Econometría Bollerslev GARCH 1986

7/25/2019 Econometra Bollerslev GARCH 1986

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J o u r n a l o f E c o n o m e t r i c s 31 (1 98 6) 3 0 7 - 3 2 7 . N o r t h - H o l l a n d

G E N E R A L IZ E D A U T O R E G R E S S I V E C O N D I T IO N A L

H E T E R O S K E D A S T I C I T Y

T im B O L L E R S L E V *

University of California at San D iego L a Jolla CA 92093 USA

Institute o f Economics University of Aarhus Denmark

Received M ay 1985, f ina l vers ion rece ived Feb rua ry 1986

A n a tu r a l g en e r a l i za t io n o f t h e AR C H ( Au to r eg r e ss iv e C o n d i t i o n a l He te r o sk ed as t i c ) p r o cess

in t r o d u ced in En g le ( 1 9 8 2 ) t o a l l o w f o r p a s t co n d i t i o n a l v a r i an ces i n t h e cu r r en t co n d i t i o n a l

v a r i an ce eq u a t io n i s p r o p o sed . S t a t io n a r i t y co n d i t i o n s an d au to co r r e l a t i o n s t r u c tu r e f o r t h i s n ew

c la ss o f p a r am e t r i c m o d e l s a r e d e r iv ed . Max im u m l ik e l ih o o d e s t im a t io n an d t e s t i n g a r e a l so

co n s id e r ed . F in a l ly an em p i r i ca l ex am p le r e l a t i n g to t h e u n ce r t a in ty o f t h e i n f l a t i o n r a t e i s

p r e s e n t e d .

1 I n t r o d u c t i o n

W h i l e co n v en t i o n a l t i me s e r i e s an d eco n o me t r i c mo d e l s o p e ra t e u n d e r an

a s s u m p t i o n o f c o n s ta n t v ar ia n c e, t h e A R C H A u t o re g r es si v e C o n d i t io n a l

H e t e ro s k e d as t i c ) p ro ces s in t ro d u ced i n E n g le 1 98 2) a l l o w s t h e co n d i ti o n a l

v a r i an ce t o ch an g e o v e r t i me a s a fu n c t i o n b f p a s t e r ro r s l e av i n g t h e u n co n d i -

t i o n a l v a r i an ce co n s t an t .

T h i s t y p e o f mo d e l b eh av i o r h a s a l r ead y p ro v en u s e fu l i n mo d e l l i n g s ev e ra l

d i f fe ren t eco no m ic phen om ena . In Eng le 1982), Eng le 1983) and Engle and

K ra f t 1983) , m ode l s fo r the in f la t ion ra te a re con s t ruc ted recogn iz ing tha t the

u n ce r t a i n t y o f i n f l a t i o n t en d s t o ch an g e o v e r t i me . In Co u l s o n an d Ro b i n s

1 9 8 5 ) t h e e s t i m a t ed i n fl a ti o n v o l a ti li ty i s r e l a ted t o s o m e k ey m ac ro eco n o m i c

v a r i ab l e s . Mo d e l s fo r t h e t e rm s t ru c t u re u s i n g an e s t i ma t e o f t h e co n d i t i o n a l

v a r i an ce a s a p ro x y fo r th e r is k p remi u m a re g iv en in E n g l e, L i li en an d Ro b i n s

1 9 8 5 ) . T h e s ame i d ea i s ap p l i ed t o t h e fo re i g n ex ch an g e mark e t i n D o mo w i t z

a n d H a k k i o 1 98 5). I n W e is s 1 98 4) A R M A m o d e ls w i t h A R C H e r r o rs a re

fo u n d t o b e s u cces s fu l i n mo d e l l in g t h i r teen d i ff e r en t U .S . mac ro ec o n o m i c

t i me s e r i e s . Co mmo n t o mo s t o f t h e ab o v e ap p l i ca t i o n s h o w ev e r , i s t h e

in t roduct ion o f a ra ther a rb i t ra ry l inear dec l in ing l ag s t ruc tu re in the cond i -

* I a m g r a t e f u l t o D a v i d H e n d r y a n d R o b E n g l e f o r i n t r o d u c i n g m e t o t h i s n e w i d e a , a n d t o

R o b En g le f o r m an y h e lp f u l d i scu ss io n s . I wo u ld a l so l i k e t o t h an k Sas t r y Pan tu l a f o r su g g es t in g

t h e a l t e r n a t i v e p a r a m e t e r i z a t i o n , t w o a n o n y m o u s r e f e r e e s f o r u s e f u l c o m m e n t s , a n d K i r s t e n

S ten to f t f o r t y p in g th e m an u sc r ip t . Th e u su a l d i sc l a im er ap p l i e s .

0 3 0 4 - 4 0 7 6 / 8 6 / 3 . 5 0 1 98 6, E l s e v ie r S c ie n c e P u b l i s h er s B .V . ( N o r t h - H o l l a n d )


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3 8 T. Bollerslev Generalization of A R C H process

tional variance equation to take account of the long memory typically found in

empirical work, since estimating a totally free lag distribution often will lead

to violation of the non-negativity constraints.

In this paper a new, more general class of processes, GARCH Generalized

Autoregressive Conditional Heteroskedastic), is introduced, allowing for a

much more flexible lag structure. The extension of the ARCH process to the

GARCH process bears much resemblance to the extension of the standard

time series AR process to the general ARMA process and, as is argued below,

permits a more parsimonious description in many situations.

The paper proceeds as follows. In section 2 the new class of processes is

formally presented and conditions for their wide-sense stationarity are derived.

The simple GARCH 1,1) process is considered in some detail in section 3. It

is well established, that the autocorrelation and partial autocorrelation func-

tions are useful tools in identifying and checking time series behavior of the

ARMA form in the conditional mean. Similarly the autocorrelations and

part ial autocorrelations for the squared process may prove helpful in identify-

ing and checking GARCH behavior in the conditional variance equation. This

is the theme in section 4. In section 5 maximum likelihood estimation of the

linear regression model with GARCH errors is briefly discussed, and it is seen

that the asymptotic independence between the estimates of the mean and the

variance parameters carries over from the ARCH regression model. Some test

results are presented in section 6. As in the ARMA analogue, cf. Godfrey

1978), a general test for the presence of GARCH is not feasible. Section 7

contains an empirical example explaining the uncertainty of the inflation rate.

It is argued that a simple GARCH model provides a marginally better fit and

a more plausible learning mechanism than the ARCH model with an eighth-

order linear declining lag structure as in Engle and Kraft 1983).

2 . The GARCH p ,q ) proces s

The AR CH process introduced by Engle 1982) explicitly recognizes the

difference between the unconditional and the conditional variance allowing the

latter to change over time as a function of past errors. The statistical

properties of this new parametric class of models has been studied further in

Weiss 1982) and in a recent paper by Milhoj 1984).

In empirical applications of the ARCH model a relatively long lag in the

conditional variance equation is often called for, and to avoid problems with

negative variance parameter estimates a fixed lag structure is typically im-

posed, cf. Engle 1982), Engle 1983) and Engle and Kraft 1983). In this light

it seems of immediate practical interest to extend the ARCH class of models

to allow for both a longer memory and a more flexible lag structure.

Let

t

denote a real-valued discrete-time stochastic process, and ~b, the

information set a-field) of all information through time t. The GARCH p, q)


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7~ Bol lerslev , Generalization o f A R C H process

309

p r o c e s s ( G e n e r a l i z e d A u t o r e g r e s s i v e C o n d i t i o n a l H e t e r o s k e d a s t i c i t y ) i s t h e n

g i v e n b y 1

~ ' t l ~ t - 1 - - N 0 , h t ) , 1 )

q P

h t = o r0 + E 2

t iEt- i }- E t~ ih t- i

i = 1 i ~ l

w h e r e

= Ot0 + A ( t ) e 2 t -t- O ( t ) h t ,

p > 0 , q > 0

a o > 0 , a i >_ 0 , i = 1 . . . . q

f l , > 0 , i = 1 . . . . . p .

(2)

F o r p = 0 t h e p r o c e s s r e d u c e s t o t h e A R C H ( q ) p r o ce s s, a n d f o r p = q = 0 E i s

s i m p l y w h i t e n o i s e . I n t h e A R C H ( q ) p r o c e s s t h e c o n d i t i o n a l v a r i a n c e i s

s p e c i f i e d a s a l i n e a r f u n c t i o n o f p a s t s a m p l e v a r i a n c e s o n l y , w h e r e a s t h e

G A R C H ( p , q ) p r o c e s s a l l o w s l a g g e d c o n d i t i o n a l v a r i a n c e s t o e n t e r a s w e l l .

T h i s c o r r e s p o n d s t o s o m e s o r t o f a d a p t iv e l e a r n i n g m e c h a n i s m .

T h e G A R C H ( p , q ) r eg re ss io n m o d el is o b t a in e d b y l et ti n g t he e t ' S b e

i n n o v a t i o n s i n a l i n e a r r e gr e s si o n ,

= y , - x ; b , 3 )

w h e r e Y t i s th e d e p e n d e n t v a r i a bl e , x , a v e c t o r o f e x p l a n a t o r y v a ri a b le s , a n d b

a v e c t o r o f u n k n o w n p a r a m e t e r s . T h i s m o d e l i s s t u d i e d i n s o m e d e t a i l i n

s ec t i o n 5 .

I f a l l t h e r o o t s o f 1 - B ( z ) = 0 l ie o u t s i d e t h e u n i t c ir cl e, ( 2) c a n b e

rewr i t t en a s a d i s t r i b u t ed l ag o f p a s t e~ s ,

h t = O to(1 - B ( 1 ) ) - I + A ( L ) ( 1 - B L ) ) - t e ~

a o 1 - Bi ~ i E 2 - i ,

i ~ 1 i = l

4 )

w h i c h t o g e t h e r w i t h ( 1) m a y b e s e en a s a n i n f in i te - d i m e n s io n a l A R C H ( o o )

p r o c e s s . T h e 8 i s a r e f o u n d f r o m t h e p o w e r s e r ie s e x p a n s i o n o f D L ) =

aW e f o l lo w E n g le 1 98 2 ) i n a s su m in g th e co n d i t i o n a l d i s t r i b u t io n to b e n o r m a l , b u t o f co u r se

o th e r d i s t r i b u t io n s co u ld b e ap p l i ed a s wel l. I n s t ead o f E _ I in eq. 2 ) t h e ab so lu t e v a lu e o f e t -

m ay b e m o r e ap p r o p r i a t e i n so m e ap p f ica t io n s ; c f. McCu l lo ch 1 98 3 ).


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3 1 0 T. Bollerslev, Ge neralizat ion of A R C H process

A ( L ) ( 1 - B ( L ) ) - 1,

j = l

= B s S , - j ,

j = l

i = 1 . . . . . q ,

i = q + l . . . . .

5 )

w he re n = m in ( p , i - 1} . I t fo l lows , tha t if B(1) < 1 , 6i wi l l be dec reas ing fo r i

g r e a t e r t h a n m = m a x { p , q }. T h u s i f D (1 ) < 1, t h e G A R C H ( p , q ) p r o c es s c a n

b e a p p r o x i m a t e d t o a n y d e g re e o f a c c u r a c y b y a s t a t io n a r y A R C H ( Q ) f o r a

s u f fi c ie n t ly l a rg e v a lu e o f Q . B u t a s in t h e A R M A a n a lo g u e , th e G A R C H

p r o c e s s m i g h t p o s s i b l y b e j u s t if i e d t h r o u g h a W a l d ' s d e c o m p o s i t i o n t y p e o f

a r g u m e n t s a s a m o r e p a r s i m o n i o u s d e s c r ip t io n .

F r o m t h e t h e o r y o n f i n it e -d i m e n s io n a l A R C H ( q ) p r oc e ss e s it is t o b e

ex p ec t e d t h a t D(1 ) < 1 , o r eq u i v a l en t l y A (1 ) + B (1 ) < 1 , s u f fi ces fo r w i d e - s en s e

s t a t i o n a r i t y ; c f . M i l h o j (1 98 4) . T h i s i s i n d e ed t h e ca s e.

Th eo r em 1. Th e G A R C H p , q ) p ro cess a s d e f in ed in 1 ) a n d 2 ) i s w id e -s en se

s ta t i o n a r y w i th E e/)=0, v a r ( e / ) = a 0 ( 1 - A ( 1 ) - B (1 )) -1 a n d c o v (e / , e s ) = 0

fo r t -4: s i f and o n ly / f A(1) + B(1) < 1 .

P r o o f . S e e a p p e n d i x .

A s p o i n t e d o u t b y S a s tr y P a n t u l a a n d a n a n o n y m o u s re fe re e , a n e q u i v a l e nt

r e p r e s e n t a t i o n o f th e G A R C H ( p , q ) p r oc e ss is g iv e n b y

a n d

w h e r e

q

e2 = OtO + E Ot i e2-i + E f l j e2 - - j - E f l j l ) t - j + l )t ( 6 )

i = 1 j = l j = l

2 - h = - 1 ) h , ,

, ' = E t

i . i . d .

- N 0 , 1 ) .

(7 )

N o t e , b y d e f i n i t i o n v i s s e r i a l l y u n c o r r e l a t e d w i t h m e a n z e r o . T h e r e f o r e , t h e

G A R C H ( p , q ) p r o c e ss c a n b e i n t e r p re t e d a s a n a u t o r eg r e ss iv e m o v i n g a ve r a ge

p r o c e s s i n e 2 o f o r d e r s m = m a x ( p , q } a n d p , r e s p ec t iv e l y . A l t h o u g h a

p a r a m e t e r i z a t i o n a l o n g t h e l i n e s o f ( 6 ) m i g h t b e m o r e m e a n i n g f u l f r o m a

t h eo re t i ca l t i m e s e r i e s p o i n t o f v i ew , (1 ) an d (2 ) a r e ea s i e r t o wo rk wi t h i n

p rac t i ce .


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T. B ollersleo, Generalization o f A R C H pro cess 311

3 . T h e G A R C H I , 1) p r o ce s s

T h e s i m p l e s t b u t o f t e n v e r y u s e f u l G A R C H p r o c e s s i s o f c o u r s e t h e

G A R C H ( 1 , 1 ) p r o ce s s giv en b y (1 ) a n d

h t _ a01Xlt_ 2 + f l l h t _ l ot 0 > 0 , o/1 >_~ 0 , f l l >-~ 0 . 8 )

F ro m Th eo rem 1 , a 1 + /31 < 1 suf fi ces fo r wide-sense s t a t ionar i ty , and in

g en e ra l w e h av e : 2

Theorem 2.

and suJ~cient condition for existence of the

2 ru t h

mom ent is

g Ol l ,

B1, m) = ~ [ m ~. .~Ja ~-J p 1 ,

i =

( 1 3 )

w h e r e m = m a x ( p , q ) ,

tP i =O t i + f l i ,

i = 1 . . . . . q ,

a i = 0 f o r i > q a n d fl i =-- 0 f o r i > p . F r o m ( 1 3 ) w e g e t t h e fo l l o w i n g a n a l o g u e

t o t h e Y u l e - W a l k e r e q u a t i o n s :

On = [ .~ [01 = ~ ~ i Pn - i , R p + 1 . 1 4 )

i =

T h u s , t h e f i rs t p a u t o c o r r e l a t io n s f o r e 2 d e p e n d ' d i re c t l y ' o n t h e p a r a m e t e r s

ot1 . . . . a tq , f l l . . . . . tip , b u t g i v e n 0 p . . . . 0 p + a - m t h e a b o v e d i f f e r e n c e e q u a t i o n

u n i q u e l y d e t e r m i n e s t h e a u t o c o r r e l a t io n s a t h ig h e r l a gs . T h i s i s s i m i la r t o t h e

r e s u lt f o r th e a u t o c o r r e la t io n s f o r a n A R M A ( m , p ) p r o c es s ; c f, B o x a n d

J e n k i n s ( 1 9 7 6 ). N o t e a ls o , t h a t ( 1 4 ) d e p e n d s o n t h e p a r a m e t e r s ot . . . . . a q ,

f l l . . . . . tip o n l y t h r o u g h ~ x , . . - , ~m

L e t ~ k k d e n o t e t h e k t h p a r t i a l a u t o c o r r e l a t i o n f o r e ~ f o u n d b y s o l v i n g t h e

s e t o f k e q u a t i o n s i n t h e k u n k n o w n ~ k l , . . . , ~ k k :

k

o . = E k o . _ = 1 . . . . k . 1 5 )

i =

B y ( 1 4 ) ~ * * c u t s o f f a f t e r l ag q f o r a n A R C H ( q ) p r o c e s s

~ k k :~ O , k < q ,

( 1 6 )

= 0 ,

k > q .

T h i s i s i d e n t i c a l t o t h e b e h a v i o u r o f th e p a r ti a l a u t o c o r r e l a t io n f u n c t i o n f o r a n

A R ( q ) p r o c e s s . A l s o f r o m ( 1 4 ) a n d w e l l - k n o w n r e s u l t s i n t h e t i m e s e r i e s

l i te r a t u r e , t h e p a r t i a l a u t o c o r r e l a t i o n f u n c t i o n f o r e 2 f o r a G A R C H ( p , q )

p r o c e s s i s i n g e n e ra l n o n - z e r o b u t d i es o u t ; s e e G r a n g e r a n d N e w b o l d ( 1 97 7 ).

I n p r a c t ic e , o f c o u r s e , th e O , 's a n d O k k 'S w i ll b e u n k n o w n . H o w e v e r , t h e

s a m p l e a n a l o g u e , s a y b , , y ie l d s a c o n s i s t e n t e s t i m a t e f o r p ,, a n d O kk i s

c o n s i s t e n t l y e s t i m a t e d b y t h e k t h c o e ff ic i en t , s a y C kk , i n a k t h - o r d e r a u t o r e -


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T. Bol lers lev , General izat ion o f A R C H process

3 1 5

g r e s s i o n f o r e 2; s e e G r a n g e r a n d N e w b o l d ( 1 97 7 ). T h e s e e s t i m a t e s t o g e t h e r

w i t h t h e ir a s y m p t o t i c v a ri a n ce u n d e r t h e n u ll o f n o G A R C H 1 / T [ c f . W e i s s

( 1 9 8 4 ) a n d M c L e o d a n d L i ( 1 98 3 )] c a n b e u s e d i n th e p r e l im i n a r y i d e n ti fi c a-

t i o n s t a g e , a n d a r e a l s o u s e f u l f o r d i a g n o s t i c c h e c k i n g .

5 E s t i m a t i o n o f t h e G A R C H r e g r e s s i o n m o d e l

I n t h is s e c t i o n w e c o n s id e r m a x i m u m l ik e l i h o o d e st im a t i o n o f t h e G A R C H

r e g r e s s i o n m o d e l ( 1 ), ( 2 ), ( 3) . B e c a u s e t h e r e s u l t s a r e q u i t e s i m i l a r t o t h o s e f o r

t h e A R C H r e g r e s si o n m o d e l , o u r d i sc u s s io n w i ll b e v e r y s c h e m a t ic .

L e t z , = (1, e ,_1,2 . . . , e 2_ q, h t _ l , . . . , h t _ p ) , 6 0 = ( o r 0 , O tl , . . . , a q , f l l . . . . t i p )

a n d 0 ~ {9, w h e r e 0 = ( b , w ) a n d O i s a c o m p a c t s u b s p a c e o f a E u c l i d e a n

s p a c e s u c h t h a t e t p o s s e s s e s f in i te s e c o n d m o m e n t s . D e n o t e t h e tr u e p a r a m e -

t e r s b y 0 0, w h e r e 0 0 ~ i n t {9. W e m a y t h e n r e w r i t e t h e m o d e l a s

e t = Y t - x ~ b ,

e tlhb t_ 1 - N (0 , h t , ( 1 7 )

h t = z ;oa.

T h e l o g l i k e l ih o o d f u n c t i o n f o r a s a m p l e o f T o b s e r v a t i o n s i s a p a r t f r o m s o m e

c o n s t a n t , 4

T

LT O) = r E l , o ) ,

t = l

I t O ) = - 1 o g h , - ~tht12 1

1 8 )

D i f f e r e n t i a t i n g w i t h r e s p e c t t o t h e v a r i a n c e p a r a m e t e r y i e l d s

0 6 ~ 1 0 h { ~2 )

~ --d = i h ~ - ~ I g - 1

oo g h ; l

2

h t Oh t e t

0to 0to h t

( 1 9 )

( 2 0 )

4 F o r s i m p l i c i t y w e a r e c o n d i t i o n i n g o n t h e p r e - s a m p l e v a l u e s . T h i s d o e s o f c o u r s e n o t a f f ec t t h e

a s y m p t o t i c r e s u l t s ; c f. W e i s s (1 9 8 2 ).


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T. Bollerslev Generalizat ion of A R C H process

w h e r e

Oh t P O h t _ i

Ooa = z t + E f l i Ooa (21)

i=1

The on ly d i f fe rence f rom Engle (1982) i s the inc lus ion o f the recurs ive par t in

(21) . 5 N ote , B (1 ) < 1 guaran tees tha t (21) is s tab le . S ince the co nd i t iona l

exp ec ta t ion o f the f i r st te rm in (20) is zero , the pa r t o f F i sher s in fo rm at ion

m a t r i x co r r e s p o n d i n g t o ~0 i s co n s i s ten t l y e s ti ma t ed b y t h e s amp l e an a l o g u e o f

the l as t t e rm in (20) which invo lves f i r s t der iva t ives on ly .

D i f f e r en t i a t i n g w i t h r e s p ec t t o t h e mean p a rame t e r s y i e l d s

O lt e t x , h [ 1

+ - 1 (22)

O b = - ~ h , - ~ -h7 ,

0 2 lt ~ - - h t l x t x ~ - - l h t 2 0 h t O h t ~ 2 t t

8 b a b O b a b l h t ]

2

Oh t e2

0 x 1

Oh t

- 2 h i e t x , - ~ + ~ - 1 ~ I ~ h i ~ 1 ,

w h e r e

(23)

Oh , q P O h t - j (24)

= - - 2 E a j X t - - j E t - - j + E J~ j O b

3 b j = l j = l

A g a i n t h e s i n g l e d i f f e r en ce f ro m t h e A RCH (q ) r eg re s s i o n mo d e l i s t h e

inc lus ion o f the recurs ive par t in (24) . A cons i s t en t es t imate o f the par t o f the

i n fo rma t i o n ma t r i x co r r e s p o n d i n g t o b i s g i v en b y t h e s amp l e an a l o g u e o f

t h e f ir s t tw o t e rms i n (2 3 ) b u t w i t h e2 h t x i n t h e s eco n d t e rm r ep l aced b y i ts

expec ted va lue o f one . Th i s es t imate wi l l a l so invo lve f i r s t der iva t ives on ly .

F i n a l l y , t h e e l emen t s i n t h e o f f -d i ag o n a l b l o ck i n t h e i n fo rma t i o n ma t r i x

m ay b e s h o w n t o b e zero. Becau s e o f th i s a s y mp t o t i c i n d ep en d en c e t0 can b e

es t i m a t ed w i t h o u t l o ss o f a s y mp t o t i c e f f ic ien cy b as ed o n a co n s i s t en t e s t im a t e

o f b , an d v i ce v e r sa .

T o o b t a i n max i mu m l i k e l i h o o d e s t i ma t e s , an d s eco n d -o rd e r e f f i c i en cy , an

i t e r a t i v e p ro ced u re i s c a l l ed fo r . Fo r t h e A RCH (q ) r eg re s s i o n mo d e l t h e

m e t h o d o f s co r i n g co u l d b e ex p re s s ed in t e rms o f a s i mp l e au x il ia ry r eg re s s io n ,

b u t t h e r ecu r s i v e t e rms i n (21 ) an d (2 4 ) co m p l i ca te t h is p ro ced u re . I n s t ead t h e

5To st art u p the recursion we need pre-sample estimates for

h t

and et t < 0. A natura l choice is

the sample analogue --xr-r 2

1 Z~t_lE .


11/21

T. Bo l lers lev , Genera l iza tion o f A R C H process 3 7

Bern d t , H a l l , H a l l an d H au s man (1 9 7 4 ) a l g o r i t h m t u rn s o u t t o b e co n v en i en t .

L e t 0 i) de no te the param eter es t imates a f t e r the i th i t e ra t ion . 0 ~+1) i s then

c a l c u l a t e d f r o m

O i + l ) = o i ) + ~ i ~ O l t O lt 1 - 1 ~ - ~ ~ Ol._~t

e = t 0 0 o o , j t = l o o

w h e r e 0 1 , / 0 0 i s eva lua ted a t 0 < i), and ~ i i s a var i ab le s t ep l eng th chosen to

maximize the l ike l ihood func t ion in the g iven d i rec t ion . Note , the d i rec t ion

vec to r i s eas i ly ca lcu la ted f rom a l eas t squares regress ion o f a T 1 vec to r o f

o n e s o n O l t / O O . Also, the i terat ions for w(i~ and b (i) m a y b e c a r r i e d o u t

s ep a ra t e l y b ecau s e o f t h e b l o ck d i ag o n a l i t y i n t h e i n fo rma t i o n ma t r i x .

F r o m W ei s s (19 82 ) it f o ll o w s t h a t t h e max i mu m l ik e l ih o o d e s t i ma t e 0 r i s

s t ro n g l y co n s i s t en t f o r 0 0 an d a s y mp t o t i ca l l y n o rma l w i t h mean 0 0 an d

c o v a r ia n c e m a t r ix - - 1 = _ E ( O 2 l , / O O O O , ) - x . H o w e ve r, ~ - = F , w h e re F =

E ( ( O l t / O 0 ) ( O l t / O O ) ) , an d a co n s i s t en t e s t i ma t e o f t h e a s y mp t o t i c co v a r i an ce

ma t r i x i s t h e r e fo re g i v en b y T - l ( T _ . r = l ( O l t /O O ) ( O l t /O 0 ) ) - 1 f ro m t h e l a s t

B H H H i t e r a t i o n .

Rep l ac i n g (1 ) w i t h t h e w eak e r s e t o f a s s u mp t i o n s

E ( e , l q ,, _ l) = 0 ,

E(e~h~-Xl~b,_l) = 1, (2 5)

E(e4h /E l~p t_ l ) _


12/21

3 1 8 T. Bollerslev Generalization of AR C H process

T h e L ag ran g e m u l t i p l i e r t e s t s t a t i s t i c fo r Ho : ~ o2 = 0 i s t h en g iv en b y 6

1 t t - 1 t

~ L M = 2 f 6 Z o ( Z o Z o ) Z o f o

(27)

w h e r e

f o = ( e ~ h ; 1 - 1 . . . . eZrh~.1 - 1 ) ' ,

c9hl Ohr]

Z o = h i . . . . . ,

(28)

a n d b o t h a r e e v a l u a t e d u n d e r H 0. W h e n H 0 i s t ru e , ~ M i s a s y m p t o t i c a l l y

c h i - s q u a r e w i t h r , t h e n u m b e r o f e l e m e n t s i n ~ 02, d e g r e es o f f r e e d o m . T h i s t e s t

d i f f e r s s l i g h t l y f ro m th e s t an d a rd r e s u l t s , B reu s ch an d Pag an (1 9 7 8 ) , i n t h a t

Oht/a~o d o e s n o t s i m p l i f y w h e n t h e c o n d i t i o n a l v a r i a n c e e q u a t i o n c o n t a i n s

l ag g ed co n d i t i o n a l v a r i an ces ; c f . eq . (2 1 ) .

I t i s w e l l k n o w n t h a t b y n o r m a l i t y a n a s y m p t o t i c a l l y e q u i v a l e n t t e s t s ta t is t ic

is

~LM = T R 2,

w h e re R 2 is t h e s q u a r ed m u l t i p l e co r re l a t i o n co e f f i c ien t b e tw een f0 an d Z 0.

F r o m s e c t i o n 5 t h i s c o r r e s p o n d s t o T . R 2 f r o m t h e O L S r e g r e s s io n i n t h e f ir s t

B H H H i t e r a t i o n f o r t h e g e n e r a l m o d e l s t a r t i n g a t t h e m a x i m u m l i k e l i h o o d

e s t i m a t e s u n d e r H 0.

T h e a l t e r n a t i v e a s r e p r e s e n t e d b y z2t n e e d s s o m e c o n s i d e r a t io n . S t r a i g h t fo r -

w a r d c a l c u l a t i o n s s h o w t h a t u n d e r t h e n u l l o f w h i t e n o i s e , Z 6 Z o i s s i n g u l a r i f

b o t h p > 0 a n d q > 0 , a n d t h e re f o re a g e n er a l te s t f o r G A R C H ( p , q ) i s n o t

f e a s i b l e . I n f a c t i f t h e n u l l i s a n A R C H ( q ) p r o c e s s , Z 6 Z o i s s i n g u l a r fo r

G A R C H (r l , q + r2) a l t e rn a t i v e s , wh e re r 1 > 0 an d r 2 > 0 . I t i s a l so i n t e re s t i n g

t o n o t e t h a t f o r a n A R C H ( q ) n u l l , t h e L M t e s t f o r G A R C H ( r , q ) a n d

A R C H ( q + r ) a l t e rn a t i v e s c o i n c id e . T h i s is s im i l a r t o t h e r e s u l ts in G o d f r e y

( 1 9 7 8 ) , w h e r e i t i s s h o w n t h a t t h e L M t e s t s f o r A R ( p ) a n d M A ( q ) e r r o r s i n a

l i n e a r r e g r e s s i o n m o d e l c o i n c i d e a n d t h a t t h e t e s t p r o c e d u r e s b r e a k d o w n

w h e n a f u l l A R M A ( p , q ) m o d e l is c o n s i d e r e d . T h e s e t e s t r e s u lt s a re , o f c o u r se ,

n o t p e c u l i a r t o t h e L M t es t, b u t c o n c e r n t h e L i k e l ih o o d R a t i o a n d t h e W a l d

t e s t s a s w e ll . A f o r m a l p r o o f o f t h e a b o v e s t a t e m e n t s c a n b e c o n s t r u c t e d a l o n g

th e s am e l i n e s a s i n G o d f re y (1 97 8, 1 98 1) .

6 N o t e b e c a u s e o f t h e b l o ck d i a g o n a l it y in t h e i n f o r m a t i o n m a t r i x i n t h e G A R C H r e g re s s io n

m o d e l t h e s a m e t e s t a p p l i e s i n t h is m o r e g e n e r a l c o n t e x t.


13/21

T. Bollerslev Generalization of AR C H process 319

7 Em pirical example

T h e u n ce r t a i n t y o f i n f l a t i o n i s an u n o b s e rv ab l e eco n o mi c v a r i ab l e o f ma j o r

i m p o r t a n c e , a n d w i t h i n t h e A R C H f r a m e w o r k s e v e r a l d i f f e r e n t m o d e l s h a v e

a l re ad y bee n cons t ruc ted ; see Eng le (1982), Eng le (1983) and Engle and K raf t

(1983) . We wi l l here concen t ra te on the model in Eng le and Kraf t (1983)

w h ere t h e r a t e o f g ro w t h i n t h e i mp l i c i t G N P d e f l a t o r i n t h e U .S . i s ex p l a i n ed

i n t e rms o f i t s o w n p as t .

L e t ~ r t= 1 0 0 . I n G D t /G D t _ I) w h e r e

G t

i s the impl ic i t p r i ce def l a to r fo r

G N P. 7 S t an d a rd u n i v a r ia t e t ime s er ie s m e t h o d s l ead t o i d en t if i c at io n o f t h e

fol lowing model for ~r t :

.B

=

0.240 + 0 .552~t 1 + 0 .177 r 2 + .0 .23 2~t-3 - 0 .209rt 4 +

et,

( 0.0 80 ) ( 0 . 0 8 3 ) - ( 0 . 0 8 9 ) - (0 .0 90 ) ( 0 . 0 8 0 ) -

h t

0 . 2 8 2 . ( 2 9 )

(0 .034)

The model i s es t imated on quar t e r ly da ta f rom 1948 .2 to 1983 .4 , i . e . , a to t a l o f

143 observa t ions , us ing o rd inary l eas t squares , wi th OLS s tandard e r ro rs in

pare n the ses . 8 Th e m odel i s s t a t ionary , and non e o f the f i rs t ten au toc or re la -

t ions o r par t i a l au toco r re la t ions fo r E a re s ign i f ican t a t the 5% leve l. H ow ever ,

look ing a t the au toco r re la t ions fo r e t: i t t u rns o u t tha t the 1s t, 3 rd , 7 th , 9 th an d

10 th a l l exceed two asympto t i c s t andard e r ro rs . S imi la r resu l t s ho ld fo r the

p a r t ia l a u t o c o r r e la t io n s f o r e t . T h e L M t es t f o r A R C H ( l ) , A R C H ( 4 ) a n d

ARCH(8) a re a l so h igh ly s ign i f i can t a t any reasonab le l eve l .

Th i s l eads Eng le and Kraf t (1983) to sugges t the fo l lowing spec i f i ca t ion :

rt= 0 .138 + 0 .42 3% x+ 0 .222r t 2 + 0 .37 7~ ,_ 3 - 0 .175~/ 4 - ~ - g t ,

(0 .0 59 ) ( 0 . 0 8 1 ) - ( 0 . 1 0 8 ) - (0 .0 78 ) ( 0 . 1 0 4 ) -

8

h t

= 0 .058 ~_ 0 .802 E ( 9 -

i ) / 36e ~ _ i .

(30)

( 0 . 0 3 3 ) - ( 0 . 2 6 5 ) i = 1

T h e e s t i ma t e s a r e max i mu m l i k e l i h o o d w i t h h e t e ro s k ed as t i c co n s i s t en t s t an -

dard e r ro rs in paren theses ; see sec t ion 5 . The cho ice o f the e igh th -order l inear

d ec l i n i n g l ag s t ru c t u re i s r a t h e r ad h o c , b u t mo t i v a t ed b y t h e l o n g memo ry i n

t h e co n d i t i o n a l v a r i an ce eq u a t i o n . T h e o rd e r o f t h e l ag p o l y n o mi a l may b e

v i ew ed a s an ad d i t i o n a l p a r ame t e r i n t h e co n d i t i o n a l v a r i an ce eq u a t i o n . F ro m

7 T h e v a lu es o f GD are t a k e n f r o m t h e C i t i b a n k E c o n o m i c D a t a b a s e a n d U . S . D e p a r t m e n t o f

C o m m e r c e ,

Survey of Current Business

Vol . 64 , no . 9 , Sep tember 1984 .

g N o te , i n E n g le an d K r a f t 1 98 3 ) t h e e s t im a t io n p e r io d is 1 94 8 .2 to 1 9 80 .3 acco u n t in g f o r t h e

sm a l l d i f f e r en ces i n t h e e s t im a t io n r e sul t s.


14/21

320

T. Bollerslev Generalization of A R CH process

- - S A R C H f l, 1)

- - - - RRH [8

\

\

~ \ \ ' \ \ \

. \ \

- 4 \

g

; t

8 1 ~ 0 l~ 2 I ~ 5

Fig. 2. Lag distribution.

M i lh e j (1 98 4) th e co n d i t io n fo r ex is t en ce o f t h e f o u r th - o r d e r m o m en t o f e is

ju s t m e t . 9 N o n e o f t h e f ir s t t en au to co r r e l a t io n s o r p a r t i a l au to co r r e l a t io n s f o r

e t h t 1 / 2 r

e 2 h t ex ceed 2 /v C l~ . Th e L M te s t s t at is t ic s f o r t h e li n ea r

r e s t r i c t i o n t ak es th e v a lu e 8 .8 7 , co r r e sp o n d in g to th e 0 .7 4 f r ac t i le i n t h e X2

d i s t r ib u t io n . Ho wev e r , t h e LM te s t s t a t i s t i c f o r t h e in c lu s io n o f h t _ in the

co nd i t ion a l var ianc e equa t ion is 4 .57 , wh ich i s s ign i fican t a t the 5 leve l.

9The cond ition as discussed n footnote 3 takes the v alue 0.989 for 0.802.


15/21

T. Bollersleo Generalization of A R C H process 321

L e t u s t h e r e f o r e c o n s i d e r t h e a l t e r n a t i v e s p e c i f i c a t i o n

~r = 0 .141 + 0 .433~ _ 1 + 0 .2 29 y t_ 2 + 0 ,34 9 ~rt_ 3 - _y r 0 . 1 6 2 + e t ,

( 0.0 60 ) ( 0 . 0 8 1 ) ( 0 . 1 1 0 ) ( 0.0 77 ) ( 0 . 1 0 4 ) ` -4

h t = 0 .0 07 + 0 .13 5 e l 1 + 0 .82 9 h t 1-

( 0 .0 0 6 ) ( 0 . 0 7 0 ) - ( 0. 06 8 ) -

3 1 )

F r o m T h e o r e m 2 t h e f o u r t h - o r d e r m o m e n t o f e e x is ts . A g a i n n o n e o f t h e f ir st

t e n a u t o c o r r e l a t i o n s o r p a r t ia l a u t o c o r r e l a ti o n s f o r

t t h

t x/2 or

e E h

[ a e x c e e d

t w o a s y m p t o t i c s t a n d a r d e r ro r s. T h e L M t e st s ta t is ti c f o r t h e in c l u s io n o f th e

e i g h t h - o r d e r l i n e a r d e c l i n i n g l a g s t r u c t u r e i s 2 . 3 3 , c o r r e s p o n d i n g t o t h e 0 . 8 7

f r a c t il e i n t h e X 2 d i s tr i b u ti o n . T h e L M t e st s t at is t ic f o r G A R C H ( 1 , 2 ) , o r

l o c a l l y e q u i v a l e n t G A R C H ( 2 , 1 ) , e q u a l s 3 .8 0 a n d t h e r e f o r e is n o t s i g n if ic a n t a t

t h e 5 % l e v e l. A l s o t h e L M t e s t st a t is t ic s f o r i n c l u s i o n o f

e t _ 2 2 . . . e t _ 5 2

t a k e s

t h e v a l u e 5 . 5 8 w h i c h i s e q u a l t o t h e 0 .7 7 f r a c t i l e i n t h e X 2 d i s t r i b u t i o n .

Inf fat ion R c L t e

C o n f . ~ nt .

j t * J \

, ,

, , , a . , , :

T . . , . , . , . . . , . , . . , . . . , . . . ,

1 9 4 8 1 9 5 3 1 9 5 5 1 9 6 3 1 9 6 8 1 9 7 3 1 9 7 8 1 9 8 3 1 9 8 5

Fig . 3 . 95 con f iden ce in terva ls fo r OLS .


16/21

322

T. Bollerslev Generalization of A R C H process

I t i s a l s o i n t e re s t i n g t o n o t e t h a t t h e s a m p le co e f f i c ien t o f k u r to s i s fo r

e h l 1 /2

f r o m m o d e l ( 31 ) e q u a l s 3.8 1, w h i c h d i f fe r s f r o m t h e ' n o r m a l ' v a l u e o f

3 .0 0 b y s l i g h t l y l es s t h a n t w o a s y m p t o t i c s t a n d a r d e r r or s ,

2 2 ~ / T

= 0.82. For

m o d e l s (2 9 ) a n d (3 0) t h e co e f f i c i en t o f k u r to s i s eq u a l s 6 . 90 a n d 4 .0 7, r e s p ec -

t i v e ly . T h e s am p le co e f f i c ien t o f s k ew n es s fo r

e t h t ] /2

f r o m e a c h o f t h e t h r e e

m o d e l s , - 0 . 1 3 , 0 . 1 8 a n d 0 . 1 1 , a r e a l l w i t h i n o n e a s y m p t o t i c s t a n d a r d e r r o r ,

V / 6 / T

= 0.20.

T h e m e a n a n d m e d i a n l a g i n t h e c o n d i t i o n a l v a r i a n c e e q u a t i o n i n ( 3 1 ) a r e

es t im at ed to be 5 .848 a nd 3 .696, respe c t ive ly [cf . s~c t ion 3] , w here as in (30) the

m e a n l a g i s f o r c e d t o 3 ~ a n d t h e m e d i a n l a g t o 2 . F u r t h e r m o r e , t h e l a g

s t r u c t u r e i n t h e G A R C H ( 1 , 1 ) m o d e l c a n b e r a t i o n a l i z e d b y s o m e s o r t o f

ad ap t i v e l ea rn in g m ech an i s m . See a l s o f i g . 2 , wh e re t h e two d i f f e ren t l ag

s h a p e s a r e i l l u s t ra t e d . I n t h i s l ig h t i t se e m s t h a t n o t o n l y d o e s t h e G A R C H ( 1 , 1 )

m o d e l p r o v i d e a s l i g h tl y b e t t e r f it t h a n t h e A R C H ( 8 ) m o d e l in E n g l e a n d

K r a f t ( 19 8 3 ), b u t i t a ls o e x h ib i ts a m o r e r e a s o n a b l e l a g s tr u c t u r e .

- - I n f l a t i o n

B a t e

-----Co nf. Int

o

J ~

J l I/[ I ~ ~\ r,k~ v , v ~ / ~-~^/~(~A / J

V I , , , I

I L I \ l

Ill Ij~

1 9 4 8 1 9 5 3 1 9 5 9 1 9 6 3 1 9 9 9 1 9 7 3 1 9 7 8 1 9 9 3 1 9 9 9

Fig. 4. 95 confid ence nterva ls for GA RC H(1,1).


17/21

T. Bollerslev,Generafizationo f A R C H process 323

In f igs . 3 an d 4 the actu al inf lat ion rate , ~r,, i s gra ph ed togeth er wi th 95

as y mp t o t i c co n f i d en ce i n t e rv a l s f o r t h e o n e - s t ep -ah ead fo r ecas t e r ro r s fo r t h e

two model s (29) and (31) . From the l a t e fo r t i es un t i l t he mid- f i f t i es the

in f l a t ion ra t e wa s very vo la t i le and hard to p red ic t . Th i s i s re f lec ted in the wide

co n f i d en ce i n t e rv a l s f o r t h e G A RCH mo d e l . T h e s i x t i e s an d ea r l y s ev en t i e s ,

h o w ev e r , w e re ch a rac t e r ized b y a s t ab l e an d p red i c t ab l e i n fl a ti o n r a te , an d t h e

O L S co n f i d en ce i n t e rv a l s eems mu ch t o o w i d e . S t a r t i n g w i t h t h e s eco n d o i l

cr ises in 1974 there i s a s l ight increase in the uncertainty of the inf lat ion rate ,

a l t h o u g h i t d o es n o t co mp are i n mag n i t u d e t o t h e u n ce r t a i n t y a t t h e b eg i n n i n g

o f t h e s amp l e p e r i o d .

Appendix

A.1. Pro of of Theorem 1

T h e b as i c i d ea o f t h e p ro o f fo l l o w s t h a t o f T h eo rem 1 i n Mi l h o j ( 1 9 8 4 ) . By

d e f i n i t i o n

= ,n / ,1 / 2 i id

Et 't t '

]t-

N 0 , 1 ) . A . I )

Su b s eq u en t s u b s t i t u t i o n y i e l d s

q P

h , = + E + E , h , _ ,

i = 1 i = 1

q 2 q P )

= a 0 + } - . a s ~ , _ j a 0 + ~ 2

l i ~ t - i - j h t - i - j + E j ~ i h t - i - j

j = l i = 1 i =1

P ( q P )

+ E f l j a o + E 2 ( A .2 )

l i ~ t - i - j h t - i - j + E ~ i h t - i - j

j = l i = 1 i = 1

a o ~

M t ,k ) ,

k=O

w h e r e M t , k ) i nvo lves a l l t he t e rms o f the fo rm

q P n

1 - - I < , : ' H ,8 7 , r i , g _ , , ,

i = 1 j = l 1 : 1


18/21

324 T . B o ll er sl eo , G e n e r a l i z a ti o n o f A R C H p r o c e s s

f o r

a n d

T h u s ,

q P q

E a , + E b j = k , E a i = n ,

i = 1 j = l i = 1

1 < S 1 < 2 < . . . < S , < m a x k q , k - 1 ) q + p } .

M t , O ) = 1 ,

q P

M ( t , 1 ) = E ai~2 -i + E fl,,

i = 1 i = 1

M ( t , 2 ) E 2

l j ~ t - j E 2

O l i ' l ~ t - i - j + E f l i

j = l i = 1 i = 1

O l i | t - - i - - j i ,

i

a n d i n g e n e r a l

q P

M t , k + 1) = E af l l~ - iM t - i, k ) + E f liM t - i , k ) .

i = 1 i = 1

(A.3 )

Since ,12 is i . i .d . , the moments of M t , k ) d o n o t d e p e n d o n t , a n d i n

pa r t i cu la r

E ( M ( t , k ) ) = E ( M ( s , k ) ) f or

all

k, t , s , (A.4)

F r o m ( A .3 ) a n d ( A .4 ) w e g e t

E M t , k + l ) ) = Y [ a i+ ~=lfl E M t , k ) )

i=1 i

= a , f l i E M / ,0 ) )

i~ l i

I l i i

i i

A.5)


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T. Bollerslev Generalization of A R C H process 325

F i n a l l y b y ( A . 1 ) , ( A . 2 ) a n d ( A . 5 ) ,

i f a n d o n l y i f

o~

E(e~) =aoE(k~_~=o ( t ' k , )

= a o Y . E ( M ( t , k ) )

k = 0

1

= a o 1 - a i - B i

i ~ i = 1

q P

E a i + E E < I ,

i = 1 i = 1

2 c o n v e r g e s a l m o s t s u re ly .

n d e ,

E ( e , ) = 0 a n d c o v ( e , e s) = 0 f o r t ~ s f o ll o w s i m m e d i a t e l y b y s y m m e t r y .

A . 2 . P r o o f o f T h e o r e m 2

B y n o r m a l i ty

E ( e 2 m ) = a m E ( h ~ ) ,

w h e r e a m i s d e f i n e d i n ( 1 0 ) . T h e b i n o m i a l f o r m u l a y i e l d s

h ~ = ( ~ o + ~ : ,~ _ ~ + ~ h , _ ~ ) ~

~ n ~ 0 ~ lb ' l f ' t - t t - l

B e c a u s e

E (e~J th t -_~[~b ,_2)

= a j h t_t,

w e h a v e

E ( h : [ ~ b , _ 2 ) - -

~ h ,_ (m)n aom -, ~ (nj .)aja ~fl J .

n = 0 j = 0

L e t

w , = ( k i n h m 1

_ , , . ., . . . . . h , ) , t h e n b y ( A . 8 )

( A . 6 )

( A . 7 )

( A . 8 )

E ( w t i ~ b , _ 2 )

= d + C w , _~ ,

( A . 9 )


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326

T. Bol lers lev , Generalization o f A R C H process

w h e r e C i s a n m m u p p e r t r i a n g u l a r m a t r i x w i t h d i a g o n a l e l e m e n t s .

J R ~ I , ~ I , / ) : ~ ( l ; ] 1 r ~ j l~ i - J

j - O J / ' J ~ l t l ,

i = 1 . . . . . m . ( A . I O )

S u b s t i t u t i n g i n A . 9 ) y i el d s

E w,l~,_k_l) = I+ C+ C2+ . . . +Ck- X)d+ Cew _k.

S i n c e t h e p r o c e s s i s a s s u m e d t o s t a r t i n d e f i n i t e l y f a r i n t h e p a s t w i t h f i n i te 2 m

m o m e n t s , t h e l im i t a s k g o e s t o in f i n i t y e x i st s a n d d o e s n o t d e p e n d o n t i f a n d

o n l y i f a l l t h e e i g e n v a l u e s o f C l ie i n s i d e t h e u n i t c i r c l e ,

l i m E ( w t l ~ b t _ k _ l ) = I - C ) - l d =

E w t ) .

k ~ m

B e c a u s e C i s u p p e r t r ia n g u l a r , t h e e i g e n v a lu e s a re e q u a l t o th e d i a g o n a l

e l e m e n t s a s g i v e n i n (A . 10 ) . T e d i o u s , b u t r a t h e r s t r a i g h t f o r w a r d c a l c u l a t io n s

s h o w t h a t ~ ~ 1, El , i ) < 1 i m p l i e s /.t(O tl, i l l , i - 1 ) < 1 fo r o t1 -4- E1 ~ 1 , a n d

t ~ a , f l l, m ) < 1 su f f ic e s f o r t h e 2 m t h m o m e n t t o e x i s t ; c f. fig . 1 .

F i n a l l y ( 1 1 ) f o l lo w s f r o m ( A . 7 ) a n d ( A . 8) b y r e a r r a n g i n g t e rm s .

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Econometría Bollerslev GARCH 1986

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