GARCH_Bollerslev_1986

8/2/2019 GARCH_Bollerslev_1986

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Jo u r n a l o f Eco n o met r i c s 3 1 ( 1 9 8 6 ) 3 0 7 - 3 2 7 . No r th - Ho l l an 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 LH 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 Diego, La Jolla, CA 92093, USAInstitute of Econom ics, University of Aarhus, Denm ark

Received May 1985 , f ina l vers ion rece ived February 1986A n a t u r a l g e n e r a l i z a t i o n o f t h e A R C H ( 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 ) p r o c e s sin t r o d u ced in En g le ( 1 9 8 2 ) to a l lo w f o r p as t co n d i t io n a l v a r i an ces in th e cu r r en t co n d i t io n a lv a r i an ce eq u a t io n i s p r o p o sed . S ta t io n a r i ty co n d i t io n s an d au to co r r e l a t io n s t r u c tu r e f o r t h i s n ewc la ss o f p a r ame t r i c mo d e l s a r e d e r iv ed . Max imu m l ik e l ih o o d e s t ima t io n an d t e s t in g a r e a l soco n s id e r ed . F in a l ly an emp i r i ca l ex amp le r e l a t in g to th e u n ce r t a in ty o f t h e in f l a t io n r a t e i sp r e s e n t e d .

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

Wh i l e co n v en t i o n a l t i m e s e r i e s an d eco n o m et r i c m o d e l s o p e ra t e u n d e r ana s s u m p t i o n o f c o n s t a n t v a r i a n c e , t h e A R C H ( A u t o r e g r e s s i v e C o n d i t i o n a lHe t e ro s k ed as t i c ) p ro ces s i n t ro d u ced i n En g l e (1 9 8 2 ) a l l o ws t h e co n d i t i o n a lv a r i an ce t o ch an g e o v e r t i m e a s a fu n c t i o n b f p as t e r ro r s l eav 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 .

Th i s t y p e o f m o d e l b eh av i o r h as a l r ead y p ro v en u s e fu l i n m o d e l l i n g s ev e ra ld i f feren t econ om ic phen om ena. In Engle (1982), Engle (1983) and E ngle andKraf t (1983) , models fo r the in f la t ion ra te a re cons t ruc ted recogniz ing tha t theu 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 m e . 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 a cro eco n o m i cv a r i ab l e s . M o 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 m a t e o f t h e co n d i t i o n a lv a r i an ce a s a p ro x y fo r t h e r is k p rem i u m a re g iv en in En g l e , L il ien an d Ro b i n s(1 9 8 5 ) . Th e s am e 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 m ark e t i n Do m o wi t zan d H ak k i o (19 85 ). In W ei s s (1 98 4) AR M A m o d e l s w i t h A R C H e r ro rs a r efo u n d t o b e s u cces s fu l i n m o d e l l in g t h i r teen d i ff e ren t U .S . m acro eco n o m i ct i m e s e r i e s . Co m m o n t o m o s t o f t h e ab o v e ap p l i ca t i o n s h o wev er , i s t h ein t roduct ion of a ra ther a rb i t ra ry l inear dec l in ing l ag s t ruc ture in the condi -

* 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 oR o b En g le f o r man 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 to th an k Sas t r y Pan tu la f o r su g g es t in gt 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 nS ten to f t f o r t y p in g th e man u sc r ip t . Th e u su a l d i sc l a imer 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 i er S c i e nc e P u b l is h e r s B .V . ( N o r t h - H o l l a n d )


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308 T. Bollerslev, Generalization of A R C H process

tional variance equation to take account of the long memory typically found inempirical work, since estimating a totally free lag distribution often will leadto violation of the non-negativity constraints.In this paper a new, more general class of processes, GARCH (GeneralizedAutoregressive Conditional Heteroskedastic), is introduced, allowing for amuch more flexible lag structure. The extension of the ARCH process to theGARCH process bears much resemblance to the extension of the standardtime 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 isformally presented and conditions for their wide-sense stationarity are derived.The simple GARCH(1,1) process is considered in some detail in section 3. Itis well established, that the autocorrelation and partial autocorrelation func-tions are useful tools in identifying and checking time series behavior of theARMA form in the conditional mean. Similarly the autocorrelations andpartial autocorrelations for the squared process may prove helpful in identify-ing and checking GARCH behavior in the conditional variance equation. Thisis the theme in section 4. In section 5 maximum likelihood estimation of thelinear regression model with GARCH errors is briefly discussed, and it is seenthat the asymptotic independence between the estimates of the mean and thevariance parameters carries over from the ARCH regression model. Some testresults 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 7contains an empirical example explaining the uncertainty of the inflation rate.It is argued that a simple GARCH model provides a marginally better fit anda more plausible learning mechanism than the ARCH model with an eighth-order linear declining lag structure as in Engle and Kraft (1983).

2 . T h e G A R C H ( p , q ) p r oc es sThe ARCH process introduced by Engle (1982) explicitly recognizes the

difference between the unconditional and the conditional variance allowing thelatter to change over time as a function of past errors. The statisticalproperties of this new parametric class of models has been studied further inWeiss (1982) and in a recent paper by Milhoj (1984).In empirical applications of the ARCH model a relatively long lag in theconditional variance equation is often called for, and to avoid problems withnegative variance parameter estimates a fixed lag structure is typically im-posed, cf. Engle (1982), Engle (1983) and Engle and Kraft (1983). In this lightit seems of immediate practical interest to extend the ARCH class of modelsto allow for both a longer memory and a more flexible lag structure.Let e t denote a real-valued discrete-time stochastic process, and ~b, theinformation set (a-field) of all information through time t. The GARCH( p, q)


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7~ Bollerslev, Generalization o f A R C H process 309p 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 ng i v e n b y 1

~ ' t l ~ t - 1 - - N ( 0 , h t ) , ( 1 )q P

h t = o r0 + E 2t iEt- i "}- E t~ ih t- ii = 1 i ~ l

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

p > 0 , q > 0a 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 ss 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 ss 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 ss p ec i f i ed a s a l i n ea r f u n c t i o n o f p as t s amp le v a r i an ces o n ly , w h er eas t h eG 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 ti v e l e a r n in g m e c h a n i sm .

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 ei n n o v a t i o n s i n a l i n e a r re g r es s io 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 ia b l 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 ba 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 ns e c 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 er ew r i t t en a s a d i s t r i b u t ed l ag o f p as 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 e n a s a n i n f i n i te - d i m e n s i o 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 i e s e x p a n s i o n o f D ( L ) =

a W e f o l l o w E n g l e ( 1 98 2 ) i n a s s u m i n g t h e c o n d i t i o n a l d i s t r i b u t i o n t o b e n o r m a l , b u t o f c o u r s eo th e r d i s t r i b u t io ns c ou ld b e a pp l i e d a s we ll . I ns t e a d o f E _ I i n e q . ( 2 ) the a bso lu t e va lue o f e t -m a y b e m o r e a p p r o p r i a t e i n s o m e a p p f i c a ti o n s ; cf. M c C u l l o c h (1 98 3 ).


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3 1 0 T. Bollerslev, Generalizat ion o f A R C H processA ( L ) ( 1 - B ( L ) ) -1 ,

j = l

= B s ' S , - j ,j = l

i = 1 . . . . . q ,

i = q + l . . . . .( 5 )

w h ere n = m i n ( p , i - 1 }. I t f o l lo ws , t h a t i f B (1 ) < 1 , 6 i w i l l b e d ec rea s i n g fo r ig 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 nb 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 ti o n a r y A R C H ( Q ) f o r as 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 Hp 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 fa r g u m e n t s a s a m o r e p a r s im 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 eex 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 c es fo r w i d e - s en s es t a t i o n a r i t y ; c f . Mi l h o j (1 9 8 4 ) . Th i s i s i n d eed t h e ca s e .Th eo r em 1 . Th e GA R C H ( p , q ) p r o ces s a s d e f in ed in (1 ) a n d (2 ) i s w i d e -s en ses t a t i o n a r y w i t h E(e/)=0, v a r ( e / ) = a 0 ( 1 - A ( 1 ) - B (1 )) -1 a n d c o v ( e / , e s ) = 0fo 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 ntr 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 P Pe2 = 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 ti . 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 eG 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 gep 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 ap 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 at h eo re t i c a l t i m e s e rie s" p o i n t o f v iew , (1 ) an d (2) a r e ea s i e r t o w o rk w i t h i np r a c t i c e .


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T. Bollersleo, Generalization o f A R C H pro cess 31 1

3 . T h e G A R C H ( I , 1 ) p r o ce s sT 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 g iv en b y (1 ) a n dh 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 ing en e ra l w e h av e : 2Theorem 2.and suJ~cient condition for existence of the 2ru th moment is

g (Ol l , B1, m ) = ~ [ m ~. .~Ja '~-J p + 1 ,i =1

( 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 f li =-- 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 et 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 P n- i, R > p + 1 . ( 1 4 )i =1

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 sot1 . . . . 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 nu 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 er e s u l t f o r th e a u t o c o r r e l a t i o n s f o r a n A R M A ( m , p ) p r o c e s s ; c f, B o x a n dJ 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 es 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 :

ko . = E , k , o . _ , , . = 1 . . . . , k . ( 1 5 )

i =1

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 nA 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 sl 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 es a m p l e a n a l o g u e , s a y b , , y i e 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 sc 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. B ol lerslev , General iza tion o f AR 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 rw i t h t h e i r a s y m p t o t i c v a r ia n c e 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 so 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 . Es t im a t io 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 lI n t h i s se c t io 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 s ti m a t io 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 rt 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 t l, . . . , 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 ns 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 ,et lhbt_ 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 ec o n s t a n t , 4TLT(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 h' j2O h t Oh t e t

0to 0 to ' 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 ea 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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316 T. Bollerslev, Generalizat ion o f A R C H process

w h e r eO h t P O h t _ iOoa = z t + E f l i Ooa (21)i=1

The on ly d i f ference f rom Engle (1982) i s the inc lus ion of 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 ndi t ionalexpecta t ion of the f i r s t t e rm in (20) i s zero , the par t o f Fi sher ' s in format ionm a t r i x co r re 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 m a t ed b y t h e s am p l e an a l o g u e o fthe l as t t e rm in (20) which involves f i r s t der iva t ives on ly .

D i f f e ren t i a t i n g w i t h r e s p ec t t o t h e m ean p a ram e 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 t8 b a b ' O b a b ' l h t ]2 Oh t e2 ) 0 x 1 O h t- 2 h i e t x , - ~ + ~ - 1 ~ I ~ h i ~ 1 ,

w h e r e

(23)

O h , q P O h t - j (24)= - - 2 E a j X t - - j E t - - j + E J~ j O b3 b j = l j = lAg ai n t h e s i n g l e d i f f e ren ce f ro m t h e ARCH(q ) r eg res s i o n m o d e l i s t h einc lus ion of the recurs ive par t in (24) . A cons i s ten t es t imate o f the par t o f thei n fo rm a t i o n m a t r i x co r re s p o n d i n g t o b i s g i v en b y t h e s am p l e an a l o g u e o ft h e f ir s t two t e rm s 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 rep l aced b y i tsexpected va lue of one . This es t imate wi l l a l so involve f i r s t der iva t ives on ly .F i n a l l y , t h e e l em en 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 rm a t i o n m a t r i xm ay b e s h o w n t o b e zero. Becau s e o f th i s a s y m p t o t i c i n d ep en d en ce t0 can b ees t i m a t ed w i t h o u t l o ss o f a s y m p t o t i c e f fi ci ency b as ed o n a co n s i s t en t e s t im a t eo f b , an d v i ce v e r sa .To o b t a i n m ax i m u m l i k e l i h o o d e s t i m a t es , an d s eco n d -o rd e r e f f i c i en cy , ani t e r a t i v e p ro ced u re i s ca l l ed fo r . Fo r t h e ARCH(q ) r eg res s i o n m o d e l t h em e t h o d o f s co r i n g co u l d b e ex p res s ed in t e rm s o f a s i m p l e au x il ia ry r eg res s io n ,b u t t h e r ecu r s i v e t e rm s i n (2 1 ) an d (2 4) co m p l i ca t e th i s p ro ced u re . In s t ead t h e

5To star t u p the recursion we need pre-sample estimates for h t and et , t < 0. A natural choice isthe sample analogue ,,--xr -r 21 Z~ t _ l E .


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T. Bo l lers lev , Genera l iza tion o f A R C H process 317Bern d t , Ha l l , Ha l l an d Hau s m an (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 te r the i th i t e ra t ion . 0 ~+1) i s thenc a l c u l a t e d f r o m

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


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3 18 T. Bollerslev, Generalization of A RC H process

T h e L a g r a n g e m u l t i p li e r te s t s ta t is t ic f o r H o : ~o2 = 0 i s t h e n 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 ef 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 ych i - s q u a r e w i th r , t h e n u m b e r o f e lem en t s i n ~02, d eg r ees o f fr eed o m . T h i s t e s td i f f e r s s l i g h t l y f r o m th e s t an d a r d r e s u l t s , B r eu s ch an d Pag an ( 1 9 7 8 ) , i n t h a tOht/a~o d o e s n o t s i m p l i fy 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 sl 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 ll 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 icis

~LM = T" R 2,w h er e R 2 is t h e s q u a r e d m u l t i p l e co r r e l a t i o n co e f f i c i en t b e tw een f 0 an d Z 0.F r o m s e c t i o n 5 t h i s c o r re 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 i rs tB 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 de s t im 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 z2 t 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 tf o 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 se , Z 6 Z o i s s ingu lar i fb o t h p > 0 a n d q > 0 , a n d t h e r e fo r e a g e n e r a l t e st fo r G A R C H ( p , q ) is n o tf 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 f o rG A R C H ( r l , q + r2 ) a l t e r n a t iv es , w h er e r 1 > 0 an d r 2 > 0 . I t i s a l s o i n t e r e s t i n gt 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 dA 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 i n 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 al 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 nw 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 th e L i k e l i h o o d R a t i o a n d t h e W a l dt e s t s a s w e ll . A f o r m a l p r o o f o f th 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 gth e s am e l i n es a s i n G o d f r ey (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 c k d i a g o n a l i t y i n t h e i n f o r m a t i o n m a t r i x in t h e G A R C H r e g r e s si o nm 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 i s m o r e g e n e r a l c o n te x t .


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T. Bollerslev, Generalization of A R C H process 319

7. Em pirical examp leTh 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 m i c v a r i ab l e o f m a 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 eal re ad y bee n cons t ruc ted ; see Engle (1982), Engle (1983) and Engle and K raf t(1 9 8 3 ) . We wi l l h e re co n cen t r a t e o n t h e m o d e l i n En g l e an d Kra f t (1 9 8 3 )wh ere t h e r a t e o f g ro wt h i n t h e i m p l i c i t GNP d e f l a t o r i n t h e U .S . i s ex p l a i n edi n t e rm s o f i t s o wn 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 D t i s the impl ic i t p r ice def la to r fo rG N P. 7 S t an d a rd u n i v a r ia t e t im e s er ie s m e t h o d s l ead t o i d en t if i cat io n o f t h efo l lowing model fo r ~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 .0 3 4 )

The model i s es t imated on quar ter ly da ta f rom 1948 .2 to 1983 .4 , i .e . , a to ta l o f143 observat ions , us ing ord inary l eas t squares , wi th OLS s tandard er rors inpare n the ses . 8 Th e m odel i s st a t ionary , and non e o f the f ir s t t en au toco rre la-t ions o r par t i a l au toco rre la t ions fo r E are s ign i f ican t a t the 5% level. H ow ever ,look ing a t the au toco rre la t ions fo r e t: i t tu rns o u t tha t the 1s t, 3 rd , 7 th , 9 th an d10th a l l exceed two asympto t ic s t andard er rors . S imi lar resu l t s ho ld fo r thep 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 dARCH(8) are a l so h igh ly s ign i f i can t a t any reasonable l evel .

This l eads Engle and Kraf t (1983) to sugges t the fo l lowing speci f i ca t ion : rt= 0 .1 38 + 0 .4 2 3 % x + 0 .2 22 r t 2 + 0 .3 7 7 ~ ,_ 3 - 0 .1 7 5 ~ / 4 - ~ ' - g t ,(0 .0 59 ) ( 0 . 0 8 1 ) - ( 0 . 1 0 8 ) - (0 .0 78 ) ( 0 . 1 0 4 ) -

8h t = 0 .058 ~_ 0 .802 E ( 9 - i ) / 36e ~ _ i. (30)( 0 . 0 3 3 ) - ( 0 . 2 6 5 ) i = 1Th e e s t i m a t es a r e m ax i m u 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 er rors in paren theses ; see sec t ion 5 . The cho ice o f the e igh th-order l ineard 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 m o t i v a t ed b y t h e l o n g m em o ry i nt 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 . Th e o rd e r o f t h e l ag p o l y n o m i a l m ay b ev i ewed a s an ad d i t i o n a l p a ram e 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 l u e s o f GD, are t a k e n f ro 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 fC o m m e r c e , Survey of Current Business, Vol . 64, no. 9 , September 1984.gN ote , i n E ng le a nd K r a f t (1983) t he e s t im a t ion pe r iod i s 1948 .2 t o 1980.3 a c c oun t ing f o r t hesm a l l d i f f e r e nc e s i n t he e s t im a t ion r e su l t s .


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320 T. Bollerslev, Generalization of A R C H process- - S A R C H f l , 1)- - - - R R H [ 8 )

\ \

~ \ \ ' \ \ \. \ \

" - 4 \g; t .

I 8 1 ~ 0 l~ 2 I ~ 5Fig. 2. Lag distribution.

M i lhe j (1984) the cond i t ion fo r ex is t ence o f the fou r th -o rd e r m om en t o f e isju s t me t . 9 No ne o f the f ir s t t en au toco r r e la t ions o r pa r t ia l au toco r r e la t ions fo re t h t 1 / 2 o r e 2 h t 1 exceed 2 /v C l~ . The L M te s t s t at is t ic s fo r the linea rr e s t r i c t ion t akes th e va lue 8 .87 , co r r e spond ing to the 0 .74 f r ac t i le in the X2d i s t r ibu t ion . However , t he LM te s t s t a t i s t i c fo r the inc lu s ion o f h t _ 1 in theco nd i t ion al v ar ianc e equa t ion i s 4 .57 , which is s ign i f ican t a t the 5% level.

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


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T. Bollersleo, Generalization o f A R C H process 32 1

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 ) ` -4h 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 stt 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 dt 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 ee 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 7f 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 rl 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 tt h e 5 % l e v e l. A l s o t h e L M t e s t s ta t i st i c 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 st 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 . 9 5% co n f id en ce in te r v a l s f o r OLS.


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322 T. Bollerslev, Generalization of A R C H processI t i s a l s o i n t e r e s t i n g t o n o t e t h a t t h e s am p le co e f f i c ien t o f k u r to s i s f o r

e , h l 1 /2 f r o m mo d e l ( 3 1 ) eq u a l s 3 . 8 1 , w h ich d i f f e r s f r o m th e ' n o r ma l ' v a lu e o f3 .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 o rs , 2 2 ~ / T = 0.82. Form o d e l s ( 2 9 ) a n d ( 30 ) t h e co e f f ic i en t o f k u r to s i s eq u a l s 6 . 9 0 an 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 ic i en t o f s k ew n es s f o r e t h t ] /2 f r o m e a c h o f t h e t h r e em 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 ees t im at ed to b e 5 .848 an d 3 .696, r espe ct ive ly [ cf . s~ct ion 3], wh erea s in (30) them 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 gs 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 fad ap t iv e l ea r n in g mech an i s m. See a l s o f i g . 2 , w h er e t h e tw o d i f f e r en t l ags h a p e s a r e i l l u s tr a t e d . I n t h is li g 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 ig h t ly 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 i n E n g l e a n dK r a f t ( 1 9 8 3 ) , b u t i t a l s o ex h ib i t s a mo r e r eas o n ab le l ag s t r u c tu r e .

- - I n f l a t i o n B a t e-----Co nf. Int,

o .

AJ ~

I I

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

V / I , , , , , II L I \ lIll 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% confiden ce nterva ls for GAR CH (1,1).


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T. B ollerslev,Generafizationo f A R C H process 323In f igs . 3 an d 4 the actu al inflat ion rate, ~r,, i s gra ph ed togeth er wi th 95%

as y m p t o t i c co n f i d en ce i n t e rv a l s fo r t h e o n e - s t ep -ah ead fo recas t e r ro r s fo r t h etwo models (29) and (31) . From the l a te fo r t i es un t i l the mid-f i f t i es theinf la t ion ra te wa s very vo la t i le and hard to p red ic t . Th is i s re f lec ted in the wideco n f i d en ce i n t e rv a l s fo r t h e GARCH m o d e l . Th 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 er , were 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 eOLS co n f i d en ce i n t e rv a l s eem s m u 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 lcr ises in 1974 there is a s l ight increase in the uncertainty of the inflat ion rate,a l t h o u g h i t d o es n o t co m p are i n m ag 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 go f t h e s am p l e p e r i o d .

AppendixA.1. Proo f of Theorem 1

Th e b as i c i d ea o f t h e p ro o f fo l l o ws t h a t o f Th eo rem 1 i n M i l h o j (1 9 8 4 ) . Bydef in i t ion

= ,n / ,1 /2 i idEt "'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 Ph , = + E + E # , h , _ ,i = 1 i = 1

q 2 ( q P )= a0 + } - ' . a s ~ ,_ j a0 + ~ 2l i ~ t - i - j h t - i - j + E j ~ i h t - i - jj = l i = 1 i =1P ( 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 - jj = l i = 1 i = 1

= a o ~ M(t,k),k= Ow h e r e M ( t , k ) invo lves a l l the t e rms of the fo rm

q P n1 - - I < , : ' H ,8 7 , r i , g _ , , ,i = 1 j = l 1 : 1


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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

fo r

a n d

T h u s ,

q P qE 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 PM ( t , 1 ) = E ai~2-i + E fl, ,i = 1 i = 1

M ( t , 2 ) E 2l j ~ t - j E 2O l i ' l ~ t - i - j + E f l ij = l i = 1 i = 1

O l i # | t - - i - - j i ,ia n d i n g e n e r a l

q PM ( 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

p a r t i c u l a rE ( M ( t , k ) ) = E ( M ( s , k ) ) fo r all k, t , s , (A.4)

F r o m ( A .3 ) a n d ( A .4 ) w e g e tqE ( M ( t , k + l ) ) = Y [ ai+ ~=lfl E ( M ( t ,k ) )

i=1 i

= a , + f l i E (M ( / , 0 ) )i ~ l i

I l i + ii i

(A.5)


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T. Bollerslev, G eneralization of A R C H pro cess 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 PE 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 r e 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 ro o f o f T h e o re m 2B 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 eE(e ~J th t -_~[~b ,_2 ) = aj h"t_ t,

w e h a v eE ( h : [ ~ b , _ 2 ) - - ~ h",_ (m)n ao m -, "~ (nj.)a ja~ fl J.

n = 0 j = 0

L e t w, = ( k i n hm - 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 , General iza t ion o f A R C H processw 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 th 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 e l d sE(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 in i t e 2 mm o m e n t s , t h e l im i t a s k g o e s to i n 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 do 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 le ,

l i m E ( w t l ~ b t _ k _ l ) = ( I - C ) - l d = E (w t ) .k ~ mB 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 l u e s a r e e q u a l to th e d i a g o n a le 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 ss 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 f o r o t1 -4- E1 ~ 1 , a n dt ~ ( a , f l l , m ) < 1 su f fi c 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 .

Re f e r e n c e sBerndt, E.K ., B.H. Hall , R.E. Hall and J.A, Hausm an, 1974, Estimation inference in nonlinears t ructural m odels, A nnals of Economic an d Social Measurement, no. 4, 653-665.Box, G.E.P. an d J.M. Jenkins, 1976, Time series analysis: Forecasting and con trol (Holden-Day,San Franci sco , CA) .Breusch, T.S. an d A .R. Pagan, 19 78, A simple test for heteroskedastic city and random coefficientvariation, Econometrica 46, 1287-1294.Co ulson , N.E. and R.P. Ro bins, 1985, Aggregate economic activity and the variance o f inflation:Another look, Economics Letters 17, 71-75.Dom owitz, I . and C .S. Hakkio, 19 85, Conditional variance an d the risk premium in the foreignexchange market, Journal of International Economics 19, 47-66.Engle, R.F ., 1982, Autoregressive conditional heteroskedasticity with estimates o f the variance ofU.K . inflation , Econom etrica 50, 987-1008.Engle, R .F., 1983, Estimates of the variance of U.S. inflation based o n the A RC H model, Journalof M oney C redi t and Banking 15, 286-301.Engle, R .F. a nd D. K raft , 1983, Mu ltiperiod forecast error variances of inflation estimated fromA R CH models , in: A . ZeUner , ed., App l ied time series analys is of economic da ta (Bureau ofthe Census, Washington, DC) 293-302.Engle, R .F., D. Lilien an d R . Robins, 1985, Estimation of t ime varying risk premium s in the termstructure, Discussion pa pe r 85-17 (University of California, S an D iego, CA).G arb ad e, K ., 1977, Two methods for examining the stabili ty of regression coefficients, Journal ofthe American Statist ical Association 72, 54-63.G od frey , L.J. , 1978, Testing against general autoregressive and moving average error mod els when

the regressors include lagged dependent variables, Econometrica 46, 1293-1302.G od frey , L.J. , 198 1, On the invariance o f the Lagrange m ultiplier test with respect to certainchanges in the alternative hypothesis, Econometrica 49, 1443-1455.


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T. Bollerslev, Generalization of A R C H pro cess 32 7Gr ange r , C.W.J . and A .P. Andersen , 197 8, A n in trodu ct ion to b i l inear t ime series models(Vandenho eck and Ruprech t , GSt t ingen ) .G rang er , C.W.J . an d P. N ewbold , 1977, Forecast ing economic t ime ser ies (Academic Press , NewYork).Ha rvey , A.C ., 1982, The econometric analysis of time series (Philip Allen , Oxford).M cCulloch , J .H. , 198 3, In teres t- risk sensit ive deposi t insura nce premia: A daptive con dit ionalheteroscedast ic es t imates , Discuss ion paper , June (O hio Sta te U nivers i ty , Colum bus, OH).M cLeod , A .J. and W .K. Li , 1900, Diagnost ic checking A RM A time series models us ing squared-res idual au tocorre la t ions , Journal of T ime Ser ies Analysis 4 , 269-273.Milhoj , A. , 1984, The moment s t ruc ture of ARCH processes , Research report 94 ( Ins t i tu te ofStatistics, Un ive rsity of Copenhagen, Copenhagen).Weiss, A.A. , 198 2, Asym ptot ic theory for AR CH models : Stabi li ty , e s t imation and test ing ,Discu ss ion paper 82-36 (Univers i ty of C al i forn ia , San Diego, CA).Weiss, A.A. , 1984, AR M A m odels with AR CH errors, Journa l of Time Ser ies Ana lysis 5 , 129-143.W hite , H. , 1982, Max imu m l ike lihood es t imation of misspecified m odels , E conom etr ica 50 , 1-25 .

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