Joint Extremal Behavior of Hidden and Observable Time Series with an Application to GARCH Processes
Andree Ehlert, Ulf-Rainer Fiebig, Anja Janßen, Martin Schlather
aa r X i v : . [ m a t h . P R ] M a y Joint Extremal Behavior of Hidden and ObservableTime Series with an Application to GARCH Processes
Andree Ehlert · Ulf-Rainer Fiebig · Anja Janßen · Martin Schlather
Received: date / Accepted: date
Abstract
We study the behavior of a real-valued and unobservable process ( Y t ) t ∈ Z under an extreme event of a related process ( X t ) t ∈ Z that is observable. Our anal-ysis is motivated by the well-known GARCH model which represents two suchsequences, i.e. the observable log returns of an asset as well as the hidden volatil-ity process. Our results complement the findings of Segers [J. Segers, Multivari-ate regular variation of heavy-tailed Markov chains, arXiv:math/0701411 (2007).Available online: http://arxiv.org/abs/math/0701411] and Smith [R. L. Smith,The extremal index for a Markov chain. J. Appl. Prob. (1992)] for a single timeseries. We show that under suitable assumptions their concept of a tail chain as alimiting process is also applicable to our setting. Furthermore, we discuss existenceand uniqueness of a limiting process under some weaker assumptions. Finally, weapply our results to the GARCH(1 ,
1) case.
Keywords
ARCH processes · GARCH processes · extremal index · joint extremalbehavior · multivariate regular variation · tail chain · time series Mathematics Subject Classification (2000) · Andree EhlertInstitute of Economics, Leuphana University of L¨uneburg, Scharnhorststr. 1, D-21335L¨uneburg, Germany. E-mail: [email protected] FiebigInstitute for Mathematical Stochastics, Georg-August-University G¨ottingen, Goldschmidt-str. 7, D-37077 G¨ottingen, Germany. E-mail: urfi[email protected] Janßen ( B )Department of Mathematics, SPST, University of Hamburg, Bundesstr. 55, D-20146 Hamburg,Germany. E-mail: [email protected] SchlatherSchool of Business Informatics and Mathematics, University of Mannheim, D-68131 Mannheim,Germany. E-mail: [email protected] Andree Ehlert et al. An extensive class of financial time series models is based on two interrelatedprocesses. In particular, many models include an unobservable part that reflects acertain regime or the volatility of the process. A well-known example is given bythe GARCH family. It is typically applied in order to model financial log returnswhere the unobservable volatility process drives the observable price of an asset.In the following, let ( X t ) t ∈ Z denote such a process and ( Y t ) t ∈ Z its unobservablecounterpart. Let both ( X t ) t ∈ Z and ( Y t ) t ∈ Z be univariate. A common approach forthe analysis of the extremal behavior of such interrelated processes focusses onthe joint sequence ( Z t ) t ∈ Z := ( X t , Y t ) t ∈ Z . More precisely, the process is studiedunder the condition {k Z k > x } for x → ∞ and an arbitrary norm k · k on R .The connection of this approach to the concept of multivariate regular variationhas been discussed extensively in [3]. We shall follow a more natural point of viewwhere the process ( Y t ) t ∈ Z is unobservable. That is, we analyze its limiting behaviorunder the (observable) event {| X | > x } as x → ∞ . Hence, for −∞ < m ≤ n < ∞ we focus on the limit distribution of L (cid:18) Y m x , . . . , Y n x (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) (1.1)as x → ∞ . We assume ( Y t ) t ∈ Z to be of a simple Markovian structure, i.e. Y t = Φ ( Y t − , ǫ t ) , t ∈ Z , (1.2)for some measurable mapping Φ : R × S → R and some sequence ( ǫ t ) t ∈ Z of i.i.d.innovations on a measurable space ( S , S ) . Additionally, we will require the se-quence of innovations ( ǫ t ) t>s to be independent of ( Y t ) t ≤ s for all s ∈ Z . Based on( Y t ) t ∈ Z and the innovations let the observable process be given by X t = Ψ ( Y t , ǫ t − s − , . . . , ǫ t + s + ) , t ∈ Z , (1.3)for some measurable mapping Ψ : R × S s − + s + +1 → R with s + , s − ≥
0. We willalways assume that a stationary solution to (1.2) and (1.3) exists. Now, by Ψ aswell as by s − and s + we have a simple, but flexible model for the dependencebetween ( X t ) t ∈ Z and ( Y t ) t ∈ Z . However, note that from the recursive definitionin (1.2) we may find a function e Ψ : R × S s − + s + +1 → R such that e X t := X t + s − +1 = e Ψ ( Y t , ǫ t +1 , . . . , ǫ t + s ), t ∈ Z , with s := s − + s + + 1. Hence, for ease of notation wemay in the following assume that there exists an s ≥ X t = Ψ ( Y t , ǫ t +1 , . . . , ǫ t + s ) , t ∈ Z . (1.4)We may interpret ( X t ) t ∈ Z and ( Y t ) t ∈ Z as a generalized hidden Markov modelwhich incorporates a large class of models for financial time series, cf. [9] forthe general definition. We shall discuss the GARCH(1 ,
1) process (cf. [6, 26]) as aspecific example, i.e. ζ t = σ t ǫ t +1 , t ∈ Z , (1.5)and σ t = q α + α σ t − ǫ t + β σ t − , t ∈ Z , (1.6) oint Extremal Behavior of Hidden and Observable Time Series 3 for suitable constants α > α , β ≥
0. Here, the sequence ( ζ t ) t ∈ Z is theobservable part, e.g. a model for financial log returns, and the series ( σ t ) t ∈ Z de-scribes the conditional standard deviation (volatility) of the process at time t ∈ Z .In the basic setup the innovation sequence ( ǫ t ) t ∈ Z is assumed to be i.i.d. standardnormal. Note that the above GARCH(1,1) model satisfies (1.2) and (1.4) for Φ ( x, e ) = p α + α x e + β x , Ψ ( x, e ) = xe, s = 1 . We remark that for β = 0 in (1.6) the GARCH(1 ,
1) setup includes the ARCH(1)model as a special case, cf. [15]. For further examples, cf. also Remark 1.It is well-known [2] that under quite general assumptions about the distributionof ǫ t , t ∈ Z , and about the size of the parameters α , α and β the stationary solu-tions to (1.5) and (1.6) share a common regularly varying (heavy tailed) behavior.A heavy tailed behavior of both the volatilities and the log returns is a desirablefeature of financial time series as it agrees with commonly accepted stylized facts.Accordingly, we will assume regular variation for the stationary solutions to both(1.2) and (1.4), cf. Condition 1 below. As it is not clear whether the limit in (1.1)exists we will discuss those questions in more detail in Sections 2 and 3. In Sec-tion 4 we will show that under some further assumptions the limiting distributionin (1.1) has a particularly simple form which can be seen as an extension to similarfindings in [24]. More precisely, outside of the period { , , . . . , s } our results willallow for a representation of the limit process in (1.1) as a multiplicative randomwalk, cf. Proposition 5. Heuristically, if we consider the example given by (1.5)and (1.6), this is the case since a large value of | ζ | stems most likely from a largevalue of σ as the tail of σ is heavier than the tail of ǫ . Now, for a large value of σ i , σ i +1 behaves asymptotically like q α ǫ i +1 + β σ i . At the same time, for i = 1,the distribution of ǫ is influenced by the extremal event of | ζ | = | σ ǫ | beinglarge while all future ǫ i , i ≥ , are not influenced by this condition. In Section 5 wewill analyze connections of our results with multivariate regular variation of thetime series ( X t , Y t ) t ∈ Z . The theoretical results are applied to the GARCH(1 , In the following, we will assume that the stationary distribution of Y t = Φ ( Y t − , ǫ t ), t ∈ Z , cf. (1.2), is regularly varying with index α > Condition 1.a lim x →∞ P ( | Y | > ux ) P ( | Y | > x ) = u − α , ∀ u > , lim x →∞ P ( Y > x ) P ( | Y | > x ) = p ∈ [0 , . (2.1) ⊓⊔ We will study the joint extremal behavior of (1.2) and (1.4) under the assumptionthat X shares the tail behavior of Y , i.e. there exists a constant C > x →∞ P ( | X | > x ) P ( | Y | > x ) = C. (2.2) Andree Ehlert et al.
Analogous to Condition 1.a we say that
Condition 1.b holds if the timeseries ( X t ) t ∈ Z satisfies (2.1) with X in place of Y (with a possibly differentvalue of p ). Furthermore, if both conditions and (2.2) are satisfied we will say that Condition 1 holds.
Proposition 1
Let ( Y t ) t ∈ Z and ( X t ) t ∈ Z be stationary time series given by (1.2) and (1.4) and let (2.1) and (2.2) be satisfied. Then, the family L (cid:18) Y m x , . . . , Y n x (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) , x > , of conditional distributions is tight for all −∞ < m ≤ n < ∞ .Proof Let u >
0. Then P n [ i = m (cid:26) | Y i | x > u (cid:27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! ≤ n X i = m P (cid:18) | Y i | x > u (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) ≤ n X i = m P ( | Y i | > ux ) P ( | X | > x ) = n X i = m P ( | Y | > ux ) P ( | Y | > x ) P ( | Y | > x ) P ( | X | > x ) . By (2.1) and (2.2) the r.h.s. is bounded by 2( n − m + 1) u − α C − for x large.Therefore, a weak accumulation point of the family of distributions exists. Thefollowing lemma shows, however, that it is not necessarily unique. Lemma 1
There exist time series ( Y t ) t ∈ Z and ( X t ) t ∈ Z of the form (1.2) and (1.4) such that Condition 1 is satisfied but (1.1) has more than one weak accumulationpoint.Proof Let ǫ t i.i.d. ∼ Par(1) , i.e. P ( ǫ t > x ) = x − , x ≥
1. With Φ ( Y t − , ǫ t ) = ǫ t wehave Y t = ǫ t , t ∈ Z , so Y t i.i.d. ∼ Par(1) as well. Let s = 1 and Ψ ( Y t , ǫ t +1 ) = f ( ǫ t +1 )for a continuous function f : R → R to be described below. Thus X t = f ( ǫ t +1 ) , t ∈ Z . By independence, any weak accumulation point of L (cid:0) Y x , Y x (cid:12)(cid:12) | X | > x (cid:1) equals δ × µ for some weak accumulation point µ of L (cid:0) Y x (cid:12)(cid:12) | X | > x (cid:1) , where δ a , a ∈ R , denotes the Dirac measure in a . With Y := ǫ ∼ Par(1) we will construct f suchthat L (cid:0) Yx (cid:12)(cid:12) | f ( Y ) | > x (cid:1) has a continuum of weak accumulation points.Let f ( t ) = t, t ≤
1. For the sequence z i = 5 i , i ∈ N , each interval [ z i , z i ] ismapped onto itself by f . On [4 z i , z i ] it interpolates linearly between the values z i and 5 z i , and on [3 z i , z i ] between 3 z i and z i . The function f can be extendedon each interval [ z i , z i ] such that f ([ z i , z i ]) ⊂ [ z i , z i ] and f ( Y ) ∼ Par(1). Thedetails of the construction of f are given in the Appendix. We first show thattwo different weak accumulation points exist. Since f ( Y ) is nonnegative we dropthe absolute value. Along the sequence x i = 5 i , i ∈ N , we have { f ( Y ) > x i } = { Y > x i } . Thus, for b ≥ P (cid:16) Yx i > b (cid:12)(cid:12)(cid:12) f ( Y ) > x i (cid:17) = b − . Hence, L (cid:16) Y x i (cid:12)(cid:12)(cid:12) | X | > x i (cid:17) = Par(1) for all i .Now, suppose that x i = 3 · i , i ∈ N . By construction x i < Y < x i implies f ( Y )
5, leads to a different weak limit µ c (at least along a subsequence)with µ c ((1 , b c )) = 0 and µ c ([ b, ∞ )) = b − for all b ≥ b c = c c .In order to study the properties of the limit in (1.1) in more detail we will make fur-ther assumptions about the functional form of Φ and Ψ which relate to those givenin [24]. There, the single time series ( Y t ) t ∈ Z is analyzed and both the existenceand the form of the weak limitlim x →∞ L (cid:18) Y m x , . . . , Y n x (cid:12)(cid:12)(cid:12)(cid:12) | Y | > x (cid:19) for all −∞ < m ≤ n < ∞ are discussed. Under Condition 1.a and under anadditional assumption (cf. Condition 2.a below) this so-called tail chain bearsresemblance to a multiplicative random walk. The idea behind this condition andthe following Proposition 2.3 is that for a stochastic process which behaves roughlylike Y t +1 ∼ Y t · φ ( ǫ t +1 , sign( Y t )) as | Y t | → ∞ for a suitable function φ , the wholeprocess behaves like a multiplicative random walk given an extreme event at time 0.Note that our Condition 2.a is a slightly stronger version of [24, Condition 2.2]that will allow to simplify some of our proofs in Section 4. Condition 2.a
There exists a function φ : S × {− , } → R such thatlim y →∞ Φ ( yw ( y ) , v ( y )) y = wφ ( v, sign( w ))for all w ( y ) → w ∈ R , v ( y ) → v ∈ S . Here, sign( w ) = 2 · [0 , ∞ ) ( w ) − , where {·} ( · ) denotes the indicator function. ⊓⊔ This condition allows for the case φ ( · , · ) ≡ Remark 1
If we identify ( Y t ) t ∈ Z = ( σ t ) t ∈ Z with the volatility process of a financialtime series, there exist several examples which satisfy Condition 2.a: – “standard” GARCH(1,1) models, cf. (1.6), with φ ( ǫ t , sign( σ t ))= p α ǫ t + β , – GJR-GARCH(1,1) models (cf. [17]) which reflect asymmetric behavior of thevolatility process with σ t = α + ( α + δ { ǫ t > } ) σ t − ǫ t + β σ t − . Here, φ ( ǫ t , sign( σ t )) = q ( α + δ { ǫ t > } ) ǫ t + β , – SR-SARV (stochastic volatility) models defined by ζ t = σ t ǫ t +1 and volatilitysequence σ t = α + ( γ + α σ t − ) η t + β σ t − or σ t = α + ( γ + α σ t − ) η t + β σ t − , where (( η t , ǫ t )) t ∈ Z with η t ≥ η t and ǫ t for a fixed value of t (cf. [1]). (In this case the space S of innovations e ǫ t = ( ǫ t , η t ) is to be taken as R .) Here, φ (( η t , ǫ t ) , sign( σ t )) = α η t + β or φ (( η t , ǫ t ) , sign( σ t )) = √ α η t + β , respectively. Andree Ehlert et al.
If we identify ( Y t ) t ∈ Z with a volatility sequence, then Y t ≥ φ on sign( w ) is not necessary. For general hidden Markov models, however, theextremal behavior of Y t +1 may differ for the cases Y t → ∞ or Y t → −∞ .The following proposition puts the aforementioned heuristic Y t +1 ∼ Y t φ ( ǫ t , sign( Y t )) for | Y t | → ∞ on solid ground. It is taken from [24] andwill be fundamental to our subsequent analysis. Here and in the following, “ w ⇒ ”denotes weak convergence of probability measures. Proposition 2 (cf. [24, Theorem 2.3])
Let ( Y t ) t ∈ Z (not necessarily stationary)be given by (1.2) and let Conditions 1.a and 2.a hold. Then for n ∈ N , as y → ∞ , L (cid:18) | Y | y , Y | Y | , ǫ , Y | Y | , . . . , ǫ n , Y n | Y | (cid:12)(cid:12)(cid:12)(cid:12) | Y | > y (cid:19) w ⇒ L ( Y, M , ǫ ( Y )1 , M , . . . , ǫ ( Y ) n , M n ) , with M j = h ( M j − , A j , B j ) , j ∈ N , where h : R → R , h ( y, a, b ) := y (cid:0) a (0 , ∞ ) ( y ) + b ( −∞ , ( y ) (cid:1) , and Y, M , ǫ ( Y )1 ,ǫ ( Y )2 , . . . are independent with(i) Y ∼ Par ( α ) , i.e. P ( Y > x ) = x − α , x ≥ , (ii) P ( M = 1) = p = 1 − P ( M = − ,(iii) ǫ ( Y ) i , i ∈ N , are i.i.d. with L ( ǫ ( Y )1 ) = L ( ǫ ) and ( A i , B i ) = ( φ ( ǫ ( Y ) i , , φ ( ǫ ( Y ) i , − , i ∈ N . Note that by embedding the ǫ i , i ∈ N , the formulation of Proposition 2 differsslighty from its analog in [24]. The proof is analogous to the proof of [24, Theorem2.3] and uses the continuous mapping theorem. The joint limit distribution inProposition 2 will be an important building block for the derivation of (1.1). Butin order to derive the limit (1.1) we also need to specify the behavior of ( Y n ) n ∈ Z before the extremal event {| Y | > y } . Going backwards in time, things are notas simple as before. To illustrate this, think of the process Y t = aY t − + ǫ t with a >
0. Now, a large value of | Y | may either be due to a large value of | Y − | or dueto a large value of | ǫ | . However, if we assume stationarity of ( Y t ) t ∈ Z in additionto the assumptions of Proposition 2, then note the following: For x, y ∈ R it holdsthat P (min(( xY − ) + , ( yY ) + ) > t ) = P (min(( xY ) + , ( yY ) + ) > t )for all t >
0, where ( x ) + := max( x, | x | , | y | ≤ P (min(( xY − ) + , ( yY ) + ) > t | | Y | > t )= P (min(( xY ) + , ( yY ) + ) > t | | Y | > t ) , oint Extremal Behavior of Hidden and Observable Time Series 7 where the r.h.s. converges tolim t →∞ P min (cid:18) x | Y | t Y | Y | (cid:19) + , (cid:18) y | Y | t Y | Y | (cid:19) + ! > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Y | > t ! = P (cid:0) min (cid:0) ( xY M ) + , ( yY M ) + (cid:1) > (cid:1) = Z P (cid:0) min (cid:0) ( xM ) α + , ( yM ) α + (cid:1) > u (cid:1) du = Z ∞ P (cid:0) min (cid:0) ( xM ) α + , ( yM ) α + (cid:1) > u (cid:1) du = E (cid:0) min (cid:0) ( xM ) α + , ( yM ) α + (cid:1)(cid:1) , since Y − α ∼ Unif(0 , ,
1) denotes the uniform distribution on (0 , xM ) α + ≤ t →∞ L (cid:18) Y − | Y | , Y | Y | (cid:12)(cid:12)(cid:12)(cid:12) | Y | > t (cid:19) =: L ( M − , M )exists, then by a similar reasoning it holds that E (cid:0) min (cid:0) ( xM ) α + , ( yM ) α + (cid:1)(cid:1) = E (cid:0) min (cid:0) ( xM − ) α + , ( yM ) α + (cid:1)(cid:1) (2.3)for all | x | , | y | ≤
1. By rescaling it holds for all x, y ∈ R . In fact, one can show thatfor all laws L ( M , M ) which may evolve in Proposition 2 from a stationary Markovchain ( Y t ) t ∈ Z there exists a law L ( M − , M ) satisfying (2.3) and that this law issufficient to determine the whole backward limit process which is Markovian likethe forward limit process. Details can be found in [24], we state the main definitionsand results below. A precise form of the limit process for the GARCH(1,1) case isgiven in Section 6. Definition 1 (cf. [24], Definition 4.1)
A time series ( M t ) t ∈ Z is said to be aback-and-forth tail chain with index 0 < α < ∞ and forward transition law µ ,denoted by BFTC( α, µ ), if(i) L ( M , M ) = µ with M ∈ {− , } , M ∈ R ,(ii) µ ∗ := L ( M , M − ) is adjoint to µ , i.e. E (min (( xM ) α + , ( yM ) α + )) = E (min ( xM − ) α + , ( yM ) α + )) , ∀ x, y ∈ R , (2.4)(iii.a) for all integer t ≥ x t − , x t − , . . . , L ( M t | M t − = x t − , M t − = x t − , . . . ) = L ( h ( x t − , A , B )) , (cf. Proposition 2 for the definition of h ) where A and B are independentwith L ( A ) = L (cid:18) M M (cid:12)(cid:12)(cid:12)(cid:12) M = 1 (cid:19) , L ( B ) = L (cid:18) M M (cid:12)(cid:12)(cid:12)(cid:12) M = − (cid:19) , (iii.b) for all integer t ≥ x − t +1 , x − t +2 , . . . , L ( M − t | M − t +1 = x − t +1 , M − t +2 = x − t +2 , . . . ) = L ( h ( x − t +1 , A − , B − )) , where A − and B − are independent with L ( A − ) = L (cid:18) M − M (cid:12)(cid:12)(cid:12)(cid:12) M = 1 (cid:19) , L ( B − ) = L (cid:18) M − M (cid:12)(cid:12)(cid:12)(cid:12) M = − (cid:19) . Andree Ehlert et al.
Proposition 3 (cf. [24, Theorem 5.2])
Let ( Y t ) t ∈ Z be a stationary time seriesgiven by (1.2) and let Conditions 1.a and 2.a hold. Then, for all m, n ∈ N , as y → ∞ , L (cid:18) | Y | y , Y − m | Y | , . . . , Y n | Y | (cid:12)(cid:12)(cid:12)(cid:12) | Y | > y (cid:19) w ⇒ L ( Y, M − m , . . . , M n ) , (2.5) with(i) Y ∼ Par ( α ) , independent of ( M t ) t ∈ Z ,(ii) ( M t ) t ∈ Z is a BFTC ( α, µ ) where µ = L ( M , M ) with ( M , M ) as in Proposi-tion 2. Propositions 2 and 3 show that the assumption about the asymptotic behavior of Φ leads to a very simple form of the tail process for ( Y t ) t ∈ Z . The key ingredientis the connection between the forward and the backward limit process which isstated in (2.4). Although this equation is sufficient for a unique determination of L ( M , M − ) from L ( M , M ) it appears to be cumbersome for specific applica-tions. It is shown in [24, Equation (3.4)] that one may derive the law of ( M − , M )from that of ( M , M ) by noting that E ( f ( M − /M ) | M = σ ) = P ( M = σ ) − E ( f ( M /M )( σM ) α + )+ [1 − P ( M = σ ) − E (( σM ) α + )] f (0) (2.6)for integrable functions f , and σ ∈ {− , } such that P ( M = σ ) > Ψ . Condition 2.b
There exists a function ψ : S s × {− , } → R such thatlim y → + ∞ Ψ ( yw ( y ) , v ( y )) y = wψ ( v, sign( w ))for all w ( y ) → w ∈ R , v ( y ) → v ∈ S s . ⊓⊔ This condition allows for the case ψ ≡ Condition 2 is satisfied. Note thatfor all examples given in Remark 1, Condition 2.b holds due to the multiplicativeform of X t = Ψ ( σ t , ǫ t +1 ) = σ t ǫ t +1 . In the following, we will investigate the uniqueness of the accumulation pointof (1.1) under Conditions 1 and 2. It will turn out that the behavior of the uni-variate distribution L ( Y /x | | X | > x ) as x → ∞ leads to a sufficient condition. Proposition 4
Let ( Y t ) t ∈ Z and ( X t ) t ∈ Z be stationary time series given by (1.2) and (1.4) and let Conditions 1 and 2 hold. Equivalent are oint Extremal Behavior of Hidden and Observable Time Series 9 (i) the weak accumulation point of (1.1) is unique, and L ( Y ( X )0 ) := lim x →∞ L ( Y /x | | X | > x ) has no mass in zero,(ii) there exists a weak accumulation point L ( ˆ Y ( X )0 ) of L ( Y /x | | X | > x ) with ˆ Y ( X )0 = 0 a.s. Consequently, the uniqueness of the limit in (1.1) may well be derived from any weak accumulation point. The following lemma will be used in the proof of Propo-sition 4. In addition, it is also of interest in its own right as it provides a criterionfor property (ii) of Proposition 4.
Lemma 2
Let the assumptions of Proposition 4 hold and let L ( ˆ Y ( X )0 ) be any weakaccumulation point of L ( Y /x | | X | > x ) . Then P ( ˆ Y ( X )0 = 0) = 1 − C − E ( | χ | α ) , with χ = χ ( M , ǫ ( Y )1 , . . . , ǫ ( Y ) s ) = M · ψ ( ǫ ( Y )1 , . . . , ǫ ( Y ) s , sign( M )) , where M , ǫ ( Y )1 , . . . , ǫ ( Y ) s are defined as in Proposition 2, and C is given by (2.2) .Proof For a > P (cid:16) x − | Y | > a (cid:12)(cid:12)(cid:12) | X | > x (cid:17) = P (cid:16) ( ax ) − | X | > a − (cid:12)(cid:12)(cid:12) | Y | > ax (cid:17) · P ( | Y | > ax ) P ( | X | > x ) . With x → ∞ the second term converges to C − a − α by Condition 1. For the firstterm we analyze the limit of L (cid:16) ( ax ) − | X | (cid:12)(cid:12)(cid:12) | Y | > ax (cid:17) = L (cid:18) ( ax ) − (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) ax Y ax , ǫ , . . . , ǫ s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | Y | > ax (cid:19) (3.1)as x → ∞ . By an application of the continuous mapping theorem (cf. [20, Theo-rem 4.27]) in combination with Condition 2 and Proposition 2 this converges to L ( | Y · χ | ). Since Y is Pareto distributed and independent of χ we may rule out apoint mass of | Y · χ | in 1 /a , and conclude thatlim x →∞ P (cid:16) ( ax ) − | X | > a − (cid:12)(cid:12)(cid:12) | Y | > ax (cid:17) = P (cid:16) | Y · χ | > a − (cid:17) . Therefore, for a sequence a n ց Y ( X )0 itfollows that P ( | ˆ Y ( X )0 | > a n ) = P (cid:16) | Y · χ | > a − n (cid:17) · a − αn C = P (cid:0) Y · | χ | > a − n (cid:1) P (cid:0) Y > a − n (cid:1) · C .
The result now follows with a n ց x →∞ P ( Y · | χ | > x ) P ( Y > x ) = E ( | χ | α ) , (3.2) which can be seen as a kind of extension of Breiman’s Theorem (cf. [8]) for thespecial case of Y ∼ Par( α ). It follows because P ( Y · | χ | > x ) = P ( Y · | χ | > x, | χ | ≤ x ) + P ( Y · | χ | > x, | χ | > x ), where the first term equals R x P ( Y > xz ) dP | χ | ( z ) = R x (cid:0) zx (cid:1) α dP | χ | ( z ) . The second term equals P ( | χ | > x ), since Y ≥ P ( Y ·| χ | > x ) /P ( Y > x ) = x α P ( Y · | χ | > x ) = R ∞ z α [0 ,x ] ( z ) + x α ( x, ∞ ) ( z ) dP | χ | ( z )and (3.2) follows from monotone convergence. This gives the result. Proof (Proof of Proposition 4)
We show that (ii) implies (i). Let ν and ν denotetwo weak accumulation points. In the following, let a ≥ , a , . . . , a s ∈ R , and A = A ( a , . . . , a s ) := ( a , ∞ ) × . . . × ( a s , ∞ ). We will show that ν k (( a , ∞ ) × A )and ν k ([ − a , ∞ ) × A ) do not depend on k ∈ { , } . Here, we shall use that (ii)implies C = E ( | χ | α ) by Lemma 2, which in turn implies that ν ( { } × S s ) = 0 forany weak accumulation point ν . Since the above sets form a generating π -system,any two weak accumulation points coincide. By tightness (cf. Proposition 1) thisimplies weak convergence.Consider first a > a , . . . , a s = 0 that avoid the at most countably manypoint masses of the coordinate projections of ν and ν . Then, ν k (( a , ∞ ) × A ) isthe limit of P (cid:18) Y x > a , Y x > a , . . . , Y s x > a s (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) (3.3)along a subsequence depending on k . For general x , insert | Y | x > a in (3.3). Thisprobability equals P (cid:18) Y a x > , Y a x > a a , . . . , Y s a x > a s a , | X | a x > a (cid:12)(cid:12)(cid:12)(cid:12) | Y | > a x (cid:19) · P ( | Y | > a x ) P ( | X | > x ) . By Conditions 1 and 2, and since the variables have point masses at most at zero,this converges to P (cid:18) Y M > , Y M > a a , . . . , Y M s > a s a , Y | χ | > a (cid:19) · C a − α , cf. the proof of Lemma 2.We have shown that ν k (( a , ∞ ) × A ) does not depend on k . Approximationfrom inside extends this to all a ≥ a , . . . , a s ∈ R . Replacing Y x > a by Y x < − a the same computation followed by an approximation argument showsthe same for ν k (( −∞ , − a ) × A ). Combining these two results for a = 0 with ν k ( { } × R s ) = 0 shows that ν k ( R × A ) does not depend on k ∈ { , } . Thus, thesame holds for the sets [ − a , ∞ ) × A = ( R × A ) \ (( −∞ , − a ) × A ). Remark 2
Lemma 2 shows that Prop. 4 (ii), and thus (i), holds if and only if C = E ( | χ | α ). We give some examples. Suppose that ( X t ) t ∈ Z and ( Y t ) t ∈ Z are nonnega-tive time series and X t = Y t · ψ ( ǫ t +1 ), thus χ = ψ . Then, E ( ψ ( ǫ ) α + δ ) < ∞ for some δ > C = E ( | χ | α ) < ∞ by Breiman’sTheorem (cf. [8]) and Condition 2 holds if ψ is continuous. If P ( Y > x ) ∼ c · x − α for some c >
0, then E ( ψ ( ǫ ) α ) < ∞ suffices to derive the same result (cf. e.g.[18, Lemma 2.1]). For the special case Y ∼ P ar ( α ) cf. the end of the proof ofLemma 3.2. For further generalizations of Breiman’s Theorem see [12]. oint Extremal Behavior of Hidden and Observable Time Series 11 Remark 3
By similar computations it can be shown that under the assumptionsof Proposition 4 uniqueness of the weak limit in (1.1) is also ensured by P ( M − =0) = 0, with M − as in Proposition 3. A key step in the argument shows that thiscondition implies weak convergence of L (cid:18) Y − m y , ǫ − m +1 , . . . , Y − y , ǫ , Y y , ǫ , Y y , . . . , ǫ n , Y n y (cid:12)(cid:12)(cid:12)(cid:12) | Y | > y (cid:19) as y → ∞ for all m ≥ n ≥
0. We give an example with P ( M − = 0) = 0 but P ( Y ( X )0 = 0) >
0, i.e. P ( M − = 0) = 0 may ensure uniqueness even if property(ii) in Proposition 4 fails. To this end, let Y and ǫ t , t ∈ Z , be nonnegative i.i.d.random variables with P ( Y > x ) = x − ln( x ) − for x ≥ c , where c ≈ .
02 solves x · ln( x ) = 1. With Φ ( y, v ) = y , let Y t = Y for all t ∈ Z . Then, Y − = Y implies Y · M − ∼ Y ∼ Par(1), thus P ( M − = 0) = 0. For s − = − , s + = 1 let X t = Y t · ǫ t +1 , t ∈ Z . Careful calculations show that C = lim x →∞ P ( X >x ) P ( Y >x ) = 2( c + √ c )(cf. [12] for similar arguments). But with α = 1 it holds that E ( | χ | α ) = E ( ǫ ) = R ∞ P ( ǫ > x ) dx = c + c ) = c + √ c , thus P ( Y X = 0) = 1 / While the existence of a limit in (1.1) has been analyzed in the preceding sectionwe will now deal with the particular form of the limit. For easy reference we shallintroduce the following condition.
Condition 3
There exists a random vector ( Y ( X )0 , . . . , Y ( X ) s ) such thatlim x →∞ L (cid:18) Y x , . . . , Y s x (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = L ( Y ( X )0 , . . . , Y ( X ) s ) . ⊓⊔ We assume that the limit distribution in Condition 3 is unique in order tosimplify the statement of the proposition below. Note, however, Remark 4 at theend of this section for a generalization to the case of non-uniqueness. We will useConditions 1 to 3 to derive a result for the form of the limit in (1.1) which issimilar to Proposition 3.While Conditions 1 and 2 bear a natural resemblance to the assumptions madein [24], Condition 3 is necessary to ensure that a “starting point” for a tail chainexists that covers the time span from 0 to s where the ǫ , . . . , ǫ s and therefore Y , . . . , Y s are directly influenced by the event {| X | > x } . We will see that outsideof this range the behavior of the process ( Y t ) t ∈ Z corresponds to Proposition 3. Proposition 5
Let ( Y t ) t ∈ Z and ( X t ) t ∈ Z be stationary time series given by (1.2) and (1.4) and let Conditions 1, 2 and 3 hold. Then, for all integers m ≥ and n ≥ we have lim x →∞ L (cid:18) Y − m x , . . . , Y s + n x (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = L ( Y ( X ) − m , . . . , Y ( X ) s + n ) (4.1) with ( Y ( X )0 , . . . , Y ( X ) s ) as in Condition 3, and Y ( X ) t = h ( Y ( X ) t − , A t , B t ) , t > s,Y ( X ) − t = h ( Y ( X ) − t +1 , A − t , B − t ) , t > , cf. Proposition 2 for the definition of h . Here, ( A t , B t ) , t ∈ Z , are independent,and independent of ( Y ( X )0 , . . . , Y ( X ) s ) with L ( A t , B t ) = L ( A , B ) , t ≥ , L ( A t , B t ) = L ( A − , B − ) , t ≤ − . Further, L ( A , B ) and L ( A − , B − ) are as in Definition 1. The proof is predecessed by a lemma and a corollary where we only assume thatConditions 1 and 2 hold.
Lemma 3
Let m ≥ . For any η > there is δ ( η ) > such that for x largeenough P (cid:18) | Y − m | x > η, | Y | x ≤ δ (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) < η for all δ < δ ( η ) .Proof For m = 0 the statement follows with δ ( η ) = η . So assume that m > P (cid:18) | Y | x ≤ δ, | X | > x | Y − m | x > η (cid:19) · P ( | Y − m | > ηx ) P ( | X | > x ) . The second factor converges to C − η − α by Condition 1. It suffices to show thatthe first factor becomes small for δ →
0. To this end, note that by stationarity thefirst factor equals P (cid:18) | Y m | ηx ≤ δη , | X m | ηx > η | Y | > ηx (cid:19) which by definition of X m equals P | Y m | ηx ≤ δη , (cid:12)(cid:12)(cid:12) Ψ (cid:16) ηx Y m ηx , ǫ m +1 , . . . , ǫ m + s (cid:17)(cid:12)(cid:12)(cid:12) ηx > η | Y | > ηx . We proceed as in the proof of Lemma 2. By an application of the continuous map-ping theorem with Condition 2 and Proposition 2 this converges to P ( Y | M m | ≤ δ/η, Y | χ m | > /η ) with χ m := M m ψ ( ǫ ( Y ) m +1 , . . . , ǫ ( Y ) m + s , sign( M m )) . Again, we use that the two limit random variables include Y ∼ Par( α ) as anindependent factor which excludes point masses on the positive axis. Now, the set { Y | M m | ≤ δ/η, Y | χ m | > /η } is contained in {| χ m | / | M m | > /δ } = n | ψ ( ǫ ( Y ) m +1 , . . . , ǫ ( Y ) m + s , sign( M m )) | > /δ o . For 0 < δ < δ ( η ) the probability of this event gets arbitrarily small for δ ( η )small enough. oint Extremal Behavior of Hidden and Observable Time Series 13 Corollary 1
Let m, n ≥ and f be a bounded uniformly continuous function on R n +1 with f (0 , . . . ) = 0 . For any ǫ > there is δ ( ǫ ) > such that for x largeenough E (cid:18) f (cid:18) Y − m x , . . . , Y − m + n x (cid:19) · {| Y |≤ δx } | X | > x (cid:19) < ǫ (4.2) for all < δ < δ ( ǫ ) .Proof Since f is bounded and uniformly continuous with f (0 , . . . ) = 0, there issome η > || f || ∞ · η + sup {| f ( y − m , . . . , y − m + n ) | with | y − m | ≤ η } < ǫ. Choose δ as in Lemma 3. For δ < δ split the expected value in (4.2) into two bysplitting {| Y |≤ δx } into {| Y − m | >ηx, | Y |≤ δx } + {| Y − m |≤ ηx, | Y |≤ δx } . The first expected value is bounded by || f || ∞ · η by Lemma 3, and the second bysup {| f ( y − m , . . . , y − m + n ) | with | y − m | ≤ η } . Proof (Proof of Proposition 5)
Note that the case m = 0 and n ≥ ǫ s +1 , ǫ s +2 , . . . ) isindependent of ( X , Y , . . . , Y s ) the continuous mapping theorem can be applied toderive (4.1) and leads to the multiplicative structure with independent increments.Let now m ≥ n ≥
0, and let us assume that Proposition 5 holds for( Y ( X ) − m +1 , . . . , Y ( X ) s + n ). Let f : R s + m + n +1 → R be bounded and uniformly continu-ous. We will show thatlim x →∞ E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = E (cid:16) f ( Y ( X ) − m , . . . , Y ( X ) s + n ) (cid:17) (4.3)with ( Y ( X ) − m , . . . , Y ( X ) s + n ) as defined in the statement of the proposition. Let us furtherassume that f ( x, . . . ) = 0 as soon as x = 0. Note that an arbitrary function f : R s + m + n +1 → R can be split up additively into two functions f and f with f ( x − m , . . . , x s + n ) = f (0 , x − m +1 , . . . , x s + n ) ,f ( x − m , . . . , x s + n ) = f ( x − m , . . . , x s + n ) − f ( x − m , . . . , x s + n ) , such that the second function satisfies the aforementioned assumption and the firstfunction depends merely on ( x − m +1 , . . . , x s + n ). Since the induction hypothesisimplies that (4.3) is satisfied by a function of ( x − m +1 , . . . , x s + n ) the assumptionabout the structure of f is no loss of generality.The idea of the proof is to substitute the condition {| X | > x } by a corre-sponding event in ( Y t ) t ∈ Z . Let ǫ >
0. Then, for x large enough (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) − E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) {| Y | >δx } (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ, for all 0 < δ < δ ( ǫ ), where δ ( ǫ ) is chosen according to Corollary 1. We have E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) {| Y | >δx } (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = P ( | Y | > δx ) P ( | X | > x ) E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) {| X | >x } (cid:12)(cid:12)(cid:12)(cid:12) | Y | > δx (cid:19) = P ( | Y | > y ) P ( | X | > y/δ ) E (cid:18) f (cid:18) δ Y − m y , . . . , δ Y s + n y (cid:19) { δ ·| Ψ ( Y ,ǫ ,...,ǫ s ) | >y } (cid:12)(cid:12)(cid:12)(cid:12) | Y | > y (cid:19) with the substitution y = δx . Here, the first factor converges by Condition 1.Furthermore, an application of the continuous mapping theorem in connectionwith Propositions 2 and 3 yields that the whole expression converges to δ − α C E (cid:16) f ( δY M − m , . . . , δY M s + n ) { δ ·| Y M ψ ( ǫ ( Y )1 ,...,ǫ ( Y ) s ) | > } (cid:17) with Y, ǫ ( Y ) i , i ∈ N , and M n , n ∈ Z , as in Propositions 2 and 3. Defining newvariables ( e A − m , e B − m ) with the same distribution as ( A − m , B − m ) in the state-ment of the proposition and independent of Y, ǫ ( Y )1 , . . . , ǫ ( Y ) s , M − m +1 , . . . , M s + n ,( Y t ) t ∈ Z , ( X t ) t ∈ Z the above expression equals δ − α C E (cid:18) f (cid:16) h ( δY M − m +1 , e A − m , e B − m ) , . . . , δY M s + n (cid:17) { δ ·| Y M ψ ( ǫ ( Y )1 ,...,ǫ ( Y ) s ) | > } (cid:19) by the definition of M − m . Next, note that by the continuous mapping theoremthis equalslim y →∞ δ − α C E (cid:18) f (cid:18) h (cid:18) δ Y − m +1 y , e A − m , e B − m (cid:19) , . . . , δ Y s + n y (cid:19) { δ ·| X | >y } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Y | > y ! . Replacing y by δx and again using Condition 1 this becomeslim x →∞ E f (cid:18) h (cid:18) Y − m +1 x , e A − m , e B − m (cid:19) , . . . , Y s + n x (cid:19) {| Y | >δx } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! . Since both h and f are uniformly continuous with f (0 , . . . ) = 0 and h (0 , . . . ) = 0this giveslim x →∞ E f (cid:18) h (cid:18) Y − m +1 x , e A − m , e B − m (cid:19) , . . . , Y s + n x (cid:19) {| Y |≤ δx } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! = lim x →∞ E g (cid:18) Y − m +1 x , . . . , Y s + n x (cid:19) {| Y |≤ δx } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! for the complementary expression, where g ( y − m +1 , . . . , y s + n ) := E ( f ( h ( y − m +1 , e A − m , e B − m ) , . . . , y s + n ))with g (0 , . . . ) = 0. We may thus conclude from Corollary 1 thatlim x →∞ E f (cid:18) h (cid:18) Y − m +1 x , e A − m , e B − m (cid:19) , . . . , Y s + n x (cid:19) {| Y |≤ δx } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! oint Extremal Behavior of Hidden and Observable Time Series 15 tends to 0 as δ →
0. Thus,lim x →∞ E (cid:18) f (cid:18) Y − m x , . . . , Y s + n x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = lim x →∞ E f (cid:18) h (cid:18) Y − m +1 x , e A − m , e B − m (cid:19) , . . . , Y s + n x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | X | > x ! . An application of the continuous mapping theorem in connection with the induc-tion hypothesis yields that the latter expression equals E f (cid:16) h (cid:16) Y ( X ) − m +1 , A − m , B − m (cid:17) , . . . , Y ( X ) s + n (cid:17)! , with ( A − m , B − m ) as in the statement of the proposition. Since Y ( X ) − m = h (cid:16) Y ( X ) − m +1 , A − m , B − m (cid:17) this finishes the proof. Remark 4
If ( ˆ Y ( X )0 , . . . , ˆ Y ( X ) s ) is a random vector such that for a sequence ( x n ) n ∈ N with x n → ∞ the relationlim n →∞ L (cid:18) Y x n , . . . , Y s x n (cid:12)(cid:12)(cid:12)(cid:12) | X | > x n (cid:19) = L ( ˆ Y ( X )0 , . . . , ˆ Y ( X ) s )holds instead of Condition 3 then a statement analogous to Proposition 5 holdstrue along the sequence ( x n ) n ∈ N . The existence of such sequences is guaranteedby Condition 1, cf. Proposition 1. Remark 5
In order to simplify notation (using only s instead of s − and s + ), wehave assumed that (1.4) holds instead of (1.3). However, under assumption (1.3)the statement of Proposition 5 looks very similar, cf. [19], Theorem 3.5.2, fordetails. In this chapter we will show that Condition 3 is closely related to the theoryof multivariate regular variation. In a time series context this property is wellexplored in the case of GARCH( p, q ) processes, cf. [2].From the equivalent definitions of multivariate regular variation given in theliterature we shall refer to the one used in [24]. Recall that a measurable function U : R + → R + is said to be univariate regularly varying with index α ∈ R iflim x →∞ U ( τ x ) /U ( x ) = τ α for all τ >
0. We call a random vector Z ∈ R d mul-tivariate regularly varying if there exists a univariate regularly varying function U : R + → R + with index − α and a non-degenerate, non-zero Radon measure ν on E = [ −∞ , ∞ ] d \ { } such that P ( Z ∈ x · ) /U ( x ) v ⇒ ν ( · ) , x → ∞ , (5.1)where “ v ⇒ ” stands for vague convergence (cf. [23]) in M + ( E ), the space of allnonnegative Radon measures on E . One can show that the limit measure ν isnecessarily homogeneous, i.e. that ν ( xA ) = x − α ν ( A ) holds for all x > all Borel sets A ⊂ E (cf. [23]). The measure ν and, consequently, the extremalbehavior of Z are thus completely described by the index α of regular variation,a constant c > S on S d − := { x ∈ R d |k x k = 1 } . Thelatter is the so-called spectral measure. Altogether, we have that ν (cid:18)(cid:26) x ∈ E : k x k > a, x k x k ∈ · (cid:27)(cid:19) = c · a − α · S ( · )holds for all a > ǫ t , t ∈ Z , a stationary GARCH( p, q ) process is multi-variate regularly varying, i.e. for m, n ≥ Z = ( σ − m , ζ − m , . . . , σ n , ζ n )with σ t and ζ t as defined in (1.5) and (1.6) satisfies (5.1). Furthermore, one caneasily show that the same holds for the vector ( σ − m , | ζ − m | , . . . , σ n , | ζ n | ). Now,the fact that a certain vector derived from the processes ( Y t ) t ∈ Z and ( X t ) t ∈ Z ismultivariate regularly varying will be useful in the verification of Condition 3 asis shown in the following.Let us again assume that ( Y t , X t ) t ∈ Z is stationary and given by (1.2) and (1.4).Note that Condition 3 is equivalent tolim x →∞ P (cid:18)(cid:18) Y x , . . . , Y s x (cid:19) ∈ A (cid:12)(cid:12)(cid:12)(cid:12) | X | > x (cid:19) = lim x →∞ P (cid:0)(cid:0) Y x , . . . , Y s x (cid:1) ∈ A, | X | > x (cid:1) P ( | X | > x ) (5.2)= P (cid:16)(cid:16) Y ( X )0 , . . . , Y ( X ) s (cid:17) ∈ A (cid:17) for a random vector ( Y ( X )0 , . . . , Y ( X ) s ) and for all A ∈ B s +1 such that P (( Y ( X )0 , . . . , Y ( X ) s ) ∈ ∂A ) = 0.In the following we will assume multivariate regular variation of( | X | , Y , . . . , Y s ) on C = (cid:0) ¯ R + , × ¯ R s +1 (cid:1) \ { } , and show how this concept relatesstrongly to Condition 3. By continuity from below it suffices to look at such A which are bounded away from in order to derive Condition 3 from (5.2). Theassumption of multivariate regular variation of ( | X | , Y , . . . , Y s ) guarantees theexistence of a function U : R + → R + such thatlim x →∞ P (cid:0) | X | > x, (cid:0) Y x , . . . , Y s x (cid:1) ∈ A (cid:1) P ( | X | > x )= lim x →∞ P (cid:0)(cid:0) Y x , . . . , Y s x (cid:1) ∈ A, | X | > x (cid:1) U ( x ) U ( x ) P ( | X | > x )= ν ((1 , ∞ ) × A ) ν ((1 , ∞ ) × R s +1 ) (5.3)if the denominator is positive (it is necessarily finite since (1 , ∞ ) × R s +1 is boundedaway from the origin). One easily checks that (5.3) defines a probability measurefor A ∈ B s +1 and may be set as the law of the random vector ( Y ( X )0 , . . . , Y ( X ) s )if ν ((1 , ∞ ) × R s +1 ) >
0. Because of the aforementioned homogeneity of ν we notethe equivalence ν ((1 , ∞ ) × R s +1 ) = 0 ⇔ ν (( δ, ∞ ) × R s +1 ) = 0 ∀ δ > . (5.4) oint Extremal Behavior of Hidden and Observable Time Series 17 Thus, ν ((1 , ∞ ) × R s +1 ) = 0 implies that the mass of ν is concentrated on thehyperplane { } × R s +1 . Note that this is not excluded by our definition of regularvariation. Nevertheless, since ν is non-degenerate and since the process ( Y t ) t ∈ Z isstationary we find that ν ((1 , ∞ ) × R s +1 ) = 0 ⇒ lim x →∞ P ( | Y | > x ) U ( x ) > . On the other hand, ν ((1 , ∞ ) × R s +1 ) = 0 implies thatlim x →∞ P ( | X | > x ) U ( x ) = 0 . Hence, ν ((1 , ∞ ) × R s +1 ) = 0 entails that | X | and | Y | are not tail equivalent.This contradicts Condition 1 and leads to the following proposition. Proposition 6
Let ( | X | , Y , . . . , Y s ) ∈ R + , × R s +1 be a multivariate regularlyvarying vector with index α and let Condition 1 hold. Then Condition 3 is satisfied. ,
1) Processes
In this section we will apply Proposition 5 to the special case of a GARCH(1 , x →∞ L (cid:18) ζ − m x , . . . , ζ n x (cid:12)(cid:12)(cid:12)(cid:12) ζ > x (cid:19) , m, n ∈ N , (6.1)of the tail process of ( ζ t ) t ∈ Z , where ( ζ t , σ t ) t ∈ Z are given by (1.5) and (1.6). Tothis end, we shall make use of Proposition 5 and initially focus on the tail chain ofthe volatility sequence ( σ t ) t ∈ Z conditioned on the event {| ζ | > x } . We will thenobtain the desired distribution in (6.1) by the special structure of ( ζ t ) t ∈ Z . We shallhenceforth assume that α > α > β ≥ , α + β <
1, and let ǫ t , t ∈ Z , i.i.d. standard normal suchthat there exists a strictly stationary process which satisfies the aforementioneddefinitions, cf. [22]. Recall that the case β = 0 corresponds to the ARCH(1)model.It is well known that the marginal distributions of a stationaryGARCH( p, q ) process with standard normal innovations show a regularly vary-ing behavior. In the case of a GARCH(1 ,
1) process there exists a particularlysimple characterization of the corresponding index α of regular variation of thesquared processes ( ζ t ) t ∈ Z and ( σ t ) t ∈ Z in terms of the unique positive solution to E (cid:16)h α ǫ + β i α (cid:17) = 1 (6.2)(cf. [10], Theorem 1 and Example 2).In order to apply Proposition 5 we verify that Conditions 1 to 3 are satisfiedfor a GARCH(1 ,
1) process. It follows from the aforementioned regular variationof the marginal distribution of σ (and hence also of | σ | ) thatlim x →∞ P ( | σ | > ux ) P ( | σ | > x ) = u − α , for all u > . Since ǫ is independent of σ , and all moments of | ǫ | exist we may apply Breiman’sTheorem (cf. [8]) in order to find thatlim x →∞ P ( | ζ | > x ) P ( | σ | > x ) = lim x →∞ P ( | σ ǫ | > x ) P ( | σ | > x ) = E ( | ǫ | α ) (6.3)and Condition 1 is satisfied. By the specifications of Φ and Ψ given in (1.5) and(1.6) we get thatlim x →∞ x − Φ ( x, e ) = φ ( e ) = p α e + β , lim x →∞ x − Ψ ( x, e ) = ψ ( e ) = e, (6.4)where both Φ and Ψ are only defined for x ≥
0. This shows that Condition 2 holdswhere we dropped the second argument in φ and ψ for simplicity. ConcerningCondition 3 there are several instructive arguments: First, it is a direct consequenceof Proposition 6 given the multivariate regular variation of the vector ( | ζ | , σ , σ ).Alternatively, it follows from an application of Proposition 4 in connection withLemma 2 using that C = E ( | χ | α ) by (6.3). Finally, the following lemma statesboth the existence and the specific distribution of ( σ ( ζ )0 , σ ( ζ )1 ). Lemma 4
For the stationary processes ( ζ t ) t ∈ Z and ( σ t ) t ∈ Z given by (1.5) and (1.6)there exists a random vector ( ζ ( ζ )0 , ǫ ( ζ )1 ) such that lim x →∞ L (cid:18) | ζ | x , ǫ , σ x , σ x (cid:12)(cid:12)(cid:12)(cid:12) | ζ | > x (cid:19) = L (cid:16) | ζ | ( ζ ) , ǫ ( ζ )1 , σ ( ζ )0 , σ ( ζ )1 (cid:17) , (6.5) with σ ( ζ )0 = | ζ | ( ζ ) | ǫ ( ζ )1 | , σ ( ζ )1 = | ζ | ( ζ ) | ǫ ( ζ )1 | φ ( ǫ ( ζ )1 ) , where | ζ | ( ζ ) and ǫ ( ζ )1 are independent with | ζ | ( ζ ) ∼ Par (2 α ) and ǫ ( ζ )1 is a symmet-ric random variable such that ( ǫ ( ζ )1 ) / is Gamma distributed with shape parameter α + 1 / and scale parameter 1.Proof We show thatlim x →∞ L (cid:18) | ζ | x , ǫ (cid:12)(cid:12)(cid:12)(cid:12) | ζ | > x (cid:19) = L (cid:16) | ζ | ( ζ ) , ǫ ( ζ )1 (cid:17) , where L (cid:16) | ζ | ( ζ ) , ǫ ( ζ )1 (cid:17) is according to the above proposition. The statement thenfollows from an application of the continuous mapping theorem in connection with oint Extremal Behavior of Hidden and Observable Time Series 19 Condition 2. Now, for s ≥ , t ∈ R and v ∈ {− , } we havelim x →∞ P (cid:18) | ζ | x > s, | ǫ | ≤ t, sign( ǫ ) = v (cid:12)(cid:12)(cid:12)(cid:12) | ζ | > x (cid:19) = lim x →∞ P (cid:16) | ζ | x > s, | ǫ | ≤ t, sign( ǫ ) = v, | ζ | > x (cid:17) P ( | ζ | > x )= s − α E ( | ǫ | α ) lim x →∞ P ( | σ ǫ | > sx, | ǫ | ≤ t, sign( ǫ ) = v ) P ( σ > sx )= s − α E ( | ǫ | α ) lim x →∞ Z t P ( σ > sx/u ) P ( σ > sx ) F | ǫ | ( du ) P (sign( ǫ ) = v )= s − α E ( | ǫ | α ) Z t u α F | ǫ | ( du )= s − α Z t / Γ ( α + 1 / y α − / e − y dy, where the penultimate equality follows by uniform convergence of regularly varyingfunctions, cf. [5]. The last equality holds by substitution and using that E ( | ǫ | α ) = π − / α Γ ( α + 1 / | ζ | ( ζ ) , | ǫ ( ζ )1 | and sign( ǫ ( ζ )1 ) are independent random variables with the stated distribu-tions.Having checked that all three conditions are met we may now apply Proposition5 to the GARCH(1 ,
1) setting.
Proposition 7
Let the stationary time series ( ζ t ) t ∈ Z and ( σ t ) t ∈ Z be given by (1.5)and (1.6). Then, for all m, n ∈ N , as x → ∞ , L (cid:16) σ − m x , . . . , σ n x (cid:12)(cid:12)(cid:12) | ζ | > x (cid:17) → L (cid:16) σ ( ζ ) − m , . . . , σ ( ζ ) n (cid:17) (6.6) with ( σ ( ζ )0 , σ ( ζ )1 ) as in Lemma 4, and σ ( ζ ) t = σ ( ζ ) t − A t , t ≥ , and σ ( ζ ) − t = σ ( ζ ) − t +1 A − t , t ≥ , (6.7) with A t , t ≥ , defined by A t = φ (ˆ ǫ t ) = q α ˆ ǫ t + β (6.8) for an i.i.d. sequence (ˆ ǫ t ) t ≥ independent of ( σ ( ζ )0 , σ ( ζ )1 ) with standard normal dis-tribution, and A − t , t ∈ N , are i.i.d. random variables on (0 , β − / ) (i.e. on (0 , ∞ ) for β = 0 ) independent of ( A t ) t ≥ and ( σ ( ζ )0 , σ ( ζ )1 ) with distribution function P ( A − ≤ x ) = p /π Z ∞ (cid:16) x − − β α (cid:17) / ( α z + β ) α exp (cid:18) − z (cid:19) dz (6.9) for < x < β − / . Proof
We apply Proposition 5. Since ( σ t ) t ∈ Z ≥ B t ) t ∈ Z from Proposition 5 and restrict ourselves to the distribution of ( A t ) t ∈ Z . Now,(6.8) follows from Proposition 2. With µ = L (1 , A ) we are left to show that thedistribution defined in (6.9) equals the distribution of the second component of theadjoint measure µ ∗ = L (1 , A − ) corresponding to the BFTC(2 α, µ ). Using (2.6)(cf. also (4.3) in [24]) we have that P ( A − ≤ x ) = E h { ( α ˆ ǫ + β ) − / ≤ x } ( α ˆ ǫ + β ) α i + (cid:16) − E h ( α ˆ ǫ + β ) α i(cid:17) , for a standard normal ˆ ǫ . This gives (6.9) since the second summand equals zeroby (6.2). With A t , t ≥ t ≤
0, as above the assertion follows with Proposition5.Next, we take Proposition 7 as a starting point for the derivation of (6.1). To thisend, note that by (1.6) we have L ( ǫ t ) = L (cid:18) σ t − α σ t − − β (cid:19) / α − / S t ! , t ∈ Z , (6.10)for a sequence ( S t ) t ∈ Z of i.i.d. random variables independent of ( σ t ) t ∈ Z with P ( S = 1) = P ( S = −
1) = 1 / . (6.11)Now, by an application of the continuous mapping theorem in connection withProposition 7 and (6.10) we getlim x →∞ L (cid:18) ζ − m x , . . . , ζ n x (cid:12)(cid:12)(cid:12)(cid:12) | ζ | > x (cid:19) = L (cid:16) σ ( ζ ) − m ǫ ( ζ ) − m +1 , . . . , σ ( ζ ) n ǫ ( ζ ) n +1 (cid:17) = L (cid:16) ζ ( ζ ) − m , . . . , ζ ( ζ ) n (cid:17) , (6.12)with ǫ ( ζ ) t = ( σ ( ζ ) t /σ ( ζ ) t − ) − β α ! / S ( ζ ) t , t ∈ Z , for a sequence ( S ( ζ ) t ) t ∈ Z with the same distribution as ( S t ) t ∈ Z and independentof ( σ ( ζ ) t ) t ∈ Z . Note that by the structure of ( σ ( ζ ) t ) t ∈ Z it follows that ǫ ( ζ ) t = (cid:18) A t − β α (cid:19) / S ( ζ ) t , t ≥ , ǫ ( ζ ) t = A − t − − β α ! / S ( ζ ) t , t ≤ . (6.13)Now, by (6.8) it holds that L ( ǫ ( ζ ) t ) = L ( ǫ t ), t ≥
2. Further, we have that ζ ( ζ )0 issymmetric where P ( | ζ ( ζ )0 | > y ) = P ( | ζ | ( ζ ) > y ) = y − α , y ≥ , (6.14)by Lemma 4 and the definition in (6.12). For simulation from the r.h.s. of (6.12)it is advantageous to write ζ ( ζ ) ± t = | ζ | ( ζ ) t Y i =1 | ζ ( ζ ) ± i || ζ ( ζ ) ± ( i − | sign( ζ ( ζ ) ± t ) = | ζ | ( ζ ) t Y i =1 σ ( ζ ) ± i | ǫ ( ζ ) ± i +1 | σ ( ζ ) ± ( i − | ǫ ( ζ ) ± ( i − | sign( ζ ( ζ ) ± t )for t ∈ N , such that replacing for (6.5), (6.7) and (6.13) yields the following propo-sition. oint Extremal Behavior of Hidden and Observable Time Series 21 Proposition 8
For m, n ∈ N let – A t , − m ≤ t ≤ − , distributed according to (6.9), – S ( ζ ) t , − m ≤ t ≤ n, i.i.d. with the distribution of S in (6.11), – ǫ ( ζ ) t , ≤ t ≤ n + 1 , i.i.d. standard normal – ǫ ( ζ )1 according to Lemma 4 and – | ζ | ( ζ ) ∼ Par (2 α ) be mutually independent random variables. Then, lim x →∞ L (cid:18) ζ − m x , . . . , ζ x , . . . , ζ n x (cid:12)(cid:12)(cid:12)(cid:12) | ζ | > x (cid:19) = L | ζ | ( ζ ) (cid:18) m Y i =1 A − i (cid:19) (cid:0) ( A − − m − β ) /α (cid:1) / | ǫ ( ζ )1 | S ( ζ ) − m , . . . , | ζ | ( ζ ) S ( ζ )0 , . . . , | ζ | ( ζ ) (cid:18) n Y i =1 ( α ( ǫ ( ζ ) i ) + β ) / (cid:19) | ǫ ( ζ ) n +1 || ǫ ( ζ )1 | S ( ζ ) n ! . (6.15)Finally, conditioning on ζ > x as in (6.1) instead of | ζ | > x leads to the samelimit distribution as in (6.15) but with S ( ζ )0 = 1 almost surely. Our analysis of two connected time series was originally motivated by an idea to ex-tend the approach considered in [11] for ARCH(1) processes to a simulation studyfor extremal characteristics of the more general GARCH(1 ,
1) class. Among suchextremal measures the extremal index is a well-known example. It characterizes thebehavior of extreme events in a time series, i.e. the strength of dependence betweensubsequent high-level exceedances. More precisely, let M n := max( X , . . . , X n ) for n ∈ N where ( X t ) t ∈ N is a stationary univariate process with marginal distributionfunction F . Let further ( f X t ) t ∈ N be the associated i.i.d. sequence with the samemarginal distribution F , and let accordingly g M n := max( f X , . . . , f X n ) for n ∈ N .Assume that there exists a nonnegative number θ X such that for every τ > u n ) n ∈ N such thatlim n →∞ P ( g M n ≤ u n ) = e − τ and lim n →∞ P ( M n ≤ u n ) = e − θ X · τ . Then, θ X is called the extremal index of the process ( X t ) t ∈ N , and θ X ∈ [0 , ,
1) model as defined in Section 1 we find that θ ζ = lim m →∞ θ ζ,m where θ ζ,m = lim x →∞ P (max( ζ , . . . , ζ m ) < x | ζ > x ) , m ∈ N , (7.1)cf. [14] and [13, Section 5.2]. In addition to the extremal index we shall in the following also consider twoalternative extremal characteristics that may be evaluated by the same simulationapproach. The so-called extremal coefficient function discussed in [16] is given by χ ζ ( h ) = lim x →∞ P ( ζ h > x | ζ > x ) (7.2)for h ∈ Z . Following the notion of usual autocovariances the extremal coefficientfunction gives the conditional probability of two extreme events separated by alag h ∈ Z . For two reasons we will also briefly describe a modification of thisconcept, i.e. a probability for threshold exceedances at a lag h ∈ N given that ζ is not only extreme as in (7.2), but given that ζ is also at the beginning of anextremal cluster in a time series, cf. [13, Chapter 5] for a discussion. More precisely,let γ ζ ( h ) = lim m →∞ γ ζ,m ( h ) , h ∈ N , where γ ζ,m ( h ) = lim x →∞ P ( ζ h > x | ζ > x and ζ i ≤ x, i = − m, . . . , − . (7.3)The first reason to touch on this characteristic in our study is its potential toserve as a complement to the extremal coefficient function regarding questions ofcluster structures in risk management and related applications that focus on thedevelopment of extremal events. The second reason is related to the numericalsimulation of (7.1) to (7.3) that will be based on the tail chain concept discussedin Section 6. While the evaluation of θ and the extremal coefficient function χ requires a series of runs of either the forward or the backward tail chain it isevident from (7.3) that the simulation of γ must be based on simultaneous runsof the forward and the backward tail chain at the same time. Note at this pointthat we are not aware of any general closed form solutions for (7.1) to (7.3) thatwould include the GARCH( p, q ) model parameters, not even for p = q = 1.Our simulation setup generalizes similar methods proposed by [11] and [21].The algorithm used in [11] is restricted to single time series which satisfy theassumptions of [24] that do, however, not hold for the log returns in a GARCH(1 , x → ∞ in (7.1) to (7.3) we note that as indicated above allthree characteristics can be expressed via the tail chain distribution. By simulationfrom this distribution (cf. Proposition 8) we may therefore evaluate these quantitiesby Monte Carlo estimation. In Table 1 we report the results of such a simulationstudy for θ ζ,m and γ ζ,m ( h ) , m = 500, (which we use as approximations for θ ζ and γ ζ ( h )) and χ ζ ( h ) for h = 1 , ,
3. The evaluation of probabilities is based on N = 10000 draws. We fix α + β = 0 .
99 in the table in order to reflect the stylizedfact that α + β is close to one in many applications. The last row of Table 1 ismotivated by the following example. Example 1
We fit the GARCH(1,1) model given by (1.5) to a data set of log returnsof the S&P 500 index from 01.04.80 to 30.03.10 (7569 records). The estimatedparameters [27] areˆ α = 0 . × − (10 − ) , ˆ α = 0 .
072 (0 . , ˆ β = 0 .
920 (0 . oint Extremal Behavior of Hidden and Observable Time Series 23 α β α ˆ θ ζ,m ˆ χ ζ (1) ˆ χ ζ (2) ˆ χ ζ (3) ˆ γ ζ,m (1) ˆ γ ζ,m (2) ˆ γ ζ,m (3)0.99 0 1.014 0.570 0.213 0.139 0.104 0.251 0.167 0.1250.15 0.84 1.478 0.207 0.061 0.063 0.065 0.153 0.144 0.1390.11 0.88 1.838 0.245 0.052 0.042 0.038 0.110 0.104 0.1040.09 0.90 2.203 0.304 0.045 0.035 0.034 0.089 0.085 0.0810.07 0.92 2.885 0.397 0.022 0.020 0.020 0.055 0.050 0.0530.04 0.95 5.991 0.854 0.005 0.004 0.003 0.007 0.007 0.0060.072 0.920 2.476 0.317 0.021 0.020 0.027 0.063 0.064 0.066 Table 1
Extremal measures ( m = 500) for selected GARCH(1,1) processes with α + β =0 .
99, as well as the process fitted in Example 1. The results are based on N = 10000 runs ofthe tail chain. The approximate confidence intervals are smaller than ± .
01 for all entries. of Table 1. In order to discuss the adequacy of a GARCH(1 ,
1) model with regardto the extremal behavior we compare the result of Table 1 with the so-called blocksestimator of the extremal index for the given data [28, 4]. For a block length of m = 126 and a threshold corresponding to the empirical 0.95 quantile the estima-tor yields ˆ θ = 0 .
305 (0 . , . N = 1000 independent GARCH(1,1) pro-cesses of length 7569 according to (7.4). As to the choice of the block lengthnote that extremal events occuring in two distinct blocks are assumed to be in-dependent. Here, six trading months correspond to 126 days and appear to bea reasonable order of magnitude. Given that our block length is a valid choicethe fact that the result falls within the simulated confidence interval indicates asatisfactory agreement of the data set and a GARCH(1 ,
1) model with regard totheir extremal behavior.
Appendix
Details on the construction of f in the proof of Lemma 1 Let z = z i = 5 i , i ∈ N , and I = [ z, z ] , I = [2 z, z ] , I = [3 z, z ] , I = [4 z, z ].The restrictions f i := f | I i will look as follows: f is strictly increasing from value z to 3 z , f is symmetric on I , on the left half of I it increases from 3 z to 5 z .Finally, f interpolates linearly between the values 3 z and z , and f between z and 5 z .Given the definitions of f and f , we show that f , f can be defined implicitely.For the definition of f via f − consider x ∈ [ z, z ]. The function f − has tosatisfy x ! = P ( f ( Y ) > x ) = P ( f − ( x ) < Y < f − ( x )) + P ( f − ( x ) < Y ) = f − ( x ) − f − ( x ) + f − ( x ) thus f − ( x ) ! = x + f − ( x ) − f − ( x ) =: h ( x ). Observe that h ′ ( x ) = − x + / f − ( x )) + / f − ( x )) ≤ − z ) + / z ) + / z ) <
0. Thus f − ( x )is strictly increasing and f − ( z ) = z , f − (3 z ) = 2 z by the above formula. Thisshows that f is well-defined as the inverse of f − .For the existence of f on I as described it suffices to show that h ( x ) := x − P ( f − ( x ) ≥ Y ) = x − f − ( x ) , x ∈ [3 z, z ], is strictly decreasing (note that h (5 z ) =0) which follows from h ′ ( x ) = − x + / f − ( x )) ≤ − z ) + / . z ) < References
1. Andersen, T. G.: Stochastic autoregressive volatility: a framework for volatility modelling.Math. Finance , 75–102 (1994).2. Basrak, B., Davis, R.A. and Mikosch, T.: Regular variation of GARCH processes. StochasticProcesses and their Applications , 95–115 (2002)3. Basrak, B. and Segers, J.: Regularly varying multivariate time series. Stochastic Processesand their Applications , 1055-1080 (2009)4. Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J.: Statistics of Extremes. Wiley, Chich-ester (2004)5. Bingham, N. H., Goldie, C. M. and Teugels, J. L.: Regular Variation. Cambridge UniversityPress, Cambridge (1987)6. Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econometrics , 307–327 (1986)7. Boman, J. and Lindskog, F.: Support theorems for the Radon transform and Cram´er-Woldtheorems. Technical report, KTH Stockholm (2002)8. Breiman, L.: On some limit theorems similar to the arc-sin law. Theory Probab. Appl. ,323–331 (1965)9. Carrasco, M. and Chen, X.: Mixing and moment properties of various GARCH and stochas-tic volatility models. Econometric Theory. , 17–39 (2002)10. Davis, R. A. and Mikosch, T.: Extreme value theory for GARCH processes, in: Andersen,T.G., Davis, R.A., Kreiß, J.-P., Mikosch, T. (editors): Handbook of Financial Time Series.Springer, New York, 187–200 (2009)11. De Haan, L., Resnick, S.I., Rootz´en, H. and de Vries, C. G.: Extremal behaviour of so-lutions to a stochastic difference equation with applications to ARCH processes. StochasticProcesses and their Applications , 213–224 (1989)12. Denisov, D. and Zwart, B.: On a theorem of Breiman and a class of random differenceequations. J. Appl. Prob. , 1031–1046 (2007)13. Ehlert, A.: Characteristics for Dependence in Time Series of Extreme Values. Ph.D. Thesis,University of G¨ottingen (2010)14. Embrechts, P., Kl¨uppelberg, C. and Mikosch, T.: Modelling Extremal Events. Springer,Berlin (1997)15. Engle, R.: Autoregressive conditional heteroscedastic models with estimates of the varianceof United Kingdom inflation. Econometrica , 987–1007 (1982)16. Fasen, V., Kl¨uppelberg, C. and Schlather, M.: High-level dependence in time series models.Extremes , 1–33 (2010)17. Glosten, R., Jagannathan, R. and Runkle, D.: On the relation between expected valueand the volatility of the nominal excess return on stocks. Journal of Finance , 1779–1801(1993)18. Gomes, M. I., de Haan, L. and Pestana, D.: Joint exceedances of the ARCH process. J.Appl. Prob. , 919–926. (2004) (Correction: , 1206. (2006))19. Janßen, A.: On Some Connections between Light Tails, Regular Variation and Extremes.Ph.D. Thesis, University of G¨ottingen (2010)20. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)21. Laurini, F. and Tawn, J. A.: The extremal index for GARCH(1,1) processes. Extremes,Published Online First, doi: 10.1007/s10687-012-0148-z (2012)22. Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econometric The-ory, , 318–334 (1990)23. Resnick, S. I.: Heavy-Tail Phenomena. Springer, New York (2007)24. Segers, J.: Multivariate regular variation of heavy-tailed Markov chains, Institut de statis-tique DP0703, available on arxiv.org as math.PR/0701411 (2007)25. Smith, R. L.: The extremal index for a Markov chain. J. Appl. Prob. , 37–45 (1992)26. Taylor, S.: Modelling Financial Time Series. Wiley, Chichester (1986)27. Trapletti, A. and Hornik, K.: tseries: Time Series Analysis and Computational Finance,URL http://CRAN.R-project.org/package=tseries , R package version 0.10-18 (2009)28. Wuertz, D., et. al. see the SOURCE file: fExtremes: Rmetrics - Extreme Financial Mar-ket Data, URL http://CRAN.R-project.org/package=fEXtremeshttp://CRAN.R-project.org/package=fEXtremes