Functional ARCH and GARCH Models: A Yule-Walker Approach

Conditional heteroskedastic ﬁnancial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C [0 , 1] and L 2 [0 , 1] , their probabilistic features were studied and their parameters were estimated. The projections of the operators on a ﬁnite-dimensional subspace were estimated, as were the complete operators in GARCH(1 , 1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1) . This article provides suﬃcient conditions for the existence of strictly stationary solutions, weak dependence and ﬁnite moments of ARCH and GARCH processes in various L p [0 , 1] spaces, C [0 , 1] and other spaces. In L 2 [0 , 1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a ﬁnite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modiﬁcations, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.


Introduction
Volatility, usually measured by the variance, is one of the essential objects of study of financial time series. These are often strictly stationary but conditional heteroskedastic, where latter means that the variances at any time conditioned on the past are non-constant and randomly changing. A popular model exhibiting this phenomenon is the autoregressive conditional heteroskedasticity (ARCH) model established by Engle (1982) [7], for which he was awarded the noble prize in economics in 2003. This model was extended to the generalized ARCH (GARCH) model by Bollerslev (1986) [4]. Various authors established modifications of univariate and multivariate ARCH and GARCH processes, studied their probabilistic properties and estimated their parameters. An excellent overview and applications of such processes is provided in Andersen et al. [1], Francq & Zakoïan [9] and Gouriéroux [10]. Due to a progress in processing techniques and since high-resolution tick data are accessible and can be described as functions, it seems reasonable to extend these models on infinite-dimensional spaces, enabling the analysis to be more accurate. From a mathematical point of view, such an extension is unproblematic for complete, separable metric spaces M since completeness of M implies that the Borel σ-field B(M ) is well defined and separability ensures that e. g. sums of random variables remain random variables, see Ledoux & Talagrand [21]. For a detailed introduction in Functional Data and Functional Time Series Analysis, the areas dealing with random variables resp. time series with values in an infinite-dimensional space, see Bosq [5], Ferraty & Vieu [8], Hsing & Eubank [14] and Ramsay & Silverman [22]. For a compact synopsis (in German), see Kühnert [20]. Hörmann et al. (2013) [11] made the initial step by introducing ARCH (1) processes with values in the spaces C[0, 1] and L 2 [0, 1] of continuous resp. of square-integrable real valued functions with domain [0, 1]. They established sufficient conditions for the existence of strictly stationary solutions, finite moments and weak dependence. In L 2 [0, 1] they constructed consistent estimators and stated explicit asymptotic upper bounds of the estimation errors for the shift term and the projections of the operator on finite-dimensional subspaces by assuming the operator to be an integral operator and estimating its kernel.   [2] established GARCH(1, 1) processes in C[0, 1] and in L 2 [0, 1] and found sufficient conditions for the existence of strictly stationary solutions, finite moments and weak dependence. In L 2 [0, 1] they derived a consistent least squares estimator for the projections of the parameters on finite-dimensional subspaces but stating an explicit asymptotic upper bound of the estimation errors. At last, Cerovecki et al. (2019) [6] studied L 2 [0, 1]-valued GARCH(p, q) processes for positive integers p, q. They developed sufficient conditions for the existence of strictly stationary solutions and finite moments. By a quasi-likelihood approach, the projections of the parameters on a finite-dimensional subspace and only for p = 1 = q the complete operators were estimated consistently. In both cases, no explicit asymptotic upper bound of the estimation errors was stated. [2], [6], [11] also provided simulation studies showing how their models matched with real data and illustrated possible applications. For further work dealing with functional ARCH and GARCH models, see Kokoszka et al. (2017) [19] and Rice et al. (2019) [23].
In this article, we establish ARCH(p) and GARCH(p, q) processes for all p, q ∈ N with values in L p [0, 1] with p ∈ [1, ∞), C[0, 1] and other spaces. We provide sufficient conditions for the existence of strictly stationary solutions, weak dependence and moments of these processes under mild conditions. The focus of this paper is on deducing estimators for the shift term and the complete operators of L 2 [0, 1]-valued ARCH(p) and GARCH(p, q) processes for any positive integer p, q and on deriving explicit asymptotic upper bounds of their estimation errors. We also deduce explicit asymptotic upper bounds of the estimation errors for the operators on a finite-dimensional subspace of these ARCH processes. The operator estimation is always based on Yule-Walker equations and the estimators for the GARCH operators also involve estimators for the operators of invertible, linear processes represented as inverted time series. We derive explicit asymptotic upper bounds of their estimation errors. Also, this upper bound holds for the estimation errors when estimating the operators in the associated linear process and is valid beyond the context of functional ARCH and GARCH models. All results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA processes due to their relationship.
In this paper, we use the following notation. a ∧ b := min(a, b) and a ∨ b := max(a, b) for a, b ∈ R. For functions f, g : For sequences (a n ) n∈N , (b n ) n∈N ⊆ (0, ∞), we write a n ∼ b n if an bn → 1, a n b n if a n ∼ cb n for some c = 0, a n = ω(b n ) if b n = o(a n ) (for n → ∞) and a n = Ω(b n ) if b n = O(a n ) (for n → ∞). Further, Ξ(a n , b n ) := ω(a n ) ∩ o(b n ), Ξ[a n , b n ) := Ω(a n ) ∩ o(b n ), Ξ(a n , b n ] := ω(a n ) ∩ O(b n ) and Ξ[a n , b n ] := Ω(a n ) ∩ O(b n ). By 0 V we denote the identity element of addition of a vector space V and  1] where λ is the Lebesgue-Borel measure on [0, 1], and f g denotes the pointwise product of f, g ∈ F if it is well-defined. Let (B, || · || B ), (B , || · || B ) be Banach spaces and (H, ·, · H ), (H , ·, · H ) be Hilbert spaces. On Hilbert spaces we use norms generated by inner products and say CONS for a complete orthonormal system. We endow Banach spaces (B n , || · || B n ) with the norm ||b|| 2 We write L B,B , K B,B , S B,B resp. N B,B for the space of bounded, compact, Hilbert-Schmidt resp. nuclear operators from B to B with L B = L B,B , K B = K B,B , S B = S B,B and N B = N B,B where the term operator always refers to a linear mapping. K * denotes the adjoint of an operator K ∈ L B,B and we write h ⊗ h := h, · H h for h ∈ H, h ∈ H . In all respects, we assume our random elements to be defined on some common probability space (Ω, A, P). For B-valued processes (X k ) k∈Z and (Y k ) k∈Z , X n = O P (Y n ) (for n → ∞) denotes that (X k /Y k ) k is asymptotically P-stochastic bounded. For p ∈ [1, ∞) we denote by L p B = L p B (Ω, A, P) the space of (classes of) B-valued random variables X with ν p,B (X) := (E||X|| p B ) 1/p < ∞, we call a process (X k ) k∈Z of B-valued random variables L p B -process if X k ∈ L p B for all k, and centered if E(X k ) = 0 B for all k with expectation in Bochner-integral sense, see [14], p. 40-45.
The rest of this article is organized as follows. Section 2 studies probabilistic features of our ARCH and GARCH processes. Section 3 introduces our parameter estimators and derives asymptotic upper bounds of the estimation errors. Section 4 summarizes the main results, delineates these from similar results in other papers and gives an outline for future research. Section 5 contains proofs.
holds a. s. for all k. If (ε k ) k∈Z is strictly stationary and ergodic, which is especially the case if (ε k ) k is i.i.d., then (2.8) implies that (Y k ) k is also strictly stationary and ergodic after [25], Theorem 3. 5. 8.
Though (2.10) is stricter than (2.9), it is easier to show. Furthermore, (2.10) is useful for the simulation of an initial value of F -valued ARCH and GARCH processes, as we can see in the following. Corollary 2.1. Let (2.10) hold for some n ∈ N and ν > 0. Further, defineς Based on ideas in [2], [11] and with (2.10), we derive a sufficient condition for the existence of moments and for weak dependence, to be precise L p -m-approximibility, of F -valued ARCH(p) and GARCH(p, q) processes for any p, q ∈ N. Finite moments and L p -m-approximibility are used to estimate the ARCH and GARCH parameters.

Estimation
In this section, we establish estimators for the parameters of H -valued ARCH and GARCH processes with known orders where H := L 2 [0, 1] and we deduce asymptotic upper bounds for their estimation errors. Throughout the section, we writeḢ := L 4 [0, 1], and, except for in section 3.1, we impose the following.
e. t and all k. Thus, (2.1) yields H -m-approximable. For the estimation of the operators in (2.1), we use (3.2) and we impose the following.
and there is no closed, affine subspace U H with P(ε 2 0 ∈ U ) = 1.

Preliminaries
Here, in order to estimate the parameters in (2.1), we state certain assumptions and establish various convergence results dealing with the asymptotic behaviour of estimation errors of specific eigenvalues and expected values, operators and eigenfunctions in Hilbert spaces (H, ·, · H ), (H , ·, · H ) and (H , ·, · H ). We also discuss the estimation of operators within a composition of operators.

Estimation of expected values, lag-h-covariance and other operators
Firstly, we define lag-h-covariance operators and their empirical versions.
H -valued time series and let h ∈ Z. Then, the lag-hcovariance operator of X is defined by where m 1 = m 1 (X) := E(X 1 ) and the empirical lag-h-covariance operator of X is defined bŷ The operators C 0 and C 0 are also called covariance operator resp. empirical covariance operator.
Ĉ h are bounded operators with finite-dimensional image withĈ * h =Ĉ −h for all h. Furthermore,Ĉ 0 is selfadjoint and positive semi-definite. We obtain the following convergence rates.
H -m-approximable time series. Then, is an unbiased esimator for m l = m l (X) := E(X l 1 ) for any l = 1, 2 and N ∈ N with Based on ideas in [3], for centered time series X = (X k ) k∈Z we define the operators [20], p. 56) [20], Definition and properties 4.36) Moreover, the empirical covariance operatorŝ

Estimation of eigenvalues and eigenfunctions
Here, we derive asymptotic upper bounds of the estimation errors for the eigenvalues and eigenfunctions of a compact, self-adjoint and positive semi-definite operator K ∈ K H , estimated by a sequence (K N ) N ∈N ⊆ K H of compact, self-adjoint, positive semi-definite operators, where eachK N depends on N observations of a stationary time series X = (X k ) k∈Z . Further, (k j ) j∈N resp. (k j ) j∈N are the eigenfunction sequences and (k j ) j∈N resp. (k j ) j∈N the associated w. l. o. g. monotonically decreasing eigenvalue sequences of K resp.K N .
For the derivation of upper bounds of the estimation errors for the eigenvalues and eigenfunctions, we need This is true according to [5], Because the eigenfunctions ofK N are unambiguously determined except for their sign, is used as an estimator for k j ifk j ⊥ k j a. s. holds where sgn is the signum function. According to [5], Lemma 4. 3, which can be generalized to any compact, self-adjoint and positive semi-definite operators, if the eigenspace of k j is one-dimensional, whereγ 1 := 2 √ 2γ 1 ,γ j := 2 √ 2 max(γ j−1 , γ j ) for j > 1 and γ j := (k j − k j+1 ) −1 for j ∈ N. The problem in usingk j as an estimator for k j is, thatk j ⊥ k j a. s. and thus sgn( k j , k j H ) = 0 a. s., which is needed to obtain asymptotic upper bounds of the estimation errors for the operators in the H -valued ARCH and GARCH model, is not guaranteed for all j, N. Therefore, we modifyk j in the following way. Let (h j ) j∈N be a CONS of H and let (ζ j ) j∈N be a sequence of i. i.d. and N(0, 1)-distributed random variables, independent of the observations of X. Then is well-defined for all j, N withk j ⊥ k j a. s. and in consequence sgn( k j , k j H ) = 0 a. s. Thus we usê as an estimator for k j , where (k j ) j is a CONS of H a. s. according to the spectral theorem.

Assumption 3.3.
For all j, k j = k j+1 and κ(j) = k j holds where κ : R → R is a convex function.
If K is injective and if the eigenvalues of K satisfy Assumption 3.3, then (3.23)

Some notes on estimating operators
In this paper, we estimate bounded operators B ∈ L H ,H in equations as where A ∈ L H,H and C ∈ L H,H . Identifiability of B from (3.24), that is BC =BC implying B =B, is only guaranteed if B has dense image. Further, if C is a compact operator and thus has no bounded inverse, we use the tikhonov-regularized of C in order to isolate B. Also, when estimating operators without projecting them on a finite-dimensional subspace, we impose the following Sobolev condition.
for m = m N → ∞ which we utilize in conversions in various proofs.

Estimation of δ in the functional ARCH and GARCH model
We derive an estimator of δ in H -valued ARCH(p) and GARCH(p, q) time series with p, q ∈ N from the idea of estimating δ in H -valued ARCH(1) time series in [11]. Under Assumption 3.1, taking the expected value on both sides of the right equation in (2.1), yields where α i = 0 L H = β j for i > p, j > q. Therefore, we proposê as an estimator for δ whereα i ,β j are estimators for α i , β j and wherem 2 := N −1 N i=1 X 2 i . Theorem 3.1. Let Assumption 3.1 hold. Then,δ in (3.28) satisfies (3.29)

Operator estimation in the functional ARCH model
In the following, X X X := (X k ) k∈Z is a H -valued ARCH(p) process with p ∈ N. Under Assumption 3.1, where S p,1 = C Zp(p),Zp+1 and S p = C 0;Z Z Z(p) . Since S p is injective as a consequence of Lemma 3.1 (see [20], Lemma 4.35), α [p] can be identified from (3.31) and as an estimator we thus imposê (3.32) Thereby, K ∈ N, (ϑ N ) N ∈N ⊆ N with ϑ N → 0,ĉ p,1 , ...,ĉ p,K are the eigenfunctions ofŜ p associated to the first biggest eigenvaluesĉ p,1 ≥ · · · ≥ĉ p,K and The assumption α [p] (c p,l ), c j H = 0 for all j > K, l ≤ K in (a), which is necessary for technical conversions in the proof, is weaker than to impose that α [p] and S p commute. Moreover, it is similar to the assumption used in Turbillon et al. [26] in order to estimate their MA(1) operator.

Operator estimation in the functional GARCH model
Throughout this section, X X X := (X k ) k∈Z is a H -valued GARCH(p, q) with p, q ∈ N and Z Z Z := (Z k ) k∈Z = (X 2 k − m 2 ) k∈Z the corresponding H -valued ARMA(r, q) time series with time series of innovations ν ν ν := (ν k ) k∈Z = (X 2 k − σ 2 k ) k∈Z (see p. 5). Z Z Z satisfies the following.

Derivation of the estimators for the operators in the functional GARCH model
At first, since α i := 0 L H =: β j for i > p, j > q, the representations (3.2) and (3.35) imply a. s. for all k. Moreover, if Assumptions 3.1-3.2 hold there is no closed subspace V H with P(Z 0 ∈ V ) = 1 which, following from [20], Lemma 4. 48 and Remark 4. 49, leads to (3.36) Since α i = 0 L H for i > p, (3.36) implies with s = p + q : where the solution β [q] is unique iff the image of π T [p,q] ∈ S H,H q lies dense in H q . This is impossible since H H q , why we establish estimators for β 1 , ..., β q based on the equation The following example illustrates that the image of [s,q] ∈ S H q can lie dense.

Example 3.2.
Let α i := 0 L H =: β j for i = p, j = q and let α p = β q =: γ where γ ∈ S H is an operator with dense image satisying γ = 0 L H and ||γ|| S H < 1. Then, because (3.36) implies π i = γ k for all i = p + (k−1)q for some k ∈ N and π i = 0 L H otherwise, we obtain is an element of S H q . Further,π k where k ∈ N, with what we estimate π k , stands for the k-th component of [s,q] ∈ S H q with associated eigenvalue sequence (ĝ s,q;j ) j∈N which decreases monotonically w. l. o. g. Because of (3.36), it is hence plausible to usê as an estimator for α i with i = 1, ..., p whereβ j is the j-th component ofβ [q] and whereα 1 :=π 1 .

Upper bounds of the estimation errors for the operators π i
The following Theorem states asymptotic upper bounds of the operators for invertible, linear processes represented as inverted time series. It is crucial for our derivation of asymptotic upper bounds of the estimation errors for the GARCH operators.

Theorem 3.3. Let Assumptions 3.1-3.2 and 3.5 hold. Let
At last, for all L, let Assumption 3.3 hold for the eigenvalue sequence (c L,j ) j and let (π L , (Φ L,ij ) i,j ) satisfy Assumption 3.4 for β, see Theorem 3.2. Then, for all i ∈ N :

Upper bounds of the estimation errors for the GARCH operators
Here, we need as well as which is both true after (3.43). According to Corollary 3. , (Φ s,q;ij ) i,j ) satisfy Assumption 3.4 for β where Φ s,q;ij := g s,q;j ⊗ c j . Then,

49)
and for all j = 1, ..., q :  [6] where functional ARCH(1), GARCH(1,1) resp. GARCH processes for any order were established. This paper also displays asymptotic upper bounds of the estimation errors for operators of invertible, linear processes represented as inverted time series. In Section 2, we introduce ARCH(p) and GARCH(p, q) processes for any order p, q ∈ N with values in the function spaces L p [0, 1] with p ∈ [1, ∞), C[0, 1] and others. For these processes, we present sufficient conditions for the existence of strictly stationary solutions in Theorem 2.1, and for the existence of finite moments and weak dependence in Lemma 2.1. Theorem 2.1 generalizes [6], Theorem 1 under a milder condition, [11], Theorem 2.1 and 2.3, and [2], Theorem 2.1 and 2.2. To the best of our knowledge, for functional ARCH(p) resp. GARCH(p, q) processes with p > 1 resp. p ∨ q > 1, a moment condition as (2.10) in Theorem 2.1 and Lemma 2.1 is new. In Section 3, we derive explicit asymptotic upper bounds of the estimation errors for the shift term and the operators of L 2 [0, 1]-valued ARCH(p) and GARCH(p, q) processes for all p, q ∈ N where the operators are estimated by a Yule-Walker approach. For this purpose, we establish convergence results regarding asymptotic upper bounds of the estimation errors for certain means, covariance and lag-h-covariance operators (Lemma 3.3), and eigenfunctions and eigenvalues which are also useful beyond the context of ARCH and GARCH. Theorems 3.1-3.4 present the main results of the article. Theorem 3.1 states upper bounds of the estimation errors for the shift term in the ARCH and GARCH processes for any order. Theorem 3.2 provides upper bounds of the estimation errors for the ARCH(p) operators for any p, namely for the projections of the operators on a finite-dimensional subspace in part (a), and the complete operators in part (b). A similar result as (a) for p = 1 was stated in [11] by imposing an integral operator and estimating its kernel. However, as far as we know, (a) with p > 1 and (b) are new. Theorem 3.3 is a convergence result stating explicit asymptotic upper bounds of the estimation errors for the operators of invertible, linear processes represented as inverted time series. From this, one immediately obtains a result with the same upper bounds of the estimation errors for the operators in the associated linear process, see [20], Section 4.4.1.3. Both results are valid without the context of ARCH and GARCH and they extend some results in Aue & Klepsch (2017) [3] and Klepsch & Klüppelberg (2017) [17] where a different approach was made. At last, Theorem 3.4, which is based on Theorem 3.3, provides upper bounds of the estimation errors for the complete GARCH operators. Projections of these operators on finite-dimensional subspaces were estimated in [2] for p = 1 = q by least squares estimators and in [6] for any order by a quasi-likelihood approach. Latter applied their same approach to estimate the complete operators in the case p = 1 = q.
To the best of our knowledge, estimating the complete operators in Theorem 3.4 for p ∨ q > 1 is new and explicit asymptotic upper bounds of estimation errors for complete operators of ARCH, ARMA, GARCH, invertible and linear processes as in the Theorems 3.2-3.4 have not been derived before.
We leave the investigations concerning the probabilistic properties of ARCH and GARCH processes in general, separable Banach spaces behind for future research. Estimating the orders of functional ARCH and GARCH processes is also an open problem, see Kokoszka & Reimherr [18]. Concerning parameter estimation in functional ARCH and GARCH processes, open problems are the estimation in general, separable Banach spaces, see Ruiz-Medina M. D. &Álvarez-Liébana J. [24], the asymptotic distribution of the estimations errors when estimating the parameters without projecting them on a finite-dimensional subspace, see [2] and [6] for the parameters projected on a finite-dimensinal subspace, and the asymptotic lower bounds of the estimations errors.

Proof of Theorem 3.3.
The proof is based on the proof of Theorem 3.2 with p replaced by an appropriate sequence L = L N → ∞. From the ideas in the proof of Theorem 3.2 and (see [20], Lemma 4.45)