To simplify notation, the index i of the process (Xi,t)t≥1 will be suppressed in this section. We will prove the following theorem which implies proposition 7.2 of section 7.3 and additionally gives the corresponding asymptotic result for the bias of the serial correlation coefficient:
Theorem 7.3 Letεt be an i.i.d. sequence of standard normal random variables.
Consider the AR(1)-process
Xt =ϕXt−1+εt, t≥1,
where both the coefficient ϕ ∈ (0,1) and the initial value X0 = c ∈ R are de-terministic. Then the bias and the mean squared error of the serial correlation coefficient ϕˆn = Pnt=1PnXtXt−1
CHAPTER 7. AGGREGATION AND ESTIMATION 153 The proof is split into several lemmas which will be presented in the following.
We give integral representations for the bias and MSE generalizing the results of Shenton and Johnson (1965), where only the case c= 0 is treated. First, we will concentrate on a slightly different version of the serial correlation coefficient which is given by PQn
The reason for this is, that tractable formulas for the bias and MSE of QPn
n can be derived. At the end of this section we will then show in lemma 7.4 that
Pn
Qn −ϕn−1 = XnQXn−1
n is asymptotically negligible uniformly in ϕ.
We start by recalling that the joint probability density function of (X1, . . . , Xn)T the moment generating function ofQn, which we interpret as a bivariate function of v and the AR(1) coefficient ϕ ∈ (0,1). Here, the notation R
. . . dx stands for R∞
−∞· · ·R∞
−∞. . . dx1· · ·dxnandqn =Pn−1
t=0 x2t. By partial integration and Fubini’s theorem we get for r∈ {0,1}
Twice differentiating (7.9) with respect to ϕ yields d
CHAPTER 7. AGGREGATION AND ESTIMATION 154 The crucial point is that the function mn(ϕ, v) can be calculated explicitly:
Lemma 7.1 Under the same assumptions as in theorem 7.3 and for v ≥0, the function mn(ϕ, v) is given by
and λ ≥ 1 is the greatest root of the two solutions of the quadratic equation λ2 − (1 + 2v +ϕ2)λ +ϕ2 = 0. Moreover, the function that maps λ onto the corresponding v, given by
λ7→v(λ) = (λ−1)(λ−ϕ2)
2λ ,
is one-to-one from [1,∞) onto [0,∞).
Proof: We follow the line of White (1961) and Breton and Pham (1989). The definition of the function mn(ϕ, v) can be reformulated in matrix notation as follows: Again, let x0 =c and x= (x1, . . . , xn)T, and write
CHAPTER 7. AGGREGATION AND ESTIMATION 155
The integral in (7.14) corresponds to the moment generating functionE[exp(tZ)]
at t = (ϕx0,0. . . ,0)T ∈ Rn of a n-dimensional vector Z that is normally dis-tributed with zero mean and covariance matrixD−n1(ϕ, v), the inverse ofDn(ϕ, v).
Recall that E[exp(tZ)] can be calculated as Dn(ϕ, v)12 Cramer’s rule. Now, observe that expansion of Dn(ϕ, v) along the first column shows that Dn(ϕ, v) fulfills the difference equation
Dn+1(ϕ, v) = (1 +ϕ2+ 2v)Dn(ϕ, v)−ϕ2Dn−1(ϕ, v), n≥2. (7.15) Hence, (7.14) can be written as
mn(ϕ, v) = e−vx20−ϕ
CHAPTER 7. AGGREGATION AND ESTIMATION 156 Finally, define D0(ϕ, v) = 1. Then the solution of the difference equation (7.15) is given by the following result (see e.g. Elaydi 2005): Let rn = Arn−1+Brn−2 Simple calculations then lead to the expression for Dn(ϕ, v), whereas solving the equation λ2−(1 +ϕ2+ 2v)λ+ϕ2 = 0 for v gives the claimed correspondence of λ and v. The bijectivity follows from v(λ)→ 0 as λ→ 1, v(λ)→ ∞ as λ → ∞ and dλdv(λ) = λ22ϕ−ϕ22 >0 for λ >1.
We can now use lemma 7.1 to prove uniform bounds of order O(n−1) for all terms in (7.12) and (7.13):
Lemma 7.2 Under the same assumptions as in theorem 7.3, we have
The right hand side can be estimated by 1
CHAPTER 7. AGGREGATION AND ESTIMATION 157
Since (7.17) and (7.18) are very similar, we only consider the more difficult term
d2 dϕ2
R∞
0 vmn(ϕ, v)dv:
By change of variable from v to λ this can be written as Z ∞
CHAPTER 7. AGGREGATION AND ESTIMATION 158 and
d2
dϕ2fn(ϕ, λ) = 1 ϕ
d
dϕfn(ϕ, λ) + 2ϕ2
"3
2(λ2−ϕ2)−12(λ−ϕ2)
γn(ϕ2, λ)12 +6(λ2−ϕ2)12 γn(ϕ2, λ)12 +6(λ2−ϕ2)12(λ−ϕ2) + 4(λ2−ϕ2)32
2γn(ϕ2, λ)32 γn′(ϕ2, λ) +3
2
(λ2−ϕ2)32(λ−ϕ2)
γn(ϕ2, λ)52 γn′(ϕ2, λ)2
−(λ2−ϕ2)32(λ−ϕ2)
γn(ϕ2, λ)32 γn′′(ϕ2, λ)
#(λ−1) 4λn+62 .
Moreover, if n ≥ 4, λ > 1, then |γ′n(ϕ2, λ)| ≤ 2 and |γn′′(ϕ2, λ)| ≤ 4/(λ− 1), since n(λ −1)λ−2n+1 ≤ 1 for n ≥ 4, λ > 1 respectively n2(λ−1)2λ−2n+1 ≤ 4 for n ≥ 3, λ > 1. Thus, using γn(ϕ, λ)−12 ≤ (λ − ϕ2)−12 ≤ (λ − ϕ)−12 and λ2−ϕ2 = (λ−ϕ)(λ+ϕ), one gets
1
ϕ d
dϕfn(ϕ, λ)
≤ 3(λ+ 1)32(λ−1)(λ2−ϕ2)
4λn+62 ≤ 3λ(λ+ 1)32
4λn+22 (7.20)
and d2
dϕ2fn(ϕ, λ)
≤ 12λ(λ+ 1)32 4λn+22 .
Consequently, since |gn(ϕ, λ)| ≤1, the same argument as in (7.19) leads to|I1| ≤
1
n3k with the same constant k.
The derivatives of gn(ϕ, λ) are given by d
dϕgn(ϕ, λ) = ϕc2gn(ϕ, λ)
λ−(n+ 1)ϕ2n(λ−1)λ−2n γn(ϕ2, λ)
+ λ(λ−ϕ2) +ϕ2(n+1)(λ−1)λ−2n
γn(ϕ2, λ)2 γn′(ϕ2, λ)
CHAPTER 7. AGGREGATION AND ESTIMATION 159 With the same arguments as for the derivatives of fn(ϕ, λ), one gets
For I2, combining (7.20) and (7.21), and again making the change of variable from λ tox as in (7.19) yields
Corollary 7.1 Under the same assumptions as in theorem 7.3, we have
CHAPTER 7. AGGREGATION AND ESTIMATION 160 In the remaining part of the section we will show that the uniform bounds for the bias and MSE of QPn XnXn−1/Qn will be calculated. Therefore, the joint moment generating func-tion
m∗n,ϕ(u, v) =E[exp(uXnXn−1−vQn)]
of XnXn−1 and Qn has to be considered. Fubini’s theorem and a change of order of differentiation and integration shows as above that
d
(The change of order of differentiation and integration can be justified by domi-nated integrability of the derivative.) Now, the same arguments as in lemma 7.1 imply that m∗n,ϕ(u, v) can be calculated explicitly:
Lemma 7.3 Under the same assumptions as in theorem 7.3 and for v ≥0, the function m∗n,ϕ(u, v) is given by
CHAPTER 7. AGGREGATION AND ESTIMATION 161 and λ ≥ 1 is the greatest root of the two solutions of the quadratic equation λ2−(1 + 2v+ϕ2)λ+ϕ2 = 0. Again, the function that maps λ onto v, given by
λ7→v(λ) = (λ−1)(λ−ϕ2)
2λ ,
is one-to-one from [1,∞) onto [0,∞).
Proof: Exactly as in the proof of lemma 7.1 we get m∗n,ϕ(u, v) = e−vx20−ϕ
Expansion along the first column shows that ˆDn(ϕ, v) solves the same difference equation as in (7.15):
Dˆn+1(ϕ, v) = (1 +ϕ2+ 2v) ˆDn(ϕ, v)−ϕ2Dˆn−1(ϕ, v), n≥2. (7.23)
CHAPTER 7. AGGREGATION AND ESTIMATION 162 However, the initial values are now given by ˆD0(ϕ, v) = 1 and ˆD1(ϕ, v) = p. The solution is thus given by (see e.g. Elaydi 2005)
Dˆn(ϕ, v) := ˆDn(ϕ, λ) = λn+2− ϕ2n+2λn λ2−ϕ2 ,
where λ depends onv as in the proof of lemma7.1. The proof is now finished by
the same arguments as in lemma 7.1.
With a similar approach as in lemma 7.2, we show in the next lemma that the first and second moment of XnXn−1/Qn are of order O(n−1) uniformly in ϕ.
Lemma 7.4 Under the same assumptions as in theorem 7.3, we have for n ≥4
Proof: By lemma 7.3 we get Z ∞ as in lemma 7.2, leads to
|I1| ≤
CHAPTER 7. AGGREGATION AND ESTIMATION 163 Thus, the change of variable from λ to x, given by λ = exp n+22x
as in (7.19), shows that|I1| ≤ Cn4, whereC4 is independent ofϕ, candn. The same arguments can be used to derive the estimates
|I2| ≤ c2 Z ∞
1
(λ2−ϕ2)322(λ−ϕ2) (λ−ϕ2)52
1 λn2+1dλ
≤ c2 Z ∞
1
2(λ+ 1)32 1 λn2+1dλ and
|I3| ≤ c2 Z ∞
1
(λ2−ϕ2)32 (λ−ϕ2)32
1 λn2+1dλ
≤ c2 Z ∞
1
(λ+ 1)32 1 λn2+1dλ.
The same approach as in (7.19) then leads to |I2|+|I3| ≤ Cn4c2 for a sufficiently large constant C4 that can be chosen independently of ϕ, c and n.
The proof of (7.25) is a straightforward extension taking into account the second derivatives leading to additional constant c4 as claimed in the lemma.
CHAPTER 7. AGGREGATION AND ESTIMATION 164
Chapter 8
Concluding remarks
Finally, we give some concluding remarks and suggestions for further research regarding the three statistical problems that have been investigated in this thesis.
(1) In section4.1, the problem ofM-estimation for long memory volatility mod-elsXt=εtσt has been studied. It was assumed that the autocovariances of σt andXt2 decay hyperbolically and are not summable. Under symmetry of εt, M-estimators defined by (4.4) have standard asymptotic distributions, while asymmetry implies a different behavior. Thus there is a need for sta-tistical tests of symmetry of the innovations distributions. Next, it should be possible to relax the main assumptions for the non-central limit theorem - independence ofεtandσtand linearity ofσt. For instance, one could study the case of dependent εt and ξtin (A4.1’) and (A4.2’), where εt is indepen-dent of past ξs, s ≤ t (e.g. εt = ξt). Then, the approach via cumulants and the diagram formula still lead to a representation of χk(Sn) in terms of (4.17). However, more general diagrams γ have to be considered which probably complicates the further study. Moreover, it should be investigated whether there is an analogon to the essential condition E[εjtψ(j)(Xt)] 6= 0 (see theorem 4.3) for more general models such as LARCH. Recall that in section 5.4, the condition E[Xtψ′(Xt)]6= 0 implied a non-standard limit theorem for partial sums ofψ(Xt), whereXtis a LARCH process. Finally, a question which is also interesting for homoskedastic models is the existence of Appell polynomial expansions for functions ψ that are not entire.
(2) In section 6.2, parametric estimation for LARCH processes has been stud-ied. Modified conditional maximum likelihood estimators were proposed
165
CHAPTER 8. CONCLUDING REMARKS 166 and the asymptotic properties were derived. While the unfeasible version θ(h)n (see definition6.1) has usual asymptotic properties, the feasible estima-tor θ(h,β)n (see definition 6.3) has a slow rate of convergence. In both cases, the asymptotic covariance matrix depends on the value of h. In particular, the asymptotic variances of the individual components of the estimators de-crease if h→0. On the other hand, numerical problems regarding the opti-mization in the definition ofθn(h,β) simultaneously increase. Thus, a criterion has to be defined by which the performance of θn(h,β) can be measured as a function of h. Moreover, disregarding the numerical difficulties, one should study the case h= 0, since then it could happen that the asymptotic distri-bution becomes degenerated. One should also address further modifications of ¯Ln,h(θ) to improve the estimator ¯θ(h)n . For instance, note that ¯σt2is not the conditional variance ofXtgiven X1, . . . , Xt−1. Thus it should be promising to search for a compact representation of σ(1,...,t2 −1),t :=E[σt2|X1, . . . , Xt−1] (essentially, that means the calculation of E[Xs|Xt] fors < t) and substi-tute ¯σ2t by σ(1,...,t−1),t2 in the objective function ¯Ln,h(θ). Finally, there are further obvious generalization of our results. In particular, more general parameterizations of the coefficients bj(θ) should be studied. For instance one could consider weights bj(θ) corresponding to c(1−(1−B)−d) with 0< c <1 andd∈(0,1/2) which implies bj(θ)∼c1jd−1. It is likely that the proofs still hold for hyperbolically decaying coefficients in the asymptotic sense. It should further be studied whether our strong moment assumptions up to order six can be relaxed. An alternative approach to parameter esti-mation would be a method of moments based on sample autocorrelations of squares γX2(0), γX2(1), . . .. Therefore, a limit theorem for quadratic forms of LARCH processes has to be derived. Probably, the diagram formalism mentioned in section 5.2 could be helpful in this context.
(3) In chapter 7, we introduced a new parameter estimator for a long memory process which is the aggregate of a panel of short memory random AR(1) processes. Given the estimates of the single AR(1) parameters, the long memory parameter is estimated by a Beta-type maximum likelihood esti-mator. Again, a truncation parameter h was used leading to the question of optimal choice of h. Moreover, obvious extensions of the results concern more general aggregation schemes. For instance, the gaussian assumption
CHAPTER 8. CONCLUDING REMARKS 167 could be relaxed by formulating sufficient conditions for uniform bounds of bias and MSE of the serial correlation coefficient respectively of alterna-tive estimators. In particular, the heteroskedatic LARCH case should be investigated, where aggregation also leads to long memory (see section 3.2).
CHAPTER 8. CONCLUDING REMARKS 168
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