In this section, we briefly describe the results obtained in Beran (2006) concerning asymptotic properties of location estimators for long memory LARCH processes.
For µ∈R, consider the process
Yt=µ+Xt,
where Xt = εtσt is a stationary LARCH process satisfying (A5.1) and (A5.2).
Since Xt has zero mean, one gets E[Yt] = µ, and hence we call Yt a LARCH process with location parameter µ. Given a sample Y1, . . . , Yn, the aim is to estimate µ. As already mentioned in section 4.1, the marginal distributions of volatility models are non-normal and can exhibit heavy tails and thus substi-tuting the mean by alternative location estimators is necessary. Here, we will concentrate on the class of M-estimators which were already studied in chapter 4.1. Recall that the statistical properties of M-estimators for general volatility models are not obvious. In particular, the rate of convergence and asymptotic distribution depends on the symmetry respectively asymmetry of the innovations εt. Moreover, since the stationary solution of the LARCH-equations possesses a very complicated structure, it is not clear whether the conditions of section 4.1 are fulfilled. In particular, there are no theoretical results about the properties (and existence) of the extremal index of LARCH processes. Therefore, the find-ings in Beran (2006) are a supplement and generalization of the results of section
CHAPTER 6. LARCH - STATISTICAL INFERENCE 99 4.1.
Due to the complex Volterra-representation of σt, the focus is on symmetric in-novations εt (the case of general εt has not been treated explicitly, but can be derived by the results of Berkes and Horv´ath 2003, see also the remarks below).
Just as in (4.4), a M-estimator ˆµn for the parameter µ is defined as solution of the equation
Xn t=1
ψ(Yt−µˆn) = 0, (6.1)
where ψ is an odd function, i.e. we have ψ(x) = −ψ(−x) for almost all x ∈ R, further satisfying E[ψ(Xt)] = 0 and E[ψ2(Xt)] < ∞. Recall that the latter moment assumptions are needed to ensure consistency of the M-estimator ˆµn. Indeed, one can easily show that the proof of convergence in probability, i.e.
ˆ
µn →p µ as n → ∞, is not affected by the special dependence structure of the process Xt (under standard regularity conditions on ψ, see below) and hence ˆµn
is a weakly consistent estimator of µ.
On the other hand, it is not clear whether the asymptotic behavior of ˆµn, which is again essentially determined by the (properly standardized) sum
Xn t=1
ψ(Xt), (6.2)
depends on the long memory property given in theorem 5.2. Therefore, the problem is to derive an appropriate limit theorem for (6.2). However, the results described in section 5.4 can not be applied. Giraitis et al. (2000b) only consider sums of powers of Xt, while Berkes and Horv´ath (2003) require ψ-functions with E[Xtψ′(Xt)] 6= 0. In our case, assuming ψ to be differentiable, the function xψ′(x) is odd, since the derivative ψ′(·) is even. Thus, if the distribution of εt is symmetric, the conditional distribution of εtσt given Ft−1 = σ(εs, s ≤ t−1) is symmetric as well, leading to
E[Xtψ′(Xt)] =E[εtσtψ′(εtσt)|Ft−1] = 0.
The needed asymptotic results for (6.2) are given in Beran (2006) and are based on the following set of assumptions:
(A6.1) Xt is a stationary LARCH process with coefficients bj, j ≥ 0 satisfying (A5.1) and (A5.2)
CHAPTER 6. LARCH - STATISTICAL INFERENCE 100 (A6.2) For each k ∈ N, define ξt,k, t ∈ N to be i.i.d. random variables with
characteristic function
φξ,k(z) =E[eizξt,k] = Y∞ j=1
ϕk(bjz), with
ϕk(z) =E[exp(izε1ε2· · ·εk)].
Then, as n→ ∞,
1 n max
1≤t≤nξt,k2k →0 in probability.
(A6.3) For each k ∈N,
1 nE
1max≤t≤nξt,k2k
→0, where ξt,k are defined in (A6.2).
(A6.4) εt has a symmetric distribution
(A6.5) ψ is a measurable function satisfying (A4.6)-(A4.8). Moreover, ψ can be approximated uniformly by stepwise linear functions with uniformly bounded intercept and slope (see assumption A7 in Beran 2006).
To explain the definition of the variables ξt,k, notice that their characteristic function φξ,k(z) is the same as for the linear sum P∞
j=1bjπj,k, where πj,k are i.i.d. random variables having the same marginal distribution as ε1ε2· · ·εk. The reason, why assumptions (A6.2) and (A6.3) are useful, is thatσtcontains products Ut = Qk
l=1εt−jl, 1 = j1 < · · · < jk with arbitrary k, whereas upper bounds for max1≤t≤nUt2 (and for maxima involving squares of linear combinations of Ut’s) have to be found to apply a central limit theorem. These upper bounds can then be traced back (by results concerning k-dependent random sequences) to the conditions formulated in (A6.2) and (A6.3).
Theorem 6.1 Denote B(u) standard Brownian motion on the interval [0,1].
Then, under assumptions (A6.1)-(A6.5),
√1 nv
X[nu]
t=1
ψ(Xt)→D B(u), as n tends to infinity, where v =var(ψ(Xt)).
CHAPTER 6. LARCH - STATISTICAL INFERENCE 101 Sketch of the proof: The proof can be described very briefly as follows: First, con-centrate on the mean, i.e. on the corresponding ψ-functionψ(Xt) =Xt. Though Xt is a martingale difference, the complex dependence structure of the station-ary solution of σt makes is difficult to apply a standard central limit theorem.
Therefore, a decomposition into a martingale difference with simpler dependence structure and a remainder term is used. Note that
σt−b0 = X∞
j=1
bjεi−jσi−j
which immediately follows from (5.2) by replacing Xt by εtσt, and consider the decomposition
σt=E[σt|Ft+−k] + (σt−E[σt|Ft+−k]) =:zt(k) +rt(k), with Fk+ :=σ(εj, j ≥k). This leads to
σt−b0 = X∞
j=1
bjεt−jzt−j(k) + X∞
j=1
bjεt−jrt−j(k) =: ζt(k) +Rt(k).
Then, the two conditions (i) n−1V arP[nu]
t=1εtRt(k)
→c(k, u) with limk→∞c(k, u) = 0 and (ii) Asymptotic normality of n−12 P[nu]
t=1εtζt(k)
imply the central limit theorem for Xt. While (i) follows easily by the definition of Rt(k), (ii) can be verified by means of a suitable martingale-limit theorem of Hall and Heyde (1980) since εtζt(k) is a martingale difference. The following two conditions on the asymptotic behavior of partial maxima,
(a) n1 max1≤t≤nε2tζt2(k)→0 in probability, and (b) n1E[max1≤t≤nε2tζt2(k)]→0,
have to be derived. The crucial point is that ζt(k) are k-dependent random variables and thus the results of Haiman and Habach (1999) imply that (a) and (b) remain unchanged when ζt(k) is replaced by i.i.d. random variables ˜ζt(k) having the same marginal distribution as ζt(k). By using the definition ζt(k) = P∞
j=1bjεt−jzt−j(k), one sees thatζt(k) (and thus ˜ζt(k)) is a linear combination of terms
Yt,l = X∞
j=1
βjZt−j,l,
CHAPTER 6. LARCH - STATISTICAL INFERENCE 102 where Zt,l has the same marginal distribution as ε1ε2· · · , εl+1 with arbitrary l.
(Here, βj =cbj are square-summable coefficients, where the constant c depends on the representation of ζt(k).) Again by l-dependence, it is then sufficient to prove (a) and (b) for an i.i.d. version of the process Yt,l (and Zt−j,l) which is directly implied by (A6.2) and (A6.3).
Finally, one has to ensure that the results carry over to more generalψ-functions.
Since higher order moments need not exist for LARCH processes, an Appell-polynomial expansion similarly as in section 4.1.2 can not be applied. However, it is still possible to generalize the results toψ-functions described in assumption (A6.5). Via approximation by stepwise linear functions the asymptotic behavior of max1≤t≤nψ2(Xt) can then be deduced by the properties (a) and (b). For more details, in particular convergence of finite-dimensional distributions and tightness,
see Beran (2006).
By the same arguments as in section4.1, it is then possible to apply the preceding result to M-estimation:
Corollary 6.1 Under (A6.1)-(A6.5), there exists a consistent sequence µˆn of solutions of (6.1). Moreover,
√n(ˆµn−µ)→ Nd (0, v),
where v =var(ψ(Xt))/E2[ψ′(Xt)].
This means that the long memory property in the squares of a LARCH process does not have any influence on the M-estimator ˆµn. However, one should keep in mind that the main prerequisite for the proof was symmetry of the innovations dis-tribution which implied E[Xtψ′(Xt)] = 0. On the other side, ifE[Xtψ′(Xt)]6= 0, the limit theorem 5.6 of Berkes and Horv´ath (2003) can be applied to show that the corresponding M-estimator has a slower rate of convergence of order nd−1/2, nevertheless the asymptotic distribution is still normal.
Also compare the results to the findings in section 4.1. For the models stud-ied there, the statistical properties of M-estimators depend on the Appell rank λk =E[εtψ(k)(Xt)] which seems to be related to the conditionsE[Xtψ′(Xt)] = 0 respectively E[Xtψ′(Xt)]6= 0. However, even non-normal limit distributions and slower rates of convergence n−α with α < 1/2−d can arise. In both situations, the LARCH model and the volatility models of section 4.1, it is very
impor-CHAPTER 6. LARCH - STATISTICAL INFERENCE 103 tant to know whether the innovations are distributed symmetrically to assess the statistical accuracy of M-estimators.