: tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier
Plan Model assumptions; Limit theorems for partial sums and sample covariances (effect of leverage); Tail empirical process and asymptotic normality of Hill estimator; Limiting conditional laws (regular variation on subcones); Future work, open problems etc. Rafa l Kulik and Philippe Soulier 1
Stochastic volatility with LRD Let {Z i, i Z} be an i.i.d. sequence with regularly varying tails: lim x + P(Z 0 > x) x α L(x) = β, lim x + P(Z 0 < x) x α L(x) = 1 β, (1) where α > 0, L is slowly varying at infinity, and β [0, 1]. Under (1), if moreover E [ Z 0 α ] =, then Z 1 Z 2 is regularly varying and (Davis and Resnick, 1986) lim x + P(Z 0 > x) P(Z 0 Z 1 > x) = 0, lim x + P(Z 1 Z 2 > x) P( Z 1 Z 2 > x) = β 2 + (1 β) 2. Rafa l Kulik and Philippe Soulier 2
We will further assume that {X i } is a stationary zero mean unit variance Gaussian process which admits a linear representation with respect to an i.i.d. Gaussian white noise {η i } with zero mean and unit variance, i.e. X i = c j η i j, j=1 c 2 j = 1. (2) j=1 We assume that the process {X i } either has short memory, in the sense that its covariance function is absolutely summable, or exhibits long memory with Hurst index H (1/2, 1), i.e. its covariance function {ρ n } satisfies ρ n = cov(x 0, X n ) = c j c j+n = n 2H 2 l(n), (3) j=1 where l is a slowly varying function. Rafa l Kulik and Philippe Soulier 3
Let σ be a deterministic, nonnegative and continuous function defined on R. Define σ i = σ(x i ) and the stochastic volatility process {Y i } by Y i = σ i Z i = σ(x i )Z i. (4) Long Memory Stochastic Volatility (LMSV) model: where {η i } and {Z i } are independent. Model with leverage: where {(η i, Z i )} is a sequence of i.i.d. random vectors. For fixed i, Z i and X i are independent, but X i may not be independent of the past {Z j, j < i}. Due to Breiman s lemma, if E[σ α+ɛ (X)] <, then lim x + P(Y 0 > x) P(Z 0 > x) = lim x + P(Y 0 < x) P(Z 0 < x) = E[σα (X 0 )]. (5) Rafa l Kulik and Philippe Soulier 4
Define Partial sums S p,n (t) = [nt] i=1 Y i p, a n = inf{x : P( Y 0 > x) < 1/n}. For any function g such that E[g 2 (η 0 )] < and any integer q 1, define J q (g) = E[H q (η 0 )g(η 0 )], where H q is the q-th Hermite polynomial. The Hermite rank τ(g) of the function g is the smallest positive integer τ such that J τ (g) 0. Let R τ,h be the so-called Hermite process of order τ with self-similarity index 1 τ(1 H). Let τ p = τ(σ p ) be the Hermite rank of the function σ p. Rafa l Kulik and Philippe Soulier 5
Theorem 1. Under the above assumptions, 1. If p < α < 2p and 1 τ p (1 H) < p/α, then a p n (S p,n ne[ Y 0 p ]) D L α/p, (6) where L α/p is a totally skewed to the right α/p-stable Lévy process. 2. If p < α < 2p and 1 τ p (1 H) > p/α, then n 1 ρ τ p/2 n (S p,n ne[ Y 0 p ]) D J τ p (σ p )E[ Z 1 p ] R τp,h. (7) τ p! 3. If p > α, then a p n S p,n D Lα/p, where L α/p is a positive α/p-stable Lévy process. Rafa l Kulik and Philippe Soulier 6
Sample covariances Define Ȳp,n = n 1 n i=1 Y i p and the sample covariances ˆγ p,n (s) = 1 n n ( Y i p Ȳp,n)( Y i+s p Ȳp,n). i=1 Furthermore, define the function K p,s by K p,s(x, y) = m p σ p (x)e[σ p (κ s ζ + c s η 0 + ς s y) Z 0 p ] m p E[σ p (X 0 )σ p (X s ) Z 0 p ], where m p = E[ Z 0 p ], κ s = s 1 j=1 c2 j, ς2 s = j=s+1 c2 j, ζ and η 0 are two independent standard Gaussian random variables. Also, define b n = inf{x : P( Y 0 Y 1 > x) 1/n}. The form of b n is clear in LMSV case, but may be more involved in case of leverage. Rafa l Kulik and Philippe Soulier 7
Theorem 2. Let τ p(s) be the Hermite rank of the bivariate function K p,s, with respect to a bivariate Gaussian vector with standard marginal distributions and correlation ςs 1 γ s. If p < α < 2p and 1 τ p(s)(1 H) < p/α, then nb p n (ˆγ p,n (s) γ p (s)) d L s, where L s is a α/p-stable random variable. If p < α < 2p and 1 τ p(s)(1 H) > p/α, then ρ τ p (s)/2 n (ˆγ p,n (s) γ p (1)) d G s, where the random vector G s is Gaussian if τ p(s) = 1. Rafa l Kulik and Philippe Soulier 8
Comments: The Hermite rank of K p,s may be different in LMSV situation and the case of leverage. Results are also formulated in a multivariate set-up. Dichotomous behaviour according to the interplay between tails and memory. For partial sums and infinite variance transformations of LRD Gaussian sequences, see Davis (1983), Sly and Heyde (2008). For sample covariances of LRD linear processes with heavy tailed innovations, see Kokoszka and Taqqu (1996), Horvath and Kokoszka (2008). For weakly dependent heavy tailed stochastic volatility models without leverage, see Davis and Mikosch (2001). Proofs use point process techniques and martingale part/long memory part decomposition. Several new technical results on tail behaviour of products, beyond Breiman s lemma. We refer to Kulik and Soulier (2012a). Rafa l Kulik and Philippe Soulier 9
Tail empirical process Let u n, n 1, u n. Under our model assumptions T n (x) := F (u n + u n x), x 0, n 1, (8) F (u n ) satisfies T n (x) T (x) = (1 + x) α. Define (see Rootzén (2009), Drees (2000)), T n (s) = 1 n F (u n ) n 1 {Yj >u n +u n s}, j=1 and e n (s) = T n (s) T n (s), s [0, ). (9) Recall that τ p = τ(σ p ). Rafa l Kulik and Philippe Soulier 10
Theorem 3. Consider the model without leverage. (i) If n F (u n )ρ τ 1 n 0 as n, then n F (u n ) e n ( ) converges weakly in D([0, )). to the Gaussian process W T ( ), where W is the standard Brownian motion. (ii) If n F (u n )ρ τ 1 n as n, then ρ τ 1/2 n e n ( ) converges weakly in D([0, )) to the process (E[σ α (X 1 )]) 1 J τp (σ p )T ( )R τp,h(1). Comments For u n big, long memory does not play any role. However, if u n is small, long memory comes into play and the limit is degenerate. One can replace T n with T in the definition of the tail empirical process, provided a second order condition is fulfilled. If the dependence is strong enough, then the limit is degenerate (cf. Dehling and Taqqu 1989). Rafa l Kulik and Philippe Soulier 11
Tail empirical process with random levels Let U(t) = F (1 1/t), where F is the left-continuous inverse of F. Define u n = U(n/k). If F is continuous, then n F (u n ) = k. Define ˆT n (s) = 1 k n 1 {Yj >Y n k:n (1+s)}, j=1 ê n(s) = ˆT n (s) T (s), s [0, ). Theorem 4. Under the conditions of Theorem 3, together with a second order assumption, we have: kê n converges weakly in D([0, )) to B T, where B is the Brownian bridge (regardless of the behaviour of kρ q n). Rafa l Kulik and Philippe Soulier 12
Applications to Hill estimator A natural application of the asymptotic results for the tail empirical process ê n is asymptotic normality of the Hill estimator of the extreme value index γ defined by ˆγ n = 1 k k log i=1 ( Yn i+1:n Y n k:n ) = 0 ˆT n (s) 1 + s ds. Corollary 5. Under the assumptions of Theorem 4, k(ˆγ n γ) converges weakly to the centered Gaussian distribution with variance γ 2. For estimation of the tail index in long memory models see also Abry, Pipiras, Taqqu (2007), Beran, Das (2012). Rafa l Kulik and Philippe Soulier 13
Nice Hill plot Estimated tail index 1.6 1.8 2.0 2.2 Estimated tail index 0 1 2 3 4 0 200 400 600 800 1000 Number of extremes 0 200 400 600 800 1000 Number of extremes Figure 1: Hill estimator: α = 2 and Pareto iid (left panel), σ = 0.05 (right panel) Rafa l Kulik and Philippe Soulier 14
Less nice Hill plot Estimated tail index 2.0 2.2 2.4 2.6 2.8 Estimated tail index 0 1 2 3 4 0 200 400 600 800 1000 Number of extremes 0 200 400 600 800 1000 Number of extremes Figure 2: Hill estimator: α = 2 and Pareto iid (left panel), σ = 1 (right panel) Rafa l Kulik and Philippe Soulier 15
Limiting conditional laws In SV model exceedances over a large threshold are asymptotically independent. More precisely, for any positive integer m, and positive real numbers x, y, lim n np(y 0 > a n x, Y m > a n y) = 0. (10) However, there is a spillover from past extreme observations onto future values in the sense that lim P(Y m y Y 0 > t) = E[σα (X 0 )F Z (y/σ(x m ))] t E[σ α (X 0 )]. (11) We may be also interested in other extremal conditioning events. Rafa l Kulik and Philippe Soulier 16
To give a general framework for these conditional distributions, we introduce a modified version of the extremogram of Davis and Mikosch (2009). For fixed positive integers h < m and h 0, Borel sets A R h and B R h +1, we are interested in the limit denoted by ρ(a, B, m): γ(a, B, m) = lim t P((Y m,..., Y m+h ) B (Y 1,..., Y h ) ta). (12) The set A represents the type of events considered. For instance, if we choose A = {(x, y, z) [0, ) 3 x + y + z > 1}, then for large t, {(Y 2, Y 1, Y 0 ) ta} is the event that the sum of last three observations was extremely large. The set B represents the type of future events of interest. Rafa l Kulik and Philippe Soulier 17
We will consider regular variation on subcones using a concept of a hidden regular variation (Resnick 2007, 2008). We are interested in cones C such that there exists an integer β C and a Radon measure ν C on C such that, for all relatively compact subsets A of C with ν C ( A) = 0, lim t P((Z 1,..., Z h ) ta) ( F Z (t)) β C = ν C (A). (13) The number β C corresponds to the number of components of a point of a relatively compact subset A of the cone C that must be separated from zero. Rafa l Kulik and Philippe Soulier 18
Proposition 6. Consider the model without leverage. Under general conditions on cones and moments, there exists an integer β C and a Radon measure ν C on C such that (13) holds and for all relatively compact sets A C with ν C ( A) = 0, for m > h and h 0, and for any Borel measurable set B R h +1, we have lim t P(Y 1,h ta, Y m,m+h B) ( F Z (t)) β C = = E [ ν C (σ(x 1,h ) 1 A)P(Y m,m+h B X ) ]. It follows the extremogram can be expressed as γ(a, B, m) = E [ ν C (σ(x 1,h ) 1 A)P(Y m,m+h B X ) ] E[ν C (σ(x 1,h ) 1 A)]. (14) Rafa l Kulik and Philippe Soulier 19
Example 1 Fix some positive integer h and consider the cone C = (0, ] h. Then β C = h and the measure ν C is defined by ν C (dz 1,..., dz h ) = α h h i=1 z α 1 i dz i. Consider the set A defined by A = {(z 1,..., z h ) R h + z 1 > 1,..., z h > 1}. Then for m > h, and B R h +1, Proposition 6 yields lim P(Y m,m+h t B Y 1 > t,..., Y h > t) = [ h ] E i=1 σα (X i )P(Y m,m+h B X ) = [ h ]. E i=1 σα (X i ) Rafa l Kulik and Philippe Soulier 20
Example 2 Consider C = [0, ] [0, ] \ {0}. Then β C = 1 and the measure ν C is ν C (dz) = α{z α 1 1 d 1 δ 0 (dz 2 ) + δ 0 (dz 1 )z α 1 2 dz 2 }, where δ 0 is the Dirac point mass at 0. Consider the set A defined by A = {(z 1, z 2 ) R 2 + z 1 + z 2 > 1}. If E[σ α+ɛ (X 1 )] < for some ɛ > 0, then Proposition 6 yields lim P(Y m,m+h t B Y 1 + Y 2 > t) = = E [ P(Y m,m+h B X )(σ α (X 1 ) + σ α (X 2 )) ] E[σ α (X 1 )] + E[σ α (X 2 )]. Rafa l Kulik and Philippe Soulier 21
Estimation An estimator ˆγ n (A, B, m) is naturally defined by ˆγ n (A, B, m) = n r=1 1 {Y r,r+h 1 Y (n:n k) A}1 {Yr+m,r+m+h B} n r=1 1 {Y r,r+h 1 Y (n:n k) A}, where k is a user chosen threshold and Y (n:1) Y (n:n) increasing order statistics of the observations Y 1,..., Y n. are the Rafa l Kulik and Philippe Soulier 22
Estimation-ctd. Theorem 7. Consider the model without leverage. Assume that A is a relatively compact subcone of C. Let k/n 0, n(k/n) β C and assume that the bias is negligible. 1. If n(k/n) β Cρ τ(a,b) n 0, then n(k/n) β C µc (A){ˆγ n (A, B, m) γ(a, B, m)} converges to a centered Gaussian distribution. 2. If n(k/n) β Cρ τ(a,b) n, then ρn τ(a,b)/2 {ˆγ n (A, B, m) γ(a, B, m)} converges weakly to a distribution which is non-gaussian except if τ(a, B) = 1. Rafa l Kulik and Philippe Soulier 23
Open problems Estimation of memory parameter for volatility models using wavelet methods: (cf. Moulines, Roueff, Taqqu 2008-2011). Inference in traffic models (Fay, Mikosch and Samorodnitsky 2006, Fay, Roueff and Soulier 2007). Inference in continuous models, like (FI)CARMA driven by a Lévy noise (cf. Brockwell, Davis, Stelzel, Fasen, Klueppelberg...). Semiparametric methods: logperiodogram regression and local Whittle estimation: Robinson (1995) for long memory Gaussian processes, Moulines, Hurvich, Soulier (2003) for stochastic volatility models. Nothing known for infinite variance linear processes with long memory. Rafa l Kulik and Philippe Soulier 24