THE PREDICTIVE DENSITY OF A GARCH(1,1) PROCESS. Contents

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1 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS K. ABADIR, A. LUATI, P. PARUOLO Absrac. Tis paper derives e predicive probabiliy densiy funcion of a GARCH(,) process, under Gaussian or Suden innovaions. Te analyic form is novel, and replaces curren meods based on approximaions and simulaions. Please noe a e paper is sill in a preliminary version, some pars need o be compleed some o be wrien: May 4, 07 Conens. Inroducion. Te predicive densiy 3. Gaussian innovaions Main resuls Limi case Illusraions 6 4. Suden innovaions Main resuls Limi case Illusraions 8 5. Empirical applicaion 8 6. Conclusions 8 References 8 Appendix 9 A recursive formula for e condiional variance 9 Proof of Teorem 9 Proof of Corollary 3 Dae: May 4, 07, file: GD-paper-curren.ex.

2 K. ABADIR, A. LUATI, P. PARUOLO. Inroducion GARCH models are widely employed in financial applicaions. One noable applicaion is e calculaion of Value a Risk (VaR) for asses and porfolios, for one or more ime periods. Te predicive probabiliy densiy funcion (pdf) or cumulaive densiy funcion (cdf) of e GARCH is e saisical ool a e basis of e calculaion of VaR. However, as remarked in Andersen, Bollerslev, Criso ersen, and Diebold (006) page 8, no analyical form of e predicive densiy as been derived o dae beyond e -sep-aead case, wose form is given by assumpion; is laer disribuion is ofen aken o be Gaussian. Tis paper gives e firs resuls in is area, by deriving e analyical form of e -sep-aead predicive densiy of a Gaussian and Suden GARCH(,) processes. Te relaion beween e -sep-aead and e -sep-aead predicive pdf is also linked o e di erence beween e condiional and e marginal disribuion of financial reurns. Te empirical uncondiional disribuion of reurns is ofen found o be lepokuric; is is compaible wi a Gaussian predicive -sep-aead pdf, see e.g. Tsai (00) Tsay (00) page 3 secion 3.5 or Andersen, Bollerslev, Criso ersen and Diebold (006), Andersen, Bollerslev, Criso ersen, and Diebold (006) secion 3.6. Tis paper derives e analyical form of e -sep-aead predicive pdf. Tis allows one o corroborae wen and by ow muc is pdf di ers from is approximaions currenly in use, see Andersen, Bollerslev, Criso ersen, and Diebold (006) page 80 for a lis of ese, and ow muc fa-ailedness is implied by GARCH. See also Baillie and Bollerslev (99) on momens of e predicive densiy of a GARCH(,). Tis paper employs a recurrence equaion for e condiional variance, a albei similar in spiri o Nelson (990), is novel. A relaed issue is e one of calculaion of VaR for a -ime periods invesmen plan, wic involves e sum of n consecuive reurns generaed by a GARCH process, see Alexander, Lazar, and Sanescu (03). See also lieraure on limi disribuion of random auoregressions, by Mikosc, Sarica and Davis. Consider e asymmeric GARCH(,). Te predicive densiy x = ", = x, := x<0. were 0 and A is e indicaor funcion for even A. Noe a = ( )x wen x < 0 and = x wen x 0. Tis paper derives e predicive probabiliy densiy funcion of x condiional on informaion available a ime 0, as summarized via. Te sequence {" } is assumed o be i.i.d., cenered around zero and wi sperical pdf f " (s) =g(s ), i.e. wose densiy depends only on s ; noe a is implies a e pdf of " is symmeric. One as, seing x := (x,...,x ) 0 and denoing one realizaion of i by

3 u := (u,...,u ) 0 were THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS 3 f x (u) = Y apple = u g is a funcion of e pas of x (or is realizaion u). Te following ransformaion will be considered, z := x wi & := sgn(" ) = sgn(x ) because > 0. Le z := (z,...,z ) 0, & := (&,...,& ) 0 and denoe e se of all possible & by S,#S =. In e following, densiies are firs compued condiionally on & and laer ey are marginalized wi respec o i. Te noaion f? a ( ) is used for e pdf f a & ( s), were a can be a scalar or a vecor. Condiionally on &, e pdf of z is f? z (w ) = f x (w ) w. Hence, indicaing w := (w,...,w ) 0 a realizaion of z, one finds f? z (w) = Y = apple w w g and one wises o compue e inegral Z I := f z(w)dw?...dw = f z? (w ). () R For =,...,, consider e cange of variables y := z /( )= x /( )= Q = " /, and observe a e domain of inegraion remains R, a e inverse ransformaion is z = y /, and a e Jacobian is,were := /( Q = ). Finally inroduce e noaion y := (y,...,y,z ) 0 wi realizaion v := (v,...,v,w ) 0 ; one finds Hence, I = f y? (v) = Y w Z R = Y = apple apple v g v g v v w w g. g w dv...dv, were e Appendix sows a for, as e following expression in erms of y s (or eir realizaions v s wen compuing e inegral), = (y ) (y )... (y ) n = (y ) (y )... (y ) io, () wi iniial condiion = 0 0x 0, measurable wi respec o e informaion se a ime 0. In e paper, wo opions are considered for e pdf of ": e N(0,) Gaussian case wi f " (s) =g(s )=( ) exp( s /) and e sandardised Suden ( ) disribuion wi > caracerised by f " (s) =g(s )=a (s /( )) p p and a := /. 3. Gaussian innovaions Tis secion conains e derivaion for e Gaussian innovaion case, as well as an illusraion of e obained resuls.

4 4 K. ABADIR, A. LUATI, P. PARUOLO 3.. Main resuls. Te main resuls are summarised in Teorem below. Before saing e main eorems, an following auxiliary assumpion is presened; is is needed in e proofs of e main resuls. Assumpion. Wen =3, le := (, were ) := p 4 ; (3) for >3 le max(, ). Te form of (, ) as a funcion of (, ) is illusraed in Fig. I can be noed a sup (, ) (, )=lim # (, )= p , as >. Figure. (, ) as a funcion of and.0 b s 0 3 w As depends on, one can express e resricion on in equaion (3) as a funcion of e raio. b.0 Figure. (, ) as a funcion of e raio /( ) wêhs L Teorem below repors e predicive densiy in e Gaussian case. In e saemen of e eorem, is e confluen ypergeomeric funcion of e second kind, also known as

5 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS 5 Tricomi funcion, see Gradseyn and Ryzik (007) 9.. Moreover, e following definiion is used in summaions: a r := ( r r N r/ N. Teorem (GARCH(,) predicive densiy for e Gaussian case). Assume a " are i.i.d. N(0, ) and le Assumpion old. Ten one as, for, f z (w ) = 3 were c j := P ss c j,,s, c r,,& := Xa r ax r k k =0 k =0 Y = X ( ) j j w j c j, (4) f x (u ) = f z (u ) u, (5) a r K X 3 k =0,r 3 r r k K ; k k r K 3 k K S b,r K 3 ; b r b K b wi & := (&,...,& ) 0, b :=, K := P i= k i and S = P i= (i )k i, and empy sums (respecively producs) are undersood o be equal o 0 (respecively equal o ). Recall finally a in (6) is a funcion of &, so a only e erms in e second line in (6) depend on &. (6) Proof. See Appendix. Noe a in e case wen =, eq. (4) olds for any value of, wile for = 3 i olds if and only if. For >3, e validiy of e (4) is guaraneed by e su cien condiion max(, ), wic is, owever, no necessary. Te line of proof of Teorem is e following: for = e inegral is solved by subsiuion, using e following equaliy Z R exp pv v ( v) j k dv = p ; j k; p > 0, wic is e inegral represenaion of e confluen ypergeomeric funcion of e second kind, or Tricomi funcion, see Gradseyn and Ryzik (007), In addiion, for 3, subsequen (negaive) binomial expansions of expression () for is ensured by e inequaliy see Lemma 5 in e Appendix. X i= i are required, wose validiy apple, (7) Immediae consequences of Teorem are colleced in e following corollary.

6 6 K. ABADIR, A. LUATI, P. PARUOLO Corollary 3 (Cdf and momens). Te predicive cdfs of z and x are given by X F z (w ) = 3 ( ) j j j w j c j, ( 3 P ( ) j F x (u ) = j(j) uj c j u 0 F x ( u ) u < 0, wi momens E(x m )=E(zm )=m 3 were c m,,s are defined in (6). 5 m X c m,,s, m =,,... ss Noice a c m,,s are made of finie sums exending o m, involving e Tricomi funcions wic are no e logarimic case for e momens calculaions. In fac, m implies a is a finie sum. ; 3 m k; 3.. Limi case. (To be compleed) = = p m k mk m k; F m k; m p k m k m mk Xk m k j m k j j j k {0,,...,m} Te limi represenaion of e random variable is in Francq and Zakoian (00) Teorem. page 4. Te ail beaviour of e limi disribuion is reviewed in Davis and Mikosc (009) Illusraions. Some sandardised densiies of x are ploed in Fig. 3, 4 and 5 for =,, 3. Figure 3. Predicive densiies f x (u ) for sandardised x, =,, 3, in blue, red and green respecively, =0.5, =0., =0.7, 0 = ; x 0 =, =0. ( = is Gaussian wi variance = ) = = =

7 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS 7 Figure 4. Rig ail of f x (u ) for sandardised x, =,, 3, in blue, red and green respecively, =0.5, =0., =0.7, 0 = ; x 0 =, =0. ( = is sandard Gaussian) = = = Figure 5. Rig ail of f x (u ) for sandardised x, =,, 3, in blue, red and green respecively, =0.5, =0., =0.7, 0 = ; x 0 =, =0. ( = is sandard Gaussian) = = = Suden innovaions Tis secion conains e derivaion for e Suden innovaion case, as well as an illusraion of e obained resuls. 4.. Main resuls. Te main resuls are summarised in Teorem 4 below. In e saemen of e eorem, F is e Gauss ypergeomeric funcion, see Abadir (999), secion 3, and B(p, q) is e Bea funcion.

8 8 K. ABADIR, A. LUATI, P. PARUOLO Teorem 4. Assume a " are (normalised) suden random variables wi degrees of freedom, and le Assumpion old. Ten one as, for, X f z (w ) = a j j w j `j, (8) were `j := P ss ` j,,s, `r,,& : = Xa r k =0 k =0 f x (u ) = f z (u ) u, (9) ax r k a r K X 3 k =0 j k j k j K 3 k k K Y S s jk q q B(j K q, ) q= F j K q,j K q s jk j K B(j K, ),jk q ; s q F j K,j K,jK ; wi & := (&,...,& ) 0, s := ( ) /( ), K := P i= k i and S = P i= (i )k i,and empy sums (respecively producs) are undersood o be equal o 0 (respecively equal o ). s Proof. Te proof is done bu needs o be wrien in a compac form. 4.. Limi case. To be compleed Illusraions. To be compleed. 5. Empirical applicaion In is secion e formulae in Secion 3 and 4 are applied o compue e ail probabiliy of a given GARCH(,). Tis could be compared wi e precision obainable via Mone Carlo ecniques in finie compuaional ime. 6. Conclusions Tis paper presens e explici form of e predicive densiy of a GARCH(,) process. Tis can be used o evaluae probabiliy of ail evens or of quaniles a may be of ineres for value a risk calculaions. References Abadir, K. M. (999) An inroducion o ypergeomeric funcions for economiss. Economeric Reviews, 8(3), Alexander, C., E. Lazar, and S. Sanescu (03) Forecasing VaR using analyic iger momens for {GARCH} processes. Inernaional Review of Financial Analysis, 30,

9 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS 9 Andersen, T., T. Bollerslev, P. F. Crisoffersen, and F. X. Diebold (006) Volailiy and correlaion forecasing. in Handbook of Economic Forecasing, Volume, ed. by G. Ellio, C. W. Granger, and A. Timmermann. Elsevier. Baillie, T. R., and T. Bollerslev (99) Predicion in dynamic models wi imedependen condiional variances. Journal of Economerics, 5, 9 3. Davis, R., and T. Mikosc (009) Exreme Value Teory for GARCH Processes. in Handbook of Financial Time Series, ed. by D. R. K. J.-P. Andersen, T.G., and T. Mikosc, pp Springer. Francq, C., and J.-M. Zakoian (00) GARCH models. Wiley. Gradseyn, I., and I. Ryzik (007) Book of Tables of inegrals, series, and producs. 7 ed., Academic Press. Mood, A. M., F. A. Graybill, and D. C. Boes (974) Inroducion o e Teory of Saisics, 3rd Ediion. Mc Graw-Hill. Nelson, D. (990) Saionariy and persisence in e GARCH (,) model. Economeric Teory, 6, Tsay, R. S. (00) Analysis of financial ime series. Wiley, 3rd edn. Appendix A recursive formula for e condiional variance. Observe a for one as = (y ) were y := " /. Hence = (y ) (y )... (y ), were = 0 0x 0 is measurable wi respec o e informaion se known a ime 0. Alernaively one can wrie = (y ) (y )... (y ). (0) For examples, see Supplemenary Maerial. Proof of Teorem. Te proof of Teorem is based on e following Lemmaa. Lemma 5 (Condiions on ). Assume a max{, }, were = (, Ten e following olds for any j ) = p 4 j X i= Proof. of Lemma 5. For j = e inequaliy () reads i 0 apple j. () Solving e quadraic on e l..s. for one finds wo roos, =( p 4 )/( ) < 0 and =( p 4 )/( ) > 0, so a e quadraic is non-negaive for < or for >. Because < 0, is olds only wen. Tis proves a () is valid for j = for and a foriori also for max{, }.

10 0 K. ABADIR, A. LUATI, P. PARUOLO An inducion approac is used for j>. Assume a () is valid for some j = j 0 and max{, }; i can en be sown a () is valid also replacing j wi j. To see is, ake () for j = j 0 and muliply by j 0 X i=.onefinds i Because, one as ( ) apple, so a, j 0 X i= i apple apple j 0. j 0 X i= i apple j 0. P Rearranging j0 i= i P as j0 i i=, one finds a () olds also for j = j 0. Te inducion sep ence proves a () olds for any j if max{, }. Lemma 6 (Coe equals c j,,s as defined in (6). ciens c j ). Assume a () olds for apple j apple ; en Z " Y exp R # X v ( ) j = Proof of Lemma 6. Rewrie () using () and seing r = j as Z Y R = = dv p v () apple exp v ( dv )r p. (3) v For = is expression equals A (r) were Z apple r A (r) = exp v (v ) R v dv Z apple r = r exp v v v R Z apple r = exp b (b b) r (b) bd R Z apple r = b r exp b ( ) r d R were b :=, := v b,0<<, dv = bd. Hence r A (r) = b r p,r 3 ; b wic follows from (Erdelyi e. al., 953, p. 55 equaion ()), (a, c; x) = (a) Z 0 exp { x} a ( ) c a d, dv e confluen ypergeomeric funcion of second kind or Tricomi funcion, wi a =, c = r 3, x = b, b = saisifying Re{a} > 0, Re{x} > 0. Tis sows a for =, A ( j) =c j,,s. Nex consider e case = 3, were 3 = (v ) (v ),

11 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS and one wises o expand 3 r. Consider e inequaliy <, and e associaed quadraic equaion =0in wi soluions and, see (3). One as a for can expand one finds > 0, wic ensure a <. Hence for one 3 r as 3 r = X r j ( v ) r j (v ) j Similarly, for case >3, one can wrie () as using e following recursions r j. = 0 (v ) a (4) a := (v ) a j := j (v j ) a j j =3,...,. (5) In is noaion a represens e inner-mos parenesis in (), a 3 e wo inner-mos pareneses in (), ec, up o = a 0. I can be sown a condiion (8) implies a j apple a j (6) in (5); in order o prove is, one can sar from j = and proceed o sow a is olds for j =. Le now r = j and apply subsequen binomial expansions o powers of a 0,a,...,a 3 in ( )r from (4) and (5) one finds ( )r =( 0 (v ) a ) r r = k ( v ) r k a r k = = = = k =0 k k k =0 k =0 k k =0 k =0 k k =0 k =0 r r k r r k r K X 3 k =0 k k k k k ( v ) r k k k ( v ) r k k a r k k k k k ( v ) r k ( v ) r k k a r k k r r k k k r K 3 k Y K S = ( v ) r K a r K, were e upper summaion is exended o wen r is no an ineger; convergence of e series is guaraneed by (). Subsiuing (7) in (3) and inegraing, one finds X X X r r k r K 3 K Y S I A (r K ) (8) k =0 k =0 k =0 k k k and Z apple I := exp v ( v ) r K v R dv = p =,r 3 K ; (7) b.

12 K. ABADIR, A. LUATI, P. PARUOLO Nex e proof of Teorem is presened. Proof of Teorem. Te inegral o be solved is p Z " f z? (w )= /w exp ( ) R Expand exp( w /( )) = P ( w /) j j f? z (w )= p w were c j,,& equals Z " exp R ( ) X = v w j and noe a X ( w /) j j # X v ( ) j = # Y = dv p v. (9) c j,,&, (0) Y = dv p v () wic, by Lemma 6, also equals e expression (6). Marginalizing wi respec o & all elemens in S f z (w ) = X equally likely, one finds f z (w s) = ss 0 p = w X ( w /) j ( ) j X p w ss ( ) c j,,& ss X ( w /) j j A = p w c j,,s ( ),being X ( w /) j were c j := P ss c j,,s. Finally one needs o prove (5). Consider e ransformaion eorem for z = x ; from sandard resuls, see e.g. Mood, Graybill, and Boes (974) Example 9 page 0, one as p f z (w )= p f x ( w ) p f x ( p w ) (w w w 0), were ( ) is e indicaor funcion. Because, by symmery, one as f x ( p w )=f x ( p w ), is expression simplifies ino f z (w )= p w f x ( p w )(w 0), or, leing u indicae p w, and solving for f x (u ), one finds f x (u )= u f z (u ), wic is (5). Noe a e expression wi e absolue value is also valid for u = p w. Proof of Corollary 3 Proof. of Corollary 3. Te c.d.f is found by inegraing ermwise e pdf. Te momens are derived as follows. From (9) one sees a E? z (w m )= p Recall a ( ) Z R Z exp R w m w apple w exp Y dw = w m dw = m apple exp v m j v dv. c j

13 THE PREDICTIVE DENSITY OF A GARCH(,) PROCESS 3 so a E? z (w m )= 3 m Proceeding as in (8) one finds m Z Y apple exp v R E? z (w m )= 3 m = m c m,,s m v dv. and ence E z (w m )= 5 m 3 m X c m,,s. ss