Tail empirical process for long memory stochastic volatility models Rafa l Kulik and Philippe Soulier Carleton University, 5 May 2010 Rafa l Kulik and Philippe Soulier
Quick review of limit theorems and LRD Assume that X i, i 1, is a stationary Gaussian process with covariance cov(x i, X j ) = ρ i j = i 2H 2 L 0 (j i), where H [1/2, 1) is the Hurst exponent and L 0 ( ) is a slowly varying function. Then [nt] 1 n H X i C(H)B H (t), L 0 (n) where i=1 B H (t) = (Γ(2H + 1) sin(πh))1/2 Γ(H + 1/2) Q (1) t (x; H)B(dx), and (B(t), < t < ) is a Brownian motion. Rafa l Kulik and Philippe Soulier 1
Moreover, let q be the Hermite rank of G( ), i.e. q := min{r 1 : J(r) = E[H r (X 1 )G(X i )] 0}, where H r ( ) is the rth Hermite polynomial, then if 1 q(1 H) > 1/2, then 1 n 1 q(1 H) L q 0 (n) [nt] (G(X i ) E[G(X i )]) C(H, q)l q (t), i=1 where L q ( ) is called Hermite-(Rosenblatt) process L q (t) := Q (q) t (x 1,..., x k ; H)B(dx 1 ) B(dx k ). Rafa l Kulik and Philippe Soulier 2
if 1 q(1 H) < 1/2, then n rate of convergence. Furthermore, recall that the limiting behaviour of max(x 1,..., X n ) is as if X i, i 1, were independent, identically distributed. Rafa l Kulik and Philippe Soulier 3
Stochastic volatility with LRD Assume that Z, Z i, i 1, is a sequence of i.i.d. random variables and that X i, i 1, is a stationary Gaussian process with covariance cov(x i, X j ) = ρ i j = i 2H 2 L 0 (i), where H [1/2, 1) is the Hurst exponent and L 0 ( ) is a slowly varying function. The case H = 1/2 can be formally thought as the case of weakly dependent Gaussian sequences. We assume that Z i, i 1, and X i, i 1, are independent. The stochastic volatility process is defined as Y i = σ(x i )Z i, where σ( ) is a nonnegative function. We note, in particular, that if E[Z1] 2 < and E[Z 1 ] = 0, then Y i, i 1, are uncorrelated. Rafa l Kulik and Philippe Soulier 4
Tail behaviour We will assume that for some α (0, ), F Z (z) = P(Z > x) = x α l(x), (1) where l is a slowly varying function. Having (1) and E[σ α+ɛ (X 1 )] <, F (x) = P(Y 1 > x) = P(σ(X 1 )Z 1 > x) E[σ α (X 1 )]P(Z 1 > x), as x. Consequently, with u n, n 1, u n, then T n (x) := F (u n + u n x), x 0, n 1, (2) F (u n ) satisfies T n (x) T (x) = (1 + x) α. Rafa l Kulik and Philippe Soulier 5
Partial sums and maxima The limiting behaviour of max{σ(x 1 )Z 1,..., σ(x n )Z n } is the same as in case of i.i.d. rvs with the same marginal distribution as σ(x 1 )Y Z. For α (1, 2) 1 n α l(n) [nt] i=1 where L α ( ) is a Lévy process. Y i fidi C(σ( ), α)l α (t), Rafa l Kulik and Philippe Soulier 6
For α (2, 4), and if α/2 > 2H 1, 1 n α/2 l(n) [nt] i=1 ( Y 2 i E[Y 2 i ] ) fidi C(σ( ), α)l α/2 (t), For α (2, 4), and if α/2 < 2H 1, 1 n 2H 1 L 2 0 (n) [nt] i=1 ( Y 2 i E[Y 2 i ] ) fidi C(σ( ), α, H)L 2 (t). Rafa l Kulik and Philippe Soulier 7
Tail empirical process 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, ). (3) Rafa l Kulik and Philippe Soulier 8
Tail empirical process - limiting behaviour Theorem 1. Let q be the Hermite rank of G(x) = σ α (x) with q(1 H) 1/2 + some technical assumptions. (i) If n F (u n )ρ q 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 q n as n then ρ q/2 n e n converges weakly in D([0, )) to the process (E[σ α (X 1 )]) 1 J(q)T L q. Rafa l Kulik and Philippe Soulier 9
Comments The meaning of the above result is that 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. Furthermore, small and big depends on the relative behaviour of the tail of Y 1 and the memory parameter. Note that the condition n F (u n )ρ q n implies that 1 2q(1 H) > 0, in which case the partial sums of the subordinate process {G(X i )} weakly converge to the Hermite process of order q. The two cases will be referred to as the limits in the i.i.d. zone and in the LRD zone. The result should be extendable to general, not necessary Gaussian, long memory linear sequences. Rafa l Kulik and Philippe Soulier 10
Case (i) guarantees Leadbetter s condition D (u n ). Therefore the condition C3 of Rootzén (2009) (2009) is fulfilled and in principle our result in the i.i.d. zone could be concluded from that paper, provided we can verify also Rootzen s conditions C1, C2 and β(u n ) mixing, which could be more difficult, than proving convergence via our approach. One can replace T n with T in the definition of the tail empirical process, provided a second order condition is fulfilled. 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 Here we consider the practical process ê n(s) = ˆT n (s) T (s), s [0, ). Rafa l Kulik and Philippe Soulier 12
Tail empirical process with random levels - result Theorem 2. Under the conditions of Theorem 1, together with a second order assumptions, 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). The behaviour described in Theorem 2 is quite unexpected, since the process with estimated levels Y n k:n has a faster rate of convergence than the one with the deterministic levels u n. A similar phenomenon was observed in the context of LRD based empirical processes with estimated parameters. Rafa l Kulik and Philippe Soulier 13
Applications to Hill estimator A natural application of the asymptotic results for tail empirical process ê n is the 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 Since γ = 0 (1 + s) 1 T (s) ds, we have ) = 0 ˆT n (s) 1 + s ds. ˆγ n γ = 0 ê n(s) 1 + s ds. Corollary 3. Under the assumptions of Theorem 2, k(ˆγ n γ) converges weakly to the centered Gaussian distribution with variance γ 2. Rafa l Kulik and Philippe Soulier 14
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 15
Terrible 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 16
Comments The model considered is σ(x i ) = exp(σx i ), where X i, i 1, are LRD standard normal. We may observe that for small volatility parameter σ there is not too much difference between i.i.d. Pareto-based Hill estimator and those for stochastic volatility models. However, if σ becomes bigger, estimation is completely inappropriate and is as bad for very strong memory as for i.i.d. case. Rafa l Kulik and Philippe Soulier 17