A Quadratic ARCH( ) model with long memory and Lévy stable behavior of squares

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1 A Quadratic ARCH( ) model with long memory and Lévy stable behavior of squares Donatas Surgailis Vilnius Institute of Mathematics and Informatics onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 1 / 30

2 Contents 1. Stylized facts of nancial returns 2. GARCH, ARCH( ) and Linear ARCH (LARCH) 3. Sentana s Quadratic ARCH (QARCH) 4. LARCH + ( ): existence, uniqueness, Volterra representation 5. Leverage and long memory 6. Limit of partial sums of squares: dichotomy between FBM and Lévy 7. Some proofs: diagrams, Hamilton cycles, principle of conditioning 8. Open problem - what next? Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 2 / 30

3 1. Stylized facts of nancial (daily) returns returns X t = log(p t /p t 1 ) are uncorrelated: corr(x t, X s ) 0 (t 6= s) squared and absolute returns have long memory: corr(x 2 t, X 2 s ) 6= 0, corr(jx t j, jx s j) 6= 0 (jt sj = ) heavy tails: EX 4 t = conditional mean µ t = E [X t jf t 1 ] 0, conditional variance σ 2 t = E Xt 2 jf t 1 randomly varying (conditional heteroskedasticity) leverage e ect: past returns and future volatilities negatively correlated: corr X s, σ 2 t < 0 (s < t) volatility clustering Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 3 / 30

4 2. GARCH, ARCH(infty) and Linear ARCH (LARCH) GARCH(p, q) : X t = σ t ε t, σ 2 t = ω + p i=1 β i σ 2 t i + q i=1 α i X 2 t i, ω 0, α i 0, β i 0, p, q = 0, 1,..., (ε t ) iid, E ε t = 0, E ε 2 t = 1 ARCH( ) : X t = σ t ε t, σ 2 t = ω + α i Xt 2 i, i=1 GARCH(p, q) : Engle(1982), Bollerslev(1986), Bougerol and Pickard(1992),..., Teräsvirta(2007, review) ARCH( ): Giraitis, Kokoszka and Leipus(2000), Kazakeviµcius and Leipus(2002,2003),..., Giraitis et al.(2007, review) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 4 / 30

5 2. GARCH, ARCH(infty) and Linear ARCH (LARCH) 9 stationary solution of ARCH( ) with EXt 2 < () i=1 α i < 1 ARCH( ) does not allow for long memory in Xt 2 (ε t ) symmetric =) no leverage Linear ARCH (LARCH)( ) (Robinson(1991), Giraitis et al.(2000,2004), Berkes and Horváth(2003)): X t = σ t ε t, σ 2 t = ω + i=1 a 2 i < 1, ω 6= 0, a i 2 R, (ε t ) iid(0, 1)! 2 a i X t i, i=1 a i ci d 1 (i!, 9 c 6= 0, d 2 (0, 1/2) (e.g., FARIMA(0, d, 0)) allows for long memory in Xt 2 and the leverage e ect Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 5 / 30

6 2. GARCH, ARCH(infty) and Linear ARCH (LARCH) partial sums of Xt 2 of LARCH( ) may converge to fractional Brownian motion (FBM) (provided EXt 4 < ) volatility σ t = ω + i=1 a i X t i not separated from zero (bad for QMLE) and can assume negative values with positive probability stationary solution σ t of LARCH( ) admits an orthogonal Volterra expansion in ε s, s < t convergent in L 2 : σ t = 1 + k=1 = : at S ε S S s k <...<s 1 <t a t s1 a s1 s 2 a sk 1 s k ε s1...ε sk S = fs k,..., s 1 g Z, a S t := a t s1 a s1 s 2 a sk 1 s k, ε S := ε s1 ε sk Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 6 / 30

7 3. Sentana s Quadratic ARCH (QARCH) Sentana(1995): Generalized Quadratic ARCH(GQARCH(p, q)): σ 2 t = ω + p i=1 a i X t i + p i,j=1 a ij X t i X t j + q i=1 b i σ 2 t i, ω, a i, a ij, b i real parameters conditions guaranteeing the existence of stationary solution with µ t = E [X t jf t 1 ] = 0, σ 2 t = E Xt 2 jf t 1 0 a.s. su cient condition for stationarity: p i=1 a ii + q i=1 b i < 1 nests a variety of ARCH models can be expressed as random coe cient VAR no explicit solution in general limited to short memory models X t, σ 2 t Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 7 / 30

8 4. LARCH+(infty) Goal: to construct a stationary process (X t ) with µ t = E [X t jf t 1 ] = 0, σ 2 t = E Xt 2 jf t 1 = ν 2 + ω + where υ, ω, a i are real parameters, i=1 a 2 i particular case of Sentana s QARCH ν = 0 corresponds to LARCH( ) ν > 0 : < i=1 a i X t i! 2, (1) conditional variance separated from 0: σ 2 t ν 2 > 0 a.s. the construction below can be extended to include general linear drift µ t = E [X t jf t 1 ] = µ + i=1 c i X t i (Giraitis and Surgailis(2002)) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 8 / 30

9 4. LARCH+(infty): de nition (η t, ζ t ) : a sequence of iid vectors, E η t = E ζ t = 0, E η 2 t = E ζ2 t = 1, ρ = E η t ζ t (2) De nition LARCH + ( ) equation: X t = κη t + ζ t where parameters κ 2 R and ρ 2 [ ω 2 R in (1) by a i X t i, (3) i=1 1, 1] in (2) are related to ν 0 and κρ = ω, κ 2 = ω 2 + ν 2. onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 9 / 30

10 4. LARCH+(infty): solution Solution Solution of LARCH + ( ) equation (3): X t = κ η t + ζ t where: k=0 u<s k <...<s 1 <t a t s1 a sk 1 s k a sk uζ s1 ζ sk η u!! = κ η t + ζ t au,t S ζ S η u, (4) u<t S (u,t) a S u,t : = a t s1 a s1 s 2 a sk 1 s k a sk u, ζ S := ζ s1 ζ sk, S = fs k,..., s 1 g, u < s k <... < s 1 < t onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 10 / 30

11 4. LARCH+(infty): existence and uniqueness in L2 Notation: F t := σ fη s, ζ s, s tg, A t := i=1 a i X t i. Theorem (1) Let κ 6= 0. A L 2 -bounded causal solution (X t ) of LARCH + ( ) equation (3) exists i kak 2 := aj 2 < 1. j=1 In the latter case,such a solution is unique, strictly stationary and is given by the Volterra series (4) convergent in L 2. Moreover, X t = σ t ε t, (5) q where σ t = ν 2 + (ω + A t ) 2 and where (ε t, F t ) form a stationary martingale di erence sequence with E [ε t jx s, s < t] = 0 and E ε 2 t jx s, s < t = 1. (6) onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 11 / 30

12 4. LARCH+(infty): general properties & examples (5)-(6) follow from µ t = E [X t jx s, s < t] = κe [η t ] + E [ζ t ] A t = 0, σ 2 t = E Xt 2 jx s, s < t = κ 2 E η 2 t + E ζ 2 t A 2 t + 2κE [η t ζ t ]A t = κ 2 + A 2 t + 2κρA t = ν 2 + ϖ 2 + A 2 t + 2ϖA t = ν 2 + (ϖ + A t ) 2. EXt 2 = χ 2 / 1 kak 2 GLARCH + (1,1): σ 2 t = ν 2 + (ϖ + A t ) 2, A t = αa t 1 + βx t 1, α 2 + β 2 < 1 GLARCH + (0, d, 0): σ 2 t = ν 2 + (ϖ + A t ) 2, A t = c (1 L) d X t 1, d 2 (0, 1/2), c 2 < Γ 2 (1 d)/γ (1 d) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 12 / 30

13 5. LARCH+(infty): leverage and long memory Everywhere below: EX 4 t =, E jx t j 3 < (most interesting case) Su cient condition (Giraitis et al.(2004)): m 1/3 3 kak kak < 1, m 4 =, (7) m p := max (E jη 0 j p, E jζ 0 j p ), kak p := fi=1 ja i j p g 1/p Leverage e ect: past returns and future volatilities negatively correlated Leverage function (Giraitis et al.(2004)): L t s := cov X s, σ 2 t = EXs Xt 2 (s < t) satis es linear equation with Hilbert-Schmidt operator (assuming EX 3 t = 0): L t = 2ωσ 2 a t + at 2 0<s<t sl s + 2a t a t+s L s (8) s>0 Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 13 / 30

14 5. LARCH+(infty): leverage and long memory Theorem (2) Assume (7) and E η 3 t = E ζ3 t = 0. (i) (Leverage property) Let ϖa 1 < 0, k 1. Then L i < 0, i = 1,..., k. (ii) (Long memory) Let ωa i 0, i = 1,..., k for some a i ci d 1 (i!, 9c 6= 0, d 2 (0, 1/2)) (9) Then L t c(d)t d 1 (t! ). (10) proof uses equation (8) for the leverage function nite 4th moment not required long memory asymptotics of cov(x p 0, X t p ) (p = 2, 3,...) under suitable moment conditions onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 14 / 30

15 6. Limit of sums of squares: between FBM and Lévy (X t ) : long memory LARCH + ( ) process with in nite 4th moment as in Thm 2 Problem Limit distribution of partial sums process [nτ] t=1 X t 2 EXt 2, τ 2 [0, 1] Assume conditions: and E ζ 4 + ζ 2 η 2 < (11) P η 2 > x c 1 x α (x!, 9α 2 (1, 2), c 1 > 0) (12) η 2 t E η 2 t belong to the domain of attraction of antisymmetric α-stable law Example of correlated (η, ζ) satisfying (11)-(12): ζ N(0, 1), η = ζ p ξ, ξ? ζ, P (ξ > x) cx α, E ξ = 1, E p ξ < 1 onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 15 / 30

16 6. Limit of sums of squares: between FBM and Lévy Theorem (3) Assume (7), (9), (11), (12) and ρ = E ηζ 6= 1. Then: (i) If d +.5 > 1/α, then n d.5 [nτ] t=1 (ii) If d +.5 < 1/α, then Xt 2 EXt 2 ) FBMτ (d +.5) (13) n 1/α [nτ] t=1 Xt 2 EXt 2 ) Lévyτ (α) (14) FBM τ (d +.5): a fractional BM with variance c(d)τ2d +1 Lévy τ (α) : a homogeneous α-stable Lévy process with zero mean and skewness parameter β = 1 onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 16 / 30

17 6. Limit of sums of squares: between FBM and Lévy ) : -di convergence d +.5 = 1/α (0 < d <.5, 1 < α < 2) : critical line d alpha d = 0 : a i ci d 1 = ci 1 or i=1 ja i j < (short memory) + (7),(11),(12): only α stable Lévy limit for partial sums of expected X 2 t partial sums of X t s tend to a standard BM ((X t )=martingale di erences with nite variance) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 17 / 30

18 7. Idea of the proof of Thm 3 Recall (κ = 1) : X t = η t + ζ t u<t au,t S ζ S η u = η t + η u S (u,t) u<t au,t S ζ S ζ t, S (u,t) The squared process Xt 2 = Yt D + 2Y O t splits into two parts: Yt D : η u1 = η u2 : "diagonal part" with in nite variance: tends to Lévy Y O t : η u1 6= η u2 : "o -diagonal part" with nite variance: tends to FBM Y D t : = (η 2 t E η 2 t ) + Y O t : = ( u<t S (u,t) u<t +ζ 2 t η u2 η u1 u 2 <u 1 <t (η 2 u E η 2 t )( au,t S ζ S ζ t ) 2, S (u,t) a S u,t ζ S ζ t ) 2 + η t ζ t u<t S 1 (u 1,t) S 2 (u 2,t) au,t S ζ S η u S (u,t) a S 1 u 1,t ζ S 1 a S 2 u 2,t ζ S 2 + E η 2 t Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 18 / 30

19 7. Proof of Thm 3: main steps: tep 1 n d.5 [nτ] t=1 (Y O t EY O t ) ) FBM τ (d +.5) (similar to Giraitis et al.(2000) for LARCH( )) tep 2 n t=1 Y D t = n u=1(η 2 u E η 2 u )Z u + o p n 1/α, where Z u := 1 + t>u ( ) 2 au,t S ζ S ζ t S (u,t) is a strictly stationary process with nite variance and anti-predictable (recall (ζ t ) and (η t ) are not independent as sequences) tep 3 n 1/α n u=1(η 2 u E η 2 u )Z u ) Lévy τ (α) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 19 / 30

20 Step 3 is based on the principle of conditioning due to Jakubowski (1986), which allows to replace (Z t ) by a similar process but independent of (η t ) Step 2 is technically most di cult Uses a bound for 4th mixed moment of Volterra series, written as a sum of diagrams (Giraitis et al. (2000)) over a table I = I (k) 4 having 4 rows I 1, I 2, I 3, I 4 of respective length k 1, k 2, k 3, k 4 = 0, 1,...; (k) 4 := (k 1,..., k 4 ) A diagram γ determines the type of diagonal in the sum Σ u,0,t : = a S 1 u,ta S 2 u,ta S 3 u,0 as 4 u,0 E ζs 1[ftg ζ S2[ftg ζ S3[f0g ζ S 4[f0g S 1 S 2 S 3 S 4 = (k) 4 γ2γ(k) 4 µ γ a (S ) 4 u,(t) (S ) 4 γ where µ γ = E ζ S 1[ftg ζ S 2[ftg ζ S 3[f0g ζ S 4[f0g = 0 unless all elements of 4 sets S 1 (u, t), S 2 (u, t), S 3 (u, 0), S 4 (u, 0) are coupled Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 20 / 30

21 7. Proof of Thm 3: diagrams: Goal: to obtain the right upper bound of Σ u,0,t :for any (k) 4 = (k 1,..., k 4 ), jkj := k k 4, and any diagram γ 2 Γ (k) 4, as u < 0 > t! + (S ) 4 γ a (S ) 4 u,(t) C kak jkj jt uj 2d 2 juj 2d 2 (15) u s 1 s 2 s s q t s 1,..., s q : coupled elements of S 1,..., S 4 Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 21 / 30

22 7. Proof of Thm 3: diagrams: If rows 1 and 2 (block 1) and rows 3 and 4 (block 2) are not connected, the lhs of (15) is dominated by a product of two convolutions: at 2 s 0...a2 u<s 0 k 0 <...<s1 0 <t 1 s 0 k 0 u a 2...a 2 s " u<s " k " <...<s 1 " <0 1 s k" " u = (a2 ) k 0 t u(a 2 ) k " u In the above special case ( block-unconnected diagram ) the bound (15) is easy the general case of block-connected" diagram γ 2 Γ (k) 4 can be reduced to the above special case by graph-theoretical argument Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 22 / 30

23 7. Proof of bound (15): idea: Eulerian cycle Idea: decoupling of diagrams. Recall the notation: au,t S := a t s1 a s1 s 2 a sk 1 s k a sk u, a (S ) 4 := a S 1 u,(t) The last product can be written as a (S ) 4 u,(t) a e = a e all edges yellow edges 1 2 yellow edges a 2 e + u,ta S 2 u,ta S 3 u,0 as 4 u,0. ae 2 blue edges a e blue edges! u s 1 s 2 s s q t Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 23 / 30

24 8. A clt for martingale transform Theorem (4) Let U i = V i ξ i (i = 1,..., n) where: (a) (ξ i ) 2DAN(α, β) iid, zero mean, 1 < α < 2, β 2 [ 1, 1], (b) (V i ) predictable, stationary, ergodic, and E jv 0 j r < (9 r > α). Then where c := (E jv 0 j α ) 1/α, [nτ] n 1/α U i ) c Lévy τ α, β 0, i=1 β 0 := βe jv 0 j α sign(v 0 ) /E jv 0 j α similar result as if (V i ) were iid (Breiman s lemma) condition (b): V i s have lighter tails than ξ i s the opposite case: Leipus et al. (2005, 2006): more di cult onatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 24 / 30

25 8. A clt for martingale transform easy case: when (ξ i ) and (V i ) are mutually independent The Principle of Conditioning (Jakubowski (1986)): allows to replace dependent (ξ i )and (V i ) by independent ones and having the same marginals recent developments of the conditioning approach in limit theorems: Peccati, Taqqu, Tudor, Nualart,... Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 25 / 30

26 9. Open problem: what next? Limit distribution of sample autocovariances: γ n,x (j) = 1 n γ n,x 2 (j) = 1 n n j i=1 n j i=1 (X i X n ) X i+j X n, (Xi 2 Xn 2 ) Xi+j 2 Thm 3 solves the limit of γ n,x (0) X 2 n, j = 0, 1,..., m Sample autocovariances of linear processes with in nite variance: Davis and Resnick(1986) Sample autocovariances of short memory GARCH: Davis and Hsing(1995), Davis and Mikosch(1998), Davis and Resnick(1996) Appell polynomials of long memory MA: Vaiµciulis(2003) Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 26 / 30

27 References(1) Giraitis, L., Robinson, P.M., Surgailis, D. (2000) A model for long memory conditional heteroskedasticity. Ann. Appl. Probab. 10, Giraitis, L., Surgailis, D. (2002) ARCH-type bilinear models with double long memory. Stoch. Process..Appl. 100, Giraitis, L., Leipus, R., Robinson, P.M., Surgailis, D. (2004) LARCH, leverage and long memory. J. Financial. Econometrics 2, Jakubowski, A. (1986) Principle of conditioning in limit theorems for sums of random variables. Ann. Probab. 14, Robinson, P.M. (1991) Testing for strong serial correlation and dynamic heteroskedasticity in multiple regression. J. Econometrics 47, Sentana, E. (1995) Quadratic ARCH models. Rev. Econ. Studies 3, Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 27 / 30

28 References(2) Berkes, I., Horváth, L.(2003) Asymptotic results for long memory LARCH sequences. Ann. Appl. Probab. 13, Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, Bougerol, P., Picard, N. (1992) Stationarity of GARCH processes and some nonnegative time series.j. Econometrics 52, Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK in ation. Econometrica 50, Davis, R.A., Resnick, S.I. (1986) Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14, Davis, R.A., Hsing, T. (1995) Point process and partial sum convergence for weakly dependent random variables with in nite variance. Ann. Probab. 23, Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 28 / 30

29 References(3) Davis, R.A., Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist.26, Davis, R.A., Resnick, S.I. (1996) Limit theory for bilinear processes with heavy tailed noise. Ann. Appl. Probab.6, Giraitis, L., Kokoszka, P.S., Leipus, R. (2000) Stationary ARCH models: dependence structure and central limit theorem. Econometric. Th. 16, Kazakeviµcius, V., Leipus, R. (2002) On stationarity in the ARCH( ) model. Econometric Th. 18, Mikosch, T., St¼aric¼a, C. (2003) Long-range dependence e ects and ARCH modeling. In: Long-Range Dependence. Theory and Applications (eds. P. Doukhan, G. Oppenheim, M.S. Taqqu), pp Birkhäuser, Boston. Mikosch, T. (2003) Modelling dependence and tails of nancial time series. In: Extreme Values in Finance, Telecommunications and the Environment (eds. B. Finkelstädt, H. Rootzén), pp Chapman and Hall. Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 29 / 30

30 References(4) Leipus, R., Paulauskas, V., Surgailis, D. (2005) Renewal regime switching and stable limit laws. J. Econometrics 129, Leipus, R., Paulauskas, V., Surgailis, D. (2006) Random coe cient AR(1)process with heavy tailed renewal switching coe cient and heavy tailed noise. J. Appl. Probab. 43, Teräsvirta, T. (2007) An introduction to univariate GARCH models. Preprint. Vaiµciulis, M. (2003) Convergence of sums of Appell polynomials with nite variance. Lithuanian J. Math. 43, Donatas Surgailis (Vilnius Institute of Mathematics A Quadratic and Informatics ARCH( ) ) model with long memory and Lévy stable behavior of squares 30 / 30