A LARCH(∞) Vector Valued Process

Q: What is the inverse of the GARCH() model?

Note that [GKL00] and [GS02] prove that the stationary ARCH(∞) model (Example 4), is not long range dependent in the previous sense; more precisely the partial sums process, normalized with √ n, converges to theBrownian Motion.

Q: What is the case for the bivariate model?

An (empirically) interesting case for the bivariate model (Xt, Yt) is obtained with the assumption that the (ζ1,t, ζ2,t) are cross-correlated.

Q: What are the conditions that imply the weak dependence of a stationary solution to (1)?

These weak dependence conditions imply various limit theorems both for partial sums processes and for the empirical process (see [DL99], [DD03] and [DL02]).

Question

1. What have the authors contributed in "A larch(∞) vector valued process" ?

2. What is the lp existence condition for the bilinear case?

3. What is the generalization of the class of multivariate ARCH() processes?

4. What is the GARCH(p, q) model in example 3?

Accepted Answer

The purpose of this chapter is to propose a unified framework for the study of ARCH ( ∞ ) processes that are commonly used in the financial econometrics literature.. The authors extend the study, based on Volterra expansions, of univariate ARCH ( ∞ ) processes by Giraitis et al. [ GKL00 ] and Giraitis and Surgailis [ GS02 ] to the multi-dimensional case.

Accepted Answer

2. Lp existence conditions are obtained in [GS02] for the bilinear case if p = 4; the method is based on the diagram formula and does not extend simply to this vector valued case.

Accepted Answer

This generalizes the class of multivariate ARCH(∞) processes, defined in the p-dimensional case as:Rt = Σ 1 2 t εt ,where Rt is a p–dimensional vector, Σt is a p× p positive definite matrix, and εt is a p–dimensional vector.

Accepted Answer

[GS02] prove that the corresponding partial sums process converges to the fractional Brownian Motion with normalization ≫ √n. • The GARCH(p, q) models in example 3, are always weakly dependent, in the sense of [DL99].

Accepted Answer

2. • If Eξ0 3= 0 (as for the case of the bilinear model in [GS02]), the authors may write Xt = ∆Mt + Eξ0 ( a+ ∑∞ j=1 ajXt−j ) where∆Mt = (ξt − Eξt)a+∞ ∑j=1ajXt−jis a martingale increment.

Accepted Answer

A final comment concerns the analogue for powers of Xt which, if suitably normalized, are proved to converge to some higher order Rosenblatt process in [GS02] for the bilinear case.

Accepted Answer

The bilinear model of Example 2 is shown in [GS02] to display the double long memory property when the series {αj} and {βj} are not summable but satisfy the condition∞ ∑j=1(α2jEζ 2 0 + β 2 j)< 1 .As a particular case, the squares of the LARCH(∞) process in Example 1 display long–range dependence.

Accepted Answer

Note that [GKL00] and [GS02] prove that the stationary ARCH(∞) model (Example 4), is not long range dependent in the previous sense; more precisely the partial sums process, normalized with √ n, converges to theBrownian Motion.

Accepted Answer

An (empirically) interesting case for the bivariate model (Xt, Yt) is obtained with the assumption that the (ζ1,t, ζ2,t) are cross-correlated.

Accepted Answer

These weak dependence conditions imply various limit theorems both for partial sums processes and for the empirical process (see [DL99], [DD03] and [DL02]).

A LARCH(∞) Vector Valued Process

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Most frequently asked questions

1. What have the authors contributed in "A larch(∞) vector valued process" ?

2. What is the lp existence condition for the bilinear case?

3. What is the generalization of the class of multivariate ARCH() processes?

4. What is the GARCH(p, q) model in example 3?

5. What is the simplest way to prove the GS02 theorem?

6. What is the convergent function of the bilinear model?

7. What is the bilinear model of Example 2?

8. What is the inverse of the GARCH() model?

9. What is the case for the bivariate model?

10. What are the conditions that imply the weak dependence of a stationary solution to (1)?

Citations

Weakly dependent chains with infinite memory

Asymptotic normality of the Quasi Maximum Likelihood Estimator for multidimensional causal processes

An invariance principle for weakly dependent stationary general models

ARCH(∞) Models and Long Memory Properties

The notion of $\psi$-weak dependence and its applications to bootstrapping time series

References

Multivariate Simultaneous Generalized ARCH

The variation of certain speculative prices

The Variation of Certain Speculative Prices

Martingale Limit Theory and Its Application

Stationarity of Garch processes and of some nonnegative time series

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