NBER WORKING PAPER SERIES A MULTIPLE INDICATORS MODEL FOR VOLATILITY USING INTRA-DAILY DATA. Robert F. Engle Giampiero M. Gallo

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1 NBER WORKING PAPER SERIES A MULTIPLE INDICATORS MODEL FOR VOLATILITY USING INTRA-DAILY DATA Robe F. Engle Giampieo M. Gallo Woking Pape p:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massacuses Avenue Cambidge, MA 0138 Novembe 003 Wiou implicaing, we would like o ank Maco J. Lombadi, Neil Sepad, Sepen Taylo and Timo Teäsvia and paicipans a e confeence "New Fonies in Financial Volailiy" in Floence, May 003, fo many insigful commens and suggesions. Te views expessed eein ae ose of e auos and no necessaily ose of e Naional Bueau of Economic Reseac. 003 by Robe F. Engle and Giampieo M. Gallo. All igs eseved. So secions of ex, no o exceed wo paagaps, may be quoed wiou explici pemission povided a full cedi, including noice, is given o e souce.

2 A Muliple Indicaos Model fo Volailiy Using Ina-Daily Daa Robe F. Engle and Giampieo M. Gallo NBER Woking Pape No Novembe 003 JEL No. C, C3, C53 ABSTRACT Many ways exis o measue and model financial asse volailiy. In pinciple, as e fequency of e daa inceases, e qualiy of foecass sould impove. Ye, ee is no consensus abou a ue o "bes" measue of volailiy. In is pape we popose o joinly conside absolue daily euns, daily ig-low ange and daily ealized volailiy o develop a foecasing model based on ei condiional dynamics. As all ae non-negaive seies, we develop a muliplicaive eo model a is consisen and asympoically nomal unde a wide ange of specificaions fo e eo densiy funcion. Te esimaion esuls sow significan ineacions beween e indicaos. We also sow a one-mon-aead foecass mac well (bo in and ou of sample) e make-based volailiy measue povided by an aveage of implied volailiies of index opions as measued by VIX. Robe F. Engle Depamen of Finance Sen Scool of Business New Yok Univesiy, Solamon Cene 44 Wes 4 See, Suie New Yok, NY and NBER engle@sen.nyu.edu Giampieo M. Gallo Dipaimeno di Saisica G. Paeni Univesiá di Fienze Viale Mogagni Fienze, Ialy gallog@ds.unifi.i

3 1. Inoducion Models o descibe and pedic financial asse volailiy abound. In pacice, in addiion o a model s capabiliy o epoduce sylized facs in obseved ime seies and exibi desiable saisical popeies, e ulimae way o evaluae a model is is usefulness as a ool in many aeas suc as deivaive poducs picing, isk evaluaion and edging, pofolio allocaion, and e deivaion of Value a Risk measues. Ye, e concep of volailiy iself is somewa elusive, as many ways exis o measue i and ence o model i (cf. e suvey by Andesen e al., 00). In ecen imes, e availabiliy of ula-ig fequency daa and e wok done on em as sed new lig on e concep of volailiy: as a mae of fac, daa sampled a egula ina-daily inevals can be summaized ino a measue called ealized volailiy wic unde some assumpions is a consisen esimao of e quadaic vaiaion of e undelying diffusion pocess. Suc a measue was widely adoped as a age of foecas accuacy, bu e dependence of e measue upon e fequency of obsevaion of e daa makes i difficul o come o clea conclusions. In pinciple, e volailiy measues deived fom ula-ig fequency daa sould pove o be moe accuae, ence allowing fo foecas efficiency gains. Neveeless, e measues a make moe inensive use of suc daa ae pone o all sos of micosucue poblems. Bid-ask bounce (Roll, 1984), sceen figing (Zou, 1996), pice disceeness, iegula spacing of quoes and ansacions can all bias volailiy esimaes. Even if e gowing lieaue on ealized volailiy as deliveed pomising esuls (Andesen e al., 000; Bandoff-Nielsen and Sepad, 00), wa is of inees is e appopiae way o povide accuae foecass in e medium-o-long un, and e poblem emains open as o wee daily o ina-daily models delive e mos successful answe. In e fis caegoy one can menion ose models favoing e exisence of long memoy o ig pesisence in e pocess of volailiy, suc as e daily componen model by Engle and Lee (1999), o e Facionally Inegaed GARCH (FIGARCH Baillie e al., 1996; o FIEGARCH, Bolleslev and Mikkelsen, 1996) o Long Memoy Socasic Volailiy Models (LMSV Beid e al., 1998; Deo and Huvic, 1999). Ina-daily models ae moe ecen: Gose and Kone (1996) adop a signal plus noise model o esimae a pesisen componen as in Engle and Lee (1999); Andesen and Bolleslev (1997) sow ow accuacy of volailiy foecass can be impoved if

4 one moves o analyzing five-minue euns; Engle (000) deives a measue of volailiy fom ansacion daa. Te appoac wic we will pusue ee is one in wic seveal measues of volailiy can be joinly used o see wee diffeen feaues of obseved ime seies can delive an enicmen of volailiy foecasing fo e medium un. As a mae of fac, nex o e adiional volailiy modelling fom daily euns measued as e log-diffeence of closing pices, we can conside absolue euns on wic consideable modeling effo is pesen in e lieaue (Taylo, 1986; Ding, Gange and Engle, 1993; Gange and Sin, 000) and, wi e aleady menioned povisos, ealized volailiy as e sandad deviaion of ina-daily euns obseved a egula inevals. Fuemoe, i as long been ecognized a e spead beween e iges ecoded daily pice and e lowes ecoded daily pice is a funcion of e volailiy duing e day and, as suc, can lead o an impovemen of e volailiy esimaes. Many auos (Taylo, 1987, Gallan e al., 1999, Cou, 001, Alizade e al., 00, Band and Diebold, 00) ave devoed consideable aenion o e infomaional conen of ange daa exending e elaionsip i as o e volailiy paamee in a geomeic Bownian moion conex (cf. e ealy papes by Pakinson, 1980, Gaman and Klass, 1980, and Beckes, 1983), and compaing is pesisence caaceisics wi e ones of daily euns (Bunei and Lildold, 003). I sould be sessed a e ee vaiables ave diffeen feaues elaive o one anoe: e main diffeence is a e daily eun uses infomaion abou e closing pice of e pevious ading day, wile e ig-low spead and e ealized volailiy ae measued on e basis of wa is obseved duing e day, e fome aking all ade infomaion ino accoun, e lae being buil on e basis of quoes sampled a discee inevals. Tus, a zeo eun is no necessaily infomaive abou wa appened duing e day, and, by e same oken, a ig eun may signal ig volailiy duing e day wile i may jus be due o an opening pice muc diffeen fom e closing pice e pevious day bu vey close o e closing pice of e same ading day, wi a small ig-low spead. Also, noe a e same value of ealized volailiy may coespond o diffeen values of e ange since we could bo ave ecoded bo ig and low values faily fa apa duing e day wi a smoo ansiion of pice movemens beween e wo, o some vivacious pice swings concenaed in a so peiod of ime. Fo ese easons, ey ae all poenially useful. 3

5 In e pesen pape we exploi e fac a absolue euns, daily ange and ealized volailiy ae vaiables evolving as funcions of e undelying ime-vaying volailiy and all exibi e usual condiional pesisence of financial ime seies. Tey can eac be consideed as indicaos of volailiy and since ey ae all non-negaive-valued ey can be modelled wi a mulivaiae exension of e Muliplicaive Eo Model suggesed by Engle (00). Eac indicao is modeled as a GARCH-ype pocess possibly augmened wi weakly exogenous vaiables. Tis esimao is obus o a ange of eo disibuion assumpions. We sow a is ee-vaiable model possesses ineesing popeies in a e foecass fo eac indicao ae augmened by e pesence of e oe indicaos lagged and by asymmeic effecs fom e diecion of pice movemens. Te model can be solved dynamically fo muli-peiod foecass and e dynamic inedependence accouns fo a subsanial depaue fom e sandad GARCH pofile of dynamic foecass. We can use ou ee equaion model o pedic muli-sep volailiy. Te model is esimaed wi daily index daa ove a elaively long sample peiod. A caeful specificaion seac selecs models fo eac equaion. We calculae -sep volailiy foecass and compae ese wi one mon opion-implied volailiies as measued by e VIX. We examine e foecas pefomance bo in and ou of sample and esimae a combinaion of foecas equaion o see wee e foecass deived in ou famewok ave a significan impac in pedicing e VIX. Te appoac is quie diffeen fom a of Blai e al. (001) wo also analyze volailiy foecasing and VIX bu inseing i as a weakly exogenous vaiable in e condiional vaiance equaion fo e S&P100. Te eade sould expec e following: in Secion we discuss e Muliplicaive Eo Model, wi some geneal consideaions abou e esimao and is popeies, and abou e specificaions of e ee models adoped. Daa issues and model selecion pocedues ae descibed in deail in Secion 3. Model pefomance is analyzed in Secion 4 focusing on e caaceisics of e model in muli-sep-aead foecasing. In ode o sow e usefulness of ou foecass wen compaed o a measue based on implied volailiies epesening make evaluaion of volailiy, in Secion 5 we build muli-sep-aead volailiy foecass and we use em as egessos in a combinaion of foecas egession wee e (log of e) volailiy index VIX is e foecas age. Concluding emaks follow. 4

6 . Te Model We assume a e evoluion of a non-negaive valued pocess x, can be descibed by a Muliplicaive Eo Model (MEM, as discussed by Engle, 00). Ta is, x is e poduc of a ime vaying scale faco (wic depends upon e ecen pas of e seies) and a sandad posiive valued andom vaiable. Suc a specificaion was adoped in Engle and Russell (1998) fo duaions, Manganelli (000) fo volume ansacion daa and Cou (00) fo ig-low ange. Te scale faco is idenified as e condiional mean if e eo disibuion is assumed o ave uni mean. In geneal, eefoe, wee µ x = µε, ε I ~ iid... D( 1, ϕ) 1 can be ae flexibly specified as p µ = ω + α x + β µ + γ ' i i j j i= 1 j= 1 q Fue ems can signal e dependence of e seies on weakly exogenous vaiables (summaized in e veco z ) included in e infomaion se available a ime 1. Te condiions o ensue saionaiy and posiive means fo all possible ealizaions ave been discussed in Engle (00). z (1) (). Te densiy of e eo em ε was lef unspecified us fa. Wile in geneal one sould specify e ue DGP, i may be possible o find obus specificaions. We now conside e family of gamma densiies a ave been used fo ACD models. 1 α 1 ε f ( ε I 1) = ε exp α Γ( αβ ) β. Te popey of uni mean 1 1 implies a β =, namely, α 1 α α 1 1 α α 1 α x f( ε I 1) = α ε exp ( αε) f( x I 1) = α x µ exp α (3) Γ( α) Γ( α) µ 1 Suc a condiion is no esicive a all: conside a if ε wee o be suc a E( ε ) = αβ 1, i could be * * * * wien as βε wi E( ε ) = 1 and β would be a muliplicaive consan o be absobed by µ. 5

7 Te pocess would ave condiional expecaion Ex ( I 1) = µ and condiional vaiance µ Va( x I 1) =. A compaison beween vaious densiies coesponding o diffeen coices α of α subjec o e consain is epoed in Figue 1: as well known, e coice of α= 1, gives e pdf of a ci-squae wi one degee of feedom, wile α = 1 yields e uni exponenial. In geneal, values of α< 1 amoun o aibuing moe weig o exemely small o lage values of e andom vaiable wile values of α > 1 geneae a ump saped densiy wic appoaces e nomal disibuion fo lage values of α. Figue 1. Compaison among diffeen gamma densiies subjec o e consain αβ = 1 Le us suppose fo e momen a e only paamees of inees ae e ones defining µ (le us call em θ ), and le us look a e coesponding log-likeliood funcion: e elevan objec fo esimaion is given by T x log L = consan α log µ + = 1 µ (4) 6

8 wee e consan depends only on e x s and α. I is clea a in maximizing is funcion wi espec o θ, e value assumed by α is ielevan, as e fis ode condiions mus obey T x µ µ 0 = = 1 µ θ. (5) Even fom a numeical poin of view, an ieaive pocedue is unaffeced (in a Newon-ype meod e α ems pesen in e invese Hessian and gadien would cancel eac oe) esuling in exacly e same esimaes (apa fom ounding off eos wen numeical deivaives ae used). Fuemoe, ese fis ode condiions coincide wi e ones deived fom an auxiliay model a specifies e squae oo of e vaiable of inees, x, as e poduc of e squae oo of e scale faco µ and a alf Gaussian eo em ν, namely, x = µν ν I 1 ~ alf Gaussian (sandad). (6) Tis model as e clea advanage of being able o exploi any GARCH sofwae wic can esimae e paamees θ in µ fom a model fo x wi no equaion fo e mean (as done fo e ACD model by Engle and Russell, 1998; cf. also Engle, 00). Clealy, e second ode condiions would diffe, since e Hessian will be popoional o α (e.g. fo e exponenial case i would be wice e Hessian fo e ci-squae), and, coespondingly, e esimaed paamee vaiance-covaiance maices. Noe, oweve, a (exploiing e esuls of Bolleslev and Wooldidge, 199 and Lee and Hansen, 1994) e obus vaiance covaiance maix, compued as e poduc 1 1 Va( ϑ) = H OPG H (wi H e Hessian maix and OPG e maix of e oue poducs of e gadiens) povides e obvious benefi of making is discepancy ielevan since e α 's cancel ou: Va( ϑ) = H α= 1α OPGα = 1 H α= 1 = H expopg exp H α α 1 exp (7) Te song lesson we lean fom is discussion is a in e absence of jusificaions on e mos appopiae disibuion o adop fo e eo em in an MEM, e avenue of deiving is paamee values oug e esimaion of an auxiliay vaiance equaion fo e posiive squae 7

9 oo of e vaiable of inees wi a GARCH specificaion and nomally disibued eos is saigfowad and povides Quasi Maximum Likeliood esimaos, ence consisen and asympoically nomal as discussed by Engle (00), building on esuls by Lee and Hansen (1994). Howeve, if we know a a Gamma disibuion assumpion fo ε is appopiae, en e same pocedue delives consisency and efficiency fo e esimaos if α is known. Even if α is no known, using obus sandad eos sields agains e specific sape of e Gamma disibuion. Inoducing α among e paamees o be esimaed would povide infomaion on e sape of e disibuion of e eo em ε (some empiical esuls ae discussed by Engle, 00) wic would be useful fo simulaing fuue values o scenaio analysis bu would no ave any impac on e values of e esimaes of θ, no on ei sandad eos as Cov( θα, ) = 0. Moe specifically fo e case a and ee, le us indicae e daily closing pice as C, and calculae e daily euns as = log( C / C 1). Le us us conside squaed euns, modeled as an MEM = l og( / ) H = I iid D ε, ε 1 ~... 1, ϕ ; ( wee is e condiional mean of. Te squae, l, of e ig-low ange is defined as l H L ), wee and L ae e iges, especively, e lowes ecoded pices duing e day, and is modeled as: = η η I ϕ ( ) l, 1 ~ i.. i d. D 1, ; (9) and e squae of e ealized volailiy, dv, defined as e sandad deviaion of obseved euns ove J subpeiods wiin e day, beween make opening ime and make closing ime v v = ζ, ζ I 1 ~ iid... D 1, ϕ ( v ) (8) (10) 3, As esablised by Ding, Gange and Engle (1993), absolue euns exibi ig levels of seial coelaion and can emselves be adoped as e fis indicao of volailiy. We pefe o ink of e squae oo of as absolue euns even if e coice beween wiing (6) fo absolue euns o fo euns (wi Gaussian innovaions) is ielevan fom a numeical poin of view, since e fis-ode condiions would no cange. 3 Te meis of e coice of e appopiae J o balance esimaion accuacy and micosucue pifalls ae discussed in Andesen e al. (000): empiically, 5 minues inevals wok in a foeign excange famewok (Andesen e al., 001), 15 minues inevals ae adoped by Scwe (1998), bu we sill lack a eoy of wa is e opimal leng o coose. In is conex we coose J=78 and we adoped e meod discussed by Zou (1996) on aw daa o puge bid-ask bounce effecs. 8

10 Pu ogee, we can wie is ivaiae MEM as a sysem of equaions ε ε η l = η I 1 ~.. i i d. D ( ι, Φ ) (11) v ζ ζ v wee indicaes e Hadamad poduc, ι is a uni (3x1) veco, and Φ is a 3x3 maix epesening, especively, e mean veco and e vaiance-covaiance maix of e innovaion ems. Te mulivaiae esimaion poblem becomes a seies of univaiae poblems wen i is assumed a e covaiance maix is diagonal. Tee is no loss of consisency if is assumpion is false bu ee can be a loss of efficiency. We can now un o e deails of e MEM specificaion. Le us conside e fis MEM and wie e basic fom of e model fo e scale faco = ω + α 1+ β 1 (1) In ode o ake asymmeic eacions o socks, is base model can be expanded, eie by wiing a esold-ype model involving e cusomay dummy vaiable fo negaive euns d ( 0) = I <, o even adding lagged euns ω α 1 β 1 γ 1d 1 1, as = (13) as a diffeen way of accouning fo asymmey (as in e APARCH model, Ding, Gange, and Engle, 199) given a ey mainain ei sign, = ω + α + β + γ d + δ (14) Te main quesion we wan o addess a is sage is wee e inclusion of e lagged vaiables l and, wic ae pa of e infomaion se I, adds significan explanaoy 1 powe o e specificaion fo. If so, en some of e infomaion conained in ese indicaos elaes o a vaiabiliy in euns wic canno be accouned fo by consideing jus squaed euns and possible asymmeic effecs. By e same oken, a way o accoun fo fue asymmeic effecs wen euns in e pevious day ae negaive is o include also e poduc of e indicaos by e same dummy vaiable d accoun can be wien as ( ) 1 v A model wic would ake all ese effecs ino = ω + α + β + γ d + δ + ϕ l + ϑ l d + ψ v + λ v d (15) 9

11 wee a subscip was added o e coefficiens indicaing a ey efe o e specificaion fo euns. Te augmenaion of e GARCH(1,1) model in is case includes e six vaiables 1 1 1, d, l, l d, v, v d wic ae all known a ime A simila appoac can be followed fo e oe wo indicaos. Consideing again l v = η, and picking six elevan vaiables fom, namely,, l d,, d,, d 1 1, one can wie a full specificaion fo as ( ) I 1 1 = ω+ α 1+ β 1 + δ 1+ γ 1 1+ ϕ 1+ ϑ 1 1+ ψ 1+ λ 1 d 1. v v = ωv+ αv 1+ βv 1 + δv 1+ γv 1 1+ ϕd 1+ ϑv 1 1+ ψv 1+ λd 1 d l l d d v v (16) Analogously, fo e ealized daily volailiy indicao we sa fom v 1 and add e coesponding six vaiables, v 1d 1, 1, 1, 1d 1, l 1, l 1d, o e base specificaion of a GARCH(1,1) o ge ( ) v = v ζ v v d d l l (17) Te quesion a we will discuss in e following secion is based on e caaceisics of e daa a one analyzes, namely, wee adding e lagged indicaos adds subsanial explanaoy powe o e basic GARCH specificaion. In fac, i may appen a e MEM fo one indicao does no equie e infomaion se o be augmened, wile ee may be significan effecs fo anoe. 3. Te Daa, Paamee Esimaion and Model Selecion Fo ou empiical models, we will use daa on e Sandad and Poo 500 index fom Januay 4, 1988 o July 14, 1999 (730 obsevaions); we leave e las 17 obsevaions fo ou-of-sample foecas compaison puposes. We build seies fo absolue euns, ig-low ange and ealized volailiy fo is peiod: since e consuced vaiables ave quie diffeen scales elaive o one anoe, we escale all ee vaiables so a ey ae all expessed in annual pecenage ems and ey sae e in-sample quadaic mean of obseved euns. Some descipive saisics ae epoed in Table 1 and sugges a e absolue euns and e daily ange ave moe caaceisics in common wi one oe an eac of em does wi daily volailiy. Howeve, 10

12 a moe complee picue is obained if one looks also a e scae-plos in Figue and e coelaion coefficiens beween e ee seies epoed in e las lines of Table 1. Figue. Te Sandad and Poo 500 Index. Scae-plos beween absolue euns, daily ange and ealized volailiy. Sample Peiod: Jan.4, 1988 o Dec. 31, Daily Range Realized Volailiy Realized Volailiy Absolue Reuns Absolue Reuns Daily Range Table 1. S&P500 Absolue euns, daily ange and ealized volailiy. Summay saisics. Absolue Reuns Daily Range Realized Volailiy Mean Median Maximum Minimum Sd. Dev Quadaic Mean Skewness Kuosis Coelaions wi absolue euns wi daily ange 0.80 All vaiables ae expessed in ems of pecenage annual ems and ey sae e same quadaic mean. Sample peiod: Jan. 4, 1988 Dec.30, Te absolue euns lie almos invaiably below e daily ange, signaling a e days in wic e pevious day closing pice is lowe (ige) an e lowes (iges) inadaily pice ae infequen. A moe dispesed paen is sown beween e absolue euns and e ealized volailiy as efleced also by e coelaion coefficien of Te fac a e coelaions ae 11

13 elaively ig bu no nealy uniy signifies a e indicaos ae diffeen fom one anoe. Tese peliminay sylized facs coupled wi e caaceisics of pesisence exibied by e absolue eun, e daily ange and e ealized volailiy seies (cf. Figue 3), mean a an effo aimed a modeling e ineacions beween ese ee vaiables in a condiional conex seems pomising. FIGURE 3 (a e end of e pape) abou ee Fo eac of ese MEMs, we envisage e inoducion of any of up o six weakly exogenous vaiables in addiion o a GARCH(1,1)-ype model wic is kep as e base specificaion in ode o avoid coefficien idenificaion poblems. Fo eac indicao, eefoe, we ae esimaing 6 =64 models anging fom e mos geneal foms (15), (16) and (17) down o e base specificaion kep as a bencmak. Te wo model selecion saegies a we will adop and compae ae: 1. a geneal-o-specific saegy weeby we sa puning e coefficiens a appea o be saisically insignifican (using Bolleslev and Wooldidge obus sandad eos) in e mos geneal expessions and go on o seac down o e level wee all coefficiens ae significan and. e smalles value of e Scwaz Infomaion Cieia (BIC) among e 64 models. Te cosen models fom e geneal-o-specific selecion pocedue ae epoed in Table. Table. S&P500 - Geneal-o-specific Model Selecion. Equaions fo e ime-vaying componen in e equaions fo e squaes of absolue euns, ig-low ange, and daily ealized volailiy. Sample peiod Jan 4, 1988 Dec. 30, 1997 (Robus -saisics in small fon unde e paamee value). = l = + l v 1 = - v v d + v v l l d

14 Te esuls sow a, wen evaluaed in ems of coefficien significance, e inclusion of oe vaiables in e infomaion se appeas o add explanaoy powe in eac of e expessions. Te model fo absolue euns includes an asymmeic effec capued by (e em d seems o be less impoan), and bo e squaed daily ange and e squaed daily volailiy. Te model fo e ig-low ange seems o be e mos pasimonious wi e pesence of jus an asymmeic esponse of o lagged values of e euns. Ineesingly, e one model wic aacs e iges numbe of significan vaiables is e model fo e daily ealized volailiy in wic ee appea o be asymmeic effecs fom all vaiables, as well as lagged squaed euns and lagged squaed daily ange. Some diagnosics (epoed in e op panel of Table 3) sow a ee ae no majo specificaion poblems: fo efeence we give e values of e BIC and e esimaed log-likeliood, as well as e esuls of an ARCH() es (5% ciical value = 5.99) and e Ljung-Box es Q(1) fo e squaed esiduals (5% ciical value = 1.03) Table 3: S&P500 - Diagnosics on Seleced Models. Sample peiod Jan 4, 1988 Dec. 30, BIC ARCH() Q(1) LOGLIK Geneal-o-Specific Model Selecion Absolue euns Daily Range Realized Volailiy Smalles BIC Model Selecion Absolue euns Daily Range Realized Volailiy Te specificaion seac guided by e lowes value of e BIC gives e esuls we pesen in Table 4. Te model seleced fo e daily ange is e same as befoe. Fo absolue euns, e model seleced ee does no conain lagged squae daily volailiy; e values of e coefficiens do no cange muc, so e pofile of foecass beween e wo models will diffe mainly because of e impac of is vaiable. Te model fo e daily volailiy ee 13

15 does no include e ems involving e daily ange, and e values of e coefficiens ae faily diffeen. Again (boom panel of Table 3) no majo poblems ae signaled by e esidual diagnosics. Table 4. S&P500 - Smalles BIC Model Selecion. Equaions fo e squae of e ime-vaying componen in absolue euns, ig-low ange, and daily ealized volailiy. Sample peiod Jan 4, 1988 Dec. 30, (Robus -saisics in small fon unde e paamee value) 1 = l = l = v v d v v Model Pefomance Te ee models us esimaed could be used sepaaely fo one-sep-aead pedicions using e esimaed coefficiens and e acual value of e ig-and side vaiables: e pefomance of ese single-equaion models could be individually evaluaed in elaionsip o e pefomance of e coesponding GARCH models. Te inees of wa is being done ee, oug, lies in e fac a e ee equaions ogee can be seen as a sysem wic can be used as a ool fo muli-sep foecasing fo medium oizons. Le us conside e lef-and side as a ee dimensional veco, and conside a a ime T+1 we ave wee * A ω A l l d v v d (18) T+ 1 T * v T+ 1 T = T+ 1 T = ω + T T T T T T T T T T T v T+ 1 T ω v (,,,,,,,, ) is a 3 by 9 maix wic includes e coefficiens on e vaiables e value of wic is known a ime T. To foecas e vaious fuue second-ode momens condiional on infomaion a ime T fo mauiies geae an 1, we need o subsiue e ig-and side vaiables wi ei condiional expecaion as of ime T. Fo a geneic oizon k, we will ave 14

16 E ( ) = 0, E E E E E T T+ k 1 T( T+ k 1) = T+ k 1 T, T( lt+ k 1) = T+ k 1 T 1 T( lt+ k 1dT+ k 1) = T+ k 1 T v T( vt+ k 1) = T+ k 1 T 1 v d v T( T+ k 1 T+ k 1) = T+ k 1 T and, eefoe e expession (18) is subsiued by A (19) T+ kt ω T+ k 1 T T+ kt = T+ kt = ω + T+ k 1 T = ω + A T+ k 1 T v + ω v T kt T+ k 1 T v Te dynamic popeies of e esimaed sysem can eefoe be evaluaed by examining e caaceisic oos of e maix A. In pinciple, ee could be sable complex conjugae oos a would give e sysem some dampened cyclicaliy in e foecasing. Fo e case a and e oos fo e wo ses of esimaes ae given as in Table 5. Table 5. Caaceisic Roos of e Maix A Model selecion Geneal-o-specific Smalles BIC Te oos ae faily simila acoss e wo selecion pocedues: one can eefoe expec a e wo muli-equaion models will povide diffeen foecasing pofiles fo e so oizon, wile ey will end o be vey simila in e medium o long un. Fo is eason we will epo e gaps jus fo e models seleced accoding o e smalles BIC cieion. To evaluae e pefomance of e models le us fis conside e pofile of e ou-of-sample foecass geneaed by e sysem saing on Jan., 1998, e fis day afe e esimaion sample, and vaying e saing dae. We ake e obseved values as saing values fo day T (coesponding o Dec. 30, 1997 and foecasing fom Jan, 1998 o July 13, 1998), T+4 (coesponding o Jan 8, 1998), and so on unil e las saing value wic coesponds o Ma. 6, 1998 fo a foecasing oizon going fom Ma.9, 1998 o Sep. 14, In Figue 4 we 15

17 supeimpose vaious pofiles of e foecass fo diffeen saing peiods and fo a oizon of 13 peiods aead. Te foecass fo e daily ange indicao exibi e ypical monoonic pofile of a GARCH(1,1) model. Tis is no e case fo e cuves coesponding o foecass obained fom e oe wo indicaos: i is ineesing o noe a e foecass poduced by ou model may ave a non-monoonic beavio wi ove- (o unde-) sooing of ei long em (uncondiional) values a inemediae oizons. FIGURE 4 (a e end of e pape) abou ee Le us now conside e em sucue of volailiy fo an asse (o an index) a a given mauiy T+ k as e squae oo of e cumulaed sum of j-sep-aead foecass geneaed by expession (18), wen j=1, o (19), wen j is beween 1 and k (cf. Engle and Paon, 001). Tis coesponds o evaluaing e squae oo of e cumulaed sum of e expeced values of eac volailiy indicao a any ime beween 1 and k. As k inceases e ems of e sum will end o epea emselves. We us ave k k T+ jt E TT+ j j= 1 j= 1 a ν T+ kt k k ν T+ kt = + = T jt ET T+ j = 1 = 1 v j j ν T+ kt k k v + + T jt E TvT j j= 1 j= 1 l (0) Fo easons a will be cleae in e nex secion wen we discuss e compaison of ese foecass wi e VIX volailiy index, we coose a oizon k equal o, a is, a one-mon aead foecass. In Figue 5 we epo e values of e cumulaive volailiy foecas as e (squae oo of e) aveage of 1-sep, -seps,, -seps aead ou-of-sample foecass obained by e geneal-o-specific ee-equaion sysem (18) and (19) elaive o e sandad GARCH(1,1) specificaion. As one would expec, fo e equaion e diffeences beween e values obained wi ou model and wi e sandad base specificaion ae quie minimal and can be ascibed o e pesence of lagged euns in e fome specificaion. Fo e oe wo indicaos e esuls sow a e esimaes obained wi ou model ae geneally ige and moe pesisen an e esimaes obained wi e base specificaion wen e absolue 16

18 eun indicao is used weeas e evese is ue fo e daily ealized volailiy equaion. A emakable diffeence is obseved fo esimaes on o abou Augus 31, 1998 fo wic e base specificaion povides muc ige esimaes. Wee is signals excessive volailiy foecass by e lae se of models is an issue a equies a ype of moe specific evaluaion. Wa we need is an oveall compaison of e wo ses of foecass gauged in efeence o a make-based volailiy measue suc as e volailiy index VIX. We un o is in e nex secion. FIGURE 5 (a e end of e pape) abou ee 5. A Volailiy Index as a Foecas Tage Te Cicago Boad of Opions Excange (CBOE) is e wold s lages opions excange wee sandadized sock and index opions ae aded. 4 Among ese opions, CBOE offes an (Ameican-syle) index opion on e S&P 100 index called OEX, e value of wic is esablised as 100 imes e cuen value of e index (e.g. on July 5, 001 e index was and ence e dolla value of e index was $61,395). OEX opions ae e mos acively aded on e CBOE and ae vey liquid. One ineesing feaue of OEX is a in 1993 e CBOE ceaed a volailiy index called VIX, wic is calculaed as a weiged aveage of e implied volailiies fom eig a-e-money call and pu opions on e OEX wic ave an aveage ime o mauiy of 30 days. Te VIX index eefoe measues e implied volailiy of a ypoeical opion a is a-e-money and as 30 days o expiaion. 5 Te beavio of e seies fom Jan. 4, 1988 o Dec. 14, 1998 is sown in Figue 6. Sandad Augmened Dickey Fulle ess sow a e uni oo ypoesis is ejeced, aloug e degee of pesisence in e seies is vey ig. Fo e puposes of is pape, e analysis of VIX in efeence o volailiy esimaes wi a muliple indicao condiional model is elevan because we can use e value of VIX as a age fo ou volailiy esimaes. Unde seveal auxiliay assumpions, e opimal foecas of volailiy fom pas pices sould mac e Black-Scoles a-e-money implied volailiy. Howeve, infomaion suc as focoming announcemens known o ades bu no o e 4 Fo moe deailed infomaion cf. e CBOE web sie 5 Te value of e index is eckoned o measue invesos feas. A vey ig value signals beaisness as downwad isk is peceived o be ige an upwad isk. Caiss look a VIX fo possible end evesals. 17

19 economeicians may lead o episodes of posiive o negaive discepancies. Te -peiod-aead oizon fo e em sucue of volailiies was cosen o ensue compaibiliy wi e oizon consideed wen VIX is consuced. 6 Figue 6. Te CBOE VIX Index: Jan. 4, 1988 o Dec. 14, We will eefoe examine wo main issues: wee e volailiy foecass povided by eac se of MEMs (base and sysem specificaions) and gouped accoding o e indicao ey efe o conibue o e explanaion of e beavio of e VIX index bo in and ou of sample. We will us es wee e coefficiens of e foecass (gouped by indicao) ae joinly equal o zeo in a egession famewok; wee e volailiy foecass povided by e MEMs fo eac indicao wi a base specificaion and wi a sysem specificaion elp in explaining e beavio of e VIX index bo in and ou of sample. We will us es wee e coefficiens of eac goup of foecass (base and sysem) ae joinly equal o zeo in a egession famewok. In ode o answe ese quesions we an wo ses of egessions eac ove wo sepaae peiods (esimaion sample and ou-of-sample). Te slope coefficien and inecep may adjus fo e diffeences beween e S&P500 and e S&P100 opions and fo poenial, un-modeled 6 We ae no ovely concened wi e fac a VIX Is calculaed on e S&P100 index wile we ae calculaing volailiy measues on e S&P500. Te wo indices ave a ig coelaion (in e sample peiod of inees ee 18

20 volailiy isk pemia. In is conex, we use bo e foecass povided by e models seleced accoding o e smalles BIC cieion and o geneal-o-specific model selecion pocedue, since ey povide paially diffeen oucomes. Two models wee esimaed, bo using e log of VIX as e dependen vaiable and e log of e one mon volailiy foecass accoding o e base specificaion, and e sysem MEMs as independen vaiables. Te diffeence beween e wo is a e second includes an AR(1) coecion. I is no supising a ee would be seial coelaion in ese esimaes. Diffeences beween e opimal economeic foecas of volailiy and implied volailiies would naually aise fom e educed infomaion se used by e economeician. Fo example, an upcoming elecion would be incopoaed in VIX bu no GARCH and is would pesis fo many days. Te esuls ae sown in Table 7 wee we epo e esuls of e in-sample and ou-of-sample analysis wi e op panel efeing o e smalles BIC selecion cieion and e boom one o e esuls obained wi e geneal-o specific appoac. Te esuls ae sow a, wi a few excepions, all coefficiens ae individually saisically significan. One noices a in all specificaions e consan em is igly significan and posiive, signaling a e volailiy esimaes ogee somewa undeesimae e volailiy level. Tis could jus be due o e fac a ou volailiy esimaes ae efeed o a wide-based index aking e aveage ove a lage numbe of socks o i could be due o e Ameican opion pemium in e OEX conac o i could be due o a volailiy isk pemium. Te specificaion wic seems o ave bee popeies is one in wic an AR(1) coecion is inoduced wic ineesingly, keeps e join explanaoy powe of e vaiables we deived. Beside an obvious impovemen in e R and in e value of e likeliood funcion, e diagnosics sow e expeced impovemen since e esidual auocoelaion and ARCH effecs ae educed. Te conibuion of vaious ses of foecass is assessed by means of HAC Wald F-ess and epoed in e lowe pa of eac panel. We es wee eac foecas as an impac by ype of indicao (ows labeled absolue euns, daily ange o ealized volailiy) o by ype of meod (base specificaion o sysem specificaion) by means of Wald-ype ess. Tee is no clea indicaion of a bee in-sample model pefomance as a consequence of e selecion cieion adoped. In bo ses of esuls one noices a some negaive coefficiens appea: concenaing on e ig-low ange, fo abou 98% fo e euns and 99% fo e euns squaed, wi ig coss-coelaions). One sould keep in mind a e S&P500 is an aveage ove a lage numbe of socks and ence as a poenially lowe volailiy. 19

21 example, oe ings being equal, an incease in e foecas of e volailiy value fo is ype of indicao would imply a educion in e value of VIX. Howeve, i is unlikely a oe ings ae equal. Te same kind of analysis may be caied ove o an ou-of-sample combinaion of foecass execised ove e peiod Januay, 1998 o Novembe 10, 1998, in wic e same vaiable (log(vix)) is egessed on a consan and on e (logs of) -day-aead volailiy foecass obained fom e same models, keeping e coefficiens fixed a ei esimaed in-sample values. Te esuls, again boken down beween lowes BIC models and geneal-o-specific models, ae pesened in e las columns unde e eading Ou of Sample. Once again, e AR(1) coecion acieves a subsanial educion in e values of e esidual diagnosics ess and, again, e significance of eac se of foecass is mainained judging fom e F-ess. Te lowes BIC povides a sligly bee pefomance: all paamees ae significan. Some feaues of is combinaion of foecass can also be assessed gapically (Figue 7) since i seems a e iges vaiance appeas o coespond o peiods in wic e VIX index is ige and moe volaile. Figue 7. Te CBOE VIX Index: Ou of sample combinaion of foecass Jan., 1998 o Nov. 10, 1998 Smalles BIC Models wi AR(1) coecion (Coefficiens in Table 7, las column) : : :10 Residual Acual Fied 0

22 Table 7. Regession esuls log(vix) egessed on a consan and e -day-aead foecass of volailiy obained using e base and e sysem specificaions wi and wiou an AR(1) coecion. In sample Ou of sample Lowes BIC w/ AR(1) w/ AR(1) Consan (.15) (1.88) (3.4) (.0) Absolue Reun Muli-sep Aveage Volailiy (Base specificaion) Muli-sep Aveage Volailiy (Sysem specificaion) Diagnosics Robus Wald ess of join zeo coefficiens Geneal o Specific Muli-sep Aveage Volailiy (base specificaion) Muli-sep Aveage Volailiy (Sysem specificaion) Diagnosics Robus Wald ess of join zeo coefficiens (10.57) (9.44) (.56) (.49) Daily Range (1.60) (4.30) (3.83) (10.87) Realized Volailiy (0.5) (-1.5) (-3.01) (-7.61) Absolue Reun (-.53) (-0.1) (-4.09) (-7.84) Daily Range (-3.01) (-3.04) (0.74) (-0.5) Realized Volailiy (3.7) (.69) (3.84) (.03) AR(1) (146.36) (9.14) R-squaed TR fo AR(4) ** 9.1 ** ** 4.17 ** TR fo ARCH(4) ** ** ** 3.1 * Abs. euns ems ** 49.6 ** 7.47 ** ** Daily Range ems 3.8 * ** 8.41 ** ** Realized Volailiy ems 7.37 ** 4.9 ** 9.31 **.11 Base ems ** ** 3. * 9.59 ** Sysem ems ** ** ** ** Consan (3.8) (4.3) (3.5) (.98) Absolue Reun (9.94) (10.9) (1.88) (.67) Daily Range (4.16) (5.5) (.61) (7.40) Realized Volailiy (0.31) (-13.61) (-0.98) (-5.09) Absolue Reun (-5.5) (-0.74) (-.99) (-6.93) Daily Range (-1.06) (-1.30) (.48) (.6) Realized Volailiy (5.00) (4.87) (7.44) (9.19) AR(1) (137.97) (37.7) R-squaed TR fo AR(4) ** 8.47 ** ** 1.90 TR fo ARCH(4) ** 11.9 ** ** 6.87 ** Abs. euns ems 55.8 ** ** 3.54 * 7.57 ** Daily Range ems ** ** 4.87 ** 6.08 ** Realized Volailiy ems ** ** ** ** Base ems ** 7.6 ** 7.13 ** 10.8 ** Sysem ems 8.78 ** ** 0.45 ** 34. ** Robus -values unde e esimaed coeffcien values. * = significan a 5%; ** = significan a 1%. 1

23 6. Conclusions Te moivaion fo is pape sems fom e consideaion a vaious volailiy indicaos used in e lieaue (absolue daily euns, daily ig-low ange and ina-daily ealized volailiy) may pesen feaues wic sould be joinly combined in a dynamic model o enance e infomaion conen of individual measues. We ave adoped a novel appoac, called Muliplicaive Eo Model (Engle, 00), wic is suied o model e condiional beavio of posiively valued vaiables coosing a convenien GARCH-ype sucue o model pesisence. I uns ou a a wide vaiey of eo assumpions can be accommodaed wi a sandad easyo-use pocedue and infeence conduced on e basis of e obus vaiance covaiance maix. Fo e poblem a and e cosen specificaion fo e MEM is mulivaiae and is suiable o be dynamically solved fo so o medium ange foecasing oizons. Fo e daa a and (Sandad and Poo 500), we sow a e appoac is ewading in a e eained specificaions fo eac indicao ae augmened by e pesence of lagged values of oe indicaos. In paicula, we obain e esul a daily ange and euns ave explanaoy powe fo ealized volailiy, and daily ange is e indicao wi e mos pasimonious model. We evaluae e pefomance of is model by poducing -sep-aead volailiy foecass fo eac of e ee indicaos and by using em in a egession famewok o deec ei explanaoy powe fo an index of make volailiy suc as e VIX index. Te esuls sow a e model pefoms well, wi significan pesisence of e VIX index being capued by is foecass bo individually and gouped eie by ype of indicao and by ype of specificaion adoped. Te esimao us deived is efficien wen e coice of eo disibuion is wiin e class of a Gamma disibuion: i eains consisency wen oe moe flexible and ime-vaying densiy specificaions may be moe appopiae and eseac is unde way o idenify suc densiies. By e same oken, in e pesen conex we adoped a diagonal sucue fo e vaiance covaiance maix of e eo ems, and fue efficiency gains could be acieved by consideing moe complex sucues of coelaions among e eo ems, paalleling e lieaue on mulivaiae GARCH models.

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26 OTHER FIGURES FROM THE TEXT Figue 3. Te S&P500 Index: Absolue euns, Daily Range, and Realized Volailiy Sample Peiod: 1/4/1988 1/30/ Absolue Reuns Daily Range Realized Volailiy 5

27 Figue 4. Te S&P500 Index: Ou-of-sample Volailiy Foecass: Abs. Reuns, Hig-Low, Daily Volailiy. Vaious saing daes /08/98 03/09/98 05/05/98 07/01/98 08/7/ Absolue euns /08/98 03/09/98 05/05/98 07/01/98 08/7/98 6

28 Figue 5. Te S&P500 Index: One mon-aead em sucue of volailiy: Abs. Reuns, Hig-Low, Daily Volailiy. Ou of Sample - Jan., 1998 o Nov. 10, : : : : : :10 AbsRe - Sysem AbsRe - GARCH HILO - Sysem HILO - GARCH : : :10 DVOL - Sysem DVOL - GARCH 7