Table Of Content

A nonlinear model for long memory conditional heteroscedasticity Paul Doukhan1, Ieva Grublyte˙1,2 and Donatas Surgailis2 5 1 October 20, 2015 0 2 1 Université de Cergy Pontoise, Départament de Mathématiques, 95302 Cedex, France t c 2 Vilnius University, Institute of Mathematics and Informatics, 08663 Vilnius, Lithuania O 8 1 Abstract. Wediscussaclassofconditionallyheteroscedastictimeseriesmodels ] T satisfying the equation rt = ζtσt, where ζt are standardized i.i.d. r.v.’s and the S conditional standard deviation σ is a nonlinear function Q of inhomogeneous t . h linear combination of past values r ,s < t with coefficients b . The existence of s j t a stationarysolutionr withfinitepthmoment,0 < p < isobtainedundersome t m ∞ conditions on Q,b and the pth moment of ζ . Weak dependence properties of [ j 0 2 rt are studied, including the invariance principle for partial sums of Lipschitz v functions of r . In the case whenQ is the squareroot of a quadratic polynomial, t 5 we prove that r can exhibit a leverage effect and long memory, in the sensethat 9 t 0 the squared process r2 has long memory autocorrelation and its normalized 0 t 0 partial sums process converges to a fractional Brownian motion. . 2 0 Keywords: ARCH model, leverage, long memory, Donsker’s invariance princi- 5 ple 1 : v i X 1 Introduction r a A stationary time series r ,t Z is said conditionally heteroscedastic if its conditional t { ∈ } variance σ2 = Var[r r ,s < t] is a non-constant random process. A class of conditionally t t| s heteroscedastic ARCH-type processes is defined from a standardized i.i.d. sequence ζ ,t t { ∈ Z as solutions of stochastic equation } r = ζ σ , σ = V(r ,s < t), (1.1) t t t t s where V(x ,x , ) is some function of x ,x , . 1 2 1 2 ··· ··· The ARCH( ) model corresponds to V(x ,x , ) = a+ ∞ b x2 1/2, or ∞ 1 2 ··· j=1 j j ∞ (cid:0) P (cid:1) σ2 = a+ b r2 , (1.2) t j t−j j=1 X 1 where a 0,b 0 are coefficients. j ≥ ≥ TheARCH( )modelincludesthewell-knownARCH(p)andGARCH(p,q)modelsofEngle ∞ [13] and Bollerslev [6]. However, despite their tremendous success, the GARCH models are not able to capture some empirical features of asset returns, in particularly, the asymmetric or leverage effect discovered by Black [5], and the long memory decay in autocorrelation of squares r2 . Giraitis and Surgailis [16] proved that the squared stationary solution { t} of the ARCH( ) model in (1.2) with a > 0 always has short memory, in the sense that ∞ ∞ Cov(r2,r2)< . (However, for integrated ARCH( ) models with ∞ b = 1,b j=0 0 j ∞ ∞ j=1 j j ≥ 0 and a = 0 the situation is different; see [19].) P P The above shortcomings of the ARCH( ) model motivated numerous studies proposing ∞ alternative forms of the conditional variance and the function V(x ,x , ) in (1.1). In 1 2 ··· particular, stochastic volatility models can display both long memory and leverage except that in their case, the conditional variance is not a function of r ,s < t alone and therefore s it is more difficult to estimate from real data in comparison with the ARCH models; see Shephard and Andersen [31]. Sentana [30] discussed a class of Quadratic ARCH (QARCH) models with σ2 beinga general quadratic form in lagged variables r , , r . Sentana’s t t−1 ··· t−p specification of σ2 encompasses a variety of ARCH modelsincludingthe asymmetric ARCH t modelofEngle[14]andthe‘linearstandarddeviation’modelofRobinson[27]. Thelimiting case (when p = ) of the last model is the LARCH model discussed in [15] (see also [16], ∞ [4], [17], [33]) and corresponding to V(x ,x , ) = a+ ∞ b x , or 1 2 ··· j=1 j j ∞ P σ = a+ b r , (1.3) t j t−j j=1 X where a R,b R are real-valued coefficients. [15] proved that the squared stationary j ∈ ∈ solution r2 of the LARCH model with b decaying as jd−1,0 < d < 1/2 may have long { t} j memory autocorrelations. The leverage effect in the LARCH model was discussed in detail in [17]. On the other hand, volatility σ (1.3) of the LARCH model may assume negative t values, lacking some of the usual volatility interpretation. The present paper discusses a class of conditionally heteroscedastic models (1.1) with V of the form ∞ V(x ,x , )= Q(a+ b x ), (1.4) 1 2 j j ··· j=1 X where Q(x),x R is a (nonlinear) function of a single real variable x R which may ∈ ∈ be separated from zero by a positive constant: Q(x) c > 0, x R. Linear Q(x) = x ≥ ∈ corresponds to the LARCH model (1.3). Probably, the most interesting nonlinear case of Q in (1.4) is Q(x)= c2+x2, p where c 0 is a parameter. In the latter case, the model is described by equations ≥ 2 r = ζ σ , σ = c2+ a+ b r . (1.5) t t t t t−s s s (cid:16) Xs<t (cid:17) 2 Note σ c 0 in (1.5) is nonnegative and separated from 0 if c > 0. Particular cases of t ≥ ≥ volatility forms in (1.5) are: σ = c2+(a+br )2 (Engle’s [14] asymmetric ARCH(1)), (1.6) t t−1 p b p 2 σ = c2+ a+ r , (1.7) t t−j v p u j=1 u (cid:0) X (cid:1) t ∞ σ = a+ b r (Q(x) = x ), (1.8) t j t−j | | j=1 (cid:12) X (cid:12) (cid:12) (cid:12) σ = c2+(a+b((1 L)−d 1)r )2. (1.9) t t − − q In (1.6)-(1.9), a,b,c are real parameters, p 1 an integer, Lx = x is the backward t t−1 ≥ shift, and (1 L)−dx = ∞ ϕ x , ϕ = Γ(d+j)/Γ(d)Γ(j +1),ϕ = 1 is the fractional − t j=0 j t−j j 0 integration operator, 0 < d < 1/2. The squared volatility (conditional variance) σ2 in P t (1.6)-(1.9)and (1.5)is aquadratic form in lagged returnsr ,r , and hencerepresent t−1 t−2 ··· particular cases of Sentana’s [30] Quadratic ARCH (QARCH) model with p = . It ∞ should be noted, however, that the first two conditional moments do not determine the unconditional distribution. Particularly, (1.1) with (1.5) generally is a different process from Sentana’s [30] QARCH process, the latter being defined as a solution of a linear random-coefficient equation for r in contrast to the nonlinear equation in (1.1). See also t { } Example 2 below. Let us describe the main results of this paper. Section 2 obtains sufficient conditions on Q,b and µ := E ζ p for the existence of stationary solution of (1.1)-(1.4) with finite j p 0 | | | | moment E r p < , p > 0. We use the fact that the above equations can be reduced to t | | ∞ the ‘nonlinear moving-average’ equation X = b ζ Q(a+X ) t t−s s s s<t X for linear form X := b r in (1.4), and vice-versa. Section 3 aims at providing weak t s<t t−s s dependence properties of (1.1) with V in (1.4), in particular, the invariance principle for P Lipschitzfunctionsof r and X , undertheassumptionthatb aresummableanddecay t t j { } { } asj−γ withγ > 1. Section4discusseslongmemorypropertyofthequadraticmodelin(1.5). For b βjd−1, j , 0< d < 1/2 as in (1.9), we prove that the squaredprocess r2 has j ∼ → ∞ { t} long memory autocorrelations and its normalized partial sums process tend to a fractional Brownian motion with Hurst parameter H = d + 1/2 (Theorem 10). Finally Section 5 establishes the leverage effect in spirit of [17], viz., the fact that the ‘leverage function’ h := Cov(σ2,r ),j 1 of model (1.5) takes negative values provided the coefficients a j t t−j ≥ and b have opposite signs. All proofs are collected in Section 6 (Appendix). j Notation. In what follows, C,C(...) denote generic constants, possibly dependent on the variables in brackets, which may be different at different locations. a b (t ) is t t ∼ → ∞ equivalent to lim a /b = 1. t→∞ t t 3 2 Stationary solution This section discusses the existence of a stationary solution of (1.1) with V of (1.4), viz., r = ζ Q a+ b r , t Z. (2.10) t t t−s s ∈ s<t (cid:0) X (cid:1) Denote X := b r . (2.11) t t−s s s<t X Then r in (2.10) can be written as r = ζ Q(a + X ) where (2.11) formally satisfies the t t t t following equation: X = b ζ Q(a+X ). (2.12) t t−s s s s<t X Below we give rigorous definitions of solutions of (2.10) and (2.12) and a statement (Propo- sition 3) justifying (2.12) and the equivalence of (2.10) and (2.12). In this section we consider a general case of (2.10)-(2.12) when the innovations may have infinite variance. More precisely, we assume that ζ ,t Z are i.i.d. r.v.’s with finite t { ∈ } moment µ := E ζ p < , p > 0. We use the following moment inequality. p t | | | | ∞ Proposition 1 Let Y ,j 1 be a sequence of r.v.’s such that E Y p < for some p > 0 j j { ≥ } | | ∞ and the sum on the r.h.s. of (2.13) converges. If p > 1 we additionally assume that Y j { } is a martingale difference sequence: E[Y Y , ,Y ] = 0, j = 2,3, . Then there exists j 1 j−1 | ··· ··· a constant K depending only on p and such that p ∞ ∞ E Y p, 0 < p 2, E Y p K j=1 | j| ≤ (2.13) j p (cid:12)Xj=1 (cid:12) ≤ P ∞j=1(E|Yj|p)2/p p/2, p > 2. (cid:12) (cid:12) (cid:0)P (cid:1) Remark 1 In the sequel, we shallrefer to K in (2.13) as the Rosenthal constant. For p 0 < p 1 and p = 2, inequality (2.13) holds with K = 1, and for 1 < p < 2, it is known as p ≤ von Bahr and Esséen inequality, see [34], which holds with K = 2. For p > 2, inequality p (2.13) is a consequence of the Burkholder and Rosenthal inequality (see [7], [29], also [18], Lemma 2.5.2). Ose¸kowski [26] proved that K1/p 2(3/2)+(1/p)(p +1)1/p 1+ p , in p ≤ 4 log(p/2) 1/4 particular, K 27.083. See also [23]. (cid:0) (cid:1) 4 ≤ Let us give some formal definitions. Let = σ(ζ ,s t),t Z be the sigma-field gener- t s F ≤ ∈ ated by ζ ,s t. A random process u ,t Z is called adapted (respectively, predictable) s t ≤ { ∈ } if u is -measurable for each t Z (respectively, u is -measurable for each t Z). t t t t−1 F ∈ F ∈ Define ∞ b p, 0< p < 2, B := j=1| j| (2.14) p P ∞ b2 p/2, p 2.  j=1 j ≥ (cid:0)P (cid:1) Definition 2 Let p > 0 be arbitrary. (i) By Lp-solution of (2.10) we mean an adapted process r ,t Z with E r p < such t t { ∈ } | | ∞ that for any t Z the series b r converges in Lp and (2.10) holds. ∈ s<t t−s s P 4 (ii) By Lp-solution of (2.12) we mean an predictable process X ,t Z with E X p < t t { ∈ } | | ∞ such that for any t Z the series b ζ Q(a+X ) converges in Lp and (2.12) holds. ∈ s<t t−s s s P Let Q(x),x R be a Lipschitz function, i.e., there exists Lip > 0 such that ∈ Q Q(x) Q(y) Lip x y , x,y R. (2.15) | − | ≤ Q| − | ∈ Note (2.15) implies the bound Q2(x) c2+c2x2, x R, (2.16) ≤ 1 2 ∈ where c 0, c Lip and c can bechosen arbitrarily close to Lip ; in particular, (2.16) 1 ≥ 2 ≥ Q 2 Q holds with c2 = (1+ǫ2)Lip2,c2 = Q2(0)(1+ǫ−2), where ǫ > 0 is arbitrarily small. 2 Q 1 Proposition 3 Let Q be a measurable function satisfying (2.16) with some c ,c 0 and 1 2 ≥ ζ be an i.i.d. sequence with µ = E ζ p < and satisfying Eζ = 0 for p > 1. In t p 0 0 { } | | | | ∞ addition, assume B < . p ∞ (i) Let X be a stationary Lp-solution of (2.12). Then r := ζ Q(a+X ) is a stationary t t t t { } { } Lp-solution of (2.10) and E r p C(1+E X p). (2.17) t t | | ≤ | | Moreover, for p > 1, r , ,t Z is a martingale difference sequence with t t { F ∈ } E[r ]= 0, E[r p ] = µ Q(a+ b r )p. (2.18) t t−1 t t−1 p t−s s |F | | |F | | | s<t (cid:12) X (cid:12) (ii) Let r be a stationary Lp-solution of (2.10). Then X in (2.11) is a stationary t t { } { } Lp-solution of (2.12) such that E X p CE r p. (2.19) t t | | ≤ | | Moreover, for p 2 ≥ ∞ E[X X ] = Er2 b b , t = 0,1,.... (2.20) t 0 0 t+s s s=1 X Remark 2 Let p 2 and µ < , then by inequality (2.13), r being a stationary p t ≥ | | ∞ { } Lp-solution of (2.10) is equivalent to r being a stationary L2-solution of (2.10) with t { } E r p < . Similarly, if Q and ζ satisfy the conditions of Proposition 3 and p 2, 0 t | | ∞ { } ≥ then X being a stationary Lp-solution of (2.12) is equivalent to X being a stationary t t { } { } L2-solution of (2.12) with E X p < . 0 | | ∞ Thefollowingtheoremobtainsasufficientconditionin(2.21)fortheexistenceofastation- ary Lp-solution of equations (2.10) and (2.12). Condition (2.21) involves the pth moment of innovations, the Lipschitz constant Lip , the sum B in (2.14) and the Rosenthal constant Q p K in (2.13). Part (ii) of Theorem 4 shows that for p = 2, condition (2.21) is close to p optimal, being necessary in the case of quadratic Q2(x) = c2+c2x2. 1 2 5 Theorem 4 Let the conditions of Proposition 3 be satisfied, p > 0 is arbitrary. In addition, assume that Q satisfies the Lipschitz condition in (2.15). (i) Let p K µ Lip B < 1. (2.21) p| |p Q p Then there exists a unique stationary Lp-solution X of (2.12) and t { } C(p,Q)µ B E X p | |p p , (2.22) t p | | ≤ 1 K µ Lip B − p| |p Q p where C(p,Q)< depends only on p and c ,c in (2.16). 1 2 ∞ (ii) Assume, in addition, that Q2(x) = c2+c2x2, where c 0,i = 1,2, and µ = Eζ2 = 1. 1 2 i ≥ 2 0 Then c2B < 1 is a necessary and sufficient condition for the existence of a stationary 2 2 L2-solution X of (2.12) with a = 0. t { } 6 Remark 3 Condition (2.21) agrees with the contraction condition for the operator defined by the r.h.s. of (2.12) and acting in a suitable space of predictable processes with values in Lp. For the LARCH model, explicit conditions for finiteness of the pth moment were obtained in [15], [17] using a specific diagram approach for multiple Volterra series. For larger values of p> 2, condition (2.21) is preferable to the corresponding condition (2p p 1)1/2 µ 1/pB1/p < 1, p = 2,4,6, , (2.23) − − | |p p ··· in([15],(2.12))fortheLARCHmodel,sincethecoefficient(2p p 1)1/2growsexponentially − − 1/p with p in contrast to the bound on K in Remark 1. See also ([20], sec. 4.3). On p the other hand for p = 4 (2.23) becomes √11 µ 1/4B1/2 < 1 while (2.21) is satisfied if | |4 2 1/4 1/4 1/2 1/4 1/2 K µ B 27.083µ B <1, see Remark 1, which is worse than (2.23). 4 | |4 2 ≤ | |4 2 Example 1 (The LARCH model) Let Q(x) = x and ζ be a standardized i.i.d. se- t { } quence with zero mean and unit variance. Then (2.12) becomes the bilinear equation X = b ζ (a+X ). (2.24) t t−s s s s<t X Thecorrespondingconditionallyheteroscedasticprocess r = ζ (a+X ) inProposition3(i) t t t { } is the LARCH model discussed in [15], [17] and elsewhere. As shown in ([15], Thm.2.1), equation (2.24) admits a covariance stationary predictable solution if and only if B = 2 ∞ b2 < 1. Note the last result agrees with Theorem 4 (ii). A crucial role in the study j=1 j of the LARCH model is played by the fact that its solution can be written in terms of the P convergent orthogonal Volterra series ∞ X = a b b ζ ζ . t t−s1··· sk−1−sk s1··· sk Xk=1sk<·X··<s1<t Except forQ(x) = x, in other cases of (2.12)includingtheQARCHmodelin(1.5), Volterra series expansions are unknown and their usefulness is doubtful. 6 Example 2 (Asymmetric ARCH(1)) Consider the model (1.1) with σ in (1.6), viz. t r = ζ c2+(a+br )2 1/2, (2.25) t t t−1 (cid:0) (cid:1) where ζ are standardized i.i.d. r.v.’s. By Theorem 4 (ii), equation (2.25) has a unique t { } stationary solution with finite variance Er2 =(a2 +c2)/(1 b2) if and only if b2 < 1. t − In parallel, consider the random-coefficient AR(1) equation r = κε +bη r , (2.26) t t t t−1 where (εt,ηt) are i.i.d. random vecetors with zeroemean Eεt = Eηt = 0 and unit variances { } E[ε2] = E[η2]= 1 and κ,b are real coefficients. As shown in Sentana [30] (see also Surgailis t t [32]), equation(2.26)hasastationarysolutionwithfinitevarianceunderthesamecondition b2 < 1as(2.25). Moreover, ifthecoefficientsκandρ:= E[ε η ] [ 1,1]in(2.26)arerelated t t ∈ − to the coefficients a,c in (2.25) as κρ = a, κ2 = a2+c2, (2.27) then the processes in (2.25) and (2.26) have the same volatility forms since σ2 := E[r2 r ,s < t] = κ2+2κbρ+b2r2 t t| s t−1 = c2+(a+br )2 t−1 e e e e agrees with the corresponding expression σt2 = c2+(a+brt−e1)2 in the case of (1.6). A natural question is whether the above stationary solutions r and r of (2.25) t t { } { } and (2.26), with parameters related as in (2.27), have the same (unconditional) finite- dimensional distributions? As shown in ([32], Corollary 2.1), the answer is poesitive in the case when ζ and (ε ,η ) are Gaussian sequences. However, the conditionally Gaussian t t t { } { } case seems tobetheonlyexception andingeneral theprocesses r and r have different t t { } { } distributions. This can be seen by considering the 3rd conditional moment of (2.25) e E[r3 r ]= µ c2+(a+br )2 3/2 (2.28) t| t−1 3 t−1 (cid:0) (cid:1) which is an irrational function of r (unless µ = Eζ3 = 0 or b = 0), while a similar t−1 3 0 moment of (2.26) E[r3 r ] = κ3ν +3bκ2ν r +3b2κν r2 +b3ν r3 (2.29) t| t−1 3,0 2,1 t−1 1,2 t−1 0,3 t−1 is a cubic polynomeiael in rt−1, where νi,j :=eE[εi0η0j]. Moreoever, (2.28) heas a constant sign independentof r while the sign of the cubic polynomial in (2.29) changes with r ranging t−1 t from to if the leadeing coefficient b3ν =0. 0,3 ∞ −∞ 6 Using the last observation we can prove that the bivariate distributions of (r ,er ) and t t−1 (r ,r ) are different under general conditions on the innovations and the parameters of t t−1 the two equations. The argument is as follows. Let b > 0,c > 0,µ > 0,ν = Eη3 > 0. 3 0,3 0 Aessueme that ζ has a bounded strictly positive density function 0 < f(x) < C,x R and 0 ∈ 7 (ε ,η ) has a bounded strictly positive density function 0 < g(x,y) < C,(x,y) R2. The 0 0 ∈ above assumptions imply that the distributions of r and r have infinite support. Indeed, t t by (2.25) and the above assumptions we have that P(r > K) = P(c2 +(a+br )2 > t R t−1 (K/y)2)f(y)dy > 0 for any K > 0 since lim P(c2 +e (a + br )2 > (K/y)2) = 1. y→∞ Rt−1 Similarly, P(r > K) = P(r > (K κx)/by)g(x,y)dxdy > 0 and P(r < K) > t R2 t−1 t − − 0,P(r < K) > 0 for any K > 0. Since h(x) := µ c2+(a+bx)2 3/2 1 for all x > K t R 3 − ≥ | | and any sufficeiently large K > 0e, from (2.28) we obtain that for any K > 0 (cid:0) (cid:1) e Er31(r > K) = Eh(r )1(r > K)> 0 and t t−1 t−1 t−1 Er31(r < K) = Eh(r )1(r < K)> 0. (2.30) t t−1 − t−1 t−1 − Ontheotherhand,sinceh(x) := κ3ν +3bκ2ν x+3b2κν x2+b3ν x3 1forx > K and 3,0 2,1 1,2 0,3 ≥ h(x) 1 for x < K and K large enough, from (2.29) we obtain that for all sufficiently ≤ − − e large K > 0 e Er31(r > K) = Eh(r )1(r > K)> 0 and t t−1 t−1 t−1 Er31(r < K) = Eh(r )1(r < K)< 0. (2.31) te t−e1 − e et−1 et−1 − Clearly, (2.30) ande(2.31e) imply that the beivaeriate deistributions of (rt,rt−1) and (rt,rt−1) are different under the stated assumptions. e e Formodels(2.25)and(2.26),wecanexplicitlycomputecovariancesρ(t) = cov(r2,r2),ρ(t) = t 0 cov(r2,r2) and some other joint moment functions, as follows. t 0 Let µ = Eζ3 = 0,µ = Eζ4 < and m := Er2, m (t) := Er2r , m (t) := Er2r2,t e0. 3 0 4 0 ∞ 2 0 3 t 0 4 t 0 ≥ Thene e m = (a2+c2)/(1 b2), m (0) = 0, 2 3 − m (1) = E[((a2 +c2)+2abr +b2r2)r ] = 2abm +b2m (0) = 2abm , 3 0 0 0 2 3 2 m (t) = E[((a2 +c2)+2abr +b2r2 )r ] = b2m (t 1) = = b2(t−1)m (1) 3 t−1 t−1 0 3 − ··· 3 2ab(a2 +c2) = b2(t−1), t 1. (2.32) 1 b2 ≥ − Similarly, m (0) = µ E[((a2+c2)+2abr +b2r2 )2] 4 4 −1 −1 = µ (a2+c2)2+(2ab)2m +b4m (0)+2b2(a2+c2)m , 4 2 4 2 { } m (t) = E[((a2+c2)+2abr +b2r2 )r2] = (a2+c2)m +b2m (t 1), t 1 4 t−1 t−1 0 2 4 − ≥ resulting in µ ((a2+c2)2+((2ab)2 +2(a2+c2)b2)m ) 4 2 m (0) = , (2.33) 4 1 µ b4 4 − 1 b2t m (t) = m (a2+c2) − +b2tm (0), t 1, 4 2 · 1 b2 4 ≥ − and ρ(t) = (m (0) m2)b2t, t 0. (2.34) 4 − 2 ≥ 8 In a similar way, when the distribution of ζ is symmetric one can write recursive linear 0 equations for joint even moments E[r2p(0)r2p(t)] of arbitrary order p = 1,2,... involving E[r2l(0)r2p(t)],1 l p 1 and m (0) = E[r2k(0)],1 k 2p. These equations can be 2k ≤ ≤ − ≤ ≤ explicitly solved in terms of a,b,c and µ ,1 k 2p. 2k ≤ ≤ A similar approach can be applied to find joint moments of the random-coefficient AR(1) processin(2.26),withthedifferencethatsymmetryof(ε ,η )isnotneeded. Letm := Er2, 0 0 2 t m (t) := E[r2r ], m (t) := E[r2r2] and ρ(t) := Cov(r2,r2), ν := E[εiηj]. Then 3 t 0 4 t 0 t 0 i,i 0 0 e e e m2 = eκe2/(1e b2), e e e e e − m (0) = E[(κε +bη r )3] = κ3ν +3κb2ν m +b3ν m (0), 3 0 0 −1 3,0 1,2 2 0,3 3 e m (1) = E[(κ+2κρbr +b2r2)r ] = 2κρbm +b2m (0), 3 0 0 0 2 3 e e e e m (t) = E[(κ2+2κρbr˜ +b2r˜2 )r˜ ] = b2m (t 1) = = b2(t−1)m (1), t 2, e 3 e t−1 e et−1 0 e 3 −e ··· 3 ≥ aned e e m (0) = E[(κε +bη r )4] 4 0 0 −1 = κ4ν +6κ2b2ν m +4κb3ν m (0)+b4ν m (0), 4,0 2,2 2 1,3 3 0,4 4 e e m (1) = E[(κε +bη r )2r2] = κ2m +2κρbm (0)+b2m (0) 4 t t 0 0 2 3 4 e e e m (t) = E[(κε +bη r )2r2] = κ2m +b2m (t 1), t 2, e 4 t tet−1e 0 e 2 e 4 − e ≥ leading to e e e e e κ3ν +3κb2ν m 3,0 1,2 2 m (0) = , 3 1 ν b3 0,3 − m (t) = b2(t−1)(2κρbm +eb2m (0)) t 1. e 3 2 3 ≥ κ4ν +6κ2b2ν m +4κb3ν m (0) 4,0 2,2 2 1,3 3 m (0) = , (2.35) e4 e 1 ν eb4 0,4 − 1 b2t e e me (t) = m κ2 − +b2t(m (0)+2κρm (0)/b), t 1, 4 2 1 b2 4 3 ≥ (cid:16) − (cid:17) and e e e e ρ (t) = b2(t−1)ρ (1), t 1, 4 4 ≥ ρ (1) = 2ρκbm (0)+b2(m (0) m2). e4 e3 4 − 2 Then if ν3,0 = ν1,2 = 0 weehave m3(0) = 0eand ρ4(t) =e(m4(0)−e m22)b2t; moreover, m2 = m2 in view of (2.27). Then ρ (t) = ρ (t) is equivalent to m (0) = m (0), which follows from 4 4 4 4 e e e e e µ4e= ν0,4 = ν4,0 and 6ν2,2 =eµ4(4ν12,1 +2), (2.36) see (2.34), (2.33), (2.35). Note that (2.36) hold for centered Gaussian distribution (ε ,η ) 0 0 with unit variances Eε2 = Eη2 = 1. 0 0 9 3 Weak dependence Various measures of weak dependence for stationary processes y = y ,t Z have been t t { } { ∈ } introducedintheliterature,seee.g. [9]. Usually,thedependencebetweenthepresent(t 0) ≥ and the past (t n) values of y is measured by some dependence coefficients decaying t ≤ − { } to 0 as n . Thedecay rate of these coefficients plays a crucial role in establishing many → ∞ asymptotic results. The classical problem is proving Donsker’s invariance principle: [nτ] 1 (y Ey ) σB(τ), in the Skorohod space D[0,1], (3.37) t √n − → t=1 X where B = B(τ),τ [0,1] is a standard Brownian motion. The above result is useful { ∈ } in change-point analysis (Cso¨rgo˝ and Horva´th [8]), financial mathematics and many other areas. Further applications of weak dependence coefficients include empirical processes [12] and the asymptotic behavior of various statistics, including the maximum likelihood estimators. See Ibragimov and Linnik [24] and the application to GARCH estimation in [25]. The present Section discusses two measures of weak dependence - the projective weak dependencecoefficients ofWu[35]andtheτ-dependencecoefficients introducedinDedecker and Prieur [10], [11] - for stationary solutions r , X of equations (2.10), (2.12). We t t { } { } show that the decay rate of the above weak dependence coefficients is determined by the decay rate of the moving average coefficients b . j 3.1 Projective weak dependence coefficients Let us introduce some notation. For r.v. ξ, write ξ := E1/p ξ p, p 1. Let y ,t Z p t k k | | ≥ { ∈ } be a stationary causal Bernoulli shift in i.i.d. sequence ζ , in other words, t { } y = f(ζ ,s t), t Z, t s ≤ ∈ where f : RN R is a measurable function. We also assume Ey = 0, y 2 = Ey2 < . → 0 k 0k2 0 ∞ Introduce the projective weak dependence coefficients ω (i; y ) := f (ξ ) f (ξ′) , δ (i; y ) := f(ξ ) f(ξ′) , (3.38) 2 { t} k i 0 − i 0 k2 2 { t} k i − i k2 where ξ := ( ,ζ ,ζ ,ζ , ,ζ ), ξ′ := ( ,ζ ,ζ′,ζ , ,ζ ), f (ξ ) := E[f(ξ )ξ ] = i ··· −1 0 1 ··· i i ··· −1 0 1 ··· i i 0 i | 0 E[y ] is the conditional expectation and ζ′,ζ ,t Z are i.i.d. r.v.s. Note the i.i.d. i|F0 { 0 t ∈ } sequences ξ and ξ′ coincide except for a single entry. Then ω (i; y ) δ (i; y ),i 0 i 2 { t} ≤ 2 { t} ≥ and condition ∞ ω (k; y ) < (3.39) 2 t { } ∞ k=0 X guarantees the weak invariance principle in (3.37) with σ2 := Cov(y ,y ), see Wu j∈Z 0 j [35]. The last series absolutely converges in view of (3.39) and the bound in ([35], Thm. 1), P implying Cov(y ,y ) ∞ ω (k; y )ω (k+j; y ),j 0. | 0 j | ≤ k=0 2 { t} 2 { t} ≥ P 10