The tail empirical process for long memory stochastic volatility models with leverage

: We consider tail empirical processes of long memory stochas- tic volatility models with heavy tails and leverage. We study the limiting behaviour of the tail empirical process with both ﬁxed and random lev- els. We show a dichotomous behaviour for the tail empirical process with ﬁxed levels, according to the interplay between the long memory parame- ter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not aﬀected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic mean estimators. The paper can be viewed as an extension of [21]; while the presence of leverage in the model cre- ates additional theoretical problems, the limiting behaviour remains un-changed.


Introduction
The tail empirical process (TEP) is an important tool used in nonparametric estimation of extremal quantities, like the Hill estimator of the index of regular variation or various risk measures. In this article, we consider a long memory stochastic volatility model with leverage. The model consists of two building blocks: volatility and noise. Informally speaking, by leverage we mean that volatility and noise are not independent. This model is of interest in finance. Our goal is to study weak convergence for the tail empirical processes associated with heavy tailed long memory stochastic volatility sequences with leverage. These results are not only of theoretical interest, but are applicable to different statistical procedures based on intermediate extremes. A similar problem was studied in the case of independent, identically distributed random variables in [15], or for weakly dependent sequences in [14], [13], [26], [24]. In [21] the authors considered heavy tailed, long memory stochastic volatility (LMSV) models and obtained asymptotic results for tail empirical processes. This was extended later on to the multiparameter situation in [22]. However, in the latter two articles leverage was excluded, greatly simplifying theoretical considerations. As evidenced in [23], the presence of long memory, heavy tails and leverage may affect the limiting behaviour of relevant statistics.
It turns out that in the present setting, leverage does not affect the limiting behaviour of the tail empirical process, and hence the results are comparable to those in [21] where leverage is not present. The limiting behaviour depends only on the interplay between the tail index α and the strength of long memory. However, it should be pointed out clearly that the extension from models without leverage to those with leverage is highly nontrivial from a theoretical point of view. In [21] the authors were able to exploit independence between volatility and noise. Here this approach is not applicable and instead we use the Doob decomposition of the tail empirical process into martingale and long memory parts. This makes the proof of tightness technically very involved.
Furthermore, as in [21], for applications we must replace unobservable quantiles with appropriate order statistics. It turns out that the limiting behaviour of the resulting TEP with random levels is not affected by either long memory or leverage. This, through integral functionals, allows us to obtain limiting results for different estimators of the tail index, including the classical Hill estimator (see [10] for results in the i.i.d. case) or the more general class of harmonic mean estimators (see [2] again for results in the i.i.d. case). A comprehensive numerical study of different estimators of the tail index in the case of long memory and heavy tails (but no leverage) is presented in [25].
In summary, our contribution in this paper is twofold. From a theoretical point of view, our most important contribution is the proof of weak convergence of the tail empirical process (with fixed and random levels) in the presence of heavy tails, long memory and leverage. Due to the complicated dependence structure of the process, the proof is not at all straightforward. From a practical point of view, the key result is that the asymptotic behaviour of the TEP with random levels is unaffected by the presence of long memory and/or leverage in the model, and so in applications the log returns may be handled exactly as if they were i.i.d. heavy-tailed random variables. This greatly enhances the utility of the LMSV model with leverage considered here.
The rest of the paper is organized as follows. In Section 2 we introduce the model and state all relevant assumptions. In Section 3 we state our main result on convergence of the tail empirical process with fixed levels (Theorem 3.2). This theorem is complemented by the corresponding result for random levels (Theorem 3.8). In Section 4 we prove weak convergence of integral functionals of the tail empirical process, which provides a unified approach to central limit theorems for estimators of the tail index (Theorem 4.2). In Section 5, we discuss the role of the various assumptions and provide an illustrative example. We end with a brief conclusion and directions for further research in Section 6. Proofs are given in Section 7 and relevant technical details on regular variation and long memory sequences can be found in Appendix A.

Long memory stochastic volatility model with leverage
One of the common features of financial data is that returns are uncorrelated, but their squares, or absolute values, are (highly) correlated, a property referred to as long memory. A second widely accepted feature is that log-returns are heavy tailed, in the sense that some moment of the log-returns is infinite. A third important feature is leverage. In the financial time series context, leverage is understood as negative dependence between previous returns and future volatility; see (12). The leverage effect has been well documented in the economic literature. As stated in [16], any mathematical model approximating the evolution of asset price should be able to generate the leverage effect. See more details in Section 2.2 below.
Motivated by these empirical findings, one of the common modelling approaches is to represent log-returns {X j } as a stochastic volatility sequence j } is the conditional variance or, more generally, a certain process which stands as a proxy for the volatility. In such a process, long memory can only be modelled through the sequence {σ j }, while the tails can be modelled either through the sequence {Z j } or through {σ j }, or both. The well known GARCH processes belong to this class of models. The volatility sequence {σ j } is heavy tailed unless the distribution of Z 0 has finite support, and leverage can be present. However, long memory of squares cannot be modelled by the GARCH processes, since such models (under the appropriate technical conditions) are mixing. See [17,Chapter 3].
Consequently, the so-called long memory stochastic volatility (LMSV) model was introduced in [5]. An overview of such models is given in [11] and [12]. In the classical LMSV model, {Z j } is a sequence of i.i.d. standard normal random variables, independent of the volatility sequence {σ j }, assumed to be of the form σ j = exp(Y j ), where {Y j } is a long memory Gaussian sequence. However, the independence assumption excludes the possibility of modelling leverage effects. We thus consider the long memory stochastic volatility model with leverage: We make the following assumptions.

Assumptions
A(i) The sequence {Y j } is strictly stationary and ergodic long memory Gaussian, that is where { j } is a sequence of i.i.d. standard normal random variables and Note that a and Y are slowly varying functions at infinity such that: where B(a, b) denotes the Beta function and 0 < d < 1 2 is referred to as the long memory parameter (for details, see [1]). Furthermore, we assume that {( j , Z j )} is a sequence of i.i.d. random vectors. For each fixed j, j and Z j may be dependent, but due to the construction above, the random variables Y j and Z j are independent. However, there can be dependence between the sequences {Z j } and {Y j }, allowing for leverage in the model. See more discussion in Section 2.2. A(ii) The random variables Z j are i.i.d. with tail distribution functionF Z : where α, c * > 0 and η * (·) is either nonnegative or nonpositive, regularly varying at infinity with index −κ, κ > 0. It is also assumed that η * is bounded -that is, there exists β > 0 such that ∀ x > 0, We shall refer to a function of the form (2) as second-order regular varying at infinity with parameters −α, −κ and rate function η * . This set of functions is denoted by 2RV ∞ (−α, −κ, η * ). A(iii) The function φ is a nonnegative measurable function and φ(Y ) is not equal to 0 with probability one. We denote by m the Hermite rank of φ α .
(For more details on Hermite rank, see [1] pg. 108.) A(iv) Let k n → ∞ be an increasing sequence of positive integers such that k n /n → 0 and let u n be defined by u n =F −1 X (k n /n), whereF −1 X is the inverse function of the tail distribution functionF X of X. (As will be argued below,F X is continuous). For ease of notation, we suppress dependence of k n on n, which is the standard practice in the extreme value literature. A(v) For all n ≥ 1, define {a n,m } and {b n,m } as follows: We assume that a n,m η * F −1 X (k/n) = a n,m η * (u n ) → n→∞ 0 .
Long memory stochastic volatility models with leverage 3457 A(vi) For α, β, κ > 0 as above, there exists > 0 such that For clarity throughout the remainder of the article, when referring to the long memory stochastic volatility model with leverage, we suppose that all the assumptions A(i) to A(vi) are satisfied. However, some results do not require all the assumptions. See more discussion in Section 5.
Remark 2.1. The elementary consequence of A(ii) is that the second-order regular variation of Z also implies thatF Z is regularly varying at infinity with index −α -that is uniformly on compact subsets of (0, ∞). Furthermore, by (5a), an application of Breiman's Lemma (see [6]) gives Therefore,F X is also regularly varying at infinity with index −α and MoreoverF Z (x)/F X (x) is bounded away from zero and infinity, that is, there exists λ > 0 such that for all x > 0, Furthermore, A(ii) implies thatF Z is continuous. So isF X , by (7). Finally, as will be shown by eqs. (29a) and (29b), (4) controls bias. In particular, for any τ 0 > 0 we have Let {G j } be the minimal filtration generated by the independent and identically distributed random vectors {( j , Z j )}, that is As a consequence, X j is G j -adapted, Y j is G j−1 -measurable and we have

Leverage
In the financial literature, leverage is understood as an asymmetric behaviour of stock prices. More specifically, increase in volatility is (negatively) associated with a movement of stock prices. The original modelling approach to leverage is due to [19]. The authors consider the model normal vectors with mean zero, unit variance and correlation ω. Writing the logarithm of the volatility as where μ, ν are constants. Thus, the expected log-volatility is a linear function of X j whenever ω = 0. Of course, our model extends the AR(1) assumption on the sequence {Y j } allowing for long memory.
Other modeling approaches to leverage can be found e.g. in [28] or [7]. The only difference lies in alternative specifications in the equation for log σ 2 j , allowing for an additional random term.

Tail empirical process
In this section, we present two main results: Theorem 3.2 and Theorem 3.8. They both pertain to weak convergence of the tail empirical process of the long memory stochastic volatility model with leverage in (1). The first one deals with deterministic levels whereas the second one with random levels (order statistics). We note different limiting behaviour -in particular, while the limiting behaviour of the tail empirical process with deterministic levels could be affected by long memory, this is not the case for the process with random levels. Furthermore, neither of the tail empirical processes is affected by the presence of leverage.
Recall that k = k n → ∞ is an increasing sequence of positive integers such that k/n → 0, as n → ∞ and u n is defined by k = nF X (u n ).

Deterministic levels
Definition 3.1. Given a sequence (k n ) as above, the empirical tail distribution function of {X j } is defined as: In the sequel, let V j,n (t) := 1 {φ(Yj )Zj >unt} . We note that (cf. (8)) The tail empirical process (with deterministic levels) is defined by Our goal is to determine the asymptotic behaviour of S n under suitable normalizations. The structure of the model considered in (1) suggests the following martingale-long memory Doob decomposition: where the summands M n and L n are defined as follows: We will call M n and L n the martingale and long memory parts, respectively. To establish weak convergence of S n under suitable normalizations, we will establish weak convergence for M n and L n , suitably normalized. This will then determine the appropriate normalization for S n (cf. Theorem 3.2). The finite dimensional convergence of M n will be handled with a classical martingale central limit theorem (cf. Theorem 2.5 in [20]), while tightness requires tedious technical arguments. The process L n will be handled with a limit theorem for Hermite polynomials (cf. [1, p. 223, 229]). In what follows, define where H m denotes the Hermite polynomial of order m. (1).

Theorem 3.2. Let {X j } be the long memory stochastic volatility model with leverage given in
in D(0, ∞) equipped with the Skorokhod J 1 topology, where W (·) denotes a standard Brownian motion on (0, ∞).
The proof of this theorem can be found in Section 7.2.

Remark 3.3.
We note that leverage has no effect on the limiting distribution. Long memory affects the limiting behaviour. We have a dichotomous behaviour, according to strength of long memory (that is, the value of the parameter d that appears explicitly in the definition of b n,m ). In the long memory case, the limiting random variable, ξ m,d+1/2 (1), is Hermite-Rosenblatt with the Hurst parameter d + 1 2 (see Definition 3.24 in [1]). It is a non-Gaussian limit unless m = 1. Remark 3.4. We note that thanks to Assumption A(v), the effect of bias introduced by replacing T un (t) by T (t) is negligible (see (10)), and so the process has the same limiting behaviour as S n .
Remark 3.5. The upper quantile (u n ) depends on the unknown distribution F X and so its value is not known. This means that the empirical tail distribution function T n cannot be observed, and so we introduce a data-based version in the next section.

Random levels
In this section we consider the tail empirical process with random levels that is used to construct estimators of the tail index. Let X 1 , . . . , X n be a sample from the stochastic volatility model with leverage defined in (1). Let X (1) ≤ · · · ≤ X (i) ≤ · · · ≤ X (n) be the corresponding order statistics. Let F n,X be the usual empirical distribution function and F n, n,X (k/n) = X (n−k) , and so it is then natural to approximate u n with X (n−k) . Definition 3.6. Let {X j } be the long memory stochastic volatility model with leverage given in (1).

• The empirical tail distribution function with random levels of {X
• The tail empirical process with random levels of {X j } is We aim at studying the asymptotic behaviour of S n . First, we need an extension to Theorem 3.2, the next lemma.  (1).
Now, we state the result for random levels. Notice that by introducing random levels, the tail empirical process vanishes at 1 and ∞, which forces the limiting process to be of a bridge type. More surprisingly, the introduction of random levels appears to cause the effect of long memory to disappear. The reason for this, as will be seen in the proof, is that the limiting behaviour of S n follows informally from the continuous mapping theorem applied to S * n and X (n−k) /u n . Thanks to the degenerate structure of the limiting process for S * n (that is, a random variable scaled by a deterministic function), the long memory effect cancels out. Note however that long memory does play a role in assumption (4) which controls bias. This can affect the choice of k; see Example 5.1.
Once again, the presence of leverage does not affect the limit.
in D(0, ∞) equipped with the Skorokhod J 1 topology. The limiting process We refer to Section 7.3 for the proof of this result.

Integral functionals
The power of weak convergence theory comes from the fact that many diverse results emerge as corollaries of a basic convergence theorem. As we shall see in Theorem 4.1, our main convergence Theorem 3.8 can be extended to integral functionals of the tail empirical process with random levels. This in turn yields a unified approach to establishing weak convergence of estimators of the tail index (cf. Section 4.1). In what follows, r denotes a nonnegative integer.  (1).
The proof (see Section 7.3) is not trivial due to the fact that the infinite integral is not a continuous functional.

Tail index estimation
We consider the long memory stochastic volatility model with leverage defined in (1). Since the tail distribution of X is regularly varying with index −α, then this raises the question of estimating the index of regular variation α. Therefore, we restrict our attention to the harmonic moment estimators ( HME) γ r,k of order r of γ := 1/α. We aim at studying their asymptotic normality. For this purpose, we begin with their construction. Recalling that T (t) = t −α , we have for r ≥ 0, If ζ r,k denotes an estimator of ζ r , then the plug-in method and (16) yield To derive the estimators of γ = 1/α, we solve for 1/α in (21) and obtain Thanks to the plug-in method, we derive the HMEs below: • The HME that corresponds to r = 1 is the Hill estimator of γ = 1/α.
• The HME that corresponds to r = 2 is the t-Hill estimator of γ, that is The main result of this section, the asymptotic normality of γ r,k , is a simple application of the delta method and Theorem 4.1.

Theorem 4.2.
Let {X j } be the long memory stochastic volatility model with leverage in (1).
where N is a standard normal random variable.

Comments and example
• The Gaussian assumption on Y j can be easily replaced by assuming that Y j is an infinite order moving average process. Instead of using Hermite polynomials, convergence of the long memory part will be concluded using tools such as Appell polynomials or a version of martingale approximation. See [1, Section 4.2.5]. • We excluded the case of d = 0 which yields short memory. It is justified in [8] that in the case of short memory, the stochastic volatility sequence {X j } is mixing and limiting results for tail empirical processes can be concluded from [26]. For the tail empirical process with deterministic levels, instead of A(ii), only regular variation is needed, while the moment conditions (5a)-(5b) can be replaced with a weaker assumption, E ((φ(Y )) α+ ) < ∞ for some > 0, in order to guarantee that the tail functionF X is regularly varying. For the tail empirical process with random levels and for the Hill estimator, a version of second-order regular variation is needed. Our method of proof uses second-order regular variation for the tail empirical process with deterministic levels. Possibly, with another method of proof, this could be avoided, although second-order regular variation will likely still be needed for the tail empirical process with random levels. • More specifically, for finite dimensional convergence of the martingale part (Proposition 7.1) the moment condition (5a) is not needed, but secondorder regular variation plays a crucial role in the proof. The bias condition (4) is not used. • Lemma 7.2 does not require any distributional assumption on Z. Also, the moment conditions are not needed. Lemma 7.3 requires the moment assumption (5a). Only regular variation of Z is needed, second-order regular variation is not required. Lemma 7.4 again requires only (5a). In summary, tightness of the martingale part (Proposition 7.5) requires only regular variation and the moment condition (5a). • Thus, weak convergence of the martingale part requires all assumptions except for (4). • Weak convergence of the long memory part (Proposition 7.6) needs (5b) and second order regular variation with (4).
1. Notice thatF Z fulfills (2) and (3). In fact, for all x ≥ 1, Therefore,F Z ∈ 2RV ∞ (−α, −α(δ − 1), η * ), where the rate function is defined by Note that η * is nonnegative, regularly varying at infinity with index −α(δ− 1) and is bounded on [1, ∞) by β = α(δ − 1). 2. Finally to get a sense of (4), denote by c a generic, nonnegative constant that can be different at each appearance. We will assume for simplicity that m = 1 and we will ignore the slowly varying function in the definition of b n,1 . Recall also that Z and X are tail equivalent and k = nF X (u n ). Thus, in order to verify (4) is suffices to show that a n,1 η * (F −1 X (k/n)) → 0 , or equivalently X (y) ∼ cy −1/α as y → 0 and η * is regularly varying with index −α(δ − 1), this in turn is equivalent to Thus we get the following restrictions on the choice of k: The first restriction is the same as in i.i.d. case, while the second condition (stemming from long memory) may be more restrictive for small d. In summary, if k = o(n 1− 1 2δ−2 ), then bias is negligible.

Conclusion
In this article, we have considered the heavy-tailed long memory stochastic volatility model with leverage given in (1). We have studied the limiting behaviour of the tail empirical process with both fixed and random levels (Theorems 3.2 and 3.8). We have shown a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter d and the tail index α; leverage does not play a role in the limiting results, but makes the proofs technically involved. On the other hand, the tail empirical process with random levels is unaffected by either long memory or leverage. Further, we have proven the weak convergence of the corresponding integral functionals (Theorem 4.1). The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic mean estimators. Consequently, all HMEs of the tail index of {X j } remain valid for this model and have the same asymptotic behaviour as in the case of i.i.d. observations (Theorem 4.2). There are several directions for further research on long memory stochastic volatility models with leverage. For example, the asymptotic behaviour of estimators of risk measures will be affected by long memory, since the scaled intermediate order statistics are affected by long memory (cf. Lemma 3.7). Other topics of interest include: detection of changes in the model and bootstrap techniques for the model.

Proofs
In this section, we gather the proofs of all our results. Recall that k = nF X (u n ). Throughout the subsequent proofs, we define ρ n := X (n−k) u n ·

Some technical results
We start with some technical results used in the proofs.
Potter's bounds Regular variation ofF Z andF X yields different versions of Potter's bounds (cf. [27], [4, p. 25]). We state them the way they are used in this paper. First, for all > 0, there exists D( ) > 1 such that ∀ x ≥ 1, t > 0, Further ∀ C > 1, > 0, there exists δ = δ(C; ) ≥ 0 such that for x ≥ δ, t > 0, Notice that (9) in conjunction with (25) yield Derivatives It follows from (2) and (3) that there exists M > 0 such that for all x ≥ 1 and t > 0, As a consequence, for all x ≥ 1 and t > 0, The bound in (26b) is transferred to T x (t). Indeed, we have By taking the derivative with respect to x and using (27), we obtain Therefore, (26b) in conjunction with (9) yield that Thus by (5a), Second-order regular variation of X By (7), regular variation of Z is transferred to X. The same applies to second-order regular variation [21, p.117]. Indeed, we haveF X ∈ 2RV ∞ (−α, −κ, η), with the rate functionη defined as Bias Convergence in (6) or (8) induces bias in statistical inference. This bias is controlled with help of second-order regular variation. From [21] we have that for all > 0, there exist C( Thus, (4) and (29a) imply that for all τ 0 > 0, By (28) and (29b), the above bound also holds when J un is replaced with T un andF Z withF X . Therefore, (10) is justified.

Proof of Theorem 3.2: TEP with deterministic levels
Recall that the proof of Theorem 3.2 is based on the martingale-long memory Doob decomposition (13). Therefore, Propositions 7.1, 7.5 and 7.6 below will prove Theorem 3.2.

Weak convergence of the martingale part
In this subsection, we state and prove in where W is a standard Brownian motion.
is a martingale difference sequence. Now by Theorem 2.5 in [20], to prove (32) it is sufficient to show that for all t 1 , . . . , t m , > 0, It follows from the definition of j M * n that Consequently, to establish (34) it is sufficient to prove We start with (35a). Note that (11) in conjunction with (6) yield Ergodicity, Slutsky's Theorem and (7) On the other hand, by (7), showing that B n p − −−− → n→∞ 0, is equivalent to proving Now, we deal with (35b). By (11), (6) and (24), we have ∀ δ > 0, where C(δ), B(δ), R(δ), K(t i , δ), G(t q , δ) and I(t i , t q , δ) are constants depending on t i , t q and δ but not on n. By regular variation of (X j ) j and (Z j ) j , ergodicity and Slutsky's Theorem, we conclude that (35b) holds. This concludes the proof of (33a). It remains to show (33b). We observe that So, for arbitrary > 0, 1 {| j M * n (t)|> } = 0, for all n sufficiently large. Thus, (33b) is proven.

Tightness
We consider for ease of notation the following setup: for all 0 < s < t < ∞. We state and prove Lemmas 7.2 to 7.4. These results will serve as ingredients for the proof of tightness in Proposition 7.5.

Lemma 7.2.
Let {X j } be the long memory stochastic volatility model with leverage given in (1). Then, where C 4 is a constant defined in Rosenthal's inequality (cf. [18, p. 23-24]) and λ is as in (9).
Proof. Since { j M n } is a martingale difference sequence, then Rosenthal's inequality ( [18, p. 23-24]) holds and there exists a constant C 4 such that Note that if V is a nonnegative random variable, then for any σ-field F, we have This along with stationarity of ( j M * n (s, t)) j yield On the other hand, we have First, stationarity of ( j M * n (s, t)) j , Jensen's inequality, and (38) yield Second, Cauchy-Schwartz's inequality and stationarity of ( j M * n (s, t)) j yield Therefore, using (9), it follows that for n ≥ 1, Thus, (39) and (40) imply that (37) holds. (1). Then for all 0 < a ≤ s, t ≤ b < ∞, there exists a positive constant C a,b,4 such that

Lemma 7.3. Let {X j } be the long memory stochastic volatility model with leverage given in
Proof. The mean value theorem in conjunction with (26b) yield that there exists s < τ = τ (ω) < t, such that: The constant is finite by (5a). Hence, (37) becomes: Again by the mean value theorem, there exists τ * ∈ (s, t) such that: Hence, (41) follows by taking Note now that the process M * n defined in (36d) satisfies where M • n and M •• n are two non-increasing processes such that ∀ t > 0, Proof. It follows from (6) that where the last inequality holds by (9). The mean value theorem and (26b) yield that there exists Consequently, we have max 0≤i≤ln−1 and the latter expression converges to zero in probability, by the Law of Large Numbers and (5a). For i = l n , since M •• n is monotone and b < a + ln+1 nF X (un) , we obtain By the same argument as above the last term converges to zero in probability.
We wrap up this subsection with the statement and proof of Proposition 7.5.
Proposition 7.5. Let {X j } be the long memory stochastic volatility model with leverage in (1). The process M n is tight in D(0, ∞).

Long memory part
This subsection deals with the weak convergence of the long memory process L n defined in (14b). Proposition 7.6. Let {X j } be the long memory stochastic volatility model with leverage given in (1).

Conclusion of proof of Theorem 3.2
Propositions 7.1 and 7.5 imply weak convergence of the martingale part, Proposition 7.6 gives weak convergence of the long memory part. The final statement of Theorem 3.2 comes from comparing the rates of convergence of the martingale and long memory parts.

Proof of Theorems 3.8 and 4.1: TEP with random levels
In this section we present the proofs of Theorems 3.8 and 4.1.
The "martingale-long memory Doob decomposition" in (13) yields Since weak convergence to a continuous limit implies uniform convergence on compact sets, then by virtue of (31) and (18), we conclude that  (L n,j (ρ n t) − T (t)L n,j (ρ n )) .