TIME DEPENDENT CONDITIONAL HETEROSKEDASTICITY Professor Richard Baillie, (March 004 Iniially ime series economeric work emphasized modeling and esing relaionships in he condiional mean of a variable. However, a number of ineresing economic relaionships and heories are concerned wih he second momens and paricularly he condiional variance of a process. While i is naural o consider ime dependencies wihin he condiional mean of a process, i is also possible for he condiional variance o exhibi similar characerisics. The characerisics of asse pricing daa are ypically raher unineresing in he condiional mean, since he firs differences of he logged price series are usually close o being uncorrelaed. Hence he coninuously compounded rae of reurns are uncorrelaed or unpredicable, so ha he asse prices are approximaely maringale difference sequences. This propery was recognized by Bachelier (900 and subsequen auhors such as Mandelbro (963 noed he furher sylized fac ha changes in asse prices having pronounced volaile and ranquil periods, so ha heir volailiy was ime dependen. A hird sylized fac was ha he reurns series exhibied exreme non-normaliy and excess kurosis. The las wo properies of volailiy and excess kurosis of he reurns densiy are boh feaures of he class of ARCH models originally developed by Engle (98 and currenly widely employed in many empirical applicaions. In order o moivae he underlying ideas of Auoregressive Condiional Heeroskedasiciy (ARCH processes i is firs worhwhile considering he problem of predicing he fuure level of he mean of a random variable which is recorded from ime series daa. The relaive success of forecasing from any dynamic economeric model essenially comes from he use of he condiional mean raher han he uncondiional mean. To illusrae his, consider he simple scalar firs order auoregression i.e., he AR( model: ( y = φy + ε
where E( ε = 0, ha E( ε = σ and E( εε = 0 for s. Since s φε = 0 y = + φy, i follows 0 = 0 E( y = φ E( ε + φ E( y = 0 since i is assumed ha he iniial value, y 0 = 0 and also ha 0 E y ( = σ ( φ. Raher han use he uncondiional mean of 0, an efficien forecasing procedure would require he use of he condiional mean of E( y Ω = + y, where Ω is he se of all relevan, worhwhile informaion available a ime. In he case of he AR( process in equaion (, Ey = E( y Ω = φ y. + + Clearly his forecas will be ime dependen as he curren informaion se, refleced in y will change over ime. The condiional variance of he y process will be Var y E y E y + = ( + +, = [ ] E φy + ( ε φy + = E ( ε + = σ. so ha he condiional variance will be consan over ime. Time Dependen Condiional Momens One of he grea insighs in he pioneering work of Engle (98 was o hink in erms of
condiional second momens raher han uncondiional momens. Before his, here were various precursors which aemped o inroduce ime dependence ino variances. For example, Mandelbro (963a calculaed recursive esimaes of he variance over ime, while Klein (977 consruced rolling moving averages around sample means. However, he aracion of he ARCH mehodology is ha i develops a coheren modeling mehodology around he simple, bu insighful observaion ha condiional variances may change over ime. The basic feaure of Engle's (98 Auoregressive Condiional Heeroskedasiciy (ARCH model, is ha he condiional variance of y is allowed o be ime dependen. Hence, ( ( ( Var y = E ε = σ, + + + and in Engle's (98 original formulaion, he informaion was resriced o conain curren and lagged squared innovaions and also a group of exogenous variables z ; hence σ = f( ε, ε,, ε, z. + q The mos general model considered by Engle (98 was he regression model wih condiionally normally disribued errors and ARCH(q errors, where σ + is a linear funcion of he las q innovaions. Then, (3 y = x β + ε, ' (4 ε Ω N(0, σ (5 q = σ = ω+ α ε. 3
In he ARCH regression model, he variance of he curren disurbance depends on he magniude of he lagged errors and no on heir signs. Hence large errors of eiher sign end o be followed by large errors of eiher sign. The ARCH(q model in (5 will have a memory lasing q periods, so ha curren volailiy will only depend on he magniude of he las q errors. An alernaive way of expressing he ARCH process in (3 and (4 is (6 ε zσ =, where σ is a ime varying posiive, and measurable funcion of he ime - informaion se, Ω ; he random variable Ez ( = 0, is uncorrelaed and Var( z =. While he ε process is serially uncorrelaed, he σ process is changing over ime. Hence, (7 E ε Ω = σ E z Ω = 0 and (8 Var ε Ω = σ Var z Ω = σ, so ha he ARCH process in ε arises as a rescaled normal innovaion sequence. Maringale wih ARCH The characerisics of asse prices are ypically raher unineresing in he condiional mean, since heir reurns, i.e. he differenced logarihm of prices are usually close o being serially uncorrelaed. Hence he coninuously compounded rae of reurns are uncorrelaed or unpredicable, so ha he asse prices are approximaely maringale difference sequences, as was firs recognized by Bachelier (900. As noed previously, mos nominal asse reurns have he disinc sylized facs of: 4
(i being close o being serially uncorrelaed, (ii having periods of pronounced volailiy and ranquiliy, which suggess ha volailiy is auocorrelaed and hence ime dependen, and (iii he uncondiional disribuion of asse reurns are non Gaussian wih excess kurosis. The las wo feaures are capured by he class of Auoregressive Condiional Heeroskedasiciy (ARCH processes, originally developed by Engle (98. In order o focus he discussion of ARCH, i will be assumed ha he condiional mean of he dependen variable is serially uncorrelaed and ha all he dynamics occur in he condiional variance. This is in accord wih many siuaions in finance and is a simpler in which o consider he properies of he ARCH model. Hence, he model in equaions (3 hrough (5 can be wrien as y = ε = zσ z iid...(0,, q = + y = σ ω α Uncondiional Momens of ARCH Processes I is firs convenien o recall he Law of Ieraed Expecaions. If Ω and Ω are wo informaion ses of random variables and Ω is a subse of Ω, hen for any random variable y, (9 E(y Ω = E[E(y Ω Ω ]. Since Ω is a larger se of informaion han Ω i follows ha condiioning on Ω is irrelevan. 5
For example, if Ω = ( y, y, and Ω = y k y k (,, and if Ω is he null se, hen E( y = E E( y Ω. For example, o derive he mean and variance of he uncondiional disribuion of he ARCH (q process in (4 and (5; E( y = E E( y Ω = 0, and (0 E( y = E E( y Ω = σ, and for he ARCH(q process, E y E y q ( = ω+ α Ω = q ( = ω+ α E( y Ω = E y q ( = ω+ α E E( y Ω Ω = E y q ( = ω+ α E E( y Ω = E y 6
Alhough y is serially uncorrelaed and a maringale sequence, so ha i is unpredicable in is condiional mean; a necessary requiremen for he saionariy of he process is for is uncondiional homoskedasiciy o be finie and consan. Then, E( y = σ, for = 0,,,..., where σ is a finie and posiive consan and is known as he uncondiional homoskedasiciy. = q + = σ ω α σ and hence, ( σ = ω q = α. Hence a necessary condiion for he saionariy and exisence of finie uncondiional variance of he ARCH(q process is ha q = α <. The ARCH modeling approach indicaes ha poenially very volaile economic and financial ime series can arise from processes wih changing condiional variances, which a he same ime have consan uncondiional variances. Previous lieraure which aemped o model changes in uncondiional heeroskedasiciy was generally problemaic and difficul o specify. ARCH Models Imply Excess Kurosis in he Uncondiional Densiy Some resuls for he uncondiional momens of he ARCH( process are due o Engle 7
(98. For he ARCH( process wih ω > 0 and α > 0, Engle (98 showed ha he m'h uncondiional momen exiss if and only if m m ( α Π( <. = Hence for E( y and 4 ( E y o exis, necessary condiions are ha α < and 3α < respecively. A proof of his is given in Appendix A. Saionariy of ARCH Processes Engle (98 also showed ha he ARCH(q process is covariance saionary if ω > 0, α,, 0 α q and all he roos of α ( L lie ouside he uni circle. Furhermore he saionary variance is given by (. A proof is given in Appendix B. Generalized ARCH Processes The applicaion of he ARCH model o pracical problems iniially had problems concerning he esimaion of parameers which saisfied he non negaiviy condiions, and o a lesser exen he saionariy condiion. For example, Engle (983 imposed a se of linearly declining weighs on he ARCH coefficiens, so ha α ( q+ α = q ( q+ α = 0, oherwise. for = 0,,,...(q-, An alernaive parameerizaion due o Geweke (987, 988 was o consider a Bayesian approach which inroduced a prior disribuions ha guaraneed he parameer esimaes o lie in a cerain feasible region. The Generalized ARCH, or GARCH process inroduced by Bollerslev (986 has he 8
desirable feaure of Parameer parsimony, which invariably considerably eases problems of parameer esimaes lying in infeasible regions. The Generalized ARCH, or GARCH(p,q process is, ( q p = + y + = = σ ω α β σ so ha he ARCH(q is appended wih p lagged condiional variance erms. The above GARCH(p,q process can be expressed as (3 σ = ω+ α( Ly + β( L σ where α( L = α L, q = p β( L = β L ; and ω > 0, α 0, for =,,...q; and β 0 for = =,,...p. On aking ieraed expecaions hroughou (, q p ( σ = ω+ α ( + β ( σ = = E E y E and since E( σ = E( y = σ, hen + + (4 σ ω = q p α β = = Or, 9
σ = ω α( β( saionary is, Similarly, a necessary condiion for he GARCH (p, q process o be covariance (5 q p. = = 0 < ( α + β < Infinie ARCH Represenaion of he GARCH Process The GARCH(p, q process in ( can be expressed as, [ β( L ] σ ω α( L y = +, and if all he roos of [ β ( L ] expressed as an infinie order ARCH process, lie ouside he uni circle, hen he GARCH (p, q process can be [ ] [ ] ( L ( L ( L y σ = β ω+ β α, ( ( Ly (6 [ ] σ = β ω + δ, δ( L = β( L α( L and is an infinie order power series in he lag operaor where [ ] The GARCH(, Process The simple GARCH(, process has become he mos widely used ARCH model, and 0
has been found o provide a good represenaion of a wide variey of volailiy processes. y = ε = σ z, z iid...(0,, σ = ω + + αy + βσ, Resricions: ω > 0, α 0 and β 0 ensure non-negaive condiional variances. 0 < ( α + β < ensures saionariy and finie variance of uncondiional reurns. E( y = ω σ = α β, Bollerslev (986 has also derived necessary and sufficien condiions for he exisence of he m'h momen of he GARCH(, process, which is denoed by m m µα (, β, m = C aαβ m <, = 0 m m! where C =, a 0 = and a =Π(i, for =,,... The m'h momen of he! ( m! i= uncondiional disribuion of ε is expressed by he recursive formula, m m n m n m E( y = am αn E( y ω Cm nµ ( α, β, n µ ( α, β, m n= 0 [ ] In paricular if,
(8 3 α + αβ + β <, hen he fourh order momen exiss and he momens of he uncondiional densiy are given by (9 E( y = ω σ = α β, and (0 4 E( y = 3 ω ( + α + β ( α β( β αβ 3 α. The coefficien of excess kurosis, κ is defined as 4 { E( y 3 ( } E y ( κ = E y ( For he normal densiy κ = 0 and for he GARCH(, process, ( κ 6 α ( β αβ 3 α =. and given he exisence of he fourh momen i can be seen ha κ > 0 which implies ha he GARCH (, process gives rise o excess kurosis, i.e. i is lepokuric. ARMA Represenaion for Squared Reurns from a GARCH(, Process The linear GARCH (p, q process can be easily reparameerized o obain a paricularly ineresing represenaion as an ARMA process in y. To moivae his idea, consider he
GARCH (, process σ = ω + + αy + βσ, and on adding y o boh sides of he equaion, y = ω+ ( y σ + αy + βσ, + + + Then y = ω+ ( α + β y + ( y σ β( y σ. + + + or, (3 y = ω+ ( α + β y + v βv, + + where (4 v y σ =, and is a whie noise process, and can be regarded as he innovaion in he condiional variance process, since by definiion (5 E v Ω = E ( y σ Ω = 0 + + +, Since E v + Ω = 0, i follows ha v + is herefore a maringale. Also, v + is serially 3
uncorrelaed since E v iv Ω = 0 + +, for any i, > 0. However, (6 Var v + Ω = E v + Ω 4 4 = E ( y+ σ+ y+ + σ+ Ω However, σ + is known a ime + since i is included in he informaion se Ω. Then, Varv = Ey σ Ey + σ 4 4 + + + + + Under condiional normaliy, Ey = 3σ and hen 4 4 + + (7 Varv = σ 4 + + Hence v + possesses a form of heeroskedasiciy and is bounded in he range where is known as he "suppor" of he disribuion. σ + (,, σ + Represenaion for Squared Reurns from a GARCH(p, q Process A similar parameerizaion exiss for he GARCH(p, q process, q p = + y + = = σ ω α β σ and on adding y o boh sides again, i can be seen ha he GARCH(p, q process reduces o an ARMA(m, p process in y, where m= max( p, q. 4
(8 q p = ω + ( α + β + β = =. y y v v A simple derivaion is available in erms of lag operaor noaion, σ = ω+ α( Ly + β( L σ, where q α( L = α and = p β( L = β. Then on rewriing he GARCH(p,q process as, = y = ω + α( L y + β( L σ, hence, [ α( L β( L ] y = ω + ( y σ β( L ( y σ and [ α( β( ] ω ( β( L L y = + L v, and hence y is ARMA(m, p, where m= max( p, q. Bollerslev (988 has used his represenaion as a means of implemening he Box Jenkins model idenificaion, or model selecion, sraegy based on he auocorrelaion and parial auocorrelaion funcions of resuls are analogous o idenifying he orders of an ARMA(m,p process in he condiional mean. The disribuion of he sample auocorrelaion funcion of usual Barle formula is inappropriae due o he non i.i.d. naure of he y. The y is quie complicaed and he y series. 5
Predicion in Models wih GARCH Innovaions In many pracical siuaions i is of ineres o make predicions in models wih ime dependen heeroskedasiciy. The presence of ARCH effecs will have implicaions for forming confidence inervals of predicions of he condiional mean and also for predicing fuure volailiy. For example, consider he ARMA(p,q model wih GARCH(, disurbances, ϕ( Ly = θ( L ε. ε = σ z σ = ω+ αε + βσ. Baillie and Bollerslev (99 have considered he properies of predicion from he above model and he regression model wih ARMA-GARCH disurbances; and hey derive he minimum MSE predicor of he linear GARCH (p,q process. For he saionary GARCH(, case, he predicor of fuure volailiy s periods ahead is, (9 s E ( ( σ+ s σ α β σ+ σ = + +, which implies ha he opimal predicion is based on he average, or uncondiional volailiy, plus an adusmen erm of geomerically declining weigh on he las "surprise" in he variance. The "surprise" in his conex is he disance of he las observed condiional variance from is average value of σ. Predicions of volailiy are imporan when conducing inference concerning predicions of he fuure mean. The s-sep ahead predicion MSE for predicing he condiional mean of he ARMA(p, q process is, 6
(30 where s s, ψi Eσ+ i i= MSE( y =, (3 y = ψε = is he Wold Decomposiion, i.e. infinie order moving average represenaion of he condiional mean of he process. Predicions of fuure volailiy have o be made in order o calculae confidence inervals for predicions of he condiional mean. Clearly in he case of condiional homoskedasiciy, he predicion MSE reduces o he usual formula, (3 s s, = i i= MSE( y σ ψ In opions pricing, forecass are generally required of fuure volailiy. In order o ascerain more abou he properies of he condiional variance process, i is useful o use he fac ha σ can be regarded as a regular covariance saionary sochasic process. Then σ will also possess a Wold decomposiion represenaion in erms of he curren and lagged innovaions in he condiional variance, namely v. Then, (33 σ ξ v = =, which can be used o deermine he properies of he s sep ahead predicion error for he condiional variance. See Baillie and Bollerslev (99 for furher deails. 7
Inegraed GARCH In many empirical sudies i is ofen found ha he sum of he parameers in a GARCH(, is close o one. Hence Engle and Bollerslev (986 suggesed ( y σ = ω + + α + α σ Or, σ = ω + ( + β y + βσ The uncondiional variance is undefined (i.e. infinie, for he IGARCH model, while he s sep ahead predicions are ( s Eσ = ω+ σ, + s + so here is a direc analogy wih he Random Walk wih drif in he condiional mean model. The opimal predicion is he curren value of he process plus a linearly increasing erm. This is one reason why his GARCH model is known as being inegraed. The IGARCH(, is an ARIMA(0,, in squared reurns y = ω + v βv, + + ( Ly = ω + ( βlv + + In a lo of applicaions, β is close o 0.8, he ypical choice of smooher consan in Exponenial Smoohing. In general, he GARCH(p, q process reduces o an IGARCH(p, q process when he 8
following facorizaion occurs, so ha [ α β ] ( ( L ( L = L Φ ( L, where Φ ( L is a polynomial of degree m- in he lag operaor and has all is roo ouside he uni circle. Then, he IGARCH(p, q process is, ( ( ε ω [ β( ] L Φ L = + L v. Nelson (990 has shown he IGARCH process is srongly saionary, bu no weakly saionary. On saring wih a coninuous ime diffusion process, which is ofen specified in finance heory, aking discree observaions a finer and finer sampling inervals leads o a discree ime volailiy process which ends o IGARCH wih a zero inercep. This is an ineresing relaionship beween wo previously differen specificaions. Exponenial GARCH Nelson (99 inroduced he EGARCH process o allow for asymmeric effecs beween volailiy and shocks, which can accoun for leverage. The EGARCH(, model is: ln( σ+ = ω + αz + γ z E z + β ln( σ, and does no require any non negaiviy resricions on he parameers. The variable gz ( = αz + γ z Ez has a mean of zero and is serially uncorrelaed. This funcion is piecewise linear in z since i can 9
be wrien as, gz ( = ( α + γ ziz ( > 0 + ( α γ ziz ( < 0 γez where I( z > 0 is he sandard indicaor funcion and is one for all posiive z and zero oherwise. The EGARCH(, model implies ha a negaive reurn has an effec ( α γ on he log of he condiional variance, while a posiive reurn has effec of ( α + γ on he log of he condiional variance. GJR Asymmeric GARCH Glosen, Jagannahan and Runkle (993 modified he sandard GARCH(, o allow for he parameer of lagged squared reurns o depend on he sign of he shock, so ha ( ( σ = ω + αy I[ y > 0] + + γ y I[ y < 0] + βσ Resricions: 0 ω >, [ α γ ] ( + / 0and β > 0 ensure non-negaive condiional variances. The furher resricion ha α + γ + β < ensures saionariy and finie variance of uncondiional reurns. ω E( ε = σ =, α + γ β 0
Long Memory ARCH, or FIGARCH Baillie, Bollerslev and Mikkelsen (996 inroduced he Fracionally Inegraed GARCH model known as FIGARCH, which incorporaes long memory in he condiional variance process. The FIGARCH(,d,0 model is, σ = ω+ [ βl ( L ] y + βσ d + + and d is he long memory volailiy parameer. The process has impulse response weighs of ω σ = + λ( Ly β + where k λ( L = λk L, k = 0 and for high lags λ k k d which is essenially he long memory propery, or "Hurs effec" of hyperbolic decay. The aracion of he FIGARCH process is ha for 0 < d <, i is sufficienly flexible o allow for inermediae ranges of persisence. Squared reurns are an ARFIMA(0,d, process
( L d y+ ω v+ βv = +, where v ε σ y σ, = = is a whie noise process. To see he relaionship beween he represenaions, noe ha ( L d y = ω + ( βl v = ω + ( βl( y σ + + + + and on re-arranging we ge σ = ω+ [ βl ( L ] y + βσ d + + Generalizaion o higher order processes is sraighforward. For example he ofen used FIGARCH(,d, model, ( φl( L d y+ = ω+ v+ βv which can be expressed as σ = ω + [ βl ( + φl( L ] y + βσ d and his will also have he same hyperbolic rae of decay a high lags, i.e. λ k k d. The reason for he long memory feaure is ha he condiional variance, σ, has a slow hyperbolic rae of decay in erms of lagged squared innovaions. The associaed impulse response weighs also exhibi quie persisen hyperbolic decay. The FIGARCH(,d,0 process can also be expressed as, ω σ = + λ( L ε, β
where Γ ( k + δ ( β ( δ λk = Γ( k Γ( δ k, and for large lags k, β λk = Γ( δ k δ, which generaes slow hyperbolic rae of decay on he impulse response weighs. The heoreical properies of he FIGARCH(0,d, process are discussed in some deail by Baillie, Bollerslev and Mikkelsen (996. They noe ha he process is sricly saionary and ergodic for 0 δ. Then shocks o he condiional variance will ulimaely die ou in a forecasing sense. Componen GARCH Ding and Granger (996 suggesed, σ = γσ + ( γ σ +, +, + σ = α y + ( + α σ,, σ = ω + + α y + β σ,, he weighed sum of an IGARCH and GARCH model. Jones, Lamon and Lumsdaine (998 applied he model o see if he shocks on paricular days associaed wih macro announcemens have differen effecs on volailiy han shocks on oher days. ARCH in Mean Models Engle, Lilien and Robins (987 have considered he ARCH in Mean model where lagged volailiy is allowed o effec he curren level of he process. A general model is of he 3
form, y = x β + γ f( σ + u, ' φ( Lu = θ( L ε, ε Ω N(0, σ where γ is known as he ARCH in Mean parameer. The choice of funcional form is fairly flexible; usually he condiional sandard deviaion, σ is used since his preserves he same scaling as he mean of he process y. Someimes he condiional variance σ iself, or ln( σ is he chosen funcion. This model can be used o represen he "marke model", where own volailiy is allowed o effec mean reurns. The original applicaion of Engle, Lilien and Robins (987 was he erm srucure of ineres raes and γ was proxying a liquidiy premium. In heir formulaion, y is he excess holding yield on long erm bonds relaive o a one period T Bill, σ is associaed as he risk premium and ε is he ex ane rae of reurn and would be expeced o be uncorrelaed in an efficien marke. Esimaion of ARCH Models Given ε = zσ and z iid(0, hen he likelihood requires evaluaion of he Jacobian ε = f ( ε =, σ σ where σ is he Jacobian. Under condiional normaliy, so ha z NID(0,, hen 4
T ε, T ln( L = ln( π (/ ln( σ + = σ In order o represen daa wih very fa ails, i.e. excess kurosis, condiional densiies of suden or power exponenial are someimes used. The sandard numerical MLE mehods can be used o find he maximum MLE, ^ ^ i i Θ =Θ H Θ s Θ ( ( as before wih ARMA models, so ha a he i'h ieraion he esimae of he parameer vecor is equal o he esimae a he previous ieraion plus an adusmen due o he Hessian pos muliplied by he score vecor. The Hessian and score vecor are boh evaluaed a he parameer esimaes obained a he previous ieraion. There are several possible procedures for evaluaing H ( θ and on deermining he sep lenghs for each ieraion. ARCH Models wih Non Gaussian Condiional Densiies Alhough he combinaion of an ARCH ype process wih a condiional normal denisy generaes an uncondiional reurns densiy wih implied excess kurosis; he degree of excess kurosis is frequenly insufficen o represen he behavior of daily or higher frequency reurns daa. For his reason, here has been some experimenaion on using condiional densiies which have greaer kurosis han he normal disribuion, since hey can generae sill greaer excess kurosis in he uncondiional reurns disribuion. In paricular, Bollerslev (987 used he suden densiy, y = z σ, where z is iid (0, σ, v, and he log likelihood is herefore, 5
v+ v v+ y L = Γ Γ v σ + σ ln( ln ln (/ ln( ln The kurosis of he suden densiy is 3( v /( v 4, so ha i is necessary for v > 4, for he condiional densiy o have a finie kurosis. Pracical Applicaion of ARCH Esimaion: o guaranee convergence o a global maximum of he log likelihood funcion i is necessary o choose good iniial sareing values for he parameer esimaes as well as using efficen numerical opimizaion rouines. Some poins worh rememebring are ha i is generally desirable o uliply reurns by 00 or 000 for ease of compuaion and o in he case of esimaion of he GARCH(, model o have α around.5 and β equal o.75 for daily daa. The inercep in he condiional variance ω, can be se o s ( - α - β, where s is he sample variance of uncondiional reurns. In he case of daily daa wih α =.5 and β =. 75, he iniial esimae of ω will be approximaely he uncondiional variance divided by 0. LM Tesing for ARCH Effecs Engle (98 also demonsraed ha a simple LM es for ARCH(q in he linear regression model, H y = x β + ε ; : ' ε Ω N(0, σ q = + = σ ω α ε 6
versus H0 : α = α = = α q = 0; can be compued as OLS residuals, ε on a consan and ε,, ε q TR from he regression of he squared. The resuling es saisic has he usual χ q. Inference in ARCH Models by QMLE The echnique of using QMLE is very useful in ARCH models when he condiional densiy may have been misspecified. Then a gaussian densiy is maximized and he subsequen QMLE has been implemened by Bollerslev and Wooldridge (990. The formal properies of MLE have only been derived for some special cases, bu simualion evidence has led researchers o believe ha hey generally possess asympoic normaliy and / T consisency. Formally, hese resuls have only been derived for he IGARCH(, model by Lee and Hansen (994 and Lumsdaine (996. Their proofs require z o be saionary and ergodic ogeher wih hree oher relaively mild condiions on he z process. Then he score vecor and Hessian are sricly saionary and ergodic, and a cenral limi heorem can be used o derive he limiing disribuion of he QMLE. Baillie, Bollerslev and Mikkelesen (996 have argued ha similar reasoning indicaes he MLE of he FIGARCH process o be consisen and asympoically normal; and hey provide deailed supporive simulaion evidence. Then, ^ / ( Θ Θ 0, ( Θ0 ( Θ0 ( Θ0 T N A B A, where A (. and B (. are he Hessian and ouer produc gradien respecively, when evaluaed a he rue parameer values Θ.. 0 Mulivariae GARCH Models 7
Suppose ε is a g dimensional vecor such ha, ε = z Ω, / where z is iid... and Ez ( = 0 and Var( z =. Then Ω is a gxg dimensional, posiive definie, ime dependen covariance marix which is measurable wih respec o he se of informaion a ime. There have been several aemps a parameerizing Ω as a mulivariae GARCH process. Recall ha for any square gxg marix A, he operaor vec(a column sacks all he elemens of A, so ha vec(a is g. For symmeric marices he relevan operaor is vech(a which sacks he lower porion of he marix A. Hence vech(a gg+ ( is in dimension. For example if, a a a A a a a 3 = 3 a a a 3 3 33 hen, [ ] ' Vec( A = a a a3 a a a3 a3 a3 a33 and [ ] ' Vech( A = a a a3 a a3 a33 One ofen used parameerizaion of he mulivariae GARCH(, process, is 8
vech( Ω = C + AVech( ε ε + Bvech( Ω ' For example, in he ' g = case, Vech( [ ω, ω, ω ] Ω =, ' Vech( ε ε = ε, ε ε, ε and on subsiuion, ω c a a a3 ε b b b3 ω ω c a a a 3 ε ε b b b 3 ω = + + ω c a3 a3 a 33 ε b3 b3 b 33 ω In many applicaions A and B are assumed o be diagonal, which implies ha boh he condiional variance processes and he condiional covariance process follow univariae GARCH(, processes. Anoher formulaion, which is slighly less racable is he formulaion, Ω = CC+ Aε ε A+ BΩ B, ' ' ' ' where C is gxg and A and B are gxg also. Equaion (50 is known as he posiive definie parameerizaion. Engle (987 and Diebold and Nerlove (989 have considered a facor ARCH process, which implies ha Ω will no have full rank. In his model he underlying idea is ha here may be a common volailiy process in a se of g asse prices, e.g. ineres raes. So one common facor explains a large amoun of each ineres rae's volailiy. The idea is very aracive economically, bu somewha difficul o es for and o esimae beyond one facor. 9
30
Appendix A: Derivaion of he m'h Momen of an ARCH( Process. I is necessary o define he r dimensional vecor ω, ' r ( r 4 ω = y, y,, y, y. For a random variable u N σ (0,, all he odd momens are zero and all he even momens follow from he resul ( r r σ r = Eu ( = Π, so ha Eu ( = σ, 4 4 Eu ( = 3σ, 6 6 Eu ( = 5σ, ec. For ease of explanaion, consider he pure ARCH( process wihou here beiing he complicaion of regressors. Then, y N σ Ω (0, σ = ω+ αy, so ha from (A m ( E y m m σ = = Π(, ( y = ω+ α Π(. = m m Ω is a m On expansion of he erm on he righ hand side i becomes clear ha E( y 3
linear combinaion of ω. Also only powers of y less han or equal o m are required. On wriing ( ω E Ω = b+ Aω, where b is an r dimensional column vecor, A is square, upper riangular marix of dimension r. Then by successive subsiuion, ( ( E ω Ω = b+ A b+ aω = b+ Ab+ A ω, and hence, ( ω ( E Ω = I + A+ A + + A b+ A ω. k k k k For he uncondiional disribuion of y o possess saionary momens, i is necessary ha all he k eigenvalues of A lie wihin he uni circle. Then lim A = 0, and k ( ω Ω ( = E I A b, lim k k hence, ( ω ( E = I A b. For example in he r = case; ω = y, y, and ' 4 3
( E y Ω = 3( ω+ αy, 4 = 3ω + 6ωαy + 3α y, 4 and 4 ( ( E y Ω 3ω 3α 6ωα y E( ω b A Ω E 4 Ω = = + ω = + E y ω 0 α y ( ω ( α ( α ω 4 E 3 ( y α 3 = = E( y ω α ω On noing ha 3 α is hree imes he variance, and since his has a coefficien 4 4 greaer han one in he expression for E( y, i follows ha ( E y is sricly greaer han ha of a normal random variable. Hence he disribuion of y possesses excess kurosis, which is anoher ineresing propery of he ARCH process; since ime dependen volailiy and excess kurosis are boh well known feaures of asse pricing daa. The proof is hen compleed by noing ha since A is upper riangular, is diagonal elemens are is eigenvalues and are given by: m m m α Π( = Πα ( = γ m = = m Noing haγ m is a produc of m facors; if he m'h facor is <, hen are he oher facors <. Then a necessary and sufficien condiion for all diagonal elemens < is ha γ m <. 33
Appendix B: Saionariy of he ARCH(q Process The ARCH(q process is covariance saionary if ω > 0 and α 0 and al he roos of α ( L lie ouside he uni circle. Then he saioanry ucondiional variance is, ( E y ω = q α = Proof: consider he vecor, ' ω = y, y,, y q, hen E ( ω ( ( E y Ω ω α α αq 0 y E y Ω 0 0 0 0 y Ω = = + + E 0 0 0 0 y q ( y + qω Then, ( ω E Ω = b+ Aω, which is a companion form represenaion. Noe ha E( y Ω = y, for =,,... and as before, ( ( k ω E Ω = I + A+ A + + A b+ A ω. k k k 34
If all he eigenvalues lie wihin he uni circle, he limi exiss and is given by ( ω Ω k = lim E ( I A b, which does no depend on he iniial condiions and is also k independen of. Equivalenly o all he eigenvalues of A lying inside he uni circle, he resul q can also be expressed as all he roos of α( L ( αl αl αql = + + + should lie ouside he uni circle. (This is exacly equivalen o an AR(p when expressed in erms of a companion form marix. On aking he firs elemen of (..4 we obain he uncondiional variance of ( E y ω =. q α = 35
Some of he Very Many References Baillie, R.T. and T. Bollerslev (99, "Predicion in dynamic models wih ime dependen condiional variances", Journal of Economerics, 5, 9-3. Baillie, R.T., T. Bollerslev and H.-O. Mikkelsen (996, "Fracionally Inegraed Generalized Auoregressive Condiional Heeroskedasiciy", Journal of Economerics, 74, 3-30. Bollerslev, T. (986, "Generalized Auoregressive Condiional Heeroskedasiciy", Journal of Economerics, 3, 307-37. Bollerslev, T (987, "A Condiional Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn", Review of Economics and Saisics, 69, 54-547. Bollerslev, T and H.-O. Mikkelsen (996, "Modeling and pricing long-memory in sock marke volailiy", Journal of Economerics, 73, 5-85. Bollerslev, T. and J.M. Wooldridge (99, "Quasi maximum likelihood esimaion and inference in dynamic models wih ime varying covariances", Economeric Reviews,, 43-7. Bollerslev, T., R.Y. Chou and K. F. Kroner (99, "ARCH models in finance: a review of he heory and empirical evidence", Journal of Economerics, 5, 5-59. Bougerol, P and N Picard (99, "Saionariy of GARCH processes and of some nonnegaive ime series", Journal of Economerics, 5, 5-7. Chou, R Y (988, "Volailiy persisence and sock valuaions: some empirical evidence using GARCH," Journal of Applied Economerics, 3, 79-94. Dros, F C and T Niman (993, "Temporal aggregaion of GARCH processes," Economerica,. Engle, R F (98, "Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of UK Inflaion", Economerica, 50, 987-008. Engle, R F and T Bollerslev (986, "Modelling he persisence of condiional variances", Economeric Reviews, 5, -50. Engle, R F and C Musafa (99, "Implied ARCH models from opions prices," Journal of Economerics, 5, 89-3. Lamoureux, C G and W D Lasrapes (990, "Persisence in variance, srucural change and he 36
GARCH model", Journal of Business and Economic Saisics, 8, 5-34. Lee, S-W W and Hansen, B. 994. Asympoic heory for he IGARCH(, quasi maximum likelihood esimaor. Economeric Theory 0: 9-5. Lumsdaine, R.L. 996. Consisency and asympoic normaliy for he quasi maximum likelihood esimaor IGARCH(, and covariance saionary GARCH(, models. Economerica 64: 573-596. Nelson, D B (990a, "Saionariy and persisence in he GARCH(, model," Economeric Theory, 6, 38-334. Nelson, D B (990b, "ARCH models as diffusion approximaions," Journal of Economerics, 45, 7-38. Nelson, D B (990c, "Condiional heeroskedasiciy in asse reurns: a new approach," Economerica, 59, 347-370. Nelson, D B and C Q Cao (99, "Inequaliy consrains in he univariae GARCH model", Journal of Business and Economic Saisics, 0, 9-35. Newey, W.K., and K.D. Wes (987, "A Simple, Posiive, Semi-Definie, Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix", Economerica, 55, 703-708. 37