly Review of Economics and Finance, 43, Conrad, J., Kaul, G., & Nimalendran, M Asymmeric predicabiliy of condiional variances. Review of Financial Sudies, 4(4), Coer, J. & Hanley, J Hedging effeciveness under condiions of asymmery. The European Journal of Finance, 18(2), Degiannakis, S., & Floros, C Hedge Raios in Souh African Sock Index Fuures. Journal of Emerging Marke Finance, 9(3), Degiannakis, S., & Floros, C Modeling CAC40 volailiy using ulra-high frequency daa. Research in Inernaional Business and Finance, 28, Ederingon, L The hedging performance of he new Fuures markes. Journal of Finance, 34, Ellio, G., Rohenberg, T. J., & Sock, J. H Efficien Tess for an Auoregressive Uni Roo. Economerica, 64, Engle, R. F., & Kroner, K. F Mulivariae simulaneous generalized ARCH. Economeric Theory, 11,

15 Ferguson, R., & Leisikow, D Are regression approach fuures hedge raios saionary?. Journal of Fuures Markes, 18(7), Floros, C., & Vougas, D. V., Hedge raios in Greek Sock Index Fuures Markes. Applied Financial Economics, 14(15), Floros, C., & Salvador, E Calendar anomalies in cash and sock index fuures: Inernaional Evidence. Economic Modelling, 37, Francq, C., & Zakoïan J.M Saionariy of Mulivariae Markov-swiching ARMA Models. Journal of Economerics, 102, Grammaikos, T., & Saunders, A Sabiliy and he Hedging Performance of Foreign Currency Fuures. Journal of Fuures Markes, 3, Hafner, R., & Wallmeier, M Volailiy as an Asse Class: European Evidence. European Journal of Finance, 13(7-8), Kroner, K. F., & Sulan, J Time varying disribuion and dynamic hedging wih foreign currency fuures. Journal of Financial and Quaniaive Analysis, 28, Lai, Y-S. & Sheu, H-J The incremenal value of a fuures hedge using realized volailiy. Journal of Fuures Markes 30:9, Lee, G.G.J Conemporary and Long-Run Correlaions: A Covariance Componen Model and Sudies on he S&P 500 Cash and Fuures Markes. Journal of Fuures Markes, 19(8), Levin, A., Lin, F., & Chu, C Uni roo ess in panel daa: asympoic and finie-sample properies. Journal of Economerics, 108, Lien, D A Noe on he Relaionship Beween he Variabiliy of he Hedge Raio and Hedging Performance. Journal of Fuures Markes, 30(11), Lien, D., Tse, Y.K., & Tsui, A Evaluaing hedging performance of he consan-correlaion GARCH model. Applied Financial Economics, 12, Lien, D., & Yang, L Alernaive selemen mehods and Ausralian individual share fuures conracs. Journal of Inernaional Financial Markes, Insiuions and Money, 14(5), Malliaris, A. G., & Urruia, J Tess of random walk of hedge raios and measures of hedging effeciveness for sock indices and foreign currencies. Journal of Fuures Markes, 11,

16 Masse, P., & Wallmeier, M A High-Frequency Invesigaion of he Ineracion beween Volailiy and DAX Reurns. European Financial Managemen, 16(3), Moschini, G. C., & Myers, R. J Tesing for consan hedge raios in commodiy markes: a mulivariae GARCH approach. Journal of Empirical Finance, 9, Park, T. H., & Swizer, L. N Time-varying disribuions and he opimal hedge raios for sock index fuures. Applied Financial Economics, 5, Phillips, P.C.B., & Perron, P Tesing for Uni Roos in Time Series Regression. Biomerika, 75, Salvador, E. & Arago, V Measuring hedging effeciveness of index fuures conracs: Do dynamic models ouperform saic models? A Regime-Swiching approach. Journal of Fuures Markes, 34(4), Schwarz, G Esimaing he Dimension of a Model. Annals of Saisics, 6, Timmermann, A., Momens of Markov swiching models. Journal of Economerics, 96(1), Tse, Y. & Williams M. R Does Index Speculaion Impac Commodiy Prices? An Inraday Analysis. The Financial Review 48, Wang, P. (2003). Financial Economerics. Rouledge. Xekalaki, E. & Degiannakis, S ARCH Models for Financial Applicaions. John Wiley and Sons, New York. Yang, M., Some properies of vecor auoregressive processes wih Markovswiching coefficiens. Economeric Theory 16,

17 Tables Table 1.Summary saisics for prices and log-reurns of spo and fuures on he seleced European indices Panel A.- Summary saisics for log-reurns Germany Unied Kingdom France Spain Spo Fuures Spo Fuures Spo Fuures Spo Fuures Mean 2.37 e e e e e e e e-04 Sandard deviaion Minimum Maximum Skewness Kurosis (excess) JB es Panel B- Saionariy es for log-reurns Germany Unied Kingdom France Spain Germany Spo Fuures Spo Fuures Spo Fuures Spo Fuures Dickey- Fuller *** *** *** *** *** *** *** *** Phillips- Perron *** *** *** *** *** *** *** *** Panel C.- Summary saisics prices Germany Unied Kingdom France Spain Spo Fuures Spo Fuures Spo Fuures Spo Fuures Mean Sandard deviaion Minimum Maximum Skewness Kurosis (levels) JB es Panel D- Saionariy es prices Germany Unied Kingdom France Spain Germany Spo Fuures Spo Fuures Spo Fuures Spo Fuures Dickey- Fuller Phillips Perron The Table shows summary saisics and saionariy ess for prices ( f ) [( 1 L) log( s ), ( 1 L) log( )] f s, and reurns of he 4 European sock indices (German DAX30, he Briish FTSE100, he French CAC40 and he Spanish IBEX35) in he spo and fuures marke. Panels A and C show he descripive saisics and he Jarque-Bera normaliy es for spo and fuures markes, respecively. Panels B and D show he saionary ess on he price and reurns series, respecively ( ***, ** and * represens rejecion of he null hypohesis a 1%, 5% and 10% levels of significance, respecively). 17

18 Table 2. Parameers esimaions of he Diag-BEKK(1,1) model. DAX30 FTSE100 CAC40 IBEX a *** ** *** *** 0 (0.0185) (0.0153) (0.0125) (0.0188) b *** ** *** *** 0 (0.0181) (0.0153) (0.0127) (0.0192) c *** ** *** *** 11 (0.0254) (0.0376) (0.0367) (0.0108) c *** *** *** *** 12 (0.0271) (0.0307) (0.0450) (0.0123) c *** *** *** *** 22 (0.0086) (0.0056) (0.0141) (0.0027) a *** *** *** *** 11 (0.0320) (0.0377) (0.0201) (0.0061) a *** *** *** *** 22 (0.0427) (0.0292) (0.0228) (0.0070) b *** *** *** *** 11 (0.0095) (0.0123) (0.0096) (0.0014) b *** *** *** *** 22 (0.0130) (0.0094) (0.0126) (0.0019) The Table shows he esimaed parameers for he model in eq.1 for he logreurns on he spo and fuures markes for he DAX30, FTSE100, CAC40 and IBEX35 indices. Sandard errors are compued using Bollerslev-Wooldridge (1992) specificaion correcing for heeroskedasiciy ( ***, ** and * represens saisical significance a 1%, 5% and 10% levels of significance, respecively). 18

19 Table 3. Uni-roo ess for HRs series Panel A. AR process H 0 : y = y + u 1 : y = ay u, where a < H + DAX30 FTSE100 CAC40 IBEX35 Saisic ADF es Criical Value Resul Canno rejec Canno rejec Canno rejec Canno rejec Saisic PP es Criical Value Resul Canno rejec Canno rejec Canno rejec Canno rejec Panel B. AR wih drif H y y + u 0 : = 1 : y = c + ay u, where < H + a and drif coefficien c DAX30 FTSE100 CAC40 IBEX35 Saisic ADF es Criical Value Resul Rejec Rejec Rejec Rejec Saisic PP es Criical Value Panel C. Trend-saionary Resul Rejec Rejec Rejec Rejec H 0 : y = y + u 1 : y = c + d + ay u, where a < H +, drif coefficien and deerminisic coefficien d DAX30 FTSE100 CAC40 IBEX35 Saisic ADF es Criical Value Resul Rejec Rejec Rejec Rejec Saisic PP es Criical Value Resul Rejec Rejec Rejec Rejec The Table shows he ADF and PP ess on he esimaed HRs using eq.2 for he spos and fuures reurns on he DAX30, FTSE100, CAC40 and IBEX35 indices (sample period: May November 2013). Each panel shows a variaion of he es in erms of he drif coefficien considered. 19

20 Table 4. RS-ADF es wih drif p 2 y = a0, s + ak s y k + bs y + u, 1, u ~ N( 0, σ s ), k= 1 Hedge raios Parameers Sae Germany UK France Spain b S = *** *** *** *** s ( ) ( ) ( ) ( ) S = *** ( ) *** ( ) *** ( ) *** ( ) a S = *** *** *** *** 0,s (5.5581) (4.6624) (5.5059) (6.5421) S = *** ( ) (0.0162) *** (6.2814) *** (6.5505) a S = * ** * ,s (1.6749) (2.1778) (1.8681) ( ) S = ( ) ( ) ( ) (0.1492) a S = ** ,s ( ) ( ) (2.0320) (0.0368) S = (0.5623) (0.2931) ( ) (0.1703) 2 S σ = *** 4.28 e-04 *** 9.02 e-04 ** 3.52 e-04 *** s (4.6984) (4.0308) (2.4834) (6.6941) S = *** (5.0392) 1.27e-05 *** (5.4779) 2.42 e-05 *** (4.4565) 1.13 e-05 *** (9.1923) P *** ( ) *** (8.5676) *** (4.0981) *** (9.3152) Q *** ( ) *** ( ) *** ( ) *** ( ) The Table shows he esimaed parameers for he RS-ADF es presened in eq.4. Dependen variables in each column represen he esimaed HRs using eq.2 for he spos and fuures reurns on he DAX30, FTSE100, CAC40 and IBEX35 indices (sample period May 2000-November 2013). Sandard errors are compued using Bollerslev-Wooldridge (1992) specificaion correcing for heeroskedasiciy ( ***, ** and * represens saisical significance a 1%, 5% and 10% levels of significance, respecively). 20

21 Table 5. RS-ADF es wih no drif p 2 y = bs y + ak s y k + u 1,, u ~ N( 0, σ s ), k = 1 Hedge raios Parameers Sae Germany UK France Spain b S = *** 9.02 e-04 *** *** *** s (3.9945) (8.4932) (7.1746) (7.0819) S = *** ( ) *** ( ) *** ( ) *** ( ) a S = ** *** 1,s ( ) (2.0078) (0.2262) ( ) S = ** ( ) ( ) ( ) ( ) a S = ** ,s ( ) (-1.069) ( ) ( ) S = ( ) ( ) ( ) ( ) 2 S σ = *** 4.64 e-04 *** 9.33 e-04 *** 3.74 e-04 *** s (4.9637) (3.7786) (2.8121) (6.9448) S = e-05 *** (5.7429) 1.37e-05 *** (5.1356) *** (4.8599) 1.16e-05 *** (9.3970) P *** ( ) *** (8.1832) *** (5.0682) *** (9.2134) Q *** ( ) *** ( ) *** ( ) *** ( ) The Table shows he esimaed parameers for he RS-ADF es presened in eq.4 bu omiing he drif componen. Dependen variables in each column represen he esimaed HRs using eq.2 for he spos and fuures reurns on he DAX30, FTSE100, CAC40 and IBEX35 indices (sample period May November 2013). Sandard errors have been correced for heeroskedasiciy ( ***, ** and * represens saisical significance a 1%, 5% and 10% levels of significance, respecively). 21

22 Table 6. RS-ADF es wih drif ADF es PP es Panel A. - Sandard uni-roo ess Hedge raios (inraday daa): Germany AR AR wih drif Trend saionary Saisic Criical Value Resul Canno rejec Rejec Rejec Saisic Criical Value Resul Canno rejec Rejec Rejec Parameers Panel B. - RS-ADF Tes Hedge raios (inraday daa) Germany S =1 S =2 b s (0.0180) (2.4421) a ,s (0.0193) (2.4855) a ,s (0.0247) (2.1608) a ,s (0.0213) (1.5695) a ,s (0.0252) (5.4937) 2 σ 3.84e-05 *** 2.85 e-04 *** s (1.02e-05) (8.86e-05) P *** (0.0020) Q *** (0.0030) Panel A shows he saisics for he ADF and he PP ess on he esimaed HRs using eq.2 for he spos and fuures reurns on he hourly DAX30 index. Each panel shows a variaion of he es in erms of he drif coefficien considered. Panel B shows he esimaed parameers for he RS-ADF es presened in eq.4 (sample period January 2000-Decemeber 2010). Sandard errors have been correced for heeroskedasiciy ( ***, ** and * represens saisical significance a 1%, 5% and 10% levels of significance, respecively). 22

23 Figure 1. Condiional variances Figures 2 This Figure plos he condiional spo ( σ ) (black line) and fuures ( σ ) variances (green line) for he 2 s, log-reurns of he DAX30, FTSE100, CAC40 and IBEX35 indices (sample period May November 2013). Figure 2. Hedge raios f, This Figure plos he esimaed HRs according o eq.2 for he spo and fuures sock indices in Germany, Unied Kingdom, France and Spain. Shaded areas correspond o periods of high volailiy based on he filered probabiliies of eq.4. 23

24 Figure 3. Filered probabiliies for low volailiy saes 1 GERMANY 1 UNITED KINGDOM 0,9 0,9 0,8 0,8 0,7 0,7 0,6 0,6 0,5 0,5 0,4 0,4 0,3 0,3 0,2 0,2 0,1 0,1 0 0 May-00 Jan-01 Sep-01 May-02 Jan-03 Sep-03 May-04 Jan-05 Sep-05 May-06 Jan-07 Sep-07 May-08 Jan-09 Sep-09 May-10 Jan-11 Sep-11 May-12 Jan-13 Sep-13 May-00 Jan-01 Sep-01 May-02 Jan-03 Sep-03 May-04 Jan-05 Sep-05 May-06 Jan-07 Sep-07 May-08 Jan-09 Sep-09 May-10 Jan-11 Sep-11 May-12 Jan-13 Sep-13 FRANCE 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 May-00 Jan-01 Sep-01 May-02 Jan-03 Sep-03 May-04 Jan-05 Sep-05 May-06 Jan-07 Sep-07 May-08 Jan-09 Sep-09 May-10 Jan-11 Sep-11 May-12 Jan-13 Sep-13 SPAIN 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 May-00 Jan-01 Sep-01 May-02 Jan-03 Sep-03 May-04 Jan-05 Sep-05 May-06 Jan-07 Sep-07 May-08 Jan-09 Sep-09 May-10 Jan-11 Sep-11 May-12 Jan-13 Sep-13 This Figure plos he probabiliy of being in a low volailiy sae [P(S =1 Ψ -1 )] for he RS-ADF es of eq.4. In hese plos we use he esimaed HRs from eq.2 using he reurns on he spo and fuures sock indices in Germany, Unied Kingdom, France and Spain as he main inpu for he regime-swiching saionariy es. Figure 4. Hedge raios for inraday daa 1.3 Hedge raios Inraday Germany This Figure plos he esimaed HRs according o eq.2 using he inraday (hourly) reurns on he spo and fuures sock indices in Germany. 24