Mixing properties of ARCH and time-varying ARCH processes

30 Pages·2012·0.24 MB·English

Mixing properties of ARCH and time-varying ARCH
processes
Piotr Fryzlewicz Suhasini Subba Rao
Abstract
There exists very few results on mixing for nonstationary processes. However, mixing
is often required in statistical inference for nonstationary processes, such as time-varying
ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic pro-
cess are derived in terms the conditional densities of the process. These bounds are used
to obtain the fi, 2-mixing and fl-mixing rates of the nonstationary time-varying ARCH(p)
process and ARCH( ) process. It is shown that the mixing rate of time-varying ARCH(p)
1
process is geometric, whereas the bounds on the mixing rate of the ARCH( ) process de-
1
pendsontherateofdecayoftheARCH( )parameters. Wementionthatthemethodology
1
given in this paper is applicable to other processes.
Key words: Absolutely regular (fl-mixing) ARCH( ), conditional densities, time-
1
varying ARCH, strong mixing (fi-mixing), 2-mixing.
2000 Mathematics Subject Classiflcation: 62M10, 60G99, 60K99.
1 Introduction
Mixing is a measure of dependence between elements of a random sequence that has a wide range
of theoretical applications (see Bradley (2007) and below). One of the most popular mixing
measures is fi-mixing (also called strong mixing), where the fi-mixing rate of the nonstationary
stochastic process X is deflned as a sequence of coe–cients fi(k) such that
t
f g
fi(k) = sup sup P(G H) P(G)P(H) : (1)
t2Z H2(cid:190)(Xt;Xt¡1;:::) j \ ¡ j
G2(cid:190)(Xt+k;Xt+k+1;:::)
1
X is called fi-mixing if fi(k) 0 as k . fi-mixing has several applications in statistical
t
f g ! ! 1
inference. For example, if fi(k) decays su–ciently fast to zero as k , then, amongst
f g ! 1
other results, it is possible to show asymptotic normality of sums of X (c.f. Davidson (1994),
k
f g
Chapter 24), as well as exponential inequalities for such sums (c.f. Bosq (1998)), asympototic
normality of kernel-based nonparametric estimators (c.f. Bosq (1998)) and consistency of change
point detection schemes of nonlinear time series (c.f. Fryzlewicz and Subba Rao (2008)). The
notion of 2-mixing is related to strong mixing, but is a weaker condition as it measures the
dependence between two random variables and not the entire tails. 2-mixing is often used in
statistical inference, for example deriving rates in nonparametric regression (see Bosq (1998)).
The 2-mixing rate can be used to derive bounds for the covariance between functions of random
variables, saycov(g(X );g(X ))(seeIbragimov(1962)), whichisusuallynotpossiblewhenonly
t t+k
the correlation structure of X is known. The 2-mixing rate of X is deflned as a sequence
k k
f g f g
fi~(k) which satisfles
fi~(k) = sup sup P(G H) P(G)P(H) : (2)
t2Z H2(cid:190)(Xt) j \ ¡ j
G2(cid:190)(Xt+k)
It is clear that fi~(k) fi(k). A closely related mixing measure, introduced in Volkonskii and
•
Rozanov (1959) is fl-mixing (also called absolutely regular). The fl-mixing rate of the stochastic
process X is deflned as a sequence of coe–cients fl(k) such that
t
f g
fl(k) = sup sup P(G H ) P(G )P(H ) ; (3)
i j i j
t2Z fHjg2(cid:190)(Xt;Xt¡1;:::) i j j \ ¡ j
fGjg2(cid:190)(Xt+k;Xt+k+1;:::)XX
where G and H are flnite partitions of the sample space ›. X is called fl-mixing if
i j t
f g f g f g
fl(k) 0 as k . It can be seen that this measure is slightly stronger than fi-mixing (since
! ! 1
an upper bound for fl(k) immediately gives a bound for fi(k); fl(k) fi(k)).
‚
Despite the versatility of mixing, its main drawback is that in general it is di–cult to derive
bounds for fi(k), fi~(k) and fl(k). However the mixing bounds of some processes are known.
Chanda(1974), Gorodetskii(1977), AthreyaandPantula(1986)andPhamandTran(1985)show
strong mixing of the MA( ) process. Feigin and Tweedie (1985) and Pham (1986) have shown
1
geometric ergodicity of Bilinear processes (we note that stationary geometrically ergodic Markov
chains are geometrically fi-mixing, 2-mixing and fl-mixing - see, for example, Francq and Zako˜‡an
(2006)). Morerecently, Tjostheim(1990)andMokkadem(1990)haveshowngeometricergodicity
2
for a general class of Markovian processes. The results in Mokkadem (1990) have been applied
in Bousamma (1998) to show geometric ergodicity of stationary ARCH(p) and GARCH(p;q)
processes, where p and q are flnite integers. Related results on mixing for GARCH(p;q) processes
can be found in Carrasco and Chen (2002), Liebscher (2005), Sorokin (2006) and Lindner (2008)
(for an excellent review) and Francq and Zako˜‡an (2006) and Meitz and Saikkonen (2008) (where
mixing of ‘nonlinear’ GARCH(p;q) processes are also considered). Most of these these results
are proven by verifying the Meyn-Tweedie conditions (see Feigin and Tweedie (1985) and Meyn
and Tweedie (1993)), and, as mentioned above, are derived under the premise that the process is
stationary (or asymptotically stationary) and Markovian. Clearly, if a process is nonstationary,
thentheaforementionedresultsdonothold. Thereforefornonstationaryprocesses,analternative
method to prove mixing is required.
The main aim of this paper is to derive a bound for (1), (2) and (3) in terms of the densities
of the process plus an additional term, which is an extremal probability. These bounds can be
applied to various processes. In this paper, we will focus on ARCH-type processes and use the
bounds to derive mixing rates for time-varying ARCH(p) (tvARCH) and ARCH( ) processes.
1
The ARCH family of processes is widely used in flnance to model the evolution of returns on
flnancial instruments: we refer the reader to the review article of Giraitis et al. (2005) for a
comprehensive overview of mathematical properties of ARCH processes, and a list of further
references. It is worth mentioning that Ho˜rmann (2008) and Berkes et al. (2008) have considered
a difierent type of dependence, namely a version of the m-dependence moment measure, for
ARCH-type processes. The stationary GARCH(p;q) model tends to be the benchmark flnancial
model. However, in certain situations it may not be the most appropriate model, for example
it cannot adequently explain the long memory seen in the data or change according to shifts
in the world economy. Therefore, recently attention has been paid to tvARCH models (see,
for example, Mikosch and Sta‚rica‚ (2003), Dahlhaus and Subba Rao (2006), Fryzlewicz et al.
(2008) and Fryzlewicz and Subba Rao (2008)) and ARCH( ) models (see Robinson (1991),
1
Giraitis et al. (2000), Giraitis and Robinson (2001) and Subba Rao (2006)). The derivations
of the sampling properties of some of the above mentioned papers rely on quite sophisticated
assumptions on the dependence structure, in particular their mixing properties.
We will show that due to the p-Markovian nature of the time-varying ARCH(p) process, the
fi-mixing, 2-mixing and fl-mixing bound has the same geometric rate. The story is difierent
for ARCH( ) processes, where the mixing rates can be difierent and vary according to the
1
3
rate of decay of the parameters. An advantage of the approach advocated in this paper is that
these methods can readily be used to establish mixing rates of several time series models. This is
especiallyusefulintimeseriesanalysis, forexample, changepointdetectionschemesfornonlinear
time series, where strong mixing of the underlying process is often required. The price we pay for
the (cid:176)exibility of our approach is that the assumptions under which we work are slightly stronger
than the standard assumptions required to prove geometric mixing of the stationary GARCH
process. However, the conditions do not rely on proving irreducibility (which is usually required
when showing geometric ergodicity) of the underlying process, which can be di–cult to verify.
In Section 2 we derive a bound for the mixing rate of general stochastic processes, in terms of
the difierences of conditional densities. In Section 3 we derive mixing bounds for time-varying
ARCH(p) processes (where p is flnite). In Section 4 we derive mixing bounds for ARCH( )
1
processes. Proofs which are not in the main body of the paper can be found in the appendix and
the accompanying technical report.
2 Some mixing inequalities for general processes
2.1 Notation
For k > 0, let Xt¡k = (X ;:::;X ); if k 0, then Xt¡k = 0. Let y = (y ;:::;y ). Let
t t t¡k • t s s 0
denote the ‘ -norm. Let › denote the sample space. The sigma-algebra generated by
1
k ¢ k
X ;:::;X is denoted as t = (cid:190)(X ;:::;X ).
t t+r Ft+r t t+r
2.2 Some mixing inequalities
Let us suppose X is an arbitrary stochastic process. In this section we derive some bounds
t
f g
for fi(k), fi~(k) and fl(k). To do this we will consider bounds for
sup P(G H) P(G)P(H) and sup P(G H ) P(G )P(H ) ;
i j i j
j \ ¡ j j \ ¡ j
H2Ftt¡r1;G2Ftt++kk+r2 fHjg2Ftt¡r1;fGig2Ftt++kk+r2Xi;j
where r ;r 0 and G and H are partitions of ›. In the proposition below, we give a
1 2 i i
‚ f g f g
bound for the mixing rate in terms of conditional densities. Similar bounds for linear processes
have been derived in Chanda (1974) and Gorodetskii (1977) (see also Davidson (1994), Chapter
4
14). However, the bounds in Proposition 2.1 apply to any stochastic process, and it is this
generality that allows us to use the result in later sections, where we derive mixing rates for
ARCH-type processes.
Proposition 2.1 Let us suppose that the conditional density of Xt+k given Xt¡r1 exists and
t+k+r2 t
denote it as fXtt++kk+r2jXtt¡r1. For · = (·0;:::;·r1) 2 (R+)r1+1, deflne the set
E = !; Xt¡r1(!) where = (” ;:::;” ); for all ” · : (4)
f t 2 Eg E f 0 r1 j jj • jg
Then for all r ;r 0 and · we have
1 2
‚
sup P(G H) P(G)P(H)
j \ ¡ j
H2Ft¡r1;G2Ft+k
t t+k+r2
2sup f (y x) f (y 0) dy +4P(Ec); (5)
• x2E ZRr2+1fl Xtt++kk+r2jXtt¡r1 j ¡ Xtt++kk+r2jXtt¡r1 j fl
fl fl
fl fl
and
sup P(G H ) P(G )P(H )
i j i j
j \ ¡ j
fHjg2Ftt¡r1;fGjg2Ftt++kk+r2Xi;j
2 sup f (y x) f (y 0) dy +4P(Ec); (6)
• ZRr2+1 x2E fl Xtt++kk+r2jXtt¡r1 j ¡ Xtt++kk+r2jXtt¡r1 j fl
fl fl
fl fl
where G and H are flnite partitions of ›. Xt¡r1. Let Wt+1 be a random vector that is
f ig f jg t t+k¡1
independent of Xt¡r1 and f denote the density of Wt+1 , then we have
t Wt+1 t+k¡1
t+k¡1
sup P(G H) P(G)P(H)
j \ ¡ j
H2Ft¡r1;G2Ft+k
t t+k+r2
r2
2 sup f (w) sup (y y ;w;x)dy dw+4P(Ec) (7)
W s;k;t s s
• x2E (y 2Rs RD j s¡1 )
Xs=0 Z s¡1 Z
and
sup P(G H ) P(G )P(H )
i j i j
j \ ¡ j
fHjg2Ftt¡r1;fGjg2Ftt++kk+r2Xi;j
r2
2 f (w) sup sup (y y ;w;x)dy dw+4P(Ec) (8)
W s;k;t s s
• (y 2Rs R x2E D j s¡1 )
Xs=0 Z s¡1 Z
5
where (y y ;w;x) = f (y w;x) f (y w;0) and for s 1
0;k;t 0 s;k;t s s;k;t s
D j ¡1 j ¡ j ‚
fl fl
fl fl
(y y ;w;x) = f (y y ;w;x) f (y y ;w;0) ; (9)
s;k;t s s;k;t s s;k;t s
D j s¡1 j s¡1 ¡ s¡1 j
fl fl
fl fl
with the conditional density of X given (Wt+1 ;Xt¡r1) denoted as f and the conditional
t+k t+k¡1 t 0;k;t
density of X given (Xt+k ;Wt+1 ;Xt¡r1), denoted as f , x = (x ;:::;x ) and
t+k+s t+k+s¡1 t+k¡1 t s;k;t 0 ¡r2
w = (w ;:::;w ).
k 1
⁄
PROOF. In Appendix A.1.
Since the above bounds hold for all vectors · (R+)r1+1 (note · deflnes the set E; see (4)), by
2
choosing the · which balances the integral and P(Ec), we obtain an upper bound for the mixing
rate.
The main application of the inequality in (7) is to processes which are ‘driven’ by the inno-
vations (for example, linear and ARCH-type processes). If Wt+1 is the innovation process,
t+k¡1
often it can be shown that the conditional density of X given (Xt+k ;Wt+1 ;Xt¡r1)
t+k+s t+k+s¡1 t+k¡1 t
can be written as a function of the innovation density. Deriving the density of X given
t+k+s
(Xt+k ;Wt+1 ;Xt¡r1) is not a trivial task, but it is often possible. In the subsequent sec-
t+k+s¡1 t+k¡1 t
tions we will apply Proposition 2.1 to obtaining bounds for the mixing rates.
The proof of Proposition 2.1 can be found in the appendix, but we give a (cid:176)avour of it here. Let
H = !; Xt¡r1(!) ; G = !; Xt+k (!) : (10)
f t 2 Hg f t+k+r2 2 Gg
It is straightforward to show that P(G H) P(G)P(H) P(G H E) P(G E)P(H) +
j \ ¡ j • j \ \ ¡ \ j
2P(Ec). The advantage of this decomposition is that when we restrict Xt¡r1 to the set (ie.
t E
not large values of Xt¡r1), we can obtain a bound for P(G H E) P(G E)P(H) . More
t j \ \ ¡ \ j
precisely, by using the inequality
inf P G Xt¡r1 = x P(H E) P(G H E) supP G Xt¡r1 = x P(H E);
x2E t \ • \ \ • x2E t \
¡ fl ¢ ¡ fl ¢
fl fl
we can derive upper and lower bounds for P(G H E) P(G E)P(H) which depend only
\ \ ¡ \
on E and not H and G, and thus obtain the bounds in Proposition 2.1.
6
It is worth mentioning that by using (7) one can establish mixing rates for time-varying linear
processes (such as the tvMA( ) process considered in Dahlhaus and Polonik (2006)). Using
1
(7) and similar techniques to those used in Section 4, mixing bounds can be obtained for the
tvMA( ) process.
1
In the following sections we will derive the mixing rates for ARCH-type processes, where one of
the challanging aspects of the proof is establishing a bound for the integral difierence in (9).
3 Mixing for the time-varying ARCH(p) process
3.1 The tvARCH process
In Fryzlewicz et al. (2008) we show that the tvARCH process can be used to explain the com-
monly observed stylised facts in flnancial time series (such as the empirical long memory). A
sequence of random variables X is said to come from a time-varying ARCH(p) if it satisfles
t
f g
the representation
p
X = Z a (t)+ a (t)X ; (11)
t t 0 j t¡j
ˆ !
j=1
X
where Z are independent, identically distributed (iid) positive random variables, with E(Z ) =
t t
f g
1 and a ( ) are positive parameters. It is worth comparing (11) with the tvARCH process used in
j
¢
the statistical literature. Unlike the tvARCH process considered in, for example, Dahlhaus and
Subba Rao (2006) and Fryzlewicz et al. (2008), we have not placed any smoothness conditions
on the time varying parameters a ( ) . The smoothness conditions assumed in Dahlhaus and
j
f ¢ g
Subba Rao (2006) and Fryzlewicz et al. (2008) are used in order to do parameter estimation.
However, in this paper we are dealing with mixing of the process, which does not require such
strong assumptions. The assumptions that we require are stated below.
Assumption 3.1 (i) For some – > 0, sup p a (t) 1 –.
t2Z j=1 j • ¡
P
(ii) inft2Za0(t) > 0 and supt2Za0(t) < 1.
(iii) Let f denote the density of Z . For all a > 0 we have f (u) f (u[1+a]) du Ka,
Z t Z Z
j ¡ j •
for some flnite K independent of a. R
7
(iv) Letf denotethedensityofZ . Foralla > 0wehave sup f (u) f (u[1+¿]) du
Z t 0•¿•aj Z ¡ Z j •
Ka, for some flnite K independent of a. R
We note that Assumption 3.1(i,ii) guarantees that the ARCH process has a Volterra expansion
as a solution (see Dahlhaus and Subba Rao (2006), Section 5). Assumption 3.1(iii,iv) is a type
of Lipschitz condition on the density function and is satisfled by various well known distribu-
tions, including the chi-squared distributions. We now consider a class of densities which satisfy
Assumption 3.1(iii,iv). Suppose f : R R is a density function, whose flrst derivative is
Z
!
bounded, after some flnite point m, the derivative f0 declines monotonically to zero and satisfles
yf0 (y) dy < . In this case
j Z j 1
R
1
sup f (u) f (u[1+¿]) du
Z Z
j ¡ j
Z0 0•¿•a
m 1
sup f (u) f (u[1+¿]) du+ sup f (u) f (u[1+¿]) du
Z Z Z Z
• j ¡ j j ¡ j
Z0 0•¿•a Zm 0•¿•a
1
a m2sup f0 (u) + u f0 (u) du Ka;
• u2Rj Z j Zm j Z j •
¡ ¢
for some flnite K independent of a, hence Assumption 3.1(iii,iv) is satisfled.
We use Assumption 3.1(i,ii,iii) to obtain the strong mixing rate (2-mixing and fi-mixing) of the
tvARCH(p) process and the slightly stronger conditions Assumption 3.1(i,ii,iv) to obtain the
fl-mixing rate of the tvARCH(p) process. We mention that in the case that X is a stationary,
t
f g
ergodic time series, Francq and Zako˜‡an (2006) have shown geometric ergodicity, which they
show implies fl-mixing, under the weaker condition that the distribution function of Z can
t
f g
have some discontinuities.
3.2 The tvARCH(p) process and the Volterra series expansion
In this section we derive a Volterra series expansion of the tvARCH process (see also Giraitis
et al. (2000)). These results allow us to apply Proposition 2.1 to the tvARCH process. We flrst
note that the innovations Zt+1 and Xt¡p+1 are independent random vectors. Hence comparing
t+k¡1 t
with Proposition 2.1, we are interested in obtaining the conditional density of X given Zt+1
t+k t+k¡1
and Xt¡p+1, (denoted f ) and the conditional density of X given Xt+k ;Zt+1 and
t 0;k;t t+k+s t+k+s¡1 t+k¡1
Xt¡p+1 (denoted f ). We use these expressions to obtain a bound for (deflned in (9)),
t s;k;t Ds;k;t
which we use to derive a bound for the mixing rate. We now represent X in terms of Z .
t t
f g f g
8
To do this we deflne
a (t)z a (t)z ::: a (t)z a (t) a (t) ::: a (t)
1 t 2 t p t 1 2 p
0 1 0 ::: 0 1 0 1 0 ::: 0 1
A (z) = B 0 1 ::: 0 C; A = A (1) = B 0 1 ::: 0 C
t t t
B C B C
BB ::: ::: ... ... CC BB ::: ::: ... ... CC
B C B C
B C B C
B 0 0 1 0 C B 0 0 1 0 C
B C B C
b (z) = (@a (t)z ;0;:::;0)0 and XAt¡p+1 = (X ;X ;::@:;X )0: A(12)
t 0 t t t t¡1 t¡p+1
Using this notation we have the relation Xt+k¡p+1 = A (Z)Xt+k¡p+b (Z). We mention the
t+k t+k t+k¡1 t+k
vector representation of ARCH and GARCH processes has been used in Bougerol and Picard
(1992), Basrak et al. (2002) and Straumann and Mikosch (2006) in order to obtain some proba-
bilistic properties for ARCH-type processes. Now iterating the relation k times (to get Xt+k¡p+1
t+k
in terms of Xt¡p+1) we have
t
k¡2 r¡1 k¡1
Xt+k¡p+1 = b (Z)+ A (Z) b (Z)+ A (Z) Xt¡p+1; (13)
t+k t+k t+k¡i t+k¡r¡1 t+k¡i t
r=0 •i=0 ‚ •i=0 ‚
X Y Y
where we set [ ¡1 A (Z)] = I (I denotes the p p dimensional identity matrix). We use
i=0 t+k¡i p p £
this expansion below.
Q
Lemma 3.1 Let us suppose that Assumption 3.1(i) is satisfled. Then for s 0 we have
‚
X = Z (Z)+ (Z;X) ; (14)
t+k+s t+k+s s;k;t s;k;t
fP Q g
wherefors = 0andn > twehave (Z) = a (t+k)+[A n¡t¡2 r A (Z)b (Z)] ,
P0;k;t 0 t+k r=0 i=1 t+k¡i t+k¡r¡1 1
(Z;X) = A k¡1A (Z)Xt¡p+1 , ([ ] denotes the flrst element of a vector).
Q0;k;t t+k i=1 t+k¡i t 1 ¢ 1 P Q
For 1 s p £ Q ⁄
• •
s¡1 p
(Z) = a (t+k +s)+ a (t+k +s)X + a (t+k +s)Z
s;k;t 0 i t+k+s¡i i k+s¡i
P
i=1 i=s
X X
k+s¡i r
a (t+k +s i)+[A A (Z) b (Z)] ;(15)
0 t+k+s¡i t+k+s¡i¡d t+k+s¡i¡r 1
¡ f g
‰ r=1 d=0 (cid:190)
X Y
9
p k+s¡i
(Z;X) = a (t+k +s)Z A A (Z)Xt¡p+1 ;
Qs;k;t i k+s¡i t+k+s¡if t+k+s¡i¡d t g 1
i=s d=0
£X Y ⁄
and for s > p we have (Z) = a (t+k+s)+ p a (t+k+s)X and (Z;X) 0.
Ps;k;t 0 i=1 i t+k+s¡i Qs;k;t ·
We note that and are positive random variables, and for s 1, is a function
s;k;t s;k;t P s;k;t
P Q ‚ P
of Xt+k (but this has been suppressed in the notation).
t+k+s¡1
PROOF. In Appendix A.2.
By using (14) we now show that the conditional density of X given Xt+k ;Zt+1 and
t+k+s t+k+s¡1 t+k¡1
Xt¡p+1 is a function of the density of Z . It is clear from (14) that Z can be expressed
t t+k+s t+k+s
as Z = Xt+k+s . Therefore, it is straightforward to show that
t+k+s Ps;k;t(Z)+Qs;k;t(Z;X)
1 y
s
f (y y ;z;x) = f : (16)
s;k;t s Z
j s¡1 (z)+ (z;x) (z)+ (z;x)
Ps;k;t Qs;k;t (cid:181)Ps;k;t Qs;k;t ¶
3.3 Strong mixing of the tvARCH(p) process
The aim in this section is to prove geometric mixing of the tvARCH(p) process without appealing
to geometric ergodicity. Naturally, the results in this section also apply to stationary ARCH(p)
processes.
In the following lemma we use Proposition 2.1 to obtain bounds for the mixing rates. It is worth
mentioningthatthetechniquesusedintheproofbelowcanbeappliedtootherMarkovprocesses.
Lemma 3.2 Suppose X isatvARCHprocesswhichsatisfles(11). Thenforany· = (· ;:::;· )
t 0 ¡p+1
f g 2
(R+)p we have
sup P(G H) P(G)P(H)
j \ ¡ j
G2F1t+k;H2Ft¡1
p¡1 k¡1 p¡1
2 sup f (z ) sup (y y ;z;x)dy dz +4 P( X · ); (17)
Z i s;k;t s s t¡j ¡j+1
• Xs=0 x2E Z Yi=1 ys¡12Rs‰ZRD j s¡1 (cid:190) Xj=0 j j ‚
10