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On stationarity and β -mixing of periodic bilinear processes
Statistics and Probability Letters 79 (2009) 79–87
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On stationarity and β -mixing of periodic bilinear processes Abdelouahab Bibi ∗ , Radia Lessak Département de Mathématiques, Université Mentouri Constantine, Algeria
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Article history: Received 5 February 2008 Received in revised form 22 June 2008 Accepted 17 July 2008 Available online 30 July 2008
MSC: primary 62M10 secondary 62M05
a b s t r a c t This paper studies some probabilistic properties of periodic bilinear processes. In these nonlinear models, the parameters are allowed to switch between different regimes. Stationarity and geometric ergodicity conditions (in periodic sense) are given under general and tractable assumptions. We use these results to give necessary and sufficient conditions for stationarity of specific periodic GARCH processes which can be written as a periodic bilinear models. Moreover, it is shown that local stationarity i.e., stationarity within each regime, is not necessary to obtain global stationarity. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Many time series data in fields such as macroeconomics, finance, biological sciences and engineering exhibit non-linearity (like occasional sharp spikes) and non-stationarity (such as the dependency of moments on the time), which cannot be explained by the standard autoregressive (AR), moving-average (MA) or by mixed autoregressive moving-average (ARMA) models. In this context, the class of bilinear (BL) models (see for a review Subba Rao and Terdik (2003) and the references therein) has attracted considerable attention in the econometric and statistical literature. Bilinear models have been shown to fit and forecast non Gaussian processes ‘‘better’’ than many linear ARMA alternatives. Most of the applications of BL models assume constant coefficients, but this assumption may not be appropriate, as an unforeseen intervention may happen (as, for example, for stock market indexes). So, time dependent BL models seem to provide a more realistic alternative to classical ARMA models, in order to focus on non-linear and non-stationary time series. These time-dependent models keep the BL form, but they allow the coefficients to vary with respect to time (see Bibi (2003) for further discussion). Our attention here is focused on periodic BL models (PBL). To our best knowledge, in spite of the well-known periodic structure of many time series (see, for a review Franses (1996)), no work has investigated on PBL models yet. The main purpose of this paper is therefore to generalize some theoretical probabilistic properties of some PBL models, given recently by Bibi and Gautier (2006) and Bibi and Moon-Ho (2006), and of course, our model includes the classical ARMA, PARMA models and a large class of periodic GARCH (PGARCH) models (see Section 4) which play an important role in econometrics and in financial mathematics. An overview of the paper is as follows. Having defined periodic bilinear processes in Section 2, we introduce the concept of periodic generalized autoregressive representation, and we investigate the fundamental properties related with this representation such that the sufficient conditions ensuring the existence and the uniqueness of strict and second order stationarity solutions satisfying this representation. Section 3, conditions for geometric ergodicity and for β -mixing are given. In Section 4, applying the conditions for periodic bilinear processes on the class of periodic GARCH models leads to conditions for stationarity, which in many cases are weaker than the ones already established by Bibi and Aknouche (2007).
∗
Corresponding author. E-mail addresses: [email protected] (A. Bibi), [email protected] (R. Lessak).
0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.07.024
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A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
Some notations are used throughout the paper: I(n) is the n × n identity matrix and I∆ (.) denotes the indicator function of the set ∆. O(k,l) denotes the matrix of order k × l, whose elements are zeros; for simplicity we set O(k) := O(k,k) and O(k) := O(k,1) . Vec(M ) is the vector obtained from a matrix M by setting down the columns of M underneath each other.
γ mij , and γ ∈]0, 1], define a matrix operation |M |γ as |M |γ := mij . It is obvious that γ P P γ γ γ MX ≤ |M |γ X for all X ∈ Rm and Mi ≤ i i |Mi | where M ≤ N means mij ≤ nij for all i and j and N = nij . ⊗ is the usual Kronecker product of matrices and M ⊗r = M ⊗ M ⊗ · · · ⊗ M r-time. If M is squared matrix then ρ (M ) represents For any n × m matrix M =
the maximum modulus of the eigenvalues of the matrix M. 2. Periodic bilinear models Suppose that (Xn )n∈Z , Z = {0, ±1, ±2, . . .} is a process defined on a probability space (Ω , =, P ) with finite second-order moment and generated by a general bilinear model with time-varying coefficients i.e., Xn = a0,n +
p X
ai,n Xn−i +
i=1
q X
bj,n en−j +
Q P X X
j =0
cij,n Xn−i en−j , n ∈ Z
(2.1)
i=1 j=1
where (en )n∈Z is an independent and identically distributed (i.i.d.) process defined on the same probability space (Ω , =, P ), such that E log+ (e2n ) < +∞ where log+ x = max {log x, 0}, x > 0 and el is independent of Xk , k < l. The coefficients ai,n 0≤i≤p , bj,n 0≤j≤q and cij,n 1≤i≤P ,1≤j≤Q switch between s-regimes, i.e., ai,n := v=1 ai (v)I∆(v) (n), bj,n := Ps Ps v=1 bj (v)I∆(v) (n) and cij,n := v=1 cij (v)I∆(v) (n) with ∆(v) := {sk + v, k ∈ Z} so that by setting Xt (v) = Xst +v and et (v) = est +v when n = st + v
Xt (v) = a0 (v) +
p X
Ps
ai (v)Xt (v − i) +
i=1
q X
bj (v)et (v − j) +
j=0
Q P X X
cij (v)Xt (v − i)et (v − j)
(2.2)
i=1 j=1
which we will make heavy use of (2.2). In (2.2) the notation Xt (v) refers to Xt during the v th ‘‘season’’ or regime v ∈ {1, . . . , s} of cycle t. For convenience, Xt (i) = Xt −1 (s + i), et (i) = et −1 (s + i) if i ≤ 0. It is worth noting that when s > 1, the process is globally nonstationary, but is stationary within each period. In order to facilitate the analysis, we shall consider through this section the Representation (2.2) and we shall assume, without loss of generality, that P = p, since otherwise zeros of ai (v) and cij (v) can be filled in. To discuss the probabilistic structure of the Representation (2.2), we require a state-space version. Indeed, let us define the matrices a1 (v) 1
a2 (v) 0
A(v) =
..
. ···0
0 0
··· ··· .. .
ap (v) 0
c1j (v) 0
,
.. .
1
0
··· ···
Cj (v) =
.. .
.. .
··· ···
0
p×p
cpj (v) 0 0
p×p
and the vectors X t (v) = (Xt (v), . . . , Xt (v − p + 1)) , H = (1, 0, . . . , 0) , et (v) = a0 (v) + notation, we can write the Model (2.2) in the form 0
0
Pq
bj (v)et (v − j) H. With this
j =0
X t (v) = At (v)X t (v − 1) + et (v)
(2.3)
where At (v) := A(v) + j=1 Cj (v)et (v − j) and Xt (v) = H X t (v). In the Representation (2.3) the matrices A(v), Cj (v) are interpreted periodically in v with period s. So, the relation in (2.3) is the same as the defining equation for multivariate generalized periodic autoregressive process. Now, it is well known that in time series models with periodic coefficients, it is possible to embed seasons into a multivariate process (see Tiao and Grupe (1980)). More precisely X t t ∈Z where X t = (X 0t (1), X 0t (2), . . . , X 0t (s))0 is a generalized RCA process, i.e.,
PQ
X t = At X t −1 + η
where At , η t as
0
(2.4)
t
being a strictly stationary and ergodic pairs of random matrix and vector defined respectively by blocks t ∈Z
O(p) O(p)
. At := ..
O(p)
... ... .. . ...
O(p) O(p)
.. .
O(p)
At (1) At (2)At (1)
.. .
s−1 Y
v=0
At (s − v)
et (1) At (2)et (1) + et (2)
, sp×sp
where as usual, empty products are set equal to I(p) .
ηt := ( X k−1 s s−Y k=1
v=0
.. .
)
At (s − v) et (k)
.
sp×1
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
81
The main aim of this section is the stability of the process (Xt )t ∈Z . We give conditions which ensure the existence of the so-called strictly (respectively second-order) periodically stationary solutions to (2.2) or equivalently to (2.3) which are =(t e) -measurable (or causal) where =(t e) is the σ -field generated by et −j , j ≥ 0 . Recall here that the process (Xt )t ∈Z is said
to be strictly periodically stationary (SPS) or periodically correlated (PC) if and only if the process X t t ∈Z defined by (2.4) is strictly or second-order stationary. Also, when the process X t t ∈Z has an ergodic solution, then the corresponding SPS solution is said to be periodically ergodic (see Boyles and Gardner (1983)). 2.1. Strict periodic stationarity When we consider a periodic bilinear model as a data generating process, it is important to give conditions ensuring stationarity and ergodicity (in periodic sense) for further statistical analysis. However, there exist various results about the conditions ensuring existence of a strictly stationary and ergodic solution to (2.4). These conditions involve the notion of γ (An) for the sequence
o
o of random matrices A := (At )t ∈Z defined by γ (A) := n the Lyapunov exponent infn≥1 E
1 n
Qn−1
log
i=0
a.s
1 n
At −i = limn→∞
Qn−1
log
i=0
At −i , where k.k denotes any operator norm on the set of sp × sp
o
n
and sp × 1 matrices. Indeed, it is clear that both E log+ kA0 k and E log+ η are finite. Therefore from Bougerol and 0
Picard (1992), the unique, strictly stationary, ergodic and causal solution of (2.4) is given by Xt =
( ∞ k−1 X Y k=1
) A t −i
η t −k + η t
(2.5)
i =0
whenever γ (A) is strictly negative. Now, a simple computation shows that
O
(p)
t Y j=0
At −j
O(p) . = At .. O(p)
... ... .. .
O(p) O(p)
...
O(p)
.. .
O(p) O(p)
.. .
. ) ( s−1 tY −1 Y At −i (s − v) v=0
i=1
Therefore, because the Lyapunov exponent is independent of the a multiplicative norm it is
Q by nchoosing n norm, o o Qs−1
t
1 (s) straightforward to obtain that γ (A) ≤ γ (A) := inft >0 E t log i=1 v=0 At −i (s − v) . It is clear that
γ (s) (A) inherits the properties of the standard Lyapunov exponent, in particular the following inequality γ (s) (A) ≤ P s (s) v=1 E {log kA0 (v)k} hold. We can think of γ (A) as the Lyaponov exponent associated with the periodic sequence of random matrices A = (At )t ∈Z . We have thus shown our first result which gives a sufficient condition for strict stationary of the process X t t ∈Z .
Theorem 2.1. Suppose that γ (s) (A) < 0. Then for all t ∈ Z the Series (2.5) converges a.s and constitute the unique strictly stationary, causal and ergodic solution of (2.4). Corollary 2.1. If γ (s) (A) < 0, then for any v ∈ {1, . . . , s} and for any fixed t ∈ Z, Eq. (2.3) has a unique SPS solution given by X t (v) =
( ∞ k−1 X Y k=1
) At (v − i) et (v − k) + et (v)
(2.6)
i =0
with the above series converges absolutely a.s and the process (Xt (v))t ∈Z – defined as the first component of X t (v) t ∈Z – is the unique SPS periodically ergodic, causal solution of (2.2).
Proof. From (2.3) we have by recursion X t (v) =
Pn−1 nQk−1 k=0
Q
n −1
i=0
o
At (v − i) et (v − k) +
nQ
n−1 i=0
o
At (v − i) X t (v − n). Hence,
Q
n −1
(s) (A) < 0 and thus i=0 At (v − i) ≤ γ
1
n
→ λ < (s) 1 λ := exp γ (A) < 1 a.s as n → ∞. On the other hand, the strong law of large numbers shows that k et (v − k) = n o
Pk Pk−1 1 k−1 1
≤ k for k sufficiently j=1 et (v − j) − k j=1 et (v − j) converges a.s to 0 as k → ∞. Hence a.s, et (v − k) k k−1
nQ
o
k−1
0 k large. Thus . By the Cauchy root test, the Series (2.6) converges absolutely a.s. The i=0 At (v − i) et (v − k) ≤ k λ it is not difficult to show that a.s limn→∞
1 n
log
i=0 At (v − i)
0
remainder of statements are immediate.
Example 2.1. For PBL(1, 0, 1, 1), we obtain a1 (v) + c11 (v)et (v − 1)), hence, by the law v=0 At (s − v) = v=1 (P s of large numbers, a sufficient condition ensuring γ (s) (A) < 0 is that v=1 E {log |a1 (v) + c11 (v)e0 (v − 1)|} < 0
Qs−1
Qs
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which reduces to the classical condition when s = 1. It is worth noting that the existence of explosive regimes (i.e., E {log |a1 (v) + c11 (v)e0 (v − 1)|} > 0) does not preclude strict periodic stationarity. The following proposition give a simple sufficient condition which ensure γ (s) (A) < 0. Proposition 2.1. Let Q = 1, δ ∈]0, 1] and Γ := E
nQ
s −1 v=0
o |At (s − v)|δ . Then ρ (Γ ) < 1 implies that γ (s) (A) < 0 and thus
we have the results of Corollary 2.1. n 1/n Proof. If ρ (Γ ) < 1, there exists λ ∈]0, 1[ such that lim supn ≤ λ < 1. By independence of the sequence →∞ kΓ k
Q
Q nQ nQ o o δ
n
s−1 s−1 δ | A ( s − v)| ≥ E (et )t ∈Z , we get kΓ n k = ni=1 E
i=1 v=0 t −i v=0 At −i (s − v) . By Jensen’s inequality we have
( ( ( ) δ ) ) n s−1 n s −1 Y
Y
Y Y 1
0 > log λ ≥ lim sup log E At −i (s − v) At −i (s − v) ≥ δ inf E log i=1 v=0
i=1 v=0
n n→∞ n 1
= δγ (s) (A). The Lyapunov exponent γ (s) (A) criterion seems difficult to obtain explicitly when p > 1, however a potential method to verify whether or not γ (s) (A) < 0 is via a Monte-Carlo simulation using Eq. (2.4). This fact heavily limits the interests of the criterion in statistical applications. Indeed, the solution need to have some moments to make an estimation theory possible and Lyapunov criterion does not guarantee the existence of such moments. Therefore, in the next subsection, we give conditions ensuring the existence of moments for the strict stationary solution. 2.2. Second-order periodic stationarity (e)
In the previous subsection, conditions ensuring existence of SPS and =t -measurable solutions that are not necessarily square integrable have been established. In this subsection we are interested in second order PC solutions to (2.2) that are (e) In the special case when Q = 1 and q = 0, Bibi and Moon-Ho (2006) showed that if E {et } = 0 and also 2= t -measurable. E et = σ 2 , then PBL(p, 0, p, 1) has a PC solution whenever
λ(1) := ρ
s Y
⊗2
A
(v) + σ C1 2
⊗2
! (v) < 1.
(2.7)
v=1
In what follows, we show how to derive a sufficient condition analogous to (2.7) for the existence of PC stationary solution of general Model (2.2) or equivalently (2.3). We give the results in explicit form only in the case Q = 2, but the same arguments apply equally well for arbitrary Q as indicated below. 2.2.1. The periodic bilinear model with Q = 2 The following theorem examines conditions ensuring the existence of second-order stationary solutions for Eq. (2.4) and its properties when with Q = 2.
Theorem 2.2. Consider the Eq. (2.3) with Q = 2 and assume that E {et } = E e3t v ∈ {1, . . . , s}, let Γ (v) be the matrices A⊗2 (v) + σ 2 C1⊗2 (v) Γ (v) := σ (A(v) ⊗ C1 (v) + C1 (v) ⊗ A (v))
2
σ 2 A⊗2 (v) + κ4 C1⊗2 (v)
= 0 and E e4t = κ4 < +∞. For any
A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v) σ (C1 (v) ⊗ C2 (v) + C2 (v) ⊗ C1 (v)) 2
σ 2 (A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v))
C2⊗2 (v) O(p2 ) .
σ 2 C2⊗2 (v)
Then a sufficient condition for the existence of second-order stationary solution to (2.4) is that
λ(2) := ρ
s Y
! Γ (v)
< 1.
(2.8)
v=1
Moreover, the solution process is unique, strictly stationary, ergodic, causal and given by (2.5). Corollary 2.2. For the model PBL(1, 0, 1, 1) Xt (v) = a1 (v)Xt (v − 1) + c11 (v)Xt (v − 1)et (v − 1) + et (v). 2 2 The Condition (2.8) reduce to v=1 (a21 (v) + σ 2 c11 (v)) < 1. Notice that the existence of explosive regimes (i.e., a21 (v) + σ 2 c11 (v) > 1) does not preclude second-order stationarity.
Qs
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
83
Proof. In this case, A(v) = a1 (v), C1 (v) = c11 (v), C2 (v) = 0, so 2 a21 (v) + σ 2 c11 (v) 2 Γ (v) := 2σ c11 (v)a1 (v) 2 σ 2 a21 (v) + κ4 c11 (v)
0 0 0
0 0 0
a simple calculation shows that the non-zero eigenvalue of
Qs
v=1
Γ (v) is
Qs
v=1
2 a21 (v) + σ 2 c11 (v) . Hence the result follows.
Corollary 2.3. When Q = 1, the Condition (2.8) reduces to (2.7). Proof. In this case the matrices Γ (v) take the form A⊗2 (v) + σ 2 C1⊗2 (v) Γ (v) = σ (A(v) ⊗ C1 (v) + C1 (v) ⊗ A (v)) σ 2 A⊗2 (v) + κ4 C1⊗2 (v)
2
O(p2 ) O(p2 ) O(p2 )
O(p2 ) O(p2 ) O(p2 )
Qs Qs ⊗2 2 ⊗2 v=1 Γ (v) are the same as those of v=1 A (v) + σ C1 (v) . Hence Qs ⊗2 2 ⊗2 v=1 A (v) + σ C1 (v) . v=1 Γ (v) = ρ
and consequently the non-zero eigenvalues of
ρ
Qs
Corollary 2.4 (The PARMA models). In the linear case when the coefficients cij (v) in (2.2) are all zeros for any v ∈ {1, . . . , s}. Qs The Condition (2.8) reduce to ρ v=1 A(v) < 1. Proof. The matrices Γ (v) take the form A⊗2 (v) Γ (v) = O(p2 ) σ 2 A⊗2 (v)
O(p2 ) O(p2 ) O(p2 )
O(p2 ) O(p2 ) O(p2 )
Qs Qs Qs Qs ⊗2 ⊗2 v=1 Γ (v) are the same of v=1 A (v). Hence ρ v=1 Γ (v) = ρ v=1 A (v) = ⊗2 Qs =ρ v=1 A(v) v=1 A(v) .
so the non-zero eigenvalues of
ρ
Q s
To prove Theorem 2.2, we use the same approach as in Liu and Brockwell (1988). We first define for each v ∈ {1, . . . , s} the following Rp -valued processes
O(p) S n,t (v) = et (v) A (v)S t n−1,t (v − 1) + et (v)
if n < 0 if n = 0 if n > 0
and ∆n,t (v) = S n,t (v) − S n−1,t (v) which satisfies the equation ∆n,t (v) = At (v)∆n−1,t (v − 1) for any n ≥ 3. Let S n,t and ∆n,t := S n,t − S n−1,t be the multivariate version processes associated with S n,t (v) and ∆n,t (v) with v ∈ {1, . . . , s}. (e)
Clearly S n,t and ∆n,t are measurable with respect to =t . By the Lp -theory (p > 1) the problem of the existence of a second-
order stationary solution of (2.4) now reduces to the convergence of S n,t n≥0 to X t in L2 . The key quantity of interest in determining of L2 -convergence is E ∆n,t ∆0n,t . We shall need to evaluate the vectors of moments
2 e V n,t (v) = E ∆⊗ n,t (v) ,
2 e Dn,t (v) = E ∆⊗ n,t (v) et (v − 1) ,
2 2 e F n,t (v) = E ∆⊗ n,t (v) et (v − 1) .
Then we have
e V n,t (v) = A⊗2 (v) + σ 2 C1⊗2 (v) e V n−1,t (v − 1) + (A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v)) e Dn−1,t (v − 1) ⊗2 e + C2 (v)F n−1,t (v − 1) e Dn,t (v) = σ 2 (A(v) ⊗ C1 (v) + C1 (v) ⊗ A(v)) e V n−1,t (v − 1) + σ 2 (C1 (v) ⊗ C2 (v) + C2 (v) ⊗ C1 (v)) e Dn−1,t (v − 1) ⊗2 2 ⊗2 2 e e e F n,t (v) = σ A (v) + κ4 C1 (v) V n−1,t (v − 1) + σ (A(v) ⊗ C2 (v) + C2 (v) ⊗ A(v)) Dn−1,t (v − 1) + σ 2 C2⊗2 (v)e F n−1,t (v − 1). 0 0 0 0 e By defining V n,t (v) = V n,t (v), e Dn,t (v), e F n,t (v) we obtain V n,t (v) = Γ (v)V n−1,t (v − 1). Define V n,t := (V 0n,t (1), . . . , V 0n,t (s))0 , then we obtain the following homogeneous equation V n,t = Γ V n−s,t −1
(2.9)
where Γ is a matrix defined by blocks as (Γ )i,i := v=0 Γ (i − v) v=0 Γ (s − v) and O(3p2 ) otherwise. The necessary and sufficient condition for the homogeneous equation in (2.9) to have a stable solution, which is finite and Qs difference independent of t is that λ(2) = ρ (Γ ) = ρ v=1 Γ (v) < 1 and hence we have V n = Γ V n−s .
Qi−1
Qs−i−1
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A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
Proof of Theorem 2.2. The above discussion shows that the Condition (2.8) is sufficient for V n geometrically. More precisely, there exits a positive constant K such that
2
2
E S n,t − S n−1,t = E ∆n,t = E trace ∆n,t ∆0n,t
n
to converge to zero
n/ 2 ≤ V n ≤ K λ(2) .
This implies that, for each fixed t, S n,t n is a Cauchy sequence in L2 and thus its v th entries S n,t (v) n converges in L2 and a.s as n → ∞ and hence its limit n X t (v) is also o in L2 and satisfies the Eq. (2.3). On the other hand, by simple iteration
we can see that S n,t (v) =
Qk−1
Pn
k=1
i=0
At (v − i) et (v − k) + et (v), so X t (v) = limn→∞ S n,t (v) satisfies (2.6) and thus
X t = limn→∞ S n,t satisfies (2.4). It may be noted that S n,t (v) n is a SPS and periodically ergodic sequence and thus its limit X t (v). Hence X t t ∈Z is strictly stationary, ergodic, causal and satisfying (2.5).
As a consequence of Theorem 2.2 the following corollary. Corollary 2.5. Under the Condition (2.8), Eq. (2.3) has a unique PC solution given by X t (v) =
( ∞ k−1 X Y k=1
) At (v − i) et (v − k) + et (v)
i=0
with the above series converging in L2 and absolutely a.s and the process (Xt (v))t ∈Z – defined as the first component of X t (v) t ∈Z – is the unique PC periodically ergodic, causal solution of (2.2) with a periodic autocovariance structure in the sense that E {Xt +s } = E {Xt } and Cov (Xt +s , Xr +s ) = Cov (Xt , Xr ) for all integers t and r (see Lund and Basawa (1999)).
2.2.2. The general periodic bilinear models For arbitrary integer Q > 2, the same argument used in the previous subsection can still be applied. In this case, we assume that (et )t ∈Z satisfying the conditions
n
2Q
E et
o
< +∞ and E ert = 0 for every odd positive integer r < 2Q .
(2.10)
Hence the recursions relations S n,t (v) and ∆n,t (v) and its multivariate versions processes S n,t and ∆n,t remainder the same
as in Section 2.2.1. However, the convergence properties of the sequence S n,t can be studied using the same arguments as in case Q = 2 by introducing the following vectors
2 e V (v) = E ∆⊗ n,t (v) , (nj,)t 2 e Dn,t (v) = E et (v − j)∆⊗ j = 1, . . . , Q − 1, n,t (v) , (jk) e ⊗2 F (v) = E e (v − j)e (v − k)∆ (v) , 1 ≤ j ≤ k < Q . n ,t
t
t
n,t
Similarly, we can define the vector V n,t (v) consisting of the above moments vectors. Then it is straightforward matter as in Section 2.2.1 to write down the matrix Γ (v) such that V n,t (v) = Γ (v)V n−1,t (v − 1).
(2.11)
Exactly the same arguments used in the proof of Theorem 2.2 then give the following result. Theorem 2.3. Consider the Eq. (2.3) and assume that (et )t ∈Z satisfying the Conditions (2.10). For any v ∈ {1, . . . , s}, let Γ (v) be the matrices in Eq. (2.11). Then a sufficient condition for the existence of second-order stationary solution to (2.4) is that
λ(Q ) := ρ
s Y
! Γ (v)
< 1.
v=1
Moreover, the solution process is unique, strictly stationary, ergodic, causal and given by (2.5). 3. Geometric ergodicity and strong mixing Mixing conditions describe some type of asymptotic independence, which can be useful in proving many limit theorems e.g. for central limit theorem, law of large numbers and for sample covariance function which can be employed to derive the consistency, asymptotic normality and the law of iterated logarithm of some estimation procedures. Let X t t ≥0 be a vectorial k discrete time Markov chain with state space E ⊂ R , k ≥ 1 and with time homogeneous t-step transition probabilities t i.e., P X , A = P X t ∈ A|X 0 = X where X ∈ E, A ∈ BE and where BE is a Borel σ -field on E with P 1 X , A = P X , A . Let
R π be the invariant probability measure, on (E, BE ), i.e., ∀A ∈ BE : π (A) = P (X , A)π (dX ). Then the chain X t t ≥0 is said to
t
be geometrically ergodic, if there exists some 0 < ρ < 1 such that ∀X ∈ E, P X , . − π (.) V = o(ρ t ) and called β -mixing
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
85
R with geometric rate if there exist constants r ∈]0, 1[ and c > 0 such that βX (t ) = P t X , . − π (.) V π (dX ) ≤ cr t where k.kV is the total variation norm. Recall that a Markov chain X t t ≥0 is said to be ϕ -irreducible for some measure ϕ on (E, BE ) if
X
Pt X, A > 0
for all X ∈ E, whenever ϕ (A) > 0.
t ≥0
A particular consequence of geometric ergodicity is that the Markov chain is β -mixing with geometric rate. Perhaps one of the most well-known techniques used in establishing the strict stationarity and the geometric ergodicity of a Markov chain, is based on the drift condition developed in Meyn and Tweedie (1996), and employed for the analysis k k of stochastic stability. This condition require the existence of a function g : R → [1, ∞[, a compact K ⊂ R , a positive constants b and 0 < λ < 1 such that E g (X t )|X t −1 = X ≤ λg (X ) + bIK (X ). To derive the geometric ergodicity and β -mixing results, we need to write the model under consideration into the framework of a Markov chain. To do so we shall restrict ourselves to a particular subclass PBL (p, 0, p, q) in which cij (v) = 0 for i < j. In this case, the process X t (v) can be rewritten as X t (v) = A(v)X t (v − 1) +
q X
Cj (v)X t (v − j) et (v − j) + (b0 (v)et (v) + a0 (v)) H
(3.1)
j =1
here the matrices Cj (v) 1≤j≤q becomes
cjj (v) 0
Cj (v) :=
.. .
0
...
cpj (v)
... .. .
0
.. .
.. .
...
0
...0 . . . 0 .. , . ...0
0 0 0
j = 1, . . . , q.
0
By introducing the state vector Y t (v) = X 0t (v) , X 0t (v) et (v) , X 0t (v − 1) et (v − 1) , . . . , X 0t (v − q + 1) et (v − q + 1) , we can express (3.1) in the following equivalent form Y t (v) = Dv (et (v))Y t (v − 1) + B(et (v))
(3.2)
where A(v) A(v)x O (p) Dv (x) :=
...
C1 (v) C1 (v)x I(p)
O(p)
C2 (v) C2 (v)x O(p)
..
.
... ... ··· .. .
O(p)
I(p)
..
. ...
Cq (v) Cq (v)x O(p) ,
.. .
O(p)
(b0 (v)x + a0 (v)) H (b0 (v)x + a0 (v)) xH O(p) B(x) := .. .
.
O(p)
Hence Y t (v) is evidently a Markov chain. Lemma 3.1. Let γ (s) (|D|) be the Lyapunov exponent associated with the sequence of periodic i.i.d. random matrices |D| = (|Dt (et )|)t ∈Z . If γ (s) (|D|) < 0 then there is a δ > 0 such that E |Xt |δ < +∞
(3.3)
Proof. First we have to show that if γ (s) (|D|) < 0 then there is δ > 0 and t0 such that
( ) δ t0 s−1 Y Y
Ds−v (et −i (s − v)) E
< 1. 0 i=1 v=0
n
Q
t
o o
| D ( e s − v))| (
< 0, there is a positive integer t0 such that s −v t − i i =1 ) ) ( t ( s−1
Y 0 Y
Ds−v (et −i (s − v)) E log
<0 0
i=1 v=0
Since γ (s) (|D|) := inft ≥1 E
1 t
log
nQ
s−1 v=0
and hence
t0 ( t ( ) ) ( )t s−1 s−1 s−1
Y
Y
Y 0 Y 0
Ds−v (et −i (s − v)) = E Ds−v (et −i (s − v)) ≤ |Ds−v (µ)| E
< +∞ 0 0
i=1 v=0
v=0
v=0
(3.4)
86
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
n t o
Qt0 Qs−1 s − v)) . Since f 0 (0) < 0, f (t ) decrease in a neighborhood where µ := E {|et |}. Let f (t ) = E i= D ( e (
s −v t − i 1 v=0 0
δ of 0 and since f (0) = 1, it follows that there exists 0 < δ < 1 such that (3.4) holds. Second, we have Y t (v) ≤
δ P∞
Qk−1
δ δ k=1 i=0 |Dv−i (et (v − i))| kB(et (v − k))k + kB(et (v))k for some v ∈ {1, . . . , s} which by the independence of
δ n
δ o P∞
Qk−1
Dv (et −i (v)) and B(et −k (v)) for i < k implies E Y t (v) ≤ Bδ k=1 E i=0 |Dv−i (et (v − i))| + Bδ where Bδ := max1≤v≤s E kB(e0 (v))kδ . Using (3.4) there exist αv > 0 and 0 < βv < 1 such that
δ k−1 Y
|Ds−v (et (v − i))| ≤ αv βvk ≤ αβ k E i=0
δ where αβ k = max1≤v≤s αv βvk . This prove that E Y t (v)
n
o
< +∞ and thus (3.3).
Now, we consider the following condition [B.1] ρ
Q s−1
| D (µ)| < 1 where µ := E {|et |}. s −v v=0
From the Proposition 2.1, the Condition [B.1] ensure that γ (s) (|D|) < 0 and hence the results of Corollary 2.1 and Lemma 3.1 holds. We are now ready to state the main result of this section, which establishes the β -mixing property with exponential decay for PBL. Theorem 3.1. Consider the multivariate version process Y t t ≥0 associated with Y t (v) t ≥0 defined by (3.2) and suppose that [B.1] hold. Then if the Markov chain Y t is ϕ -irreducible, then it is geometrically ergodic and, hence, β -mixing with geometric rate.
Proof. First, note that by Condition [B.1], a unique SPS solution to the Eq. (3.2) exists. To show the geometric ergodicity, we check the three condition of Theorem 1 in Feigin and Tweedie (1985). The Lebesgue dominated convergence theorem ensure that the function E g (Y t )|Y t −1 = Y is continuous in Y for every bounded and continuous g on Rsp(q+1) and hence the Markov chain is Feller. By assumption, the chain is ϕ -irreducible, so it remains Indeed, by n to verify the drift condition. o Lemma 3.1 there is t0 and δ > 0 such that we have by recursion Y t0 (0) =
nQ
t0 i=1
Pst0
k=0
Qk−1 i =0
Ds−i (et0 −1 (s − i)) B(et0 −1 (s − k)) +
nQ
oo s−1 Y 0 (0) and v=0 Ds−v (et0 −i (s − v))
δ (
( ) δ ) t0 st0 Y s−1 k−1
Y Y X
δ
Y (0) δ ≤ D ( e ( s − v)) D ( e ( s − i )) B ( e ( s − k )) +
Y 0 (0) .
s −v t − i s − i t − 1 t − 1 t0 0 0 0
i=1 v=0
i=0 k=0 Let g be the Lyapunov function defined by g (Y ) = kY kδ + 1, then E g (Y t0 (0))|Y 0 (0) = Y ≤ α g Y + b + 1 − α
nQ
δ n o δ o P∞
k−1
Qt0 Qs−1 , b := . Set where α := E i=1 v=0 Ds−v (et0 −i (s − v)) k=0 E i=0 Ds−i (et0 −1 (s − i)) B(et0 −1 (s − k)) p(q+1) λ = α + (1 − α) /2 and let C := Y ∈ R : λg (Y ) ≤ α g (Y ) + 1 − α . Since α < 1, then α < λ < 1, C is a compact and we obtain E g (Y t0 (0))|Y 0 (0) = Y ≤ λg Y + bIC (Y ). Remark 3.1. The disadvantage of the above theorem is the ϕ -irreducibility assumption, which is generally difficult to verify with such a representation, possibly due to over-dimension of the state vector Y t . However, Tweedie (1988) (see also Liu (1992)) has extended his previous results to deal with second order stationarity by dropping off the ϕ -irreducibility requirement. 4. Strict periodic stationarity of a class of periodic GARCH processes We here consider a class of periodic GARCH (p, q) (PGARCH(p, q)) processes which can be written on the form of a periodic diagonal bilinear models. This enables us to give a necessary and sufficient conditions for strict periodic stationarity. We consider thus the process (t )t ∈Z solving t (v) = σt (v)zt (v), v ∈ {1, . . . , s} where (zt )t ∈Z is an i.i.d process. The volatility process (σt )t ≥0 is assumed to verify f (σt (v)) = a0 (v) +
p X i =1
ai (v)f (σt (v − i)) +
q X i =1
cii (v)f (σt (v − i)) h (zt (v − i))
(4.1)
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
87
2 where +f :2 R+ → R is strictly monotonic function and where h : R → R such that h (zt ) is nondegenerate and E log h (zt ) < +∞. We observe that Xt (v) = f (σt (v)) in this case is a periodic diagonal bilinear process with errors et given by et = h (zt ) thus is a special case of (3.1).
Theorem 4.1. Let γ (s) (D) the Lyapunov exponent associated with the sequence D = (Dt (h (zt )))t ∈Z . Then 1. γ (s) (D) < 0, is a sufficient condition for (t )t ∈Z to have a unique SPS and periodically ergodic solution. If a0 (v), ap (v), cqq (v) 6= 0 for all v , then γ (s) (D) < 0 is also necessary. 2. Consider the condition, for all v a0 (v) > 0, ai (v) ≥ 0,
i = 1, . . . , p,
cii (v) ≥ 0,
i = 1, . . . , q
and
h (zt (v)) ≥ 0 with E {h (zt (v))} < +∞. Then under (4.2) a sufficient condition for γ (s) (D) < 0 is ρ
Q s−1
< 1 where µ := E {h (zt )}. This at the same Q s −1 e time is necessary and sufficient for E {f (σt )} < +∞. A necessary condition for γ (s) (D) < 0 is ρ D (µ) < 1 where s −v v=0 e Dv (µ) denote the matrix obtained by replacing Av by O(p) in Dv (µ). v=0
Ds−v (µ)
(4.2)
Proof. 1. The sufficient condition is immediate. Since Dt (h (zt )) is an i.i.d. sequence, then under the condition of the theorem γ (s) (D) < 0 is necessary (see Bougerol and Picard (1992)). 2. The first result follows from the theorem of Kesten and Spitzer (1984), and the second is a consequence immediate of the positivity of the sequence Dt (h (zt )). Corollary 4.1. Under the Condition (4.2) we consider the following periodic GARCH(p, q) models: (1) : Linear PGARCH(p, q): for which f (x) = x2 , h(x) = x2 , (2) : Power PGARCH(p, q): for which f (x) = xr , h(x) = |x|r , r > 0, (3) : L-PGARCH (p, q): for which f (x) = x, h(x) = x. Then 1. γ (s) (D) < 0 is necessary and sufficient condition for (t )t ∈Z to have a unique SPS and periodically ergodic solution. Moreover, if t t ∈Z is ϕ -irreducible, then the solution process is geometrically ergodic and hence β -mixing. 2. p = q = 1, γ (s) (D) = starting point.
Ps
v=1
E {log(a1 (v)) + c11 (v)h (zt (v))} ≥ 0 implies f (σt ) → +∞ almost surely for any given
Proof. Straightforward and hence omitted.
Acknowledgements We are deeply grateful to two anonymous referees for their constructive remarks and suggestions that have led to a substantial improvement in the form and the content of the paper. References Bibi, A., 2003. On the covariance structure of the time-varying bilinear models. Stoch. Anal. Appl. 21, 25–60. Bibi, A., Gautier, A., 2006. Propriétés dans L2 et estimation des processus purement bilinéaires et strictement superdiagonaux à coefficients périodiques. Canad. J. Statist. 34 (1), 131–148. Bibi, A., Moon-Ho, R., 2006. Estimation of periodic bilinear time series models. Commun. Stat. Theory Methods 35, 1745–1756. Bibi, A., Aknouche, A., 2007. On some probabilistic properties of periodic GARCH processes. Available at: http://arxiv.org/PS_cache/arxiv/pdf/0709/0709. 2983v1.pdf. Bougerol, P., Picard, N., 1992. Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (4), 1714–1730. Boyles, R.A., Gardner, W.A., 1983. Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE Trans. Inform. Theory 29, 105–114. Feigin, P.D., Tweedie, 1985. Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6, 1–14. Franses, P.H., 1996. Periodicity and Stochastic Trends in Economic Time Series. Oxford University Press, Oxford. Kesten, H., Spitzer, F., 1984. Convergence in distribution od products of random matrices. Z. Wahrs 67, 363–386. Liu, J., 1992. On stationarity and asymptotic inference of bilinear time series models. Statist. Sinica 2, 479–494. Liu, J., Brockwell, P.J., 1988. On the general bilinear time series model. J. Appl. Probab. 25, 553–564. Lund, R.B., Basawa, I.V., 1999. Modeling and inference for periodically correlated time series. In: Ghosh, S. (Ed.), Asymptotic, Nonparametric and Time Series. Marcel Dekker, NY, pp. 37–62. Meyn, S.P., Tweedie, R.L., 1996. Markov Chains and Stochastic Stability. Springer-Verlag, London. Subba Rao, T., Terdik, G., 2003. On the theory of discrete and continuous bilinear time series models. In: Shanbhag, D.N., Rao, C.R. (Eds.), Handbook of Statistics, vol. 21. pp. 827–870. Tiao, G.C., Grupe, M.R., 1980. Hidden periodic autoregressive-moving average models in time series data. Biometrica 67, 365–373. Tweedie, R.L., 1988. Invariant measures for Markov chains with no irreducibility assumptions. J. Appl. Probab. 25A, 275–285.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On stationarity and β -mixing of periodic bilinear processes Abdelouahab Bibi ∗ , Radia Lessak Département de Mathématiques, Université Mentouri Constantine, Algeria
article
info
Article history: Received 5 February 2008 Received in revised form 22 June 2008 Accepted 17 July 2008 Available online 30 July 2008
MSC: primary 62M10 secondary 62M05
a b s t r a c t This paper studies some probabilistic properties of periodic bilinear processes. In these nonlinear models, the parameters are allowed to switch between different regimes. Stationarity and geometric ergodicity conditions (in periodic sense) are given under general and tractable assumptions. We use these results to give necessary and sufficient conditions for stationarity of specific periodic GARCH processes which can be written as a periodic bilinear models. Moreover, it is shown that local stationarity i.e., stationarity within each regime, is not necessary to obtain global stationarity. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Many time series data in fields such as macroeconomics, finance, biological sciences and engineering exhibit non-linearity (like occasional sharp spikes) and non-stationarity (such as the dependency of moments on the time), which cannot be explained by the standard autoregressive (AR), moving-average (MA) or by mixed autoregressive moving-average (ARMA) models. In this context, the class of bilinear (BL) models (see for a review Subba Rao and Terdik (2003) and the references therein) has attracted considerable attention in the econometric and statistical literature. Bilinear models have been shown to fit and forecast non Gaussian processes ‘‘better’’ than many linear ARMA alternatives. Most of the applications of BL models assume constant coefficients, but this assumption may not be appropriate, as an unforeseen intervention may happen (as, for example, for stock market indexes). So, time dependent BL models seem to provide a more realistic alternative to classical ARMA models, in order to focus on non-linear and non-stationary time series. These time-dependent models keep the BL form, but they allow the coefficients to vary with respect to time (see Bibi (2003) for further discussion). Our attention here is focused on periodic BL models (PBL). To our best knowledge, in spite of the well-known periodic structure of many time series (see, for a review Franses (1996)), no work has investigated on PBL models yet. The main purpose of this paper is therefore to generalize some theoretical probabilistic properties of some PBL models, given recently by Bibi and Gautier (2006) and Bibi and Moon-Ho (2006), and of course, our model includes the classical ARMA, PARMA models and a large class of periodic GARCH (PGARCH) models (see Section 4) which play an important role in econometrics and in financial mathematics. An overview of the paper is as follows. Having defined periodic bilinear processes in Section 2, we introduce the concept of periodic generalized autoregressive representation, and we investigate the fundamental properties related with this representation such that the sufficient conditions ensuring the existence and the uniqueness of strict and second order stationarity solutions satisfying this representation. Section 3, conditions for geometric ergodicity and for β -mixing are given. In Section 4, applying the conditions for periodic bilinear processes on the class of periodic GARCH models leads to conditions for stationarity, which in many cases are weaker than the ones already established by Bibi and Aknouche (2007).
∗
Corresponding author. E-mail addresses: [email protected] (A. Bibi), [email protected] (R. Lessak).
0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.07.024
80
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
Some notations are used throughout the paper: I(n) is the n × n identity matrix and I∆ (.) denotes the indicator function of the set ∆. O(k,l) denotes the matrix of order k × l, whose elements are zeros; for simplicity we set O(k) := O(k,k) and O(k) := O(k,1) . Vec(M ) is the vector obtained from a matrix M by setting down the columns of M underneath each other.
γ mij , and γ ∈]0, 1], define a matrix operation |M |γ as |M |γ := mij . It is obvious that γ P P γ γ γ MX ≤ |M |γ X for all X ∈ Rm and Mi ≤ i i |Mi | where M ≤ N means mij ≤ nij for all i and j and N = nij . ⊗ is the usual Kronecker product of matrices and M ⊗r = M ⊗ M ⊗ · · · ⊗ M r-time. If M is squared matrix then ρ (M ) represents For any n × m matrix M =
the maximum modulus of the eigenvalues of the matrix M. 2. Periodic bilinear models Suppose that (Xn )n∈Z , Z = {0, ±1, ±2, . . .} is a process defined on a probability space (Ω , =, P ) with finite second-order moment and generated by a general bilinear model with time-varying coefficients i.e., Xn = a0,n +
p X
ai,n Xn−i +
i=1
q X
bj,n en−j +
Q P X X
j =0
cij,n Xn−i en−j , n ∈ Z
(2.1)
i=1 j=1
where (en )n∈Z is an independent and identically distributed (i.i.d.) process defined on the same probability space (Ω , =, P ), such that E log+ (e2n ) < +∞ where log+ x = max {log x, 0}, x > 0 and el is independent of Xk , k < l. The coefficients ai,n 0≤i≤p , bj,n 0≤j≤q and cij,n 1≤i≤P ,1≤j≤Q switch between s-regimes, i.e., ai,n := v=1 ai (v)I∆(v) (n), bj,n := Ps Ps v=1 bj (v)I∆(v) (n) and cij,n := v=1 cij (v)I∆(v) (n) with ∆(v) := {sk + v, k ∈ Z} so that by setting Xt (v) = Xst +v and et (v) = est +v when n = st + v
Xt (v) = a0 (v) +
p X
Ps
ai (v)Xt (v − i) +
i=1
q X
bj (v)et (v − j) +
j=0
Q P X X
cij (v)Xt (v − i)et (v − j)
(2.2)
i=1 j=1
which we will make heavy use of (2.2). In (2.2) the notation Xt (v) refers to Xt during the v th ‘‘season’’ or regime v ∈ {1, . . . , s} of cycle t. For convenience, Xt (i) = Xt −1 (s + i), et (i) = et −1 (s + i) if i ≤ 0. It is worth noting that when s > 1, the process is globally nonstationary, but is stationary within each period. In order to facilitate the analysis, we shall consider through this section the Representation (2.2) and we shall assume, without loss of generality, that P = p, since otherwise zeros of ai (v) and cij (v) can be filled in. To discuss the probabilistic structure of the Representation (2.2), we require a state-space version. Indeed, let us define the matrices a1 (v) 1
a2 (v) 0
A(v) =
..
. ···0
0 0
··· ··· .. .
ap (v) 0
c1j (v) 0
,
.. .
1
0
··· ···
Cj (v) =
.. .
.. .
··· ···
0
p×p
cpj (v) 0 0
p×p
and the vectors X t (v) = (Xt (v), . . . , Xt (v − p + 1)) , H = (1, 0, . . . , 0) , et (v) = a0 (v) + notation, we can write the Model (2.2) in the form 0
0
Pq
bj (v)et (v − j) H. With this
j =0
X t (v) = At (v)X t (v − 1) + et (v)
(2.3)
where At (v) := A(v) + j=1 Cj (v)et (v − j) and Xt (v) = H X t (v). In the Representation (2.3) the matrices A(v), Cj (v) are interpreted periodically in v with period s. So, the relation in (2.3) is the same as the defining equation for multivariate generalized periodic autoregressive process. Now, it is well known that in time series models with periodic coefficients, it is possible to embed seasons into a multivariate process (see Tiao and Grupe (1980)). More precisely X t t ∈Z where X t = (X 0t (1), X 0t (2), . . . , X 0t (s))0 is a generalized RCA process, i.e.,
PQ
X t = At X t −1 + η
where At , η t as
0
(2.4)
t
being a strictly stationary and ergodic pairs of random matrix and vector defined respectively by blocks t ∈Z
O(p) O(p)
. At := ..
O(p)
... ... .. . ...
O(p) O(p)
.. .
O(p)
At (1) At (2)At (1)
.. .
s−1 Y
v=0
At (s − v)
et (1) At (2)et (1) + et (2)
, sp×sp
where as usual, empty products are set equal to I(p) .
ηt := ( X k−1 s s−Y k=1
v=0
.. .
)
At (s − v) et (k)
.
sp×1
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
81
The main aim of this section is the stability of the process (Xt )t ∈Z . We give conditions which ensure the existence of the so-called strictly (respectively second-order) periodically stationary solutions to (2.2) or equivalently to (2.3) which are =(t e) -measurable (or causal) where =(t e) is the σ -field generated by et −j , j ≥ 0 . Recall here that the process (Xt )t ∈Z is said
to be strictly periodically stationary (SPS) or periodically correlated (PC) if and only if the process X t t ∈Z defined by (2.4) is strictly or second-order stationary. Also, when the process X t t ∈Z has an ergodic solution, then the corresponding SPS solution is said to be periodically ergodic (see Boyles and Gardner (1983)). 2.1. Strict periodic stationarity When we consider a periodic bilinear model as a data generating process, it is important to give conditions ensuring stationarity and ergodicity (in periodic sense) for further statistical analysis. However, there exist various results about the conditions ensuring existence of a strictly stationary and ergodic solution to (2.4). These conditions involve the notion of γ (An) for the sequence
o
o of random matrices A := (At )t ∈Z defined by γ (A) := n the Lyapunov exponent infn≥1 E
1 n
Qn−1
log
i=0
a.s
1 n
At −i = limn→∞
Qn−1
log
i=0
At −i , where k.k denotes any operator norm on the set of sp × sp
o
n
and sp × 1 matrices. Indeed, it is clear that both E log+ kA0 k and E log+ η are finite. Therefore from Bougerol and 0
Picard (1992), the unique, strictly stationary, ergodic and causal solution of (2.4) is given by Xt =
( ∞ k−1 X Y k=1
) A t −i
η t −k + η t
(2.5)
i =0
whenever γ (A) is strictly negative. Now, a simple computation shows that
O
(p)
t Y j=0
At −j
O(p) . = At .. O(p)
... ... .. .
O(p) O(p)
...
O(p)
.. .
O(p) O(p)
.. .
. ) ( s−1 tY −1 Y At −i (s − v) v=0
i=1
Therefore, because the Lyapunov exponent is independent of the a multiplicative norm it is
Q by nchoosing n norm, o o Qs−1
t
1 (s) straightforward to obtain that γ (A) ≤ γ (A) := inft >0 E t log i=1 v=0 At −i (s − v) . It is clear that
γ (s) (A) inherits the properties of the standard Lyapunov exponent, in particular the following inequality γ (s) (A) ≤ P s (s) v=1 E {log kA0 (v)k} hold. We can think of γ (A) as the Lyaponov exponent associated with the periodic sequence of random matrices A = (At )t ∈Z . We have thus shown our first result which gives a sufficient condition for strict stationary of the process X t t ∈Z .
Theorem 2.1. Suppose that γ (s) (A) < 0. Then for all t ∈ Z the Series (2.5) converges a.s and constitute the unique strictly stationary, causal and ergodic solution of (2.4). Corollary 2.1. If γ (s) (A) < 0, then for any v ∈ {1, . . . , s} and for any fixed t ∈ Z, Eq. (2.3) has a unique SPS solution given by X t (v) =
( ∞ k−1 X Y k=1
) At (v − i) et (v − k) + et (v)
(2.6)
i =0
with the above series converges absolutely a.s and the process (Xt (v))t ∈Z – defined as the first component of X t (v) t ∈Z – is the unique SPS periodically ergodic, causal solution of (2.2).
Proof. From (2.3) we have by recursion X t (v) =
Pn−1 nQk−1 k=0
Q
n −1
i=0
o
At (v − i) et (v − k) +
nQ
n−1 i=0
o
At (v − i) X t (v − n). Hence,
Q
n −1
(s) (A) < 0 and thus i=0 At (v − i) ≤ γ
1
n
→ λ < (s) 1 λ := exp γ (A) < 1 a.s as n → ∞. On the other hand, the strong law of large numbers shows that k et (v − k) = n o
Pk Pk−1 1 k−1 1
≤ k for k sufficiently j=1 et (v − j) − k j=1 et (v − j) converges a.s to 0 as k → ∞. Hence a.s, et (v − k) k k−1
nQ
o
k−1
0 k large. Thus . By the Cauchy root test, the Series (2.6) converges absolutely a.s. The i=0 At (v − i) et (v − k) ≤ k λ it is not difficult to show that a.s limn→∞
1 n
log
i=0 At (v − i)
0
remainder of statements are immediate.
Example 2.1. For PBL(1, 0, 1, 1), we obtain a1 (v) + c11 (v)et (v − 1)), hence, by the law v=0 At (s − v) = v=1 (P s of large numbers, a sufficient condition ensuring γ (s) (A) < 0 is that v=1 E {log |a1 (v) + c11 (v)e0 (v − 1)|} < 0
Qs−1
Qs
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which reduces to the classical condition when s = 1. It is worth noting that the existence of explosive regimes (i.e., E {log |a1 (v) + c11 (v)e0 (v − 1)|} > 0) does not preclude strict periodic stationarity. The following proposition give a simple sufficient condition which ensure γ (s) (A) < 0. Proposition 2.1. Let Q = 1, δ ∈]0, 1] and Γ := E
nQ
s −1 v=0
o |At (s − v)|δ . Then ρ (Γ ) < 1 implies that γ (s) (A) < 0 and thus
we have the results of Corollary 2.1. n 1/n Proof. If ρ (Γ ) < 1, there exists λ ∈]0, 1[ such that lim supn ≤ λ < 1. By independence of the sequence →∞ kΓ k
Q
Q nQ nQ o o δ
n
s−1 s−1 δ | A ( s − v)| ≥ E (et )t ∈Z , we get kΓ n k = ni=1 E
i=1 v=0 t −i v=0 At −i (s − v) . By Jensen’s inequality we have
( ( ( ) δ ) ) n s−1 n s −1 Y
Y
Y Y 1
0 > log λ ≥ lim sup log E At −i (s − v) At −i (s − v) ≥ δ inf E log i=1 v=0
i=1 v=0
n n→∞ n 1
= δγ (s) (A). The Lyapunov exponent γ (s) (A) criterion seems difficult to obtain explicitly when p > 1, however a potential method to verify whether or not γ (s) (A) < 0 is via a Monte-Carlo simulation using Eq. (2.4). This fact heavily limits the interests of the criterion in statistical applications. Indeed, the solution need to have some moments to make an estimation theory possible and Lyapunov criterion does not guarantee the existence of such moments. Therefore, in the next subsection, we give conditions ensuring the existence of moments for the strict stationary solution. 2.2. Second-order periodic stationarity (e)
In the previous subsection, conditions ensuring existence of SPS and =t -measurable solutions that are not necessarily square integrable have been established. In this subsection we are interested in second order PC solutions to (2.2) that are (e) In the special case when Q = 1 and q = 0, Bibi and Moon-Ho (2006) showed that if E {et } = 0 and also 2= t -measurable. E et = σ 2 , then PBL(p, 0, p, 1) has a PC solution whenever
λ(1) := ρ
s Y
⊗2
A
(v) + σ C1 2
⊗2
! (v) < 1.
(2.7)
v=1
In what follows, we show how to derive a sufficient condition analogous to (2.7) for the existence of PC stationary solution of general Model (2.2) or equivalently (2.3). We give the results in explicit form only in the case Q = 2, but the same arguments apply equally well for arbitrary Q as indicated below. 2.2.1. The periodic bilinear model with Q = 2 The following theorem examines conditions ensuring the existence of second-order stationary solutions for Eq. (2.4) and its properties when with Q = 2.
Theorem 2.2. Consider the Eq. (2.3) with Q = 2 and assume that E {et } = E e3t v ∈ {1, . . . , s}, let Γ (v) be the matrices A⊗2 (v) + σ 2 C1⊗2 (v) Γ (v) := σ (A(v) ⊗ C1 (v) + C1 (v) ⊗ A (v))
2
σ 2 A⊗2 (v) + κ4 C1⊗2 (v)
= 0 and E e4t = κ4 < +∞. For any
A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v) σ (C1 (v) ⊗ C2 (v) + C2 (v) ⊗ C1 (v)) 2
σ 2 (A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v))
C2⊗2 (v) O(p2 ) .
σ 2 C2⊗2 (v)
Then a sufficient condition for the existence of second-order stationary solution to (2.4) is that
λ(2) := ρ
s Y
! Γ (v)
< 1.
(2.8)
v=1
Moreover, the solution process is unique, strictly stationary, ergodic, causal and given by (2.5). Corollary 2.2. For the model PBL(1, 0, 1, 1) Xt (v) = a1 (v)Xt (v − 1) + c11 (v)Xt (v − 1)et (v − 1) + et (v). 2 2 The Condition (2.8) reduce to v=1 (a21 (v) + σ 2 c11 (v)) < 1. Notice that the existence of explosive regimes (i.e., a21 (v) + σ 2 c11 (v) > 1) does not preclude second-order stationarity.
Qs
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83
Proof. In this case, A(v) = a1 (v), C1 (v) = c11 (v), C2 (v) = 0, so 2 a21 (v) + σ 2 c11 (v) 2 Γ (v) := 2σ c11 (v)a1 (v) 2 σ 2 a21 (v) + κ4 c11 (v)
0 0 0
0 0 0
a simple calculation shows that the non-zero eigenvalue of
Qs
v=1
Γ (v) is
Qs
v=1
2 a21 (v) + σ 2 c11 (v) . Hence the result follows.
Corollary 2.3. When Q = 1, the Condition (2.8) reduces to (2.7). Proof. In this case the matrices Γ (v) take the form A⊗2 (v) + σ 2 C1⊗2 (v) Γ (v) = σ (A(v) ⊗ C1 (v) + C1 (v) ⊗ A (v)) σ 2 A⊗2 (v) + κ4 C1⊗2 (v)
2
O(p2 ) O(p2 ) O(p2 )
O(p2 ) O(p2 ) O(p2 )
Qs Qs ⊗2 2 ⊗2 v=1 Γ (v) are the same as those of v=1 A (v) + σ C1 (v) . Hence Qs ⊗2 2 ⊗2 v=1 A (v) + σ C1 (v) . v=1 Γ (v) = ρ
and consequently the non-zero eigenvalues of
ρ
Qs
Corollary 2.4 (The PARMA models). In the linear case when the coefficients cij (v) in (2.2) are all zeros for any v ∈ {1, . . . , s}. Qs The Condition (2.8) reduce to ρ v=1 A(v) < 1. Proof. The matrices Γ (v) take the form A⊗2 (v) Γ (v) = O(p2 ) σ 2 A⊗2 (v)
O(p2 ) O(p2 ) O(p2 )
O(p2 ) O(p2 ) O(p2 )
Qs Qs Qs Qs ⊗2 ⊗2 v=1 Γ (v) are the same of v=1 A (v). Hence ρ v=1 Γ (v) = ρ v=1 A (v) = ⊗2 Qs =ρ v=1 A(v) v=1 A(v) .
so the non-zero eigenvalues of
ρ
Q s
To prove Theorem 2.2, we use the same approach as in Liu and Brockwell (1988). We first define for each v ∈ {1, . . . , s} the following Rp -valued processes
O(p) S n,t (v) = et (v) A (v)S t n−1,t (v − 1) + et (v)
if n < 0 if n = 0 if n > 0
and ∆n,t (v) = S n,t (v) − S n−1,t (v) which satisfies the equation ∆n,t (v) = At (v)∆n−1,t (v − 1) for any n ≥ 3. Let S n,t and ∆n,t := S n,t − S n−1,t be the multivariate version processes associated with S n,t (v) and ∆n,t (v) with v ∈ {1, . . . , s}. (e)
Clearly S n,t and ∆n,t are measurable with respect to =t . By the Lp -theory (p > 1) the problem of the existence of a second-
order stationary solution of (2.4) now reduces to the convergence of S n,t n≥0 to X t in L2 . The key quantity of interest in determining of L2 -convergence is E ∆n,t ∆0n,t . We shall need to evaluate the vectors of moments
2 e V n,t (v) = E ∆⊗ n,t (v) ,
2 e Dn,t (v) = E ∆⊗ n,t (v) et (v − 1) ,
2 2 e F n,t (v) = E ∆⊗ n,t (v) et (v − 1) .
Then we have
e V n,t (v) = A⊗2 (v) + σ 2 C1⊗2 (v) e V n−1,t (v − 1) + (A(v) ⊗ C2 (v) + C2 (v) ⊗ A (v)) e Dn−1,t (v − 1) ⊗2 e + C2 (v)F n−1,t (v − 1) e Dn,t (v) = σ 2 (A(v) ⊗ C1 (v) + C1 (v) ⊗ A(v)) e V n−1,t (v − 1) + σ 2 (C1 (v) ⊗ C2 (v) + C2 (v) ⊗ C1 (v)) e Dn−1,t (v − 1) ⊗2 2 ⊗2 2 e e e F n,t (v) = σ A (v) + κ4 C1 (v) V n−1,t (v − 1) + σ (A(v) ⊗ C2 (v) + C2 (v) ⊗ A(v)) Dn−1,t (v − 1) + σ 2 C2⊗2 (v)e F n−1,t (v − 1). 0 0 0 0 e By defining V n,t (v) = V n,t (v), e Dn,t (v), e F n,t (v) we obtain V n,t (v) = Γ (v)V n−1,t (v − 1). Define V n,t := (V 0n,t (1), . . . , V 0n,t (s))0 , then we obtain the following homogeneous equation V n,t = Γ V n−s,t −1
(2.9)
where Γ is a matrix defined by blocks as (Γ )i,i := v=0 Γ (i − v) v=0 Γ (s − v) and O(3p2 ) otherwise. The necessary and sufficient condition for the homogeneous equation in (2.9) to have a stable solution, which is finite and Qs difference independent of t is that λ(2) = ρ (Γ ) = ρ v=1 Γ (v) < 1 and hence we have V n = Γ V n−s .
Qi−1
Qs−i−1
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Proof of Theorem 2.2. The above discussion shows that the Condition (2.8) is sufficient for V n geometrically. More precisely, there exits a positive constant K such that
2
2
E S n,t − S n−1,t = E ∆n,t = E trace ∆n,t ∆0n,t
n
to converge to zero
n/ 2 ≤ V n ≤ K λ(2) .
This implies that, for each fixed t, S n,t n is a Cauchy sequence in L2 and thus its v th entries S n,t (v) n converges in L2 and a.s as n → ∞ and hence its limit n X t (v) is also o in L2 and satisfies the Eq. (2.3). On the other hand, by simple iteration
we can see that S n,t (v) =
Qk−1
Pn
k=1
i=0
At (v − i) et (v − k) + et (v), so X t (v) = limn→∞ S n,t (v) satisfies (2.6) and thus
X t = limn→∞ S n,t satisfies (2.4). It may be noted that S n,t (v) n is a SPS and periodically ergodic sequence and thus its limit X t (v). Hence X t t ∈Z is strictly stationary, ergodic, causal and satisfying (2.5).
As a consequence of Theorem 2.2 the following corollary. Corollary 2.5. Under the Condition (2.8), Eq. (2.3) has a unique PC solution given by X t (v) =
( ∞ k−1 X Y k=1
) At (v − i) et (v − k) + et (v)
i=0
with the above series converging in L2 and absolutely a.s and the process (Xt (v))t ∈Z – defined as the first component of X t (v) t ∈Z – is the unique PC periodically ergodic, causal solution of (2.2) with a periodic autocovariance structure in the sense that E {Xt +s } = E {Xt } and Cov (Xt +s , Xr +s ) = Cov (Xt , Xr ) for all integers t and r (see Lund and Basawa (1999)).
2.2.2. The general periodic bilinear models For arbitrary integer Q > 2, the same argument used in the previous subsection can still be applied. In this case, we assume that (et )t ∈Z satisfying the conditions
n
2Q
E et
o
< +∞ and E ert = 0 for every odd positive integer r < 2Q .
(2.10)
Hence the recursions relations S n,t (v) and ∆n,t (v) and its multivariate versions processes S n,t and ∆n,t remainder the same
as in Section 2.2.1. However, the convergence properties of the sequence S n,t can be studied using the same arguments as in case Q = 2 by introducing the following vectors
2 e V (v) = E ∆⊗ n,t (v) , (nj,)t 2 e Dn,t (v) = E et (v − j)∆⊗ j = 1, . . . , Q − 1, n,t (v) , (jk) e ⊗2 F (v) = E e (v − j)e (v − k)∆ (v) , 1 ≤ j ≤ k < Q . n ,t
t
t
n,t
Similarly, we can define the vector V n,t (v) consisting of the above moments vectors. Then it is straightforward matter as in Section 2.2.1 to write down the matrix Γ (v) such that V n,t (v) = Γ (v)V n−1,t (v − 1).
(2.11)
Exactly the same arguments used in the proof of Theorem 2.2 then give the following result. Theorem 2.3. Consider the Eq. (2.3) and assume that (et )t ∈Z satisfying the Conditions (2.10). For any v ∈ {1, . . . , s}, let Γ (v) be the matrices in Eq. (2.11). Then a sufficient condition for the existence of second-order stationary solution to (2.4) is that
λ(Q ) := ρ
s Y
! Γ (v)
< 1.
v=1
Moreover, the solution process is unique, strictly stationary, ergodic, causal and given by (2.5). 3. Geometric ergodicity and strong mixing Mixing conditions describe some type of asymptotic independence, which can be useful in proving many limit theorems e.g. for central limit theorem, law of large numbers and for sample covariance function which can be employed to derive the consistency, asymptotic normality and the law of iterated logarithm of some estimation procedures. Let X t t ≥0 be a vectorial k discrete time Markov chain with state space E ⊂ R , k ≥ 1 and with time homogeneous t-step transition probabilities t i.e., P X , A = P X t ∈ A|X 0 = X where X ∈ E, A ∈ BE and where BE is a Borel σ -field on E with P 1 X , A = P X , A . Let
R π be the invariant probability measure, on (E, BE ), i.e., ∀A ∈ BE : π (A) = P (X , A)π (dX ). Then the chain X t t ≥0 is said to
t
be geometrically ergodic, if there exists some 0 < ρ < 1 such that ∀X ∈ E, P X , . − π (.) V = o(ρ t ) and called β -mixing
A. Bibi, R. Lessak / Statistics and Probability Letters 79 (2009) 79–87
85
R with geometric rate if there exist constants r ∈]0, 1[ and c > 0 such that βX (t ) = P t X , . − π (.) V π (dX ) ≤ cr t where k.kV is the total variation norm. Recall that a Markov chain X t t ≥0 is said to be ϕ -irreducible for some measure ϕ on (E, BE ) if
X
Pt X, A > 0
for all X ∈ E, whenever ϕ (A) > 0.
t ≥0
A particular consequence of geometric ergodicity is that the Markov chain is β -mixing with geometric rate. Perhaps one of the most well-known techniques used in establishing the strict stationarity and the geometric ergodicity of a Markov chain, is based on the drift condition developed in Meyn and Tweedie (1996), and employed for the analysis k k of stochastic stability. This condition require the existence of a function g : R → [1, ∞[, a compact K ⊂ R , a positive constants b and 0 < λ < 1 such that E g (X t )|X t −1 = X ≤ λg (X ) + bIK (X ). To derive the geometric ergodicity and β -mixing results, we need to write the model under consideration into the framework of a Markov chain. To do so we shall restrict ourselves to a particular subclass PBL (p, 0, p, q) in which cij (v) = 0 for i < j. In this case, the process X t (v) can be rewritten as X t (v) = A(v)X t (v − 1) +
q X
Cj (v)X t (v − j) et (v − j) + (b0 (v)et (v) + a0 (v)) H
(3.1)
j =1
here the matrices Cj (v) 1≤j≤q becomes
cjj (v) 0
Cj (v) :=
.. .
0
...
cpj (v)
... .. .
0
.. .
.. .
...
0
...0 . . . 0 .. , . ...0
0 0 0
j = 1, . . . , q.
0
By introducing the state vector Y t (v) = X 0t (v) , X 0t (v) et (v) , X 0t (v − 1) et (v − 1) , . . . , X 0t (v − q + 1) et (v − q + 1) , we can express (3.1) in the following equivalent form Y t (v) = Dv (et (v))Y t (v − 1) + B(et (v))
(3.2)
where A(v) A(v)x O (p) Dv (x) :=
...
C1 (v) C1 (v)x I(p)
O(p)
C2 (v) C2 (v)x O(p)
..
.
... ... ··· .. .
O(p)
I(p)
..
. ...
Cq (v) Cq (v)x O(p) ,
.. .
O(p)
(b0 (v)x + a0 (v)) H (b0 (v)x + a0 (v)) xH O(p) B(x) := .. .
.
O(p)
Hence Y t (v) is evidently a Markov chain. Lemma 3.1. Let γ (s) (|D|) be the Lyapunov exponent associated with the sequence of periodic i.i.d. random matrices |D| = (|Dt (et )|)t ∈Z . If γ (s) (|D|) < 0 then there is a δ > 0 such that E |Xt |δ < +∞
(3.3)
Proof. First we have to show that if γ (s) (|D|) < 0 then there is δ > 0 and t0 such that
( ) δ t0 s−1 Y Y
Ds−v (et −i (s − v)) E
< 1. 0 i=1 v=0
n
Q
t
o o
| D ( e s − v))| (
< 0, there is a positive integer t0 such that s −v t − i i =1 ) ) ( t ( s−1
Y 0 Y
Ds−v (et −i (s − v)) E log
<0 0
i=1 v=0
Since γ (s) (|D|) := inft ≥1 E
1 t
log
nQ
s−1 v=0
and hence
t0 ( t ( ) ) ( )t s−1 s−1 s−1
Y
Y
Y 0 Y 0
Ds−v (et −i (s − v)) = E Ds−v (et −i (s − v)) ≤ |Ds−v (µ)| E
< +∞ 0 0
i=1 v=0
v=0
v=0
(3.4)
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n t o
Qt0 Qs−1 s − v)) . Since f 0 (0) < 0, f (t ) decrease in a neighborhood where µ := E {|et |}. Let f (t ) = E i= D ( e (
s −v t − i 1 v=0 0
δ of 0 and since f (0) = 1, it follows that there exists 0 < δ < 1 such that (3.4) holds. Second, we have Y t (v) ≤
δ P∞
Qk−1
δ δ k=1 i=0 |Dv−i (et (v − i))| kB(et (v − k))k + kB(et (v))k for some v ∈ {1, . . . , s} which by the independence of
δ n
δ o P∞
Qk−1
Dv (et −i (v)) and B(et −k (v)) for i < k implies E Y t (v) ≤ Bδ k=1 E i=0 |Dv−i (et (v − i))| + Bδ where Bδ := max1≤v≤s E kB(e0 (v))kδ . Using (3.4) there exist αv > 0 and 0 < βv < 1 such that
δ k−1 Y
|Ds−v (et (v − i))| ≤ αv βvk ≤ αβ k E i=0
δ where αβ k = max1≤v≤s αv βvk . This prove that E Y t (v)
n
o
< +∞ and thus (3.3).
Now, we consider the following condition [B.1] ρ
Q s−1
| D (µ)| < 1 where µ := E {|et |}. s −v v=0
From the Proposition 2.1, the Condition [B.1] ensure that γ (s) (|D|) < 0 and hence the results of Corollary 2.1 and Lemma 3.1 holds. We are now ready to state the main result of this section, which establishes the β -mixing property with exponential decay for PBL. Theorem 3.1. Consider the multivariate version process Y t t ≥0 associated with Y t (v) t ≥0 defined by (3.2) and suppose that [B.1] hold. Then if the Markov chain Y t is ϕ -irreducible, then it is geometrically ergodic and, hence, β -mixing with geometric rate.
Proof. First, note that by Condition [B.1], a unique SPS solution to the Eq. (3.2) exists. To show the geometric ergodicity, we check the three condition of Theorem 1 in Feigin and Tweedie (1985). The Lebesgue dominated convergence theorem ensure that the function E g (Y t )|Y t −1 = Y is continuous in Y for every bounded and continuous g on Rsp(q+1) and hence the Markov chain is Feller. By assumption, the chain is ϕ -irreducible, so it remains Indeed, by n to verify the drift condition. o Lemma 3.1 there is t0 and δ > 0 such that we have by recursion Y t0 (0) =
nQ
t0 i=1
Pst0
k=0
Qk−1 i =0
Ds−i (et0 −1 (s − i)) B(et0 −1 (s − k)) +
nQ
oo s−1 Y 0 (0) and v=0 Ds−v (et0 −i (s − v))
δ (
( ) δ ) t0 st0 Y s−1 k−1
Y Y X
δ
Y (0) δ ≤ D ( e ( s − v)) D ( e ( s − i )) B ( e ( s − k )) +
Y 0 (0) .
s −v t − i s − i t − 1 t − 1 t0 0 0 0
i=1 v=0
i=0 k=0 Let g be the Lyapunov function defined by g (Y ) = kY kδ + 1, then E g (Y t0 (0))|Y 0 (0) = Y ≤ α g Y + b + 1 − α
nQ
δ n o δ o P∞
k−1
Qt0 Qs−1 , b := . Set where α := E i=1 v=0 Ds−v (et0 −i (s − v)) k=0 E i=0 Ds−i (et0 −1 (s − i)) B(et0 −1 (s − k)) p(q+1) λ = α + (1 − α) /2 and let C := Y ∈ R : λg (Y ) ≤ α g (Y ) + 1 − α . Since α < 1, then α < λ < 1, C is a compact and we obtain E g (Y t0 (0))|Y 0 (0) = Y ≤ λg Y + bIC (Y ). Remark 3.1. The disadvantage of the above theorem is the ϕ -irreducibility assumption, which is generally difficult to verify with such a representation, possibly due to over-dimension of the state vector Y t . However, Tweedie (1988) (see also Liu (1992)) has extended his previous results to deal with second order stationarity by dropping off the ϕ -irreducibility requirement. 4. Strict periodic stationarity of a class of periodic GARCH processes We here consider a class of periodic GARCH (p, q) (PGARCH(p, q)) processes which can be written on the form of a periodic diagonal bilinear models. This enables us to give a necessary and sufficient conditions for strict periodic stationarity. We consider thus the process (t )t ∈Z solving t (v) = σt (v)zt (v), v ∈ {1, . . . , s} where (zt )t ∈Z is an i.i.d process. The volatility process (σt )t ≥0 is assumed to verify f (σt (v)) = a0 (v) +
p X i =1
ai (v)f (σt (v − i)) +
q X i =1
cii (v)f (σt (v − i)) h (zt (v − i))
(4.1)
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87
2 where +f :2 R+ → R is strictly monotonic function and where h : R → R such that h (zt ) is nondegenerate and E log h (zt ) < +∞. We observe that Xt (v) = f (σt (v)) in this case is a periodic diagonal bilinear process with errors et given by et = h (zt ) thus is a special case of (3.1).
Theorem 4.1. Let γ (s) (D) the Lyapunov exponent associated with the sequence D = (Dt (h (zt )))t ∈Z . Then 1. γ (s) (D) < 0, is a sufficient condition for (t )t ∈Z to have a unique SPS and periodically ergodic solution. If a0 (v), ap (v), cqq (v) 6= 0 for all v , then γ (s) (D) < 0 is also necessary. 2. Consider the condition, for all v a0 (v) > 0, ai (v) ≥ 0,
i = 1, . . . , p,
cii (v) ≥ 0,
i = 1, . . . , q
and
h (zt (v)) ≥ 0 with E {h (zt (v))} < +∞. Then under (4.2) a sufficient condition for γ (s) (D) < 0 is ρ
Q s−1
< 1 where µ := E {h (zt )}. This at the same Q s −1 e time is necessary and sufficient for E {f (σt )} < +∞. A necessary condition for γ (s) (D) < 0 is ρ D (µ) < 1 where s −v v=0 e Dv (µ) denote the matrix obtained by replacing Av by O(p) in Dv (µ). v=0
Ds−v (µ)
(4.2)
Proof. 1. The sufficient condition is immediate. Since Dt (h (zt )) is an i.i.d. sequence, then under the condition of the theorem γ (s) (D) < 0 is necessary (see Bougerol and Picard (1992)). 2. The first result follows from the theorem of Kesten and Spitzer (1984), and the second is a consequence immediate of the positivity of the sequence Dt (h (zt )). Corollary 4.1. Under the Condition (4.2) we consider the following periodic GARCH(p, q) models: (1) : Linear PGARCH(p, q): for which f (x) = x2 , h(x) = x2 , (2) : Power PGARCH(p, q): for which f (x) = xr , h(x) = |x|r , r > 0, (3) : L-PGARCH (p, q): for which f (x) = x, h(x) = x. Then 1. γ (s) (D) < 0 is necessary and sufficient condition for (t )t ∈Z to have a unique SPS and periodically ergodic solution. Moreover, if t t ∈Z is ϕ -irreducible, then the solution process is geometrically ergodic and hence β -mixing. 2. p = q = 1, γ (s) (D) = starting point.
Ps
v=1
E {log(a1 (v)) + c11 (v)h (zt (v))} ≥ 0 implies f (σt ) → +∞ almost surely for any given
Proof. Straightforward and hence omitted.
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