Tail Dependence for Regularly Varying Time Series

Abstract

We use tail dependence functions to study tail dependence for regularly varying (RV) time series. First, tail dependence functions about RV time series are deduced through the intensity measure. Then, the relation between the tail dependence function and the intensity measure is established: they are biuniquely determined. Finally, we obtain the expressions of the tail dependence parameters based on the expectation of the RV components of the time series. These expressions are coincided with those obtained by the conditional probability. Some simulation examples are demonstrated to verify the results we established in this paper.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ai-Ju Shi 1, 2 and Jin-Guan Lin 1

Recommended by Ming Li

1, Department of Mathematics, Southeast University, Nanjing 210096, China
2, College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

Received 9 January 2012; Accepted 14 March 2012

1. Introduction

Copula is a useful tool for handling multivariate distributions with given univariate margins. A copula C is a distribution function, defined on the unit cube [0,1^]d , with uniform one-dimensional margins _Ui . For any (_u1 ,...,_ud )∈^[0,1]d , C(_u1 ,...,_ud )=P{_U1 ...4;_u1 ,...,_Ud ...4;_ud } ; the survival copula is C...(_u1 ,...,_ud )=P{_U1 ...5;1-_u1 ,...,_Ud ...5;1-_ud } , the joint survival function of copula C is C¯(_u1 ,...,_ud )=C...(1-_u1 ,...,1-_ud ) . Given a copula C , let [figure omitted; refer to PDF] then F is a multivariate distribution with univariate margins _F1 ,...,_Fd . On the other hand, given a distribution F with margins _F1 ,...,_Fd , there exists a copula C such that (1.1) holds. And copula C is unique if _F1 ,...,_Fd are all continuous (Sklar [1], Nelsen [2]).

In generally, copula forms a natural way to describe the dependence between series when making abstraction of their marginal distributions. Overviews of the probabilistic and statistical properties of copula are to be found in [1-6].

Tail dependence plays an important role among dependence measures due to its ability to describe dependence among extreme values (Frahm et al. [7], Resnick [8, 9], and Nikoloulopoulos et al. [10]) which is introduced by Joe [4]. The issue of tail dependence is mainly for heavy tailed phenomena, heavy tailed phenomenon in fractal time series. It is extensively studied and applied in insurance, risk management, traffic management and engineering management, and so forth. [11-27].

Researchers find various multivariate distributions with heavy tails to describe the extremal or tail dependence, see, Pisarenko and Rodkin [13], Hult and Lindskog [28], and Fang et al. [29]. Many interesting tail quantities have been derived via standard methods: coefficients of tail dependence [30-37] and tail dependence copulas (Charpentier and Segers [38]).

In this paper, we are interested in the tail behavior of the time series _X1 ,...,_Xt which have the form: [figure omitted; refer to PDF] where the scale variable R is independent of random vector (_Z1 ,...,_Zt ) . And X is multivariate regularly varying with distribution function F having copula C .

This distribution is a generalized class, including, for example, multivariate Pareto and multivariate elliptical distribution as special ones. Especially, the multivariate t distribution is included in it. As an example, we will justify the results through multivariate t copula.

In order to analyze the tail dependence behavior of (1.2), we first study the tail dependence functions via intensity measure. Then using the relation between tail dependence parameter and the tail dependence functions, we explore the explicit representations of the tail dependence parameters.

The outline of this paper is as follows. After some preliminaries about multivariate regularly varying series and dependence functions in Section 2, detailed results for the tail dependence functions are discussed in Section 3, the expressions of tail dependence parameters for RV time series are demonstrated in Section 4, and multivariate t distribution is demonstrated as an example in Section 5.

Throughout, (_X1 ,...,_Xt ) is a random vector with joint distribution function F and copula C . Minima and maxima will be denoted by ⋀ and ⋁ , respectively. The Cartesian product ^∏i=1t [_ai ,_bi ] is denoted by [a,b] for any a,b∈^R¯t .

2. Preliminaries

Definition 2.1.

The t -dimensional random vector X is said to be regularly varying with index α...5;0 if there exists a random vector Θ with values in ^...t-1 a.s., where ^...t-1 denotes the unit sphere in ^Rd with respect to the norm |·| , such that, for all u>0 , [figure omitted; refer to PDF] as x[arrow right]∞ . The symbol [arrow right]v stands for vague convergence on ^...t-1 ; vague convergence of measures is treated in detail in Kallenberg [39]. The distribution of Θ is referred to as the spectral measure of X . For further information on multivariate regular variation we refer to Resnick [8, 9].

In fact, (2.1) is equivalent to the following expression [figure omitted; refer to PDF] where μ is an intensity measure or Radon measure on R/{0} and { _an } is a sequence an of nonnegative numbers.

From the Definition 2.1, we can see that the regularly varying distribution is connected with intensity measure μ . The following lemma yields the explicit relation between them which can be found in [8].

Lemma 2.2.

Let random vector X be regularly varying with index α...5;0 and distribution function F , then it is equivalent to the following.

(1) There exists an intensity measure μ on ^Rt /{0} , such that for every Borel set B⊂^Rt /{0} bounded away from the origin that satisfies μ(∂B)=0 , [figure omitted; refer to PDF] with the homogeneous condition μ(uB)=^u-α μ(B) .

(2) There exists an intensity measure μ on ^Rt /{0} , such that [figure omitted; refer to PDF] for all continuous points x of μ . According to Lemma 2.2, one notices that for any nonnegative multivariate regularly varying random vector X , its nondegenerate univariate margins _Xi have regularly varying right tails and with the same index of X also, that is, [figure omitted; refer to PDF] where _Li (x) is a slowly varying function.

Lemma 2.3 (Breiman [40]).

Let ξ and η be two independent nonnegative random variables, η be regularly varying with index α . If there exists a γ>α , such that E^ξγ <∞ , then [figure omitted; refer to PDF] The multivariate version of the Lemma belongs to Basrak et al. [41]. It is said that, if X is regularly varying in the sense of (2.2), A is a random t×t matrix, independent of X , with 0<E^||A||γ <∞ for some γ>α , then [figure omitted; refer to PDF] where [arrow right]v denotes vague convergence on ^...t /{0} .

Definition 2.4 (Kluppelberg et al. [42]).

Let F be the distribution function of random vector X with continuous margins _Fi ,1...4;i...4;t and copula C . For any w=(_w1 ,_w2 ,...,_wt )∈^R+t , the lower dependence function is defined as [figure omitted; refer to PDF] and the upper dependence function is defined as [figure omitted; refer to PDF] The upper exponent function is defined as [figure omitted; refer to PDF] where _uS (_wS ;_CS )=_{lim x[arrow right]⁰⁺} C¯(1-x_wj ,∀j∈S)/x .

From the definition, we can verify the elementary properties listed in Proposition 2.5 of the tail dependence function. We denote _τJ =_{lim x[arrow right]^1-} P{_Fj (_Xj )>x,∀j∉J|"_Fi (_Xi )>x,∀i∈J} and _ξJ =_{lim x[arrow right]⁰⁺} P{_Fj (_Xj )<x,∀j∉J|"_Fi (_Xi )<x,∀i∈J} are the upper tail and lower dependence parameters of X , respectively, where J is a nonempty subset of I={1,...,t} . _CJ is the margin of C with component indexes in J .

Proposition 2.5.

(1) For any 1...4;i,j...4;t , [figure omitted; refer to PDF] where _Cij is the margin copula of _Xi ,_Xj .

(2) : For any nonempty J⊂I ,

[figure omitted; refer to PDF]

(3)

[figure omitted; refer to PDF]

Proof.

(1) According to the definition of _τij , we get [figure omitted; refer to PDF] similarly, [figure omitted; refer to PDF]

(2) : Note that

[figure omitted; refer to PDF] combined with (2.9), the first part is determined. The second part can be verified similarly.

(3) : We can obtained the proof only paying attention to C¯(_u1 ,...,_ut )=C...(1-_u1 ,...,1-_ut ) .

From the proposition, the upper tail dependence function of copula C is the lower one of its survival copula C... . And in most fractal time series, from the point of view of either theory or applications, people only need to understand the right tail of the data, so we focus on the upper tail function u(w;C) and coefficient _τJ in the following.

We first study the upper tail dependence function of multivariate regularly varying time series in (1.2) using the intensity measure.

3. The Upper Tail Dependence Function for RV Time Series

Theorem 3.1.

Let _X1 ,...,_Xt be RV time series with regularly varying index α , distribution function F , copula C , and the stochastic representation as (1.2). If the margins are tail equivalent as x[arrow right]∞ , then the upper tail dependence function can be written as [figure omitted; refer to PDF] and the upper exponent function can be written as [figure omitted; refer to PDF]

Proof.

For any w=(_w1 ,...,_wt )∈^R+t , [figure omitted; refer to PDF] Since every margin _Fi is regularly varying with the same index α , we obtain that [figure omitted; refer to PDF] where _Li (y) is slowing varying function. So for any _wi >0 , as x[arrow right]⁰⁺ , [figure omitted; refer to PDF] where _hi (_wi ,y)=_Li (^wi1/α y)/_Li (y)[arrow right]1 as y[arrow right]∞ . So the equation becomes [figure omitted; refer to PDF] in other words, [figure omitted; refer to PDF] Now we let _F¯i (y)=x_wi , then [figure omitted; refer to PDF] so, ^F¯i-1 (x_wi )=^{wi-1/αF¯i-1} (x_hi (_wi ,^F¯i-1 (x_wi ))) .

As x[arrow right]⁰⁺ , _hi (_wi ,^F¯i-1 (x_wi ))[arrow right]1 , so we get that [figure omitted; refer to PDF]

And since the margins are equivalent, that is, _F¯i (y)/_F¯1 (y)[arrow right]1 as y[arrow right]∞ . We have ^F¯i-1 (x)/^F¯1-1 (x)[arrow right]1 as x[arrow right]⁰⁺ (Resnick [8]). So for sufficient small x , ^F¯i-1 (x)[approximate]^F¯1-1 (x) , and z=^F¯1-1 (x) , combining (3.3) and (2.3), we obtain that [figure omitted; refer to PDF]

In order to calculate ^u* (w;C) , we recall the inclusion-exclusion formula , it says that [figure omitted; refer to PDF] is valid for any finite set I and arbitrary events _Ai , where i∈I .

Using this formula, (2.10) becomes [figure omitted; refer to PDF] By using the same method of (3.3), the following equation holds: [figure omitted; refer to PDF]

Corollary 3.2.

Under the same conditions as Theorem 3.1, the following result holds [figure omitted; refer to PDF]

Proof.

By (2.4), one can see that μ(^[0,1]c )=1 . So we can get the result immediately by letting all _wi =1, 1...4;i...4;t in (3.2).

According to Theorem 3.1 and Corollary 3.2, we can represent the intensity measure through the tail dependence function as the following Corollary.

Corollary 3.3.

Under the same conditions as Theorem 3.1, one has [figure omitted; refer to PDF]

4. The Upper Tail Dependence Parameters for Regularly Varying Time Series

According to Proposition 2.5 and Theorem 3.1, we can express the tail dependence parameters by their tail dependence functions. In this section, we will deduce the upper tail dependence parameters of time series with multivariate varying distribution in (1.2) by this method. Hereafter, we let μ be the intensity measure of R=(R,R,...,R) with copula ^CR . Where R is regularly varying at ∞ with index α , with survival function _F¯R (r)=L(r)/^rα , and L(·) is a slowly varying function. So for any nonnegative vector w=(_w1 ,...,_wt ) , we have [figure omitted; refer to PDF] by inserting _F¯R (r^⋀i=1t_wi )=L(r^...i=1t_wi )/^{(r...i=1t_wi )α} and _F¯R (r)=L(r)/^rα into the representation, then, [figure omitted; refer to PDF] Similarly, we have, [figure omitted; refer to PDF] Consequently, we get the main result as follows.

Theorem 4.1.

Let _X1 ,...,_Xt be regularly varying time series with the same regularly varying index α and the stochastic representation given in (1.2), the margins are tail equivalent as x[arrow right]∞ . If there exists a γ>α holds for 0<E(^Zi+γ )<∞ , then the upper tail dependence parameter of _X1 ,...,_Xt is [figure omitted; refer to PDF]

Proof.

We first calculate the tail dependence function of X=(R_Z1 ,...,R_Zt ) . In the following, let ^CX and ^CY be the copula of X and Y , respectively. Denote [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Note that _Yi =(_Zi+ /^{(E(Zi+α ))1/α} ) R is strictly increasing transformation of _Xi >0, for all i∈I , and the tail dependence function and the parameter are all copula properties. Hence Y and X have the same tail dependence functions. By Lemma 2.3, one can see that the marginal variables _Yi of vector Y are tail equivalent and regularly varying with the same index as X as x[arrow right]∞ . Denote the intensity measures of Y and R by μ...(·) and μ(·) , respectively. According to (2.7), [figure omitted; refer to PDF] Now by (4.6), we see that, [figure omitted; refer to PDF] combining this with (4.3), for any nonnegative w , we obtain the intensity measure given by [figure omitted; refer to PDF] Hence, we have [figure omitted; refer to PDF] Substituting this measure into (3.1), we get the upper tail dependence function of vector Y as follows: [figure omitted; refer to PDF] Since Y and X have the same tail dependence functions, we have [figure omitted; refer to PDF] By (2) in Proposition 2.5, we obtain the upper tail dependence parameters of vector X .

5. Examples

Let Z in (1.2) be Z=A(_U1 ,...,_Un ) , where A is a t× n matrix with A^AT =Σ , and Σ is a t×t semidefinite matrix, U=(_U1 ,...,_Un ) is uniformly distributed on the unit sphere (with respect to Euclidean distance) in ^Rn . We know that X conforms to an elliptical contoured distribution (Fang et al. [43]). The tail dependence of the elliptical contoured distribution has been discussed in Schmidt [33]. Here we select the t distribution to display our results in Theorem 4.1 as a special case.

If X~_{[double struck t]n} (μ,Σ,ν) , then X has the stochastic representation ([43]): [figure omitted; refer to PDF] where S~^χν2 and Z~_...n (0,Σ) are independent, μ∈^Rn .

Let R=ν/S . Then ^R2 ~IG(ν/2,ν/2) and R is regularly varying with index ν at ∞ . So the vector (_X1 ,...,_Xn ) is regularly varying according to Schmidt [33].

For the upper tail dependence that only relies on the tail behavior of the random vector, we can focus, without loss of generality, on the random vector X with zero mean vector. Furthermore, since the strictly increasing transformation of (_X1 ,...,_Xn ) does not change the copula, ^Δ-1/2 X has the same copula as X , where Σ=(_σij ) and Δ=diag (_σ11 ,_σ22 ,...,_σnn ) . Thus ^Δ-1/2 X~_{[double struck t]n} (0,^Δ-1/2 Σ^Δ-1/2 ,ν) . It is evident that ^Δ-1/2 Σ^Δ-1/2 becomes the correlation matrix of the random vector. Consequently, we may assume that the covariance matrix Σ is the correlation matrix. In this situation, all ^{Zi[variant prime]}_s have the same margins as N(0,1) . So E(^Zi+ν ) are all equal for any 1...4;i...4;n . Under these assumptions, using (4.4), we get the upper tail dependence parameter of _tn (0,Σ,ν) as [figure omitted; refer to PDF] This is coincided to the one obtained in Shi and Lin [34].

6. Simulations

In Section 4, we obtain the expressions of the tail dependence indexes about RV time series in (1.2). In Section 5, we display our result in the multivariate t distribution as example. In this Section, we will illustrate these results by some Monte Carlo simulated numerical examples. Given that ^y(1) ,^y(2) ,...,^y(m) be generated from the multivariate normal distribution _Nn (0,ρ) , then the upper tail dependence indices of _tn (_μ ,Σ,ν) can be estimated by [figure omitted; refer to PDF]

We estimate the upper tail dependence parameter of 3-dimensional t distribution under autoregressive of order 1 (AR(1)), exchangeable(EX), Toeplitz(TOEP), and unstructured(UN) correlation structure, respectively. For each correlation matrix, we first generate 80,000 pseudorandom vectors, then use (5.2) to estimate tail dependence parameter for different ν . Specifically, we do the following simulations. [figure omitted; refer to PDF]

Let J={2} and {1,2} , respectively. The corresponding upper tail dependence parameters are denoted by _τ2 and _τ12 . _Σ1 and _Σ2 are under AR(1) and EX correlation structure, respectively, the simulated values of _τ2 , _τ12 about different ν are computed and plotted in Figure 1. _Σ3 and _Σ4 are under TOEP and UN correlation structure, the corresponding results are demonstrated in Figure 2.

The estimation of _τ2 , _τ12 under AR(1) (the left one) and EX (the right one) correlation structure.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The estimation of _τ2 , _τ12 under TOEP (the left one) and UN (the right one) correlation structure.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

From the two figures, in spite of the correlation structure, _τJ decreased and approached 0 quickly as ν increased to ∞ , which is the tail dependence index for multivariate normal copula.

Many researchers try to discuss the monotonicity of the tail dependence parameter about the regular varying index. Embrechts et al. [11] proved that the tail dependence of the bivariate t distribution is decreasing about the regular varying index ν , and demonstrated that the tail dependence parameter _τ1 is decreasing in ν by numerical results. But From the right graph in Figure 2., these conclusions are not always correct when t...5;3 . [figure omitted; refer to PDF]

7. Conclusion

In the paper, we mainly study tail dependence of RV time series in (1.2). We use tail dependence function and intensity measure to express tail dependence parameters. Using tail dependence function, we do not need to consider the explicit representation of the copula. We first discuss the tail dependence function of the RV time series due to the propositions of the regularly varying function, connecting the biuniquely determined property between the tail dependence function and the intensity measure. Then we calculate the explicit formula of the upper tail dependence parameter about the RV time series under some conditions. In fact, we can obtain the extreme upper tail dependence index (Shi and Lin [34]) very similarly to Theorem 4.1, for concise, we omit it here.

Copula of continuous variables is invariant under strictly increasing transformation (Nelsen [2]). In order to obtain the tail dependence function of random vector X , we shift to solve that of Y in (4.5), which is just a strictly increasing transformation of X .

At last, we select the t distribution as a special case to display our result, they are coincided to the one given in [34]. The monotonicity of the tail dependence parameters about the regular varying index is still an open problem. Under what constraints the tail dependence parameters will be deceasing in the variation index? We are still interested in the problem. We will discuss it in the following work in details. In engineering application, when we confront fractal time series and seasonal data, we can model the tail dependence property via the tail dependence function if the data is consistent with the constraint conditions in our work.

Acknowledgment

The authors are very grateful to the referees for their comments and suggestions on the earlier version of the paper, which led to a much improved paper. This project is supported by MSRFSEU 3207011102, NSFC 11171065, and NSFJS BK2011058.

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