The relationship between stock returns and volatility in the seventeen largest international stock markets: A semi-parametric approach

Transcription

1 MPRA Munich Personal RePEc Archive The relaionship beween sock reurns and volailiy in he seveneen larges inernaional sock markes: A semi-parameric approach Dimirios Dimiriou and Theodore Simos Deparmen of Economics, Universiy of Ioannina January 011 Online a hps://mpra.ub.uni-muenchen.de/3758/ MPRA Paper No. 3758, posed 1. March 01 13:41 UTC

2 The relaionship beween sock reurns and volailiy in he seveneen larges inernaional sock markes: A semi-parameric approach. Dimiriou Dimirios 1, Simos Theodore, 1 Deparmen of Economics, Universiy of Ioannina, Universiy Campus, Ioannina, Greece Deparmen of Economics, Universiy of Ioannina, Universiy Campus, Ioannina, Greece Absrac We empirically invesigae he relaionship beween expeced sock reurns and volailiy in he welve EMU counries as well as five major ou of EMU inernaional sock markes. The sample period sars from December 199 unil December 007 i.e. up o he recen financial crisis. Empirical resuls in he lieraure are mixed wih regard o he sign and significance of he mean variance radeoff. Based on parameric GARCH in mean models we find a weak relaionship beween expeced reurns and volailiy for mos of he markes. However, using a flexible semiparameric specificaion for he condiional variance, we unravel significan evidence of a negaive relaionship in almos all markes. Furhermore, we invesigae a relaed issue, he asymmeric reacion of volailiy o posiive and negaive shocks in sock reurns confirming a negaive asymmery in all markes. JEL codes: F300, F500, G110, G100 Key Words: Risk-reurn radeoff; inernaional sock markes; semi-parameric specificaion of condiional variance. Corresponding auhor: simos@cc.uoi.gr

3 1. Inroducion An imporan opic in asse valuaion research is he radeoff beween volailiy and reurn (ypically log price changes). The heoreical asse pricing models (e.g., Sharpe, 1964; Liner, 1965; Mossin, 1966; Meron, 1973, 1980) link reurns of an asse o is own variance or o he covariance beween he reurns of sock and marke porfolio. However, wheher such a relaionship is posiive or negaive has been conroversial. As summarized in Baillie and DeGennarro (1990), mos asse-pricing models (e.g., Sharpe, 1964; Liner, 1965; Mossin, 1966; Meron, 1973) sugges a posiive radeoff for expeced reurns and volailiy. On he oher hand, here are many empirical sudies ha confirm a negaive relaionship beween reurns and volailiy: Black (1976), Cox and Ross (1976), Bekaer and Wu (000), Whielaw (000). Pindyck (1984) indicaes ha here is a srong posiive relaionship beween risk and excess reurn while Shiller (1981), Poreba and Summers (1986) as well as Bekaer and Wu (000, pp. 1) found negaive relaionship in a variey of USA marke indices. French, Schwer and Sambaugh (1987) using daa from he S&P index, find a negaive relaionship beween unexpeced volailiy and excess reurns. Similar are he findings of Shawky and Marahe (1995) which use a wo regime model. French e al. (1987) argue ha he observed negaive relaionship provides indirec evidence of a posiive relaionship beween expeced risk premium and ex-ane volailiy. In oher words, if expeced risk premiums are posiively relaed o predicable volailiy, hen a posiive unexpeced change in volailiy increases fuure expeced risk premiums and lowers curren sock prices. Campbell and Henschel (199) also poin o a posiive relaionship beween risk and reurn for USA monhly and daily reurns over he period Fama and Schwer (1977) and Nelson (1991) in various asses of USA marke find a negaive relaionship before Chan, Karolyi and Sulz (199) repor no significan relaionship in USA sock marke a he same period. Also, Harvey (1989) suggess ha he relaionship beween risk and reurn may be ime varying. These conflicing empirical resuls in he lieraure warran furher examinaion using differen and probably more appropriae economeric echniques. In his work we uilize a flexible funcional form o model condiional variance. We favour his approach, given ha esimaion via a parameric GARCH-M model is prone o model misspecificaions. Consisen esimaion of a GARCH-M model requires ha he full model is correcly specified (Bollerslev e al., 199, pp. 14). Indeed, he problem ha inferences drawn on he basis of GARCH-M models may be suscepible o model misspecificaion is well known o applied researchers. Nelson (1991, pp. 347) argues ha parameer resricions imposed by GARCH models may unduly resric he dynamics of he condiional variance process. In conras, a semi-parameric specificaion of he condiional variance allows flexible funcional forms, and herefore can lead poenially o more reliable esimaion and inference. In his paper, we propose hree parameric GARCH-M models and a semi-parameric model for esing he null hypohesis of zero GARCH in mean effec. We hen apply he above models o daily empirical daa of seveneen larges inernaional sock markes and wo world indices. We show some evidence ha a significan negaive relaionship beween sock marke reurns and marke volailiy prevails in mos major sock markes. Moreover he paper invesigaes he asymmeric impac of posiive and negaive shocks on volailiy using wo parameric models: EGARCH-M and AGARCH-M. Bollerslev e al. (199), Bollerslev, Engle, and Nelson (1994) and Henschel (1995) provide horough surveys in his area. The res of his paper is organized as follows. Secion wo discusses he parameric and semiparameric models used in he presen empirical analysis, secion hree presens he daa, in secion four we es he forecasing power of each model, in secion five we inerpre he empirical resuls and in secion six we conclude he paper.

4 . The parameric GARCH-M model wih disribued innovaions The model is specified as follows: k s s s1 y b y u (1) u (). d.(0,, v), (3) 1 q p ai 1 i 1 i1 i1. (4) The dependen variable y denoes he sock marke reurns, heir condiional variance and u he disurbance erm. We assume ha he random erm is disribued as he suden s wih ν degrees of freedom. Equaion (1) represens dynamic changes in he mean reurns, while equaion (4) describes ime variaion in he condiional variance. The informaion available a period -1 is denoed by 1. The condiional variance in (4), is modelled according o ARCH GARCH (q,p) specificaion of Bollerslev (1986). Among all he parameers o be esimaed, he mos relevan for his sudy is he parameer δ. The sign and significance of he parameer defines he relaionship beween sock marke reurns and condiional variance. Seing ai i 1 implies a highly persisen condiional variance. This model is known as he inegraed GARCH or ΙGARCH which is more likely for daily daa series..1. Parameric ΕGARCH-M model In his case he model is specified as follows: k s s s1 y b y u (5) u, (6)..(0,1), (7) 1 nd p p a0 ai 1 i i E i i i1 i1, (8) log { ( )} log The EGARCH specificaion allows he condiional variance process o respond asymmerically o posiive and negaive shocks. This is refleced in he value of he parameer produc i 1. When i 1 0( 0), he variance ends o rise (fall) when he shock is negaive (posiive)... The asymmeric GARCH-M (GJR) model In order o capure he asymmeric impac of new informaion on volailiy Engle and Ng (1993) suggesed he GJR-GARCH (q, p) model. I is specified by he following equaions: k s s s1 y b y u (9) u, (10). d.(0,, v), (11) 1 3

5 q p ai 1 D 1 1 i 1 i1 i1. (1) The condiional variance equaion (1) includes an exra erm: he dummy variable D 1, i is equal o one when 1 is negaive (good news), and equal o zero when 1 is posiive (bad news). A saisically significan parameer indicaes volailiy clusering. In case 0 here are leverage effecs while when 0 he news impac curve is symmeric, i.e. pas posiive shocks have he same impac as pas negaive shocks on oday s volailiy..3. A semi-parameric GARCH-M specificaion We consider he following semi-parameric GARCH-M model y a a y u x a u, (13) where y is he sock marke reurns, x (1, y 1, ), a ( a0, a1, )' is a vecor of parameers o be esimaed, var( y 1) is he condiional on 1 variance of y, 1 is he informaion se available up o ime -1. The error erm is a maringale difference process, i.e., Eu ( ) 0. We are ineresed in esing he null hypohesis of H : 0 versus 0 1 H : 0. The null hypohesis implies ha he condiional variance 1 does no affec he reurns y. We firs examine a simple semi-parameric GARCH model of he form mu ( ), (14) 1 1 where he funcional form of m() is no necessary specified paramerically. In case m( u ) a u he model is reduced o he sandard GARCH(1,1) specificaion. Under he 1 1 null H : 0 using (13), we obain 0 u 1 y 1 a0 a1 y. Then, we can generalize (14) as follows var( y I ) g( y, y ), (15) g( y, y ) m( y a a y ) m( u ). Expression (15) allows he condiional where variance o depend on lagged values of y. Denoing z 1 y 1 y recursively yields d d (, ) and subsiuing (15) g( z ) g( z ) g( z )... g( z )... (16) Given ha 0<γ<1, we may approximae (16) by a finie lag model of lengh d: g( z ) g( z ) g( z )... g( z ). (17) d d 4

6 Equaion (17) is a resriced addiive model wih he resricion ha he differen addiive funcions g() are proporional o each oher. The model allows lagged y sbeing included a he righ-hand side of (17). Yang (00), in a similar framework, suggess a kernel-based mehod o esimae he model. Alhough equaion (17) is only a wo-dimensional non-parameric model, i can be difficul o esimae by he popular kernel mehod, especially when d is large. Moreover, when d is large, he kernel mehod can give quie unreliable esimaes due o is failure o impose he addiive model srucure. In his work we op o esimae (17) by he non-parameric series mehod. The advanage of using a series mehod is ha he addiive proporional model srucure is imposed direcly and he esimaion is performed in one sep. To his end, le { i( y)} i 0 denoe a series-based funcion ha can be used o approximae any univariae funcion my ( ). We can use a linear combinaion of he produc base funcion o approximae g( y 1, y ), i.e. q q i0 i' 0 a ( y ) ( y ) a ( y ) ( y )... a ( y ) ( y ) ii' i s i' s s 0 s1 0q 0 s q s1 a ( y ) ( y )... a ( y ) ( y ) 10 1 s 0 s1 0q 1 s q s1... a ( y ) ( y )... a ( y ) ( y ), s=1,..., d. q0 q s 0 s1 qq q s q s1 Afer re-arranging erms, he approximaing funcion is of he form: d d s1 s1 a00 0 y s 0 y s1 a0q 0 y s q y s1 s1 s1 ( ) ( )... ( ) ( ) d d s 1 s s 0 s1 1q 1 s q s1 s1 s1 a ( y ) ( y )... a ( y ) ( y )... d d s 1 s 1 q0 q s 0 s1 qq q s q s1 s1 s1 ( 1) 1 aij i j (18) a ( y ) ( y )... a ( y ) ( y ) There are q parameers, namely γ and (, 0,..., q). Noe ha he number of parameers in model (18) does no depend on he number of lags included in he model. For example, if q is fixed, hen he number of parameers is also fixed. Therefore, we can le d as T (wih d/ T 0). Asympoically, i allows an infinie lag srucure wihou he curse of dimensionaliy problem (since q is independen of d). The esimaion procedure is as follows: Under H : 0 we obain 0 y a0 a1 y 1 u. Define y y a0 a1 y 1, hen E( y ) and 1 y v (19) where E y ( 1) 0. Using he series approximaion of in (18) we subsiue ou (19), and replacing a 0 and a 1 by he leas squares esimaors of a0 and 1 parameers γ and a s ( i, j 0,1,..., q ij in a, we can esimae he a ij s using nonlinear leas squares mehods. Alernaively, one can esimae ) by leas squares, regressing funcions, and searching over he grid [0,1] above procedure leads o a consisen esimae of he () y ( y ) on he series approximaing base for opimizing values. In order o ensure ha he g funcion, we need o le q and 5

7 q/ T 0 as T. The condiion q ensures ha he asympoic bias goes o zero. For example, if one uses power series as he base funcion, hen i is well known ha he approximaion error of using a q-h order polynomial goes o zero as q. The condiion q/ T 0 ensures ha he esimaion variance goes o zero as sample increases. See Newey (1997) and De Jong (00) for more deails on he rae of convergence of series esimaion. Le denoe he resuling nonparameric series esimaor of Eq.(13), we ge. Replacing by in y a a y, (0) u ( ) where. We hen esimae he parameers vecor a ( a0, a1, )' by leas squares. Equaion (0) conains a non-paramerically generaed regressor. Le a a a ( 0, 1, )' denoe he resuling esimaor. If one esimae he condiional variance ignoring he addiive srucure of, hen can use he resuls of Balagi and Li (001) and obain he asympoic disribuion of a from: n( a a) N(0, ), (1) where Σ is he asympoic covariance marix. Based on he leas squares esimaion of (18), he non-parameric esimaor, is consisen under he null hypohesis of H : 0. 0 Nex, we discuss he selecion of he number of lags d and he order of series approximaion q, in finie sample applicaions. For a fixed value of q, he number of parameers o be esimaed is fixed and does no depend on d. In paricular, choosing a large value of d does no lead o over fiing because he number of parameers ha need o be esimaed does no vary as d increases. Therefore, i makes sense o selec he value of q ha minimizes he AKAIKE informaion crieria. 3. Descripion of he daa To esimae he models we use daily US-dollar denominaed reurns 1 on sock indices of seveneen counries: he welve sock markes of he European Moneary Union, Ialy (ITA), Greece (GRE), Germany (GER), France (FRA), Finland (FIN), Belgium (BEL), Ausria (AUS), Ireland (IRL), Neherlands (NETH), Luxemburg (LUX), Spain (SPN), Porugal (POR), as well as he Unied Saes of America (USA), he Unied Kingdom (UK), Japan (JAP), Sweden (SWD) and Russia (RUS). We also include wo world sock-indices: he inernaional index of DaaSream s sock-exchange markes (D-W.I.) (his includes socks from he mos developed markes worldwide and i remains hrough many years one of he mos recognized indices in he world) and he M.S.C.I (Morgan Sanley Capial Inernaional World) world index, which is a sock index of he inernaional marke. The las index includes socks from 3 developed 1 As suggesed by Bekaer and Harvey (1995), calculaing he reurns in U.S. dollars eliminaes he local inflaion. 6

8 markes. I is compued since 1969 and has been a common evaluaion index of he inernaional sock-markes. The daa se sars from 4 h December of 199 and end up a 5 h December of 007, comprising observaions. The source is he DaaSream daabase. Daily reurns of each counry are calculaed using he firs logarihmic differences of general indices ha include dividends. 4. Forecasing power of he models. In his secion we empirically invesigae he four models, saed in secion, in erms of heir forecasing abiliy. The es saisic is he mean absolue error (M.A.E) which is a measuremen of he co-cumulaive forecasing error of he model. In our analysis he M.A.E is calculaed from he monhly sub-samples of observaions. Therefore, we are able o obain a M.A.E figure for each monh. The resuls are presened on able 1. We noice ha he bes forecasing model is he semi-parameric. In almos all markes he average M.A.E of he semi-parameric model is smaller of ha of he parameric models. The second bes forecasing model is EGARCH excep of Japan and Ialy where he GARCH-M and AGARCH-M models provide a smaller M.A.E. Moreover, he models GARCH-M and AGARCH-M have a similar forecasing abiliy wih AGARCH-M being slighly ahead. Table 1. Resuls of he mean of he mean absolue forecasing errors of he four models MARKET AUS BEL FIN GER GRE IRL ITA JAP LUX garch-m egarch-m agarch-m semi-par E E FRA NETH POR RUS SPN SWD UK USA D-W.I. M.S.C.I Based on he resuls repored in Table 1 we conclude ha forecasing abiliy of he semiparameric model is broadly superior. 5. Empirical resuls Noice ha he condiional variance of he reurns for each marke is calculaed using he sock reurns insead of he excess sock reurns. The excess reurn of a sock is defined as he difference beween reurn and he risk-free rae ha is dominan in he marke. Baillie and DeGennarro (1990), Nelson (1991), Choudhry (1996), Lee a all (001) agree ha using reurns is broadly equivalen wih excess reurns. In he presen sudy for he esimaion of he condiional variance we use reurns compued by log price differences. For he hree models: GARCH-M, EGARCH-M, and AGARCH-M, we decide he appropriae disribuion and number of lags p and q based on he AKAIKE crierion. So, for he models GARCH-M and AGARCH-M he bes 7

9 resuls are derived using he -suden disribuion, while for EGARCH M he normal disribuion has prevailed Inerpreaion of he empirical resuls The parameer, δ, is esimaed using he models of secion. In able we noice ha wih he GARCH-M specificaion, parameer δ is saisically significan a 1% level in Ausria, Finland, Ireland and Japan. Also we noice a negaive relaionship in Finland and Ireland and a posiive relaionship in he oher wo markes. On he oher hand, wih EGARCH-M models δ is no saisically significan a all levels. Nex we esimae he asymmeric GARCH-M model. According o he resuls of able i is obvious ha only for Ausria, Germany and Japan δ is saisical significan a 1% level. Also we observe a posiive relaionship beween he reurns and condiional variance for hese hree markes. Finally, we esimae he semi-parameric model using B-splines approximaing base funcions (Schumaker, 1981). In all markes under examinaion δ is saisically significan a 1% level (wih he excepion of Japan where he esimae of he parameer δ is saisically significan a 5%). The negaive relaionship beween reurn and condiional variance is dominan in almos all markes excep Ausria, Belgium and Luxemburg. Noice ha he empirical resuls of he parameric models are in broad agreemen wih hose of he semi-parameric model. However, he bes forecasing abiliy and parameer esimae s saisical significance of he semi-parameric model renders i more reliable. Esimaes of asymmery parameers are saed in Table 3. We noice ha asymmeric response of he condiional variance is a dominan propery of he counries under examinaion. Moreover, from Table, parameer esimaes of for all he markes, excep France, exhibi posiive signs a 1% level, confirming he exisence of a negaive asymmery. Given he fac ha he semiparameric specificaion fis beer he daa his sudy ends o suppor he claim ha volailiy is negaively correlaed wih reurns. Noice ha he above resuls are consisen wih he empirical findings of Glosen e al. (1993), Whielaw (000) and Qi Li e al. (005) bu conradic empirical findings of an insignifican relaionship repored in Baillie and DeGennarro, 1990; Nelson, 1991; Theodossiou and Lee, 1995; Choudhry, 1996; Lee e al., Conclusions In his paper we empirically invesigaed he relaionship beween expeced reurns and condiional variance in welve sock markes of he European Union as well as five large sock markes and wo world indices. Mos asse pricing models (e.g., Sharpe, 1964; Liner, 1965; Mossin, 1966; Meron, 1973) predic a posiive risk-reurn radeoff. In order o invesigae his, boh parameric and semi-parameric esimaion mehods of he condiional variance have been applied o daily daa from he above markes. Based on he semi-parameric specificaion, we find a saisically significan negaive relaionship of he risk-reurn radeoff in mos markes. There are only hree excepions: Ausria, Belgium and Luxemburg. Furhermore, we find significan evidence of a negaive asymmery in all markes confirming he previous empirical findings. Full able of resuls are available upon reques. 8

10 Table : Parameer esimaes for all four models MARKET AUS BEL FIN GER GRE IRL ITA JAP LUX garch-m δ *** *** *** *** (-sa) (14.5) (0.003) (-5.6) (0.001) (0.036) (-3.) (-0.016) (39.6) (0.016) α i +β i egarch-m δ (-sa) (-0.03) (0.087) (-0.011) (-0.00) (0.088) (0.11) (-1.7) (0.91) (-0.16) agarch-m δ *** *** *** 0.09 (-sa) (11.0) (0.01) (-0.013) (30.) (0.04) (0.531) (0.0186) -175,00 (0.0119) α i +β i κ *** 0.06*** 0.009*** *** 0.003*** *** 0.003*** *** (-sa) (.18) (3.15) (1.56) (4.6) (3.63) (4.47) (4.39) (4.67) (0.61) semipar/ric δ 0.000*** 0.06*** -6.57*** -16*** -9.6*** -13.7*** -13*** -3.6** 6.66*** (-sa) (8.6) (16.0) (-7.97) (-10.8) (-9.5) (-6.7) (-9.47) (-.3) (.91) FRA NETH POR RUS SPN SWD UK USA D-W.I. M.S.C.I (0.004) (0.0077) (0.01) (0.0067) (0.006) ( ) (0.005) (0.0) (0.003) (0.009) (0.361) (0.98) (0.046) (-0.366) (0.0019) (0.44) (-0.054) (0.18) (1.13) (0.0361) *** (0.018) (0.04) (0.0097) (0.017) -196,00 (0.035) (0.018) (0.018) (0.014 (0.01) *** *** 0.001*** *** *** *** 0.004*** 0.003*** *** (5.11) (-5.6) (1.53) (1.30) (.90) (4.69) (14.6) (8.6) (8.4) (8.39) -9.83*** -1.86*** -15.0*** -8.16*** -.0*** -8.96*** *** -16.0*** -5.4*** *** (-6.36) (-11.49) (-8.5) (-8.07) (-9.85) (-9.04) (-9.75) (-1.1) (-10.77) (-7.85) *denoes saisical significance a 10% level, * *denoes saisical significance a 5% level, * **denoes saisical significance a 1% level. 9

11 Table 3: Parameer esimaes of he EGARCH-M model. MARKET AUS BEL FIN GER GRE IRL ITA JAP LUX α i θ 1,α i = *** -0.06*** ** -0.05*** *** *** (-sa) (-.87) (-4.87) (-.31) (-4.18) (-1.8) (-1.8) (-.79) (-.93) (0.618) ** *** *** *** α θ *** (-sa) (-.10) (-3.15) (68.6) (1.63) (4.0) (7.61) * *** α 3 θ * (-sa) (-1,9) (-8.79) -11 (0.105) (0.748) (4.04) α 4 θ * (-sa) (-1.54) (-1.44) (-1.45) FRA NETH POR RUS SPN SWD UK USA D-W.I. M.S.C.I *** *** -0.05** *** -0.1 *** -0.11*** -0.06*** 0.08*** (-3.44) (-5.31) (-.08) (-0.6) ( ) (3.65) (-3.48) (-5.83) (-5.4) (-.04) -0.03*** *** -0.09*** 0.061** (1.0) (.13) (0.589) (-0.497) (7.5) (-3.95) ** (0.61) (1.94) (0.309) (-0.179) (-0.643) (-1.49) (1.17) (-1.8) *denoes saisical significance a 10% level, * *denoes saisical significance a 5% level, * **denoes saisical significance a 1% level. 10

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