Richard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen

Transcription

1 Mulivariae Regular Variaion wih Applicaion o Financial Time Series Models Richard A. Davis Colorado Sae Universiy Bojan Basrak Eurandom Thomas Mikosch Universiy of Groningen

2 Ouline + Characerisics of some financial ime series IBM reurns NZ-USA exchange rae + Models for log-reurns GARCH sochasic volailiy + Regular variaion univariae case mulivariae case + Applicaions of mulivariae regular variaion Sochasic recurrence equaions (GARCH) limi behavior of sample correlaions

3 Characerisics of Some Financial Time Series Define = ln (P ) - ln (P - ) (log reurns) heavy ailed P( > x) ~ C x α, 0 < α < 4. uncorrelaed ˆ ρ ( h) near 0 for all lags h > 0 (MGD sequence) and have slowly decaying auocorrelaions ρˆ ˆ converge o 0 slowly as h increases. ( h) and ρ ( h) process exhibis sochasic volailiy. 3

4 Log reurns for IBM /3/6-/3/00 (blue=96-98) 00*log(reurns) ime 4

5 Sample ACF IBM (a) 96-98, (b) (a) ACF of IBM (s half) (b) ACF of IBM (nd half) ACF ACF Lag Lag 5

6 Sample ACF of abs values for IBM (a) 96-98, (b) (a) ACF, Abs Values of IBM (s half) (b) ACF, Abs Values of IBM (nd half) ACF ACF Lag Lag 6

7 Sample ACF of squares for IBM (a) 96-98, (b) (a) ACF, Squares of IBM (s half) (b) ACF, Squares of IBM (nd half) ACF ACF Lag Lag 7

8 Sample ACF of original daa and squares for IBM Lag Lag ACF ACF

9 M(4)/S(4) Plo of M (4)/S (4) for IBM

10 Hill s plo of ail index for IBM (96-98, ) Hill Hill m m 0

11 500-daily log-reurns of NZ/US exchange rae ()

12 ACF of ()=log-reurns of NZ/US exchange rae ACF lag h

13 ACF( ^) ACF of () lag h 3

14 M(4)/S(4) Plo of M (4)/S (4)

15 Hill s plo of ail index Hill m 5

16 Models for Log(reurns) Basic model = ln (P ) - ln (P - ) (log reurns) = σ Z, where {Z } is IID wih mean 0, variance (if exiss). (e.g. N(0,) or a -disribuion wih ν df.) {σ }is he volailiy process σ and Z are independen. Properies: E = 0, Cov(, +h ) = 0, h>0 (uncorrelaed if Var( ) < ) condiional heeroscedasic (condiion on σ ). 6

17 Models for Log(reurns)-con = σ Z Two classes of models for volailiy: (observaion eqn in sae-space formulaion) (i) GARCH(p,q) process (General AuoRegressive Condiional Heeroscedasic-observaion-driven specificaion) σ = α0 + α - + m+ α Special case: ARCH(): p -p + β σ - + m+ β q σ -q. = ( α = α 0 Z + α α )Z 0 Z = A (sochasic recursion eqn) - + B h ρ ( h) = α, if α < / 3. 7

18 8 Models for Log(reurns)-con GARCH(,): Then follows he SRE given by Quesions: Exisence of a unique saionary soln o he SRE? Disribuional properies of he saionary disribuion? Momen properies of he process? Finie variance?., Z σ + β + α + α = α σ = σ )',, ( - σ = Y α + σ β α α β α α = σ 0 0 Z 0 0 Z Z Z

19 Models for Log(reurns)-con = σ Z (observaion eqn in sae-space formulaion) (ii) sochasic volailiy process (parameer-driven specificaion) logσ = ψ jε j, ψ j j= j= <,{ ε } ~ IID N(0, σ ) ρ 4 ( h) = Cor( σ, σ h) / EZ + 9

20 Regular Variaion univariae case Definiion: The random variable is regularly varying wih index α if P( > x)/p( >) x α and P(> )/P( >) p, or, equivalenly, if P(> x)/p( >) px α and P(< x)/p( >) qx α, where 0 p and p+q=. Equivalence: is RV(α) if and only if P( ) /P( >) v µ( ) ( v vague convergence of measures on R\{0}). In his case, µ(dx) = (pα x α I(x>0) + qα (-x) -α I(x<0)) dx Noe: µ(a) = -α µ(a). 0

21 Regular Variaion univariae case Anoher formulaion: Define he ± valued rv θ, P(θ = ) = p, P(θ = ) = p = q. Then is RV(α) if and only if or P( x, / S) P( > ) > α x P( θ S) P( x, / ) P( > ) > α x v P( θ ) ( v vague convergence of measures on S 0 = {-,}).

22 Regular Variaion mulivariae case Mulivariae regular variaion of =(,..., m ): There exiss a random vecor θ S m- such ha P( > x, / )/P( >) v x α P( θ ) ( v vague convergence on S m-, uni sphere in R m ). P( θ ) is called he specral measure α is he index of. Equivalence: P( P( ) > ) µ is a measure on R m which saisfies of x > 0 and A bounded away from 0, µ(xb) = x α µ(xa). v µ ( )

23 Regular Variaion mulivariae case Examples: Le, be posiive regularly varying wih index α. If and are iid, hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ =(0,) ) = P( θ =(,0) ) =.5 (mass on axes). Inerpreaion: Unlikely ha and are very large a he same ime.. If =, hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ = (/sqr(), /sqr()) ) =. 3

24 Regular Variaion mulivariae case Anoher equivalence? Suppose > 0. MRV all linear combinaions of are regularly varying i.e., if and only if P(c T > )/P( T > ) w(c), exiss for all real-valued c, in which case, w(c) = α w(c). ( ) rue (use vague convergence wih A c ={y: c T y > }, i.e., P( A ) T P( > ) T P( c > ) P( > ) P( > T P( > ) ) µ (A µ (A c c = = ) ) : w( c) 4

25 Regular Variaion mulivariae case ( ) esablished by Basrak, Davis and Mikosch (000) for α no an even ineger case of even ineger is unknown. Idea of argumen: Define he measures m ( )= P( )/P( T > ) By assumpion we know ha for fixed x, m (A x ) µ(a x ) {m } is igh: For B bded away from 0, sup m (B) <. Do subsequenial limis of {m } coincide? If m ' v µ and m '' v µ, hen µ ( Ax ) = µ (A x ) for all x 0. Problem: Need µ = µ bu only have equaliy on A x no a π-sysem. Overcome his using ransform heory. 5

26 Applicaions of Mulivariae Regular Variaion Domain of aracion for sums of iid random vecors (Rvaceva, 96). Tha is, when does he parial sum a n = converge for some consans a n? Domain of aracion for componenwise maxima of iid random vecors (Resnick, 987). Limi behavior of a n n = Weak convergence of poin processes wih iid poins. Soluion o sochasic recurrence equaions, Y = A Y - + B Weak convergence of sample auocovarainces. n 6

27 Poin Processes Wih IID Vecors Theorem Le { } be an iid sequence of random vecors ha are mulivariae regularly varying. Then we have he following poin process convergence N n : = n = ε / a n d N : = j= ε Pθ i i, where {a n } saisfies np( > a n ), and N is a Poisson process wih inensiy measure µ. {P i } are Poisson ps corresponding o he radial par (inensiy measure α x α (dx). {θ i } are iid wih he specral disribuion given by he MRV. 7

28 8 Applicaions sochasic recurrence equaions Y = A Y - + B, (A, B ) ~ IID, A d d random marices, B random d-vecors Examples ARCH(): =(α 0 +α -) / Z, {Z }~IID. Then he squares follow an SRE wih GARCH(,): Then follows he SRE given by. Z B, Z A, Y 0 α = = α =., Z σ + β + α + α = α σ = σ )',, ( - σ = Y α + σ β α α β α α = σ 0 0 Z 0 0 Z Z Z

29 Sochasic Recurrence Equaions (con) Regular variaion of he marginal disribuion (Kesen) Assume A and B have non-negaive enries and E A ε < for some ε > 0 A has no zero rows a.s. W.P., {ln ρ(a A n ): is dense in R for some n, A A n >0} κ0 + There exiss a κ 0 > 0 such ha E A ln A < and E min i=,...,d d j= A ij κ 0 Then here exiss a κ (0, κ 0 ] such ha all linear combinaions of Y are regularly varying wih index κ. (Also need E B κ <.) d κ 0 / 9

30 Applicaion o GARCH Proposiion: Le (Y ) be he soln o he SRE based on he squares of a GARCH model. Assume Top Lyapunov exponen γ < 0. (See Bougerol and Picard`9) Z has a posiive densiy on (, ) wih all momens finie or E Z h =, for all h h 0 and E Z h < for all h < h 0. No all he GARCH parameers vanish. Then (Y ) is srongly mixing wih geomeric rae and all finie dimensional disribuions are mulivariae regularly varying wih index κ. Corollary: The corresponding GARCH process is srongly mixing and has all finie dimensional disribuions ha are MRV wih index κ = κ. 30

31 Applicaion o GARCH (con) Remarks:. Kesen s resul applied o an ierae of Y, i.e.,. Deerminaion of κ is difficul. Explici expressions only known in wo(?) cases. ARCH(): E α Z κ/ =. α κ ~ Y = A Y + (-)m GARCH(,): E α Z + β κ/ = (Mikosch and S ric ) For IGARCH (α + β = ), hen κ = infinie variance. Can esimae κ empirically by replacing expecaions wih sample momens. m ~ B 3

32 Summary for GARCH(p,q) κ (0,): κ (,4): κ (4, ): d ( ρˆ ( h)) h=,, m ( Vh / V0 ) h=,, m ( / κ ( )) d n ρ h γ (0)( V ). ˆ h=,, m h h=,, m, ( / ) d n ρ ( h) (0)( G ). ˆ γ h=,, m h h=,, m Remark: Similar resuls hold for he sample ACF based on and. 3

33 Realizaion of GARCH Process Fied GARCH(,) model for NZ-USA exchange: 7 = σ Z, σ = (6.70) σ (Z ) ~ IID -disr wih 5 df. κ is approximaely 3.8 Realizaion of fied GARCH - - ()

34 ACF of Fied GARCH(,) Process ACF of squares of realizaion ACF of squares of realizaion ACF ACF Lag Lag 34

35 ACF of realizaions of an (ARCH) : =( ) / Z lag h lag h ACF( ^) ACF( ^)

36 Sample ACF for GARCH and SV Models (000 reps) (a) GARCH(,) Model, n= (b) SV Model, n=

37 Sample ACF for Squares of GARCH and SV (000 reps) (a) GARCH(,) Model, n= (b) SV Model, n=

38 Sample ACF for Squares of GARCH and SV (000 reps) (c) GARCH(,) Model, n= (d) SV Model, n=