Geometric ergodicity of asymmetric volatility models with stochastic parameters

In this paper, we consider a general family of asymmetric volatility models with stationary and ergodic coefficients. This family can nest several non-linear asymmetric GARCH models with stochastic parameters into its ambit. It also generalizes Markov-switching GARCH and GJR models. The geometric ergodicity of the proposed process is established. Sufficient conditions for stationarity and existence of moments have also been investigated. Geometric ergodicity of various volatility models with stochastic parameters has been discussed as special cases.


Introduction
The failure of GARCH models to account for the asymmetry and overestimation of persistence of variance has led to plenty of research in conditional heteroscedastic models.The findings of Black [4] about the asymmetric behavior of the market with respect to news resulted in number of asymmetric volatility models, such as EGARCH, GJR, AGARCH, TGARCH etc., see Engle and Ng [11] for details.Lamoureux and Lastrapes [26] and recently, Hwang and Pereira [22] have claimed that the persistence in variance may be overestimated because of the existence and failure to account for structural shifts in the volatility processes.Mikosch and Starica [30] showed that structural breaks in the unconditional variance of the GARCH volatility process can cause spurious high persistence.Moreover, these break points may be responsible for the asymmetry and long range dependence in the data set.
The immense popularity of the problem of structural breaks has resulted in several approaches of modeling the volatility with stochastically varying parameters, especially in the form of Markov-switching ARCH and GARCH models, see for example [7], [18], [16], [24], [23] and [17].Ardia [1] modified Haas' [17] model to a Markov-switching asymmetric GJR model and discussed its Bayesian estimation procedures.Bauwens et al. [2] have established the stability of a regime switching GARCH model.Also see [12], [20] and [32] and references therein for various other approaches.In all the above papers, authors consider only GARCH models with parameter variation restricted to a finite state space.
Given the changing pace of the world economy at the time of various financial crises, several authors have considered the continuous variation in the parameters of GARCH models.Dahlhaus and Subba Rao [10], Čí žek and Spokoiny [9] and Rohan and Ramanathan [33] developed inference procedures for the time varying ARCH (∞) and GARCH models.Recently, Fryzlewicz and Subba Rao [14] proved the mixing properties of a time varying ARCH model.However, the variation considered in the parameters of time varying ARCH and GARCH models is deterministic.Asymmetric volatility models address the limitations of a simple GARCH model, such as persistence and volatility clustering, and if we allow the coefficients of such models to vary stochastically according to time, then they can be better candidates for modeling volatility.In fact, Granger [15] advocated the use of stochastically varying parameter models by commenting 'Most of the time varying parameters change stochastically rather than deterministically'.In this paper, we introduce a general asymmetric volatility model with stochastically varying parameters and study its probabilistic properties.Sufficient conditions for the geometric ergodicity and stationarity of the chain associated with the model are given.The results discussed here provide a basis for a similar investigation on various other asymmetric volatility models with stochastic parameters.An important implication of the geometric ergodicity is that the law of large numbers and central limit theorems can be applied and therefore it becomes possible to develop a rigorous asymptotic estimation theory.
Establishing the probabilistic properties of GARCH type models has always been a dynamic topic of research due to the their vast applicability to financial data.Carrasco and Chen [8] provided sufficient conditions for the β-mixing and existence of higher order moments for various GARCH models as an application of a generalized random coefficient autoregressive model.Later, Francq and Zako ïan [13] proved the geometric ergodicity of a GARCH (1,1) process with rather mild conditions.Sa ïdi and Zako ïan [34] introduced a class of non-linear ARCH models and studied its probabilistic aspects.Using the similar ideas as in Carrasco and Chen [8], Meitz and Saikkonen [27] established the mixing properties of a class of Markov models and applied their results to the GARCH (1,1) model.However, all these papers study the probabilistic properties of standard GARCH models with constant coefficients.This article has a different contribution as it introduces various GARCH models with stationary and ergodic coefficients and establishes their geometric ergodicity.
The paper is organized as follows.A family of asymmetric volatility models with stochastic parameters is introduced in Section 2. In Section 3, we discuss the geometric ergodicity of the process and provide sufficient conditions for the strict and covariance stationarity.Various stochastic parameter volatility models and their probabilistic properties are also considered in Section 3.

A family of asymmetric volatility models with stochastic parameters
To define a family of asymmetric volatility models, we start from a general family of GARCH type equation for the volatility process with coefficients modeled as a Markov chain.
for all t ∈ Z, the set of all integers.
We call the elements of D as regimes.The state space D is assumed to have a metric inherited from some norm on R 2 + × (0, 1) × R × [−1, 1] × R 2 + and the Borel σ-algebra B(D).This state space ensures the positivity of the variance process in every regime.The function ∆(θ, D), θ ∈ D, D ∈ B(D) denotes the one-step Markov transition function of the chain {θ t }.
The model defined in (2.1) may be considered as a generalization of the family of volatility models given by Hentschel [21], which can nest almost all the volatility models for different parameter choices.This definition of the process also generalizes other Markov-switching models in two ways.First, it allows infinitely many regimes as opposed to only finite number of regimes in the other Markov-switching GARCH models.Second, this general model also allows for a shift in the parameters of asymmetry of both types i.e., the asymmetry due to shift, denoted by b, and that due to rotation, denoted by c, in the news impact curve, see [21].It can handle the situations, where there is asymmetry in one regime and symmetry in the other or one type of asymmetry in one regime and another type in the other.Hence this model can capture different dynamics in low and high volatility periods efficiently as it can take different asymmetric volatility models into account.

Stationarity and geometric ergodicity
Due to the stochastic nature of the parameters, it would be interesting to establish the existence of a stationary solution of the process defined in (2.1).Towards this, first we make the following assumption.
2. If the random sequence {θ t } is i.i.d., then the assumption A 1 is also necessary for the existence of a unique stationary solution (see [5]).
3. Under the i.i.d.assumption of the parameter chain {θ t }, σ λt−1 t and (A t , B t ) are independent, where Next, we consider general order moments of σ λt t+1 .The following theorem puts an upper and lower bound on the moments of σ λt t+1 and finds conditions for their existence.Theorem 3.3.Suppose that the assumption A 1 is satisfied.Then, Also, the series on the right hand sides of (3.2) (for 0 < r ≤ 1) and where Remark 3.4.
1.If any of the expectations E(ω r t ) and E(η i,t ω t−i ) r in (3.2) does not exist, then E(σ rλt t+1 ) = +∞.2. If the chain {θ t } is i.i.d., then the sufficient condition for the existence of r th moment (3.4) reduces to E(A r t ) < 1 and E(ω r t ) < ∞, which is equivalent to the condition assumed by Nelson [31] for a constant parameter stationary GARCH models.
3. If the chain {θ t } is degenerate at some constant θ, then the condition (3.4) , which is a sufficient condition for the existence of r th moment of the Hentschel [21] family of models with constant parameters.This condition further reduces to E(2αv 2 t + β) r < 1 for a simple GARCH model, which is same as the Nelson's [31] condition for existence of moments.
To establish the geometric ergodicity of the variance process defined in (2.1), we prove the same for z t = (h t , θ t ) , where h t = σ λt t+1 .Notice that v t is independent of z t−1 .Suppose that the state space of the process {z t } is given by X ⊂ R + × D, where this subset X is such that all the sets in it with h t < inf This restriction is due to the lower limit on h t established in Theorem 3.1.Since {θ t } is a vector Markov chain, there exists a measurable function F θ : D × R 7 → D such that θ t = F θ (θ t−1 , u t ), where the error term u t ∈ R 7 is i.i.d. and independent of θ t−1 and v t (according to the definition of θ t ).Let ζ t = (u t , v t ).Then, Here, independence of v t with z t−1 ensures that {z t } is a Markov chain.Hence, although the variance process alone in model (2.1) is non-Markovian, the process z t = (h t , θ t ) is Markovian, which simplifies our derivation.Now consider the following assumptions: A 2 : The random variable v t has a density g(•) with respect to Lebesgue measure on real line, which is positive and continuous everywhere.
The assumption A 2 is required for the irreducibility of z t .A 3 is used to prove its geometric ergodicity.
Suppose B(X ) denotes the Borel sigma field of X and λ * is the measure associated with the chain, such that λ * (A) is well defined for every A ∈ B(X ).Here λ * = ν L ⊗ φ is the product measure on (X , B(X )), where ν L denotes the Lebesgue measure and φ is an irreducibility measure on B(D).Denote p m (z 0 , B) = P (z t ∈ B|z t−m = z 0 ), B ∈ B(X ), z 0 ∈ X , the probability that starting with z 0 , the process moves to B in m steps.
The first major step towards establishing the geometric ergodicity of the process {z t } is to prove the irreducibility.We deal with this in the following theorem.
Theorem 3.5.Let the chain {θ t } be φ−irreducible and aperiodic.Suppose that the assumption A 2 regarding {v t } holds.Then the chain {z t } is λ * -irreducible and aperiodic.Further, there exists a probability measure Ψ on B(X ), called the maximal irreducibility measure, such that {z t } is Ψ-irreducible.
The following corollary establishes that if transitions of the chain {θ t } are continuous, then the chain {z t } is weak Feller.We also prove that the process {z t } is a T-chain.
The definitions of small sets, petite sets, Feller chain and T-chain, which are used in proving the geometric ergodicity, can be found in [29].
Corollary 3.6.(i) Suppose there exists a measurable function F θ such that θ t = F θ (θ t−1 , u t ), where {u t } is an i.i.d.sequence independent of θ t−1 , assuming values in the measurable space (D, B(D)) and F θ (•, u) is continuous for any fixed u.Then, the process {z t } is weak Feller.(ii) Further, if the support of φ has a non-empty interior, that is, the measurable space (D, B(D)) has at least one set with positive φ−measure, then the process {z t } is a Tchain.Moreover, X can be written as the union of open small sets and every compact set in X is petite.
Remark 3.7.The proof of the weak Feller property of {z t } is similar to the proof of the Proposition 6.1.2 of Meyn and Tweedie [29], in which the weak Feller property of nonlinear state space (NSS) model is proved.Compared to NSS model, which requires the assumption of smoothness of F , our assumption is weaker because we do not impose any differentiability restriction on F .In fact, F is not differentiable in our case.
Towards proving the geometric ergodicity of {z t }, we state the following lemma first.The proof of the lemma is based on the idea that under the assumption A 1 , we can find a drift function (see [29]) to show that the process (θ t , v t ) is geometrically ergodic.Lemma 3.8.Let the assumptions of Theorem 3.5 and Corollary 3.6 hold.Then, the process {(θ t , v t )} is geometrically ergodic if the assumptions A 1 to A 3 are satisfied.Now we prove the geometric ergodicity of the process z t with the help of an extra assumption ensuring the existence of a small order moment of the process (2.1).Notice that condition (3.5) in Theorem 3.9 below is sufficient for the existence of moment of order ϑ, E(h t ) ϑ < ∞ for some small ϑ > 0. Theorem 3.9.Suppose that the assumptions A 1 to A 3 along with those of Theorem 3.5 and Corollary 3.6 are satisfied.Then, the process {z t } is geometrically ergodic if the following condition holds.
where A t is as defined in Theorem 3.3.
Corollary 3.10.Let all the assumptions of Theorem 3.9 be satisfied.Then the process {z t } is strictly stationary and strong or α−mixing if its initial distribution is same as the stationary distribution.
As mentioned in Section 1, the model (2.1) nests several GARCH models with stochastic parameters (SP hereafter) and hence provides the basis for investigation of their probabilistic properties.In Table 1, we provide the constraints on the parameters of model (2.1) which lead to a specific volatility model with stochastic parameters.In this table, by equating some of the components of θ t to a constant, we mean that, they are degenerate or take a constant value with probability one.
The results of Theorems 1-4 and their corollaries are also applicable to the various GARCH models with stochastic parameters mentioned in Table 1.That is, each of the models is stationary and geometrically ergodic under the corresponding specific assumptions subject to the constraints of Table 1.The stationary solution and conditions for existence of moments for all the models can also be directly obtained using (3.1) and (3.4) under the constraints mentioned in Table 1.
Remark 3.11.SPGARCH and SPGJR models (Table 1) can be considered as the generalization of Markov-switching GARCH and GJR models with only finitely many regimes of Haas et al. [17] and Ardia [1] respectively.Remark 3.12.Meitz and Saikkkonen [28] studied the stability of a simple AR-GARCH model.However, they require stringent smoothness assumptions and hence their results are not applicable to the threshold type non-linear volatility models.Our approach differs from theirs in two ways.First, it considers several GARCH models with stochastic parameters as discussed above.Second, due to moderate assumptions, it is possible to take into account the threshold type non-linear models such as GJR and TGARCH with stochastic parameters.

Appendix: Proofs
Proof of Theorem 3.1.The variance process (2.1) can be written as, where Y t+1 = σ λt t+1 and A t , B t are as defined in Remark 3.2.Here (θ t , v t ) and hence (A t , B t ) is a stationary ergodic sequence.Therefore, the result immediately follows from Brandt ([6], Theorem 1).Now from (3.1), we can write Hence, the lower limit for σ λt t+1 follows.
Proof of Theorem 3.5.To prove the λ * −irreducibility of the process, we need to show that p k (z 0 , B) > 0 for some k ≥ 1 and ∀ z 0 ∈ X , where B is any measurable set such that B ∈ B(X ) with λ * (B) > 0. This is done by showing that starting with an arbitrary (h 0 , θ 0 ) ∈ X , any point (h, θ) ∈ B can be reached in finite number of steps with a positive probability.Denote the component sets of B as B h and B θ which are Borel sets such that B ≡ B h × B θ .The φ−irreducibility of the process {θ t } ensures that if φ(B θ ) > 0, then there exists an m ≥ 0 (depending on θ 0 and B θ ) such that ∆ m+1 (θ 0 , B θ ) > 0, where ∆ m+1 (θ 0 , B θ ) represents the (m + 1)-step transition probability of {θ t } to the set B θ , starting with θ 0 .Thus there exist intermediate Borel sets , through which the chain {θ t } transits to B starting with θ 0 .Corresponding to such sets, identify the sets B h i such that starting with z 0 , the chain {z t } can reach to z through the Borel sets B i ≡ B h i × B θ i in m + 1 steps with a positive probability.The sets B h i are defined as Therefore, the chain {z t } can transit through the following path Here each (h i , θ i ) ∈ B i , i = 1, 2, . . ., m.By the transition (h i−1 , θ i−1 ) → (h i , θ i ), we mean that the transition is from (h i−1 , θ i−1 ) ∈ B i−1 to B i .The probability of reaching z = (h, θ) ∈ B from z 0 = (h 0 , θ 0 ) in the m intermediate steps through sets B i , i = 1, 2, . . ., m is, Here, the first probability is strictly greater than zero under the assumption A 2 .In this way, each z ∈ B with λ * (B) > 0 can be reached in finite number of steps with a positive probability, i.e., p m+1 (z 0 , B) > 0. Hence the process z t is λ * −irreducible.
For B ∈ B(X ), define the probability measure Ψ as, Hence the Proposition 4.2.2 of Meyn and Tweedie ( [29], page 90) implies that {z t } is Ψ-irreducible where Ψ is a maximal irreducibility measure.Aperiodicity of the chain is obvious as {θ t } is aperiodic and corresponding to every B θ i there exist sets and B h i as shown above.
Proof of Corollary 3.6.(i) This proof is similar to the proof of the Proposition 6.1.2 of Meyn and Tweedie [29].Denote the class of all bounded continuous functions from X to R by C(X ).Using the similar notations as in Meyn and Tweedie ([29], Section 6.1), the transition probability kernel P acts on the bounded functions through the following mapping P h(z) = P (z, dy)h(y).
where h ∈ C(X ) is a bounded continuous function.Then by the definition of the mapping F (z, ζ) and the continuity of F θ , it is possible to claim that h(F (z, ζ)) is bounded and continuous.Thus it follows that (Meyn and Tweedie [29], Section 3.5.5) is a continuous function of z ∈ X .Therefore, by the definition, {z t } is a weak Feller chain.
(ii) Let z ∈ X be such that it can be reached from any z 0 ∈ X in a finite number of steps.First, we show that z is reachable, that is, every neighborhood of z is reachable with a positive probability (see [29], Section 6.1.2).Let B ∈ B(X ) be any neighborhood of z.It is shown in Theorem 3.5 that the chain {z t } is λ * −irreducible.Also, the λ * -measure of the boundary points is zero and support of φ has a non-empty interior (which implies that {θ t } is a T-chain).Hence, there exists an open ball B ⊂ B with λ * (B ) > 0, such that all z ∈ B can be reached in a finite number of steps, starting with any z 0 .This implies that for any open set B containing z, we have λ * (B) ≥ λ * (B ) > 0 and hence Ψ(B) ≥ Ψ(B ) > 0. Therefore, the point z is reachable and hence support of Ψ has a non empty interior (Meyn and Tweedie [29], Lemma 6.1.4).We have already shown that z t is a Ψ−irreducible Feller chain.Thus, {z t } is a T-chain (Meyn and Tweedie [29], Theorem 6.2.9) and hence the state space can be written as a union of open small sets (Meyn and Tweedie [29], Theorem 6.0.1).

Definition 2.1. Let
θ t = (ω t , α t , β t , b t , c t , λ t , ν t ) be a stationary and ergodic Markov chain with state space D, a measurable subset of R 2 Let { t } be a process such that E( t |F t−1 ) = 0 and E( 2 t |F t−1 ) = σ 2 t , where F t−1 = σ{ t−1 , t−2 , . ..}. Suppose {v t } is a sequence of real valued independent and identically distributed (i.i.d.) random variables independent of {θ t }, having mean 0 and

Table 1 .
Constraints on model (2.1) leading to specific stochastic parameter (SP)