Higher order autocorrelations
Giraitis et al. (2000b) also extend the diagram formalism introduced for E[σtp] to the more general case of autocovariances of integer powers cov(σtp, σt+kp ) and mixed covariances cov(σtp, σpt+k′ ), (p6= p′) and study the asymptotic behavior for k → ∞, see their theorem 2.2 and lemma 5.1. This can then be transferred to the autocovariances of Xtp as follows: For t, k >0 set yt,p:= (εpt −µp)σtp, then
Xtp =µpσtp+yt,p
and assumption (A5.2) easily implies that cov(yt,p, yt+k,p) = cov(σpt, yt+k,p) = 0.
This leads to the decomposition
cov(Xtp, Xt+kp ) =µ2pcov(σtp, σpt+k) +µpcov(σt+kp , yt,p). (5.14) Giraitis et al. (2000b) then show that σpt can be approximated by dpσt (with some constant dp), in the sense that, as k tends to infinity, the difference of the autocovariances decays sufficiently fast,
|cov(σtp, σt+kp )−d2pcov(σt, σt+k)|=o(k2d−1), (5.15)
CHAPTER 5. LARCH 91 if
bj ∼c1jd−1, as j → ∞, (5.16) where c1 >0 is a finite constant. Corollary 5.1 thus implies that cov(σpt, σt+kp ) = O(k2d−1). Finally, by provingcov(σt+kp , yt,p) =o(k2d−1), they derive the following theorem:
Theorem 5.2 Let (A5.1), (A5.2), (M′′2p) and (5.16) hold. Then, forl = 2, . . . ,2p, γXl(k) = cov(Xtl, Xt+kl )∼c2l|k|2d−1, as k → ∞,
where cl= cb10lE[Xtl].
The theorem thus states that for the uncorrelated processXt, assuming (5.16) and finiteness of respective moments, integer powers Xtp,p≥2 exhibit long memory.
Moreover, the order of asymptotic decay k2d−1 is the same for all pand only the coefficient cp varies.
Leverage
Next we address the results concerning leverage derived by Giraitis et al. (2003c).
Recall that the conditional variancevar(Xt|Ft−1) is given byσtand that leverage can be measured by the sequence
hk=cov(σt+k2 , Xt), k ≥1.
We say that Xt has leverage of order k iff hj <0 forj = 1, . . . , k. Observe that, if E[ε2t] = 1, one gets cov(Xt+k2 , Xt) = E[ε2t+kσt+k2 Xt] = cov(σt+k2 , Xt) and thus hk coincides with the cross-covariance function of Xt and Xt2. Since equation (5.2) directly involves the pastXt−j (instead of their absolute values or squares) it is clear that there is leverage potential in the LARCH model. Intuitively, it is helpful to examine (5.1) and (5.2) in the finite first order case, where (5.2) reduces to
σt =b0+b1Xt−1.
Let b0 >0 and b1 <0, then σt moves up, if Xt−1 drops down and vice versa why σt and Xt−1 are probably negatively correlated. On the other hand, if b0 >0, σt
tends to be positive and thus positive correlation ofσtandσt2is plausible resulting in a negative correlation between σt2 and Xt−1. This observation was rigorously
CHAPTER 5. LARCH 92 studied in Giraitis et al. (2003c) and is given in the next theorem. Moreover, a long memory property for ht is stated similar to theorem 5.2. Notice, that the study of ht only requires assumption (M3). Hence, one advantage of choosing ht
instead of γX2(k) as a measure of non-linear long memory inXt, is that moments of order 4 may be infinite.
Theorem 5.3 Let (A5.1), (A5.2) and (M3) (or (M′′4)) hold.
(a) Moreover let
|µ3| ≤ 2(1−5kbk22) kbk2(1 + 3kbk22). Then for any fixed k such that 1≤k≤ ∞:
(i) If b0b1 <0, b0bj ≤0, j = 2, . . . , k, then hj <0, j = 1, . . . , k;
(ii) If b0b1 >0, b0bj ≥0, j = 2, . . . , k, then hj >0, j = 1, . . . , k.
(b) Further assume that bj ∼c1jd−1 as j → ∞ for some d∈ 0,12
. Then hk ∼Ckd−1, as k → ∞,
where C = 2cb1
0 (E[Xt2])2.
Proof: See theorem 2.3 and 2.4 in Giraitis et al. (2003c).
5.4 Limit theorems
According to corollary 5.1, the volatility process σt has long memory, charac-terized by cov(σt, σt+k) ∼ c2k2d−1, if (5.16) holds. Recall that this property is implied by the fact that σt−b0 has the same second order properties as the linear process σ∗t defined by
σt∗ = X∞
j=1
bjξt−j t∈Z,
whereξtare i.i.d. random variable whose distribution is the same as the marginal distribution ofεtσt. Giraitis et al. (2000b) even prove the more general result that the partial sums Pn
t=1(σt−b0) and Pn
t=1σ∗t have the same asymptotic behavior in the sense that, suitably normalized, both converge in distribution to the same limit as n → ∞. Again, by applying decomposition (5.14) and approximation
CHAPTER 5. LARCH 93 (5.15), they deduce a limit theorem for integer powers of Xt, which is given in the next theorem. Recall that BH(u), u≥0 is a fractional Brownian motion with Hurst-indexH ∈(0,1), ifBH(u) is a zero-mean gaussian process with covariances
E[BH(u)BH(v)] = 1
2(|u|2−H +|v|2−H − |u−v|2−H), u, v ≥0.
Theorem 5.4 Let (A5.1), (A5.2) and (M′′2p) hold. Further assume that (5.16) holds.
(a) Then, as n→ ∞,
n−1/2−d X[nu]
t=1
(σt−b0)→D c1γB1−2d(u), u≥0.
(b) For j = 2, . . . ,2p, n−1/2−d
X[nu]
t=1
(Xtj−E[Xtj])→D cjγB1−2d(u), u≥0,
where γ = (d+ 2d2)−12 and cj is defined in theorem 5.2.
Proof: For convergence of finite-dimensional distributions see theorem 2.3 in Giraitis et al. (2000b). Note that an approximation by a martingale differ-ence is used via the decomposition σt = E[σt|Ft+−k] + (σt −E[σt|Ft+−k]) with Fk+ = σ(εj;j ≥ k) (the same technique is used in the analysis of M-estimation in Beran (2006), see section 6.1). The proof of tightness is given in Berkes and
Horv´ath (2003, pg. 647).
However, the strong relationship to linear processes given in (a) does not extend to general sums Pn
t=1f(σt) respectively Pn
t=1f(Xt). This is already indicated in part (b) of the preceding theorem, since for all powers Xtj the same standardiza-tion n−1/2−d and the same limit process (up to a constant) appears. This has to be compared to the asymptotic distribution of P∞
j=1f(σ∗t) with f(x) =xj which depends on the Appell-rank of f, whereas different standardization and limit processes may arise. In Berkes and Horv´ath (2003) it is shown that for LARCH processes an invariance principle holds that is different to the case of linear pro-cesses. For given f, satisfying conditions that are specified below, they introduce
CHAPTER 5. LARCH 94 the following processes, where the coefficients bj in the Wold-decomposition of σt−b0 are replaced by coefficients depending on f:
σt(f) := B1εt−1σt−1+B2εt−2σt−2+· · ·
˜
σt(f) := B˜1εt−1σt−1+ ˜B2εt−2σt−2+· · · , where
Bj =E[f′(σ0)sj], B˜j =E[f′(X0)ε0sj] and
sj = X
r≥1 j1, . . . , jr≥1 j1+· · ·jr=j
bj1· · ·bjrε−j1· · ·ε−j1−···−jr−1.
Then, they prove the following theorem:
Theorem 5.5 Let (A5.1) and (A5.2) hold with E[|εt|p] < ∞ for some p > 4.
Moreover, let (5.16) hold with
kbk22 < p−1 3(6p)3µ2p.
Let f be twice continuously differentiable with |f′′(x)| ≤ C(|x|α + 1), where 0<
α <(p−4)2/(2p). Then, Xn
t=1
(f(σt)−E[f(σt)]) = Xn
t=1
σ(ft )+Cn1/2+d+ǫζn
and
Xn t=1
(f(Xt)−E[f(Xt)]) = Xn
t=1
˜
σt(f)+Cn1/2+d+ǫηn
for some C, ǫ > 0, where E[ζn2] ≤ 1 and E[ηn2] ≤ 1. Moreover, the coefficients Bj,B˜j satisfy
Bj ∼γbj, B˜j ∼γ1bj
as j → ∞ with
γ = 1 b0
E[σtf′(σt)], γ1 = 1 b0
E[Xtf′(Xt)].
CHAPTER 5. LARCH 95 Proof: See Berkes and Horv´ath (2003, theorem 1.2).
Hence, the asymptotic distribution of Pn
t=1(f(σt)−E[f(σt)]) can be deduced from Pn
t=1σt(f), whereas one can apply theorem5.4 for the latter sum (note that the coefficientsBj and ˜Bj are asymptotically equivalent to the originalbj) leading to the following theorem which is also due to Berkes and Horv´ath (2003, theorem 1.1):
Theorem 5.6 Under the assumptions of the preceding theorem we have n−(1/2+d)
X[nu]
t=1
(f(σt)−E[f(σt)])→D γc2B1/2+d(u) (5.17) and
n−(1/2+d) X[nu]
t=1
(f(Xt)−E[f(Xt)])→D γ1c2B1/2+d(u), (5.18) where c2 is a constant that depends only on d, b0,kbk22 and c1.
Notice that we get a degenerate limit if γ respectively γ1 are zero. For ex-ample, f(x) = x yields γ1 = 0, which obviously agrees with the fact that the variance of Pn
t=1Xt is of order n. For the linear process σt∗, convergence (5.17) analogously holds (up to a non-zero constant) if E[f′(σt)] 6= 0, where the latter implies an Appell-rank of one. However, there are functions f with E[σtf′(σt)]6= 0, E[f′(σt)] = 0 and E[f(k)(σt)]6= 0, k >1, implying that the vari-ances of Pn
t=1(f(σt)−E[f(σt)]) and Pn
t=1(f(σt∗)−E[f(σt∗)]) grow at different orders and different limit processes appear. Thus, at first sight, the connec-tion to Appell polynomials is lost for LARCH processes. On the other hand, the results should be compared to section 4.1, where the link of the condition E[εtf(k)(εtσt)]6= 0 to Appell polynomials is derived for a specific class of volatil-ity processes σt. See also the concluding remarks in chapter 8.
CHAPTER 5. LARCH 96
Chapter 6
LARCH - statistical inference
There are only very few papers discussing statistical methods for LARCH cesses. The reason is probably the complicated structure of the volatility pro-cess σt, which makes standard statistical methods problematic, in particular see section 6.2. However, a suitable parameterization and corresponding statistical theory are necessary to increase practical applicability of the model.
The first paper dealing with statistical aspects for the LARCH model is Giraitis et al. (2003a), where the (non-parametric) R/S-test and related tests are studied (see also Giraitis et al. 2003b). Thus, tests for the presence of long memory are mainly considered and the special structure of the LARCH model is not taken into account. The problem of location estimation is considered in Beran (2006) and generalized to non-parametric estimation of a trend function in Beran and Feng (2007). Since the paper of Beran (2006) shows very interesting similarities (and differences) to the results derived in section 4.1, we briefly review the paper in the following section of this chapter.
A parametric treatment for LARCH processes, i.e. estimation of all coefficients bj, j ≥ 0, has not been available until the recent appearance of three working papers: Beran and Sch¨utzner (2008b), Francq and Zakoian (2008), and Tru-quet (2008). Only the first one considers the long memory case including non-summable autocorrelations of the squares. The others concentrate on the finite order LARCH model, i.e. equation (5.2) is replaced by
σt=b0+ XK
j=1
bjXt−j
97
CHAPTER 6. LARCH - STATISTICAL INFERENCE 98 for some finite K < ∞. Note that the latter equation directly implies expo-nentially decaying autocorrelations of the squares of Xt = εtσt, meaning short memory in volatility. After arguing that, for finite order LARCH, the standard conditional MLE described in section 4.2leads to inconsistent estimators, Francq and Zakoian (2008) give a comprehensive theoretical and numerical study of an alternative weighted least square estimator. On the other hand, Truquet (2008) alters the conditional MLE very similarly to the method introduced (indepen-dently) by Beran and Sch¨utzner (2008b). The main object of section 6.2 is the detailed description of the latter paper. A very simple parameterization of the coefficients bj, j ≥0 is used, still preserving the main feature, hyperbolically de-cay of the coefficients. Consistency and asymptotic normality of the modified conditional MLE are derived.