Sample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen

Transcription

1 Sample Auocorrelaions for Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen

2 Ouline Characerisics of some financial ime series IBM reurns NZ-USA exchange rae Linear processes Background resuls Mulivariae regular variaion Poin process convergence Applicaion o sample ACVF and ACF Applicaions Sochasic recurrence equaions (GARCH) Sochasic volailiy models

3 Characerisics of Some Financial Time Series Define = 00*(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 = 00*(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. 6

17 Models for Log(reurns)-con = σ Z Examples of models for volailiy: (observaion eqn in sae-space formulaion) (i) GARCH(p,q) process (observaion-driven specificaion) σ = α 0 + α - + m+ α Special case: ARCH(), p -p + β σ (ii) sochasic volailiy process (parameer-driven specificaion) logσ ρ = ψ jε j, ψ j j= j= - h ρ ( h) = α, if α < / 3. = σ σ 4 ( h) Cor(, ) / EZ + h = ( α 0 + α + m+ β - )Z q σ -q <,{ ε } ~ IID N(0, σ ).. 7

18 Model: Properies: P( >x) ~ C x α Define ρ( h) = ψ ψ / ψ. {Z }~IID, P( Z >x) ~ Cx α, 0<α<. Case α > : n / (ρ(h) ^ ρ(h)) (ρ(h+j)+ρ(h-j)-ρ(j)ρ(h)) N j, {N }~IIDN Case 0 < α < : j= = ψ jz - Linear Processes (n / ln n) /α ( ρ(h) ^ ρ(h)) (ρ(h+j)+ρ(h-j)-ρ(j)ρ(h)) S j / S 0, {S }~IID sable (α), S 0 sable (α/) j j= d j= j j+ h d j= j= j 8

19 Background Resuls mulivariae regular variaion 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- ). P( θ ) is called he specral measure α is he index of. Equivalence: There exis posiive consans a n and a measure µ, np(/ a n ) v µ( ) In his case, one can choose a n and µ such ha µ((x, ) B) = x α P( θ B ) 9

20 Background Resuls mulivariae regular variaion Anoher equivalence? MRV all linear combinaions of are regularly varying i.e., if and only if in which case, ( ) rue P(c T > )/P( T > ) w(c), exiss for all real-valued c, w(c) = α w(c). ( ) esablished by Basrak, Davis and Mikosch (000) for α no an even ineger case of even ineger is unknown. 0

21 Background Resuls poin process convergence Theorem (Davis & Hsing `95, Davis & Mikosch `97). Le { } be a saionary sequence of random vecors. Suppose (i) finie dimensional disribuions are joinly regularly varying (le (θ k,..., θ k ) be he vecor in S (k+)m- in he definiion). (ii) mixing condiion A (a n ) or srong mixing. (iii) Then exiss. If γ > 0, hen 0. ) ( lim sup lim 0 = > > y a y a P n n r k n k n α + = α θ θ θ = γ / ) ( lim ) ( 0 ) ( ) ( 0 k k j j k k k E E, : : / ij = = = ε = ε = j P i d n a n i n N N Q

22 Background Resuls poin process convergence(con) where (P i ) are poins of a Poisson process on (0, ) wih inensiy funcion ν(dy)=γαy α dy., i, are iid poin process wih disribuion Q, and Q is he weak limi of = ε ij j Q + = α θ + = α θ θ ε θ θ k k j j k k k j j k k k E I E k ) ( ) ( 0 ) ( ) ( 0 ) ( ) / ( ) ( lim ) (

23 3 Background Resuls applicaion o ACVF & ACF Se-up: Le { } be a saionary sequence and se = (m) = (,..., +m ). Suppose saisfies he condiions of previous heorem. Then Sample ACVF and ACF: ACVF ACF If E 0 <, hen define γ (h) =E 0 h and ρ (h)=γ (h)/ γ (0)., : : / ij = = = ε = ε = j P i d n a n i n N N Q 0,, ) ( ˆ = γ + = h n h h h n, ), ( ˆ ) / ( ˆ ) ( ˆ γ = γ ρ h h h h

24 Background Resuls applicaion o ACVF & ACF (i) If α (0,), hen where ( na V n h γˆ (ˆ ρ ( h)) ( h)) i= j= h= 0, l, m h=, l, m = P d ( V d ( V (0) ij ( h) ij / V (ii) If α (,4) + addiional condiion, hen i Q Q, h h ) h= 0, l, m 0 ) h=, l, m, h = 0, l, m. ( ) d nan (ˆ γ ( h) γ ( h) ( V ) h= 0, l, m h h= 0, l, m ( (ˆ ( ) ( ))) d na ρ h ρ h γ (0)( V ρ ( h) V ). n h=, l, m h 0 h=, l, m 4

25 5 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

26 Examples (ricks) GARCH(,): Sochasic Recurrence Equaions (con) = σ Z, Alhough his process does no have a -dimensional SRE represenaion, he process does. Ieraing, we have σ σ = α 0 + α - + β σ -. σ = α 0 + α = ( α Z β + β σ ) σ α = α α σ - Z - + β σ - Bilinear(): =a - +b - Z - +Z, {Z }~IID =Y - +Z, Y = A Y - + B A =a+bz,b =A Z 6

27 Sochasic Recurrence Equaions (con) Y = A Y - + B, (A, B ) ~ IID Exisence of saionary soluion E ln + A < Eln + B < inf n - E ln A A n =: γ < 0 (γ op Lyapunov exponen) Ex. (d=) E ln A < 0. Srong mixing If E A ε <, E B ε < for some ε > 0, hen he SRE (Y ) is geomerically ergodic srong mixing wih geomeric rae (Meyn and Tweedie `93). 7

28 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 / 8

29 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. 9

30 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 Sarica) For IGARCH (α + β = ), hen κ = infinie variance. Can esimae κ empirically by replacing expecaions wih sample momens. m ~ B 30

31 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

32 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 - - ()

33 ACF of Fied GARCH(,) Process ACF of squares of realizaion ACF of squares of realizaion ACF ACF Lag Lag 33

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

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

36 Sochasic Volailiy Models SVM: = σ Z (Z ) ~ IID wih mean 0 (if i exiss) (σ ) is a saionary process ( log σ is a linear process) given by logσ = j= ψ j ε j, <, ( ε ) ~ IID N(0, σ Heavy ails: Assume Z has Pareo ails wih index α, i.e., P( Z > z) ~ C z α P( > z) ~ C Eσ α z α. j= ψ j ) Then if α (0,), / α d h+ α h ( n / ln n) ρˆ ( h). σ σ σ α S S 0 36

37 Sochasic Volailiy Models (con) Oher powers:. Absolue values: α (,), Ε = Ε σ Ε Z, Ε +h = (Ε σ σ +h )(Ε Z Ε Z +h ) Cov(, +h ) = Cov(σ, σ +h )(Ε Z ) Cor(, +h ) = Cor(σ, σ +h )(Ε Z ) / EZ =0 (?). We obain n and / α ( n ln n) ( γˆ ( h) γ ( h) ) / α d h+ α h ( n / ln n) ρˆ ( h). σ σ σ d σσh + α α S S 0 S h 37

38 . Higher order: α (0,) Sochasic Volailiy Models (con) The squares are again a SV process and he resuls of he previous proposiion apply. Namely, In paricular, / α d h+ α / h ( n / ln n) ρˆ ( h). σ σ σ α / S S 0 ˆ ρ ( h) P 0. 38

39 Sochasic Volailiy Models (con) (log ) - mean for NZ-USA exchange raes log ^()

40 Sochasic Volailiy Models (con) ACF/PACF for (log ) suggess ARMA (,) model: µ= , Y =.9646Y - + ε.8709 ε -, (ε ) WN(0,4.6653).00 ACF.00 PACF model sample model sample

41 Sochasic Volailiy Models (con) The ARMA (,) model for log leads o he SV model wih = σ Z ln σ = v + ε v =.9646 v - + γ, (γ ) WN(0,.0753) (ε ) WN(0,4.43). 4

42 Sochasic Volailiy Models (con) Simulaion of SVM model: Took ε o be disribued according o log of a random variable wih 3 df (suiable normalized). ACF: abs(realizaion) ACF Lag

43 Sochasic Volailiy Models (con) ACF: (realizaion) ACF: (realizaion) 4 ACF ACF Lag Lag 43

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

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

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