1

Multivariate Regular Variation with Application to Financial Time Series Models

Richard A. DavisColorado State University

Bojan BasrakEurandom

Thomas MikoschUniversity of Groningen

2

Outline+ Characteristics of some financial time series

� IBM returns� NZ-USA exchange rate

+ Models for log-returns� GARCH � stochastic volatility

+ Regular variation� univariate case� multivariate case

+ Applications of multivariate regular variation� Stochastic recurrence equations (GARCH)� limit behavior of sample correlations

3

Characteristics of Some Financial Time Series

Define Xt = ln (Pt) - ln (Pt-1) (log returns)

• heavy tailed

P(|X1| > x) ~ C x−α, 0 < α < 4.

• uncorrelatednear 0 for all lags h > 0 (MGD sequence)

• |Xt| and Xt2 have slowly decaying autocorrelations

converge to 0 slowly as h increases.

• process exhibits ‘stochastic volatility’.

)(ˆ hXρ

)(ˆ and )(ˆ 2|| hh XX ρρ

4

Log returns for IBM 1/3/62-11/3/00 (blue=1961-1981)

1962 1967 1972 1977 1982 1987 1992 1997

time

-20

-10

010

100*

log(

retu

rns)

5

Sample ACF IBM (a) 1962-1981, (b) 1982-2000

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

(a) ACF of IBM (1st half)

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

(b) ACF of IBM (2nd half)

6

Sample ACF of abs values for IBM (a) 1961-1981, (b) 1982-2000

Lag

ACF

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

(a) ACF, Abs Values of IBM (1st half)

Lag

ACF

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

(b) ACF, Abs Values of IBM (2nd half)

7

Sample ACF of squares for IBM (a) 1961-1981, (b) 1982-2000

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

(a) ACF, Squares of IBM (1st half)

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

(b) ACF, Squares of IBM (2nd half)

8

Sample ACF of original data and squares for IBM 1962-2000

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

0 10 20 30 40

Lag

0.0

0.2

0.4

0.6

0.8

1.0

ACF

9

Plot of Mt(4)/St(4) for IBM

0 2000 4000 6000 8000 10000

t

0.0

0.2

0.4

0.6

0.8

1.0

M(4

)/S(4

)

10

Hill’s plot of tail index for IBM (1962-1981, 1982-2000)

0 200 400 600 800 1000

m

12

34

5

Hill

0 200 400 600 800

m

12

34

5

Hill

11

500-daily log-returns of NZ/US exchange rate

t

X(t)

0 100 200 300 400 500

-0.0

4-0

.02

0.0

0.02

12

ACF of X(t)=log-returns of NZ/US exchange rate

lag h

ACF

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

13

ACF of X2(t)

lag h

ACF(

|X|^

2)

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

14

Plot of Mt(4)/St(4)

t

M(4

)/S(4

)

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

15

Hill’s plot of tail index

m

Hill

0 20 40 60 80

12

34

5

16

Models for Log(returns)Basic model

Xt = ln (Pt) - ln (Pt-1) (log returns)= σt Zt ,

where• {Zt} is IID with mean 0, variance 1 (if exists). (e.g. N(0,1) or

a t-distribution with ν df.)

• {σt}is the volatility process

• σt and Zt are independent.

Properties:

• EXt = 0, Cov(Xt, Xt+h) = 0, h>0 (uncorrelated if Var(Xt) < ∞)

• conditional heteroscedastic (condition on σt).

17

Models for Log(returns)-contXt = σt Zt (observation eqn in state-space formulation)

Two classes of models for volatility:

(i) GARCH(p,q) process (General AutoRegressive ConditionalHeteroscedastic-observation-driven specification)

Special case: ARCH(1):

(stochastic recursion eqn)

. XX 2q-t

21-t1

2p-t

21-t10

2t σβ++σβ+α++α+α=σ qp mm

t2

1-tt

2t0

21-t

2t1

2t

21-t10

2t

BXA

ZXZ

Z)X(X

+=

α+α=

α+α=

.3/1 if , )( 21

h12 <= ααρ hX

18

Models for Log(returns)-cont

GARCH(2,1):Then follows the SRE given by

Questions:• Existence of a unique stationary soln to the SRE?• Distributional properties of the stationary distribution?• Moment properties of the process? Finite variance?

. XX ,ZX 21-t1

22-t2

21-t10

2tttt σβ+α+α+α=σσ=

)',X ,X( 2t

21-t

2tt σ=Y

α+

σ

βαα

βαα=

σ 00Z

XX

001ZZZ

XX 2

t0

21-t

22-t

21-t

121

2t1

2t2

2t1

2t

21-t

2t

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Models for Log(returns)-contXt = σt Zt (observation eqn in state-space formulation)

(ii) stochastic volatility process (parameter-driven specification)

)N(0, IID~}{ , ,log 222 σεψεψσ tj

jjtj

jt ∞<= ∑∑∞

−∞=−

∞

−∞=

41

22 /),( )( 2 EZCorh httX +σσ=ρ

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Regular Variation — univariate caseDefinition: The random variable X is regularly varying with indexα if

P(|X|> t x)/P(|X|>t) → x−α and P(X> t)/P(|X|>t) →p,or, equivalently, if

P(X> t x)/P(|X|>t) → px−α and P(X< −t x)/P(|X|>t) → qx−α ,where 0 ≤ p ≤ 1 and p+q=1.Equivalence:

X is RV(α) if and only if P(X ∈ t • ) /P(|X|>t)→v µ(• ) (→v vague convergence of measures on RR\{0}). In this case,

µ(dx) = (pα x−α−1 I(x>0) + qα (-x)-α−1 I(x<0)) dxNote: µ(tA) = t-α µ(A).

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Regular Variation — univariate caseAnother formulation:

Define the ± 1 valued rv θ, P(θ = 1) = p, P(θ = −1) = 1− p = q.Then

X is RV(α) if and only if

or

(→v vague convergence of measures on SS0= {-1,1}).

)(x)t |X(|

)|X|X/ x,t |X(| SPP

SP ∈θ→>

∈> α−

)(x)t |X(|

)|X|X/ x,t |X(| •∈θ→>

•∈> α− PP

Pv

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Regular Variation—multivariate caseMultivariate regular variation of X=(X1, . . . , Xm): There exists a random vector θθθθ

∈ Sm-1 such that

P(|X|> t x, X/|X| ∈ • )/P(|X|>t) →v x−α P( θθθθ ∈ • )(→v vague convergence on Sm-1, unit sphere in Rm) .

• P( θθθθ ∈• ) is called the spectral measure• α is the index of X.

Equivalence:

µ is a measure on Rm which satisfies of x > 0 and A bounded away from 0,

µ(xB) = x−α µ(xA).

)()t |(|)t ( •µ→

>•∈

vPP

XX

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Regular Variation—multivariate caseExamples: Let X1, X2 be positive regularly varying with index α

1. If X1 and X2 are iid, then X= (X1, X2 ) is multivariate regularly varying with index α and spectral distribution

P( θθθθ =(0,1) ) = P( θθθθ =(1,0) ) =.5 (mass on axes).

Interpretation: Unlikely that X1 and X2 are very large at the same time.

2. If X1 = X2, then X= (X1, X2 ) is multivariate regularly varying with index α and spectral distribution

P( θθθθ = (1/sqrt(2), 1/sqrt(2)) ) = 1.

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Regular Variation—multivariate caseAnother equivalence? Suppose X > 0.MRV ⇔ all linear combinations of X are regularly varying

i.e., if and only if

P(cTX> t)/P(1TX> t) →w(c), exists for all real-valued c,

in which case,

w(tc) = t−αw(c).

(⇒) true (use vague convergence with Ac={y: cTy > 1}, i.e.,

)(w:)A()A(

)t ()t|| (

)t |(|)t (

)t ()tA (

T

T

T cX1X

XXc

X1X

1

cc =µµ→

>>

>>=

>∈

PP

PP

PP

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Regular Variation—multivariate case(⇐ ) established by Basrak, Davis and Mikosch (2000) for α not an even integer—case of even integer is unknown.

Idea of argument: Define the measures

mt(•)= P(X∈ t•)/P(1TX> t)

• By assumption we know that for fixed x, mt(Ax) →µ(Ax)

• {mt} is tight: For B bded away from 0, supt mt(B) < ∞.

• Do subsequential limits of {mt} coincide?

If mt' →v µ1 and mt'' →v µ2, then

for all x ≠≠≠≠ 0.

Problem: Need but only have equality on Ax not a π-system.

Overcome this using transform theory.

)A()A( 21 xx µ=µ

21 µ=µ

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Applications of Multivariate Regular Variation• Domain of attraction for sums of iid random vectors (Rvaceva, 1962). That is, when does the partial sum

converge for some constants an?

• Domain of attraction for componentwise maxima of iid random vectors (Resnick, 1987). Limit behavior of

• Weak convergence of point processes with iid points.

• Solution to stochastic recurrence equations, Y t= At Yt-1 + Bt

• Weak convergence of sample autocovarainces.

∑=

−n

tna

1t

1 X

t1

1 Xn

tna=

− ∨

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Point Processes With IID VectorsTheorem Let {Xt} be an iid sequence of random vectors that are multivariate regularly varying. Then we have the following point process convergence

where {an} satisfies nP(| Xt|> an) →1, and N is a Poisson process with intensity measure µ.

• {Pi} are Poisson pts corresponding to the radial part (intensity measure α x−α−1 (dx).

• {θθθθi} are iid with the spectral distribution given by the MRV.

,::11

/t ∑∑∞

=θ

=

ε=→ε=j

Pd

n

tan iin

NN X

28

Applications—stochastic recurrence equations

Yt= At Yt-1+ Bt, (At , Bt) ~ IID,At d×d random matrices, Bt random d-vectors

ExamplesARCH(1): Xt=(α0+α1 X2

t-1)1/2Zt, {Zt}~IID. Then the squares follow an SRE with

GARCH(2,1):Then follows the SRE given by

.Z B,ZA ,XY 2t0t

2t1t

2tt α=α==

. XX ,ZX 21-t1

22-t2

21-t10

2tttt σβ+α+α+α=σσ=

)',X ,X( 2t

21-t

2tt σ=Y

α+

σ

βαα

βαα=

σ 00Z

XX

001ZZZ

XX 2

t0

21-t

22-t

21-t

121

2t1

2t2

2t1

2t

21-t

2t

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Stochastic Recurrence Equations (cont)

Regular variation of the marginal distribution (Kesten)

Assume A and B have non-negative entries and

• E ||A1||ε < 1 for some ε > 0

• A1 has no zero rows a.s.

• W.P. 1, {ln ρ(A1… An): is dense in R R for some n, A1… An >0}

• There exists a κ0 > 0 such that and

Then there exists a κ1∈ (0, κ0] such that all linear combinations ofY are regularly varying with index κ1. (Also need .)

2/

1ij1,...,di

0

0

AminE κ

κ

==≥

∑ d

d

j

∞<+κ AlnAE 0

∞<κ1|B|E

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Application to GARCHProposition: Let (Yt) be the soln to the SRE based on the squares of a GARCH model. Assume

• Top Lyapunov exponent γ < 0. (See Bougerol and Picard`92)

• Z has a positive density on (−∞, ∞) with all moments finite orE|Z|h = ∞, for all h ≥ h0 and E|Z|h < ∞ for all h < h0 .

• Not all the GARCH parameters vanish.

Then (Yt) is strongly mixing with geometric rate and all finite dimensional distributions are multivariate regularly varying with index κ1.

Corollary: The corresponding GARCH process is strongly mixing and has all finite dimensional distributions that are MRV with index κ = 2κ1.

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Application to GARCH (cont)Remarks:1. Kesten’s result applied to an iterate of Yt , i.e.,

2. Determination of κ is difficult. Explicit expressions only known in two(?) cases.

• ARCH(1): E|α1 Z2|κ/2 = 1.

α1 .312 .577 1.00 1.57κ 8.00 4.00 2.00 1.00

• GARCH(1,1): E|α1 Z2+ β1|κ/2 = 1 (Mikosch and St�ric�)

• For IGARCH (α1 + β1 = 1), then κ = 2 ⇒ infinite variance.

• Can estimate κ empirically by replacing expectations with sample moments.

t1)m-(tttm~~ BYAY +=

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Summary for GARCH(p,q)

κ∈(0,2):

κ∈(2,4):

κ∈(4,∞):

Remark: Similar results hold for the sample ACF based on |Xt| andXt

2.

,)/())(ˆ( ,,10,,1 mhhd

mhX VVh�� == →ρ

( ) ( ) .)0()(ˆ ,,11

,,1/21

mhhXd

mhX Vhn�� =

−=

κ− γ→ρ

( ) ( ) .)0()(ˆ ,,11

,,12/1

mhhXd

mhX Ghn�� =

−= γ→ρ

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Realization of GARCH Process

t

X(t)

0 100 200 300 400 500

-20

-10

010

Fitted GARCH(1,1) model for NZ-USA exchange:

(Zt) ~ IID t-distr with 5 df. κ is approximately 3.8

772.X1519.10)70.6( ,ZX 21-t

21-t

72tttt σ++=σσ= −

Realization of fitted GARCH

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ACF of Fitted GARCH(1,1) Process

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

AC F of squares o f realization 1

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

AC F of squares of rea liza tion 2

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ACF of 2 realizations of an (ARCH)2: Xt=(.001+.7 Xt-1)1/2 Zt

lag h

ACF(

|X|^

2)

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

lag h

ACF(

|X|^

2)

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

36

Sample ACF for GARCH and SV Models (1000 reps)-0

.3-0

.10.

10.

3

(a) GARCH(1,1) Model, n=10000

-0.0

6-0

.02

0.02

(b) SV Model, n=10000

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Sample ACF for Squares of GARCH and SV (1000 reps)0.

00.

20.

40.

6

(a) GARCH(1,1) Model, n=10000

0.0

0.05

0.10

0.15

(b) SV Model, n=10000

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Sample ACF for Squares of GARCH and SV (1000 reps)0.

00.

20.

40.

6

(c) GARCH(1,1) Model, n=100000

0.0

0.01

0.02

0.03

0.04

(d) SV Model, n=100000