Asymptotic behavior of the sample autocovariance and

Asymptotic behavior of the sample autocovariance and
autocorrelation function of the AR(1) process with ARCH(1)
errors
Milan Borkovec
Abstract
We study the sample autocovariance and autocorrelation function of the stationary AR(1)
process with ARCH(1) errors. In contrast to ARCH and GARCH processes, AR(1) processes
with ARCH(1) errors can not be transformed into solutions of linear stochastic recurrence
equations. However, we show that they still belong to the class of stationary sequences
with regular varying nite-dimensional distributions and therefore the theory of Davis and
Mikosch (1998) can be applied.
AMS 1991 Subject Classications: primary: 60H25
secondary: 60G10, 60J05
Keywords: ARCH model, autoregressive process, extremal index, geometric ergodicity, heavy
tail, multivariate regular variation, point processes, sample autocovariance function, strong mixing
School of Operations Research and Engineering, 232 Rhodes Hall, Cornell University, Ithaca, New York 14853,
USA, email: borkovec@orie.cornell.edu
1
1 Introduction
Over the last two decades, there has been a great deal of interest in modelling real data using time series models which exhibit features such as long range dependence, nonlinearity and
heavy tails. Many data sets in econometrics, nance or telecommunications have these common
characteristics. In particular, they appear to be reconcilable with the assumption of heavy-tailed
marginal distributions. Examples are le lengths, CPU time to complete a job or length of on/o
cycles in telecommunications and logreturns of stock indices, share prices or exchange rates in
nance.
The feature of nonlinearity can be often detected by considering the sample autocorrelation
functions (ACFs) of a time series, their absolute values and squares. The reason is the following.
A heavy tailed time series that can be represented as an innite moving average process has the
property that the sample ACF at lag h converges in probability to a constant (h) although the
mathematical correlation typically does not exist (Davis and Resnick (1985),(1986)). However,
for many nonlinear heavy tailed sequences the sample ACF at lag h converges in distribution to
a nondegenerate random variable. In Resnick and Van den Berg (1998) a test for (non)linearity
of a given innite variance time series is proposed, based on subsample stability of the sample
ACF.
The phenomenon of random limits of sample ACFs was observed rst in the context of
innite variance bilinear processes by Davis and Resnick (1996) and Resnick (1997). Davis
and Mikosch (1998) studied the weak limit behavior for a large variety of nonlinear processes
with regularly varying marginal distributions which satisfy a weak mixing condition and some
additional technical assumptions. It is shown in their article that the sample autocovariance
function (ACVF) and ACF of such processes with innite 4th but nite second moments have
a rate of convergence to the true ACVF and ACF that become slower the closer the marginal
distributions are to an innite second moment. In cases of an innite second moment, the limits
of the sample ACVF and ACF are nondeterministic. Processes which belong to the framework of
Davis and Mikosch (1998) are the ARCH(1) processes, the simple bilinear processes with lighttail noise (Basrak, Davis and Mikosch (1999)) and the GARCH(1,1) processes (Mikosch and
Starica (1999)). Finally, Davis, Mikosch and Basrak (1999) embedded the three aforementioned
processes to a larger class of processes which still satisfy the conditions for the theory of Davis
and Mikosch (1998). These processes have basically the property that they can be transformed
to solutions of multivariate linear stochastic recurrence equations. Linear stochastic recurrence
2
equations of this form were considered by Kesten (1973) and Vervaat (1979) and include the
important family of the squares of GARCH processes.
The aim of this paper is to apply the general theory of Davis and Mikosch (1998) to a
dierent type of processes with dierent structure than considered in Davis, Mikosch and Basrak (1999), namely the autoregressive (AR) processes of order 1 with autoregressive conditional
heteroscedastic one (ARCH(1)) errors. The class of AR (or more general ARMA) models with
ARCH errors were rst proposed by Weiss (1984). In the paper of Weiss, they were found to
be successful in modelling thirteen dierent U.S. macroeconomic time series. AR models with
ARCH errors are one of the simplest examples of models that can be written by a random
recurrence equation of the form
Xt = t + t "t ; t 2 N ;
(1.1)
where "t are iid innovations with mean zero, t is the conditional expectation of Xt (which may
or may not depend on t) and the volatility t describes the change of (conditional) variance.
Because of the nonconstant conditional variance models of the form (1.1) are often referred to
as conditional heteroscedastic models. Empirical work has conrmed that such models t many
types of nancial data (log-returns, exchange rate, etc.). In this paper, we concentrate on the
AR(1) process with ARCH(1) in order to have a Markov structure and hence make the model
analytically tractable. It is dened by specifying t and t as follows:
t = Xt
1
and t2 = + Xt2 1 ;
(1.2)
where 2 R and ; > 0. Note that for = 0 we get just the ARCH(1) model introduced by
Engle (1982).
The research of the sample ACVF and ACF of the AR(1) process with ARCH(1) errors
is motivated by the following. The AR(1) process with ARCH(1) errors is a natural mixture
between an AR(1) and an ARCH(1) process. Therefore results of this paper can be seen as a
generalization of results for the aforementioned two processes. The weak limit behavior of the
ARCH(1) process was studied by Davis and Mikosch (1998). For = 0, the process dened by
(1.1) and (1.2) is an AR(1) process. A summary of results about the asymptotical theory of the
sample ACFs of AR processes can be found for instance in Brockwell and Davis (1990), Chapter
7.2 and 13.3, or Embrechts, Kluppelberg and Mikosch (1997), Chapter 7.3.
AR(1) processes with ARCH(1) errors are not solutions of linear stochastic recurrence equations and there is also no obvious way how to transform them to such equations. However, we
3
show that the processes still belong to stationary weak dependent sequences which are jointly
regularly varying. One conclusion of this paper is that AR(1) processes with ARCH(1) errors
serve as one of the simplest examples of sequences which do not fulll the framework in Davis,
Mikosch and Basrak (1999) but to which the theory of Davis and Mikosch (1998) can still be
applied.
The paper is organized as follows. In Section2 we introduce the AR(1) model with ARCH(1)
errors and consider some basic theoretical properties of it. The weak convergence of the point
processes associated with the sequences (Xt ), (jXt j) and (Xt2) is investigated in Section 3. Finally,
in Section 4 we present the results concerning the weak convergence of the sample ACVF and
ACF of the AR(1) process with ARCH(1) errors, the absolute and squared values.
2 Preliminaries
We consider an autoregressive model of order 1 with autoregressive conditional heteroscedastic
errors of order 1 (AR(1) model with ARCH(1) errors) which is dened by the stochastic dierence
equation
q
Xt = Xt 1 + + Xt2 1"t ; t 2 N ;
(2.1)
where ("t ) are i.i.d. random variables, 2 R; ; > 0 and the parameters and satisfy in
addition the inequality
p
E (ln j + "j) < 0 :
(2.2)
This condition is necessary and sucient for the existence and uniqueness of a stationary distribution. Here " is a generic random variable with the same distribution as "t . In what follows we
assume the same conditions for " as in Borkovec and Kluppelberg (1998). These are the so-called
general conditions:
" is symmetric with continuous Lebesgue density p(x) ;
" has full support R ;
the second moment of " exists ;
and the technical conditions (D:1) (D:3):
(D.1) p(x) p(x0) for any 0 x < x0 .
4
(2.3)
(D.2) For any c 0 there exists a constant q = q(c) 2 (0; 1) and functions f+(c; ); f (c; ) with
f+(c; x); f (c; x) ! 1 as x ! 1 such that for any x > 0 and t > xq
p( xp+ c + t2 ) p( px + t 2 ) f+(c; x) ;
+ t
+ t
x
+
c
t
x
t ) f (c; x) :
p
p( p
)
p
(
+ t2
+ t2
(D.3) There exists a constant > 0 such that
p(x) = o(x
p
N +1++3q)=(1 q)) ;
(
as x ! 1 ;
where N := inf fu 0 ; E (j "ju ) > 2g and q is the constant in (D:2).
There exists a wide class of distributions which satisfy these assumptions. Examples are the
normal distribution, the Laplace distribution or the Students distribution. Conditions (D:1)
(D:3) are necessary for determing the tail of the stationary distribution. For further details
concerning the conditions and examples we refer to Borkovec and Kluppelberg (1998). Note
that the process (Xn)n2N is evidently a homogeneous Markov chain with state space R equipped
with the Borel -algebra. The next theorem collects some results on (Xt).
Theorem 2.1 Consider the process (Xt ) in (2.1) with ("t ) satisfying the general conditions
(2.3) and with parameters and satisfying (2.2). Then the following assertions hold:
(a) (Xt) is geometric ergodic. In particular, (Xt) has a unique stationary distribution and
satises the strong mixing condition with geometric rate of convergence X (h), h 0. The
stationary df is continuous and symmetric.
(b) Let F (x) = P (X > x); x 0; be the right tail of the stationary df and the conditions
(D:1) (D:3) are in addition fullled. Then
F (x) c x ; x ! 1 ;
where
p
p
2 j
X
j
+
E
+
X
"
)
j
X
j
"
(
+
p c = 21
p E j + "j ln j + "j
(2.4)
(2.5)
and is given as the unique positive solution to
p
E (j + "j ) = 1 :
(2.6)
Furthermore, the unique positive solution is less than 2 if and only if 2 + E ("2) > 1.
5
p
Remark 2.2 (a) Note that E (j + "j ) is a function of ; and . It can be shown that for
xed , the exponent is decreasing in jj. This means that the distribution of X gets heavier
tails when jj increases. In particular, the AR(1) process with ARCH(1) errors has for =
6 0
heavier tails than the ARCH(1) process.
(b) The strong mixing property includes automatically that the sequence (Xt ) satises the
conditions A(an). The condition A(an ) is a frequently used mixing condition in connection with
point process theory and was introduced by Davis and Hsing (1995). See (3.7) for the denition.
2
In order to investigate the limit behavior of the sample ACVF and ACF of (Xt) we dene three
auxiliary processes (Yt), (Xet ) and (Zet ) as follows: let (Yt) and (Xet) be the processes given by
the random recurrence equations
q
Yt = Yt 1 + Yt2 1 "t ; t 2 N ;
(2.7)
and
q
eXt = jXet + + Xet "tj ; t 2 N ;
where the notation is the same as in (2.1), Y = X , Xe = jX j a.s. and set
2
1
0
0
1
0
(2.8)
0
Zet = ln(Xet2) ; t 2 N :
It is easy to see that the process (Yt) is not stationary (or at least not non-trivial stationary).
However, for Yt Xt large and M 2 N xed, we will see that the sequence Yt ; Yt+1; :::; Yt+M
behaves approximately as Xt ; Xt+1; :::; Xt+M (see also Figure 1). This fact will be very important
in order to establish joint regular variation of X0; X1 ; :::; XM .
Because of the symmetry of ("t), the independence of "t and Xt 1 in (2.1) and the homogeneous Markov structure of (Xt) and (Xet ) it is readily seen that (Xet ) =d (jXtj). Studying the
process (Xet) instead of (jXt j) can be often much more convenient. In particular, since (Xet)
follows (2.8) the process (Zet ) satises the stochastic dierence equation
q
eZt = Zet + ln( + e Zet + "t) ; t 2 N ;
(2.9)
where Ze equals ln(X ) a.s.. Note that (Zet) =d (ln(Xt )). Moreover, (Zet) does not depend on
the sign of the parameter since "t is symmetric. The following lemma shows that (Zet ) can be
0
2
1
1
2
2
0
bounded above a high threshold by a random walk with negative drift. The proof of this result
6
60
60
40
40
20
20
0
0
200
0
5
400
600
800
1000
0
200
0
5
400
600
800
1000
0
0
20
20
40
40
60
60
0
10
15
20
25
30
10
15
20
25
30
Figure 1: Simulated sample path of (Xt) with initial value X0 = 50 and parameters = 0:6; = 1; = 0:6 (left)
and of (Yt ) with the same initial value and parameters (right) in the case " N (0; 1). Both simulations are based
on the same simulated noise sequence ("t). The pictures demonstrate that the processes behave similar for large
values.
can be found in Borkovec (2000) and is based basically on the recurrence structure of (Zet) in
(2.9). The result is crucial for proving Proposition 3.1 .
Lemma 2.3 Let a be large enough, Na := inf f 1 j Ze ag and Ze > a. Then
0
Zet Ze0 + Sta for any t Na a.s. ;
where Sta is random walk with negative drift given by
S0a = 0 and
Sta
= Sta
1
p
+ ln ( + e
a + "t )2
Moreover, for a " 1, we have
Sta a:s:
! St ;
7
(2.10)
p
2p e a=2"t 1
+ ln 1
:
( + e a + "t )2 f"t<0g
p
where S0 = 0 and St = St 1 + ln( + "t )2 .
2
3 Weak convergence of some point processes associated with
the AR(1) process with ARCH(1) errors
In this section we formulate results on the weak convergence of some point processes of the form
Nn =
n
X
t=1
Xtm =an ; n = 1; 2; :::;
(
(3.1)
)
where Xt(m) are random row vectors with dimension m + 1 2 N arbitrary whose components are
highly related to the AR(1) process with ARCH(1) errors (Xt) dened in the previous section
and (an) is a normalizing sequence of real-valued numbers. The main result in this section is
summarized in Theorem 3.3. The proof of this result is basically an application of the theory in
Davis and Mikosch (1998). Proposition 3.1 collects some properties of (Xt(m)) which we need for
the proof of Theorem 3.3.
We follow the notation and the point process theory in Davis and Mikosch (1998) and
Kallenberg (1983), respectively. The state space of the point processes considered is Rm+1 nf0g.
Write M for the collection of Radon counting measures on Rm+1 nf0g with null measure o. This
P n , where n 2 f1; 2; 3; :::g and
means that 2 M if and only if is of the form = 1
i
i=1 i xi
m+1
xi 2 R n f0g distinct and #fi j jxij > y g < 1 for all y > 0 .
In what follows we suppose that (Xt) is the stationary AR(1) process with ARCH(1) given
by (2.1). ("t) satises the general conditions (2.3) and (D:1) (D:3) and the parameters and
are chosen such that (2.2) holds. We start by specifying the random row vectors (X(tm)) and
the normalising constants (an) in (3.1) and by introducing some auxilary quantities in order to
be in the framework of Davis and Mikosch (1998). For m 2 N0 , dene
Xtm = (Xt; Xt ; :::; Xt m) ; t 2 Z;
p
p
p
Z m = r ; (r + " ); :::; (r + " ) Qms ( + rs "s )
(
)
(
0
)
and
+1
0
p
0
Ztm = (r + " )
(
)
0
+
1
1
tY1
0
1
=1
1
p
( + rs "s+1); :::;
s=1
8
mY
+t 1
s=1
+1
p
!
( + rs "s+1) ; t 2 N ;
Q
where rs = sign(Ys) is independent of jYs j, (Ys ) is the process in (2.7) and 0i=1 = 1 . Besides,
for k 2 N0 arbitrary but xed, dene the stochastic vectors
X mk (2k + 1) = (X mk ; X mk ; :::; Xkm )
(
)
(
)
(
)
+1
(
)
and
Z m (2k + 1) = (Z m ; Z m ; :::; Z mk ) :
(
0
(
0
)
)
(
1
(
2
)
)
Analogously to Davis and Mikosch (1998) we take j j to be the max-norm in Rm+1, i.e.
jxj = j(x ; :::; xm)j = i max
jx j :
;:::;m i
0
=0
Now we are able to dene the sequence (an) in (3.1). Let (a(nk;m) ) be a sequence of positive
numbers such that
P (jX j > a(nk;m) ) (n E (jZ(0m) (2k + 1)j )) 1 ; as n ! 1 :
(3.2)
For k = 0, we write an = an(0;m) in the following. Note that because of (2.4) one can choose
a(nk;m) as
a(nk;m) = 2 c E (jZ0(m) (2k + 1)j)
=
1
n1= ; n 1 :
(3.3)
With this notation we can state the following proposition.
Proposition 3.1 Let (Xt ) be the stationary AR(1) process with ARCH(1) given by (2.1) and
and assume that the conditions of Theorem 2.1 hold. Then
(a) (X(tm)) is strongly mixing with a geometric rate of convergence. To be more specic, there
exist constants 2 (0; 1) and C > 0 such that for any h m
sup
A2(X(sm) ; s0);B2(Xs(m) ; sh)
jP (A \ B ) P (A) P (B )j =: X m (h)
(
)
= X (h m) C h m :
(b) X(mk)(2k + 1) is jointly regularly varying with index > 0, more precisely
n P (jX(mk)(2k + 1)j > t a(nk;m) ; X(mk) (2k + 1)=jX(mk) (2k + 1)j 2 )
(3.4)
!v t E (jZ(0m)(2k + 1)j 1fZ m (2k+1)=jZ m (2k+1)j2 g)=E (jZ(0m)(2k + 1)j) ; t > 0 ;
(
0
)
(
0
)
v stands for vague convergence.
as n ! 1, where the symbol !
9
(c) Let (pn) be an increasing sequence such that
pn ! 0 and nX m (ppn) ! 0 as n ! 1 :
(
n
)
pn
(3.5)
Then for any y > 0
1
0
_
jXtm j > an y jX m j > any A = 0 :
lim lim sup P @
p!1 n!1
(
)
(
0
pjtjpn
)
(3.6)
Remark 3.2 (a) In the spirit of Davis and Mikosch (1998) the jointly regular varying property
of X mk (2k + 1) can also be expressed in the more familiar way
(
)
n P (jX(mk) (2k + 1)j > t a(nk;m) ; X(mk)(2k + 1)=jX(mk)(2k + 1)j 2 ) !v t P ( ) ; as n ! 1 ;
where P ( ) = Pe ((kk) ; :::; k(k) ) 1 , j(k) = Z(km+)j =jZ(0m) (2k + 1)j, j = k; :::; k, and dPe =
jZ0(m)(2k + 1)j=E (jZ(0m) (2k + 1)j ) dP : In the following we will basically use this notation.
(b) Due to statement (b) in Proposition 3.1 the positive sequence (an) in (3.2) with k = 0 can
be also characterized by
lim n P (jX(0m)j > an) = 1 :
n!1
Hence an can be interpreted as the (approximated) (1 n 1)-quantile of X0(m) .
(c) In the case of a strong mixing process, the conditions in (3.5) are sucient to guarantee that
(pn) is a A(an )-separating sequence, i.e.
E exp
n
X
t=1
!
f (Xtm)=an )
(
E exp
pn
X
t=1
!!kn
f (Xtm)=an )
(
! 0 ; as n ! 1 ; (3.7)
where kn = [n=pn] and f is an arbitrary bounded non-negative step function on Rm n f0g . Note
that (pn) is in the case of a strong mixing process independent of (an).
Proof. (a) This is an immediate consequence of the strong mixing property of (Xn) and
the fact that strong mixing is a property of the underlying -eld.
(b) Fix t > 0 and let > 0 be small enough such that t 2 > 0. Moreover, choose B 2
B(S (2k+1)(m+1) 1) arbitrary, where S := S (2k+1)(m+1) 1 denotes the unit sphere in R(2k+1)(m+1)
with respect to the max-norm j j. Dene B = fx 2 S j 9 y 2 B : jx yj 2=(t 2)g and
B = fx 2 B j jx yj 2=(t 2) 8y 2 S n B g. Note that B B B .
Next set Yt(m) := (Yt ; Yt+1; :::; Yt+m) ; t 2 N0 ; and Y0(m)(2k +1) = (Y0(m); Y1(m); :::; Y2(mk )), where
10
(Yt ) is the process given in (2.7). Using the denition of the process (Yt) and of the stochastic
vectors Z(tm), it can be readily seen that
Y m (2k + 1) = jX j Z m (2k + 1) :
(
0
)
(3.8)
( )
0
0
The basic idea of proving (3.4) is now to approximate X(mk) (2k + 1) by Y0(m)(2k + 1). Because
of the stationarity of (Xt) it is sucient to compare X(0m)(2k + 1) with Y0(m) (2k + 1). First we
bound the probability in the left hand side of (3.4) from above as follows.
n P jX(mk)(2k + 1)j > t a(nk;m) ; X(mk) (2k + 1)=jX(mk) (2k + 1)j 2 B
= n P jX(0m) (2k + 1)j > t a(nk;m) ; X0(m)(2k + 1)=jX(0m)(2k + 1)j 2 B
n P jX(0m) (2k + 1) Y0(m) (2k + 1)j > a(nk;m)
+ n P jX(0m) (2k + 1)j > t a(nk;m) ; X(0m) (2k + 1)=jX(0m) (2k + 1)j 2 B ;
jX0(m) (2k + 1) Y0(m)(2k + 1)j a(nk;m)
n P jX(0m) (2k + 1) Y0(m) (2k + 1)j > a(nk;m)
+ n P jY0(m)(2k + 1)j > (t ) a(nk;m) ; Y0(m) (2k + 1)=jY0(m) (2k + 1) 2 B ;
jY0(m)(2k + 1) Y0(m)(2k + 1)j a(nk;m)
=: (I1) + (I2) ;
where the last inequality follows from the fact that for any x; y 2 R(2k+1)(m+1) the inequalities jx yj a(nk;m) and jxj > ta(nk;m) imply jyj > (t )a(nk;m) , jx=jxj y=jxjj =t and
jjyj=jxj 1j =(t 2). The rest is a triangular argument.
First, we consider (I1). By the denition of the max-norm and due to the Boolean and
Markov inequalities, we derive
(I1) = n P (0smax
jX Ysj > a(nk;m) )
2k+m s
q
= n P (0smax
j + Xs
2k+m
2
q
1
p
Xs2 1j j"s j > a(nk;m) )
n P ( smax
j" j > ank;m = )
k m s
E (j"j ) ;
(2k + m + 1) nk;m
p
(a = ) (
0
)
2 +
+
(
n
E (j"j+ )
)
+
where > 0 is chosen such that
< 1 . This is possible because of the assumption
(D:3). Using (3.3) the right hand side converges to zero.
11
Now we estimate (I2). By (3.8) , we have
(I2) n P jX0j jZ(0m) (2k + 1)j > (t )a(nk;m) ; Z(0m)(2k + 1)=jZ(0m)(2k + 1)j 2 B
= n P jX0j jZ(0m) (2k + 1)j 1fZ m (2k+1)=jZ m (2k+1)j2B g > (t )a(nk;m)
(
0
)
(
0
)
Note that jX0 j and Z(0m) (2k + 1) are independent, nonnegative random variables. Moreover,
jX0j is regularly varying with index > 0 and E (jZ(0m) (2k + 1)j) < 1. Thus, a result of
Breiman (1965) yields that (I2) behaves asymptotically as
n P jX0j > (t )a(nk;m) ) E (jZ(0m)(2k + 1)j 1fZ m (2k+1)=jZ m (2k+1)j2B g
E (jZ(0m)(2k + 1)j 1fZ m (2k+1)=jZ m (2k+1)j2B )
(t )
; as n ! 1 ;
E (jZ(0m)(2k + 1)j)
where we used in the second line (3.2). Because > 0 is arbitrary and B # B as # 0 we have
found that
(
0
(
0
)
(
0
)
(
0
)
)
lim sup n P jX(mk) (2k + 1)j > t a(nk;m) ; X(mk)(2k + 1)=jX(mk)(2k + 1)j 2 n!1
t E jZ m (2k + 1)j 1fZ m
(
0
)
(
0
)
jZ m (2k + 1)j) :
m) (2k+1)j2 g )=E (
(2k+1)=jZ
(
0
( )
0
(3.9)
Next we proceed to establish that the inequality (3.9) also holds in the converse direction for
lim inf. With similar arguments as above, we have
n P jX(mk)(2k + 1)j > t a(nk;m) ; X(mk) (2k + 1)=jX(mk) (2k + 1)j 2 B
n P jY0(m)(2k + 1)j > (t + ) a(nk;m) ; Y0(m)(2k + 1)=jY0(m)(2k + 1) 2 B ;
jX(0m) (2k + 1) Y0(m)(2k + 1)j a(nk;m)
n P jY0(m)(2k + 1)j > (t + ) a(nk;m) ; Y0(m)(2k + 1)=jY0(m)(2k + 1) 2 B n P jX(0m) (2k + 1) Y0(m) (2k + 1)j > a(nk;m)
E (jZ(0m)(2k + 1)j 1fZ m (2k+1)=jZ m (2k+1)j2B g)
(t + )
; as n ! 1 :
E (jZ(0m)(2k + 1)j)
Since again > 0 is arbitrary the statement follows.
(c) We start by rewriting the probability in statement (c).
(
0
P
_
(
0
)
jXtm (2k + 1)j > an y jX m (2k + 1)j > an y
(
pjtjpn
)
)
( )
0
= P p tmax
jXtj > any 0max
jX j > an y
j 2k+1 j
n p+2k+1
+ P ptmax
j
X
j
>
a
y
max
j
X
j
>
a
y
t
n
j
n
p +2k+1
0j 2k+1
=: (J1) + (J2) :
n
12
In what follows we consider only (J1). (J2) can be treated in a similar way. First note that
(J1) k P (max
X
2 +1
j =0
2k+1
X
P
j =0
P (jXj j > an y)
pnt p+2k+1 jXt j > an y; jXj j > any)
P (jXj j > any)
P (max0j 2k+1 jXj j > any)
max
jX j > any jX0j > any
pn j t p+2k+1 j t
2(k + 1) P
2(k + 1)
jXt j > an y jX j > any
max
0
pn (2k+1)t p+2k+1
pX
+2k+1
P jXt j > an y jX0j > any :
pn (2k+1)
t=
Moreover, using again the property of conditional probability together with the stationarity of
(Xt ) and substituting t by t, we get that (J1) is bounded by
2(k + 1)
t= pn
= 2(k + 1)
P jX tj > any jX0 j > an y
(2k+1)
pn+(2
Xk+1)
t=p
pX
+2k+1
k
P jXt j > any jX0j > an y :
(2 +1)
Recalling that (Zet ) = (ln(Xet)2) =d (ln(Xt )2) it follows that the last expression can be also
expressed by
2(k + 1)
pn+(2
Xk+1)
t=p
k
P Zet > ln(any)2 Ze0 > ln(any)2 :
(2 +1)
Next, set Na = inf fs 2 N ; Zes a g as in Lemma 2.3. Choose the threshold a large enough
=4
p
p
for a xed
in order to guarantee that E ( + e a + ")2 2 e a=2"1f"<0g
p u
2 (0; 1). This is possible because of (2.2) which implies that E (j + "j ) < 1 for all u 2 (0; )
and the fact that
p
p
E ( + e a + ")2 2 e
a=2"1
f"<0g
= 4
p
! E j + "j= ; a ! 1 ;
2
by the dominated convergence theorem. We derive
k
p X
(J1) 2(k + 1)
+
2( +1)
pnX
+2k
=p
k
=1
1
pn+(2
Xk+1)
t=p (2k+1)
pn+(2
Xk+1)
t p
(2 +1) =
k
P (Zet > ln(any)2; Na = j Ze0 > ln(any)2)
P (Zet > ln(any)2; Na = j ln Ze0 > ln(any)2)
(2 +1)
13
+
1
X
pn+(2
Xk+1)
=pn+(2k+1) t=p
k
P (Zet > ln(any)2; Na = j Ze0 > ln(any)2)
(2 +1)
=: 2(k + 1) (K1 ) + (K2) + (K3)
It can be shown now (see Borkovec (2000)) that the summands (K1), (K2) and (K3 ) tend to
zero as n ! 1 and then p ! 1. The basic idea underlying this result is to use Lemma 2.3
and the fact that the expression n P (Zet > ln(any)2 j Ze0 = x) is uniformly bounded for any
n 2 N; x 2 [e n; ea ] and t 2 N . This nishes the proof.
2
Proposition 3.1 provides some properties for (Xt(m)) and (Xt(m) (2k + 1)) which are just the
required assumptions in Davis and Mikosch (1998) for weak convergence of point processes of
the form (3.1). If we dene
f = f 2 M j (fx j jxj > 1g) = 0 and (fx j x 2 S mg) > 0g)
M
f) be the Borel -eld of M
f then the following theorem is an immediate
and if we let B(M
consequence of Proposition 3.1.
Theorem 3.3 Assume (Xt) is the stationary AR(1) process with ARCH(1) errors satisfying
the conditions of Theorem 2.1. Then
1 X
1
n
X
X
X
X
Nn = Xt=an ! N =
Pi Qij ;
t=1
where Xt = X(tm) ,
(3.10)
i=1 j =1
P1 is a Poisson process on (0; 1] with intensity
i Pi
Yk
p
=1
(dy) = E (sup
( + "s ) P 1) y
k1 s=1
1
dy
P
and P is a Pareto()-distributed random variable, independent of ("s). The process 1
i=1 Pi is
P
independent of the sequence of iid point processes 1
j =1 Qij , i 1, with joint distribution Q on
f; B(M
f)), where Q is the weak limit of
(M
Ee j0(k) j
_k
j =1
jjk j 1f g
( )
+
X
jtjk
tk
( )
e k _k k =E j j
jj j
( )
0
( )
j =1
+
(3.11)
as k ! 1, and the limit exists. Ee is the expectation with respect to the probability measure dPe
dened in Remark 3.2(a).
14
Remark 3.4 Analogous results can be found for the vectors
jXtj = jXtm j = (jXtj; :::; jXt mj) and Xt = Xtm = (Xt ; :::; Xt m) ; t 2 Z; m 2 N ;
(
)
)2
(
2
+
2
2
+
by using (3.10) and the continuous mapping theorem. Thus, under the same assumptions as in
Theorem 3.3, we have
NnjXj =
and
n
X
t=1
jXt j=an !
1 X
1
X
i=1 j =1
PijQij j
XX
X
PiQij
NnX = Xt =an !
1 1
n
2
2
t=1
i=1 j =1
2
where the sequences (Pi), (Qij ) are the same as above and
jQij jl = (jQij jl ; jQij jl ; :::; jQijm jl ) ; l = 1; 2 :
(0)
(1)
(
)
Proof. The proof is simply an application of Theorem 2.8 in Davis and Mikosch (1998). The
assumptions of this theorem are satised because of Proposition 3.1. Finally, the extremal index
W
= limk!1 E j0(k) j kj=1 jj(k) j + =E j0(k) j of the AR(1) process with ARCH(1) errors is
not zero and is specied by the formula (see Borkovec (2000))
Yk
p
= E (sup ( + "i ) P 1) ;
k1 i=1
where P is a Pareto()-distributed random variable, independent of ("s). Hence the statement
follows.
2
4 Asymptotic behavior of the sample ACVF and ACF
In what follows we derive the limit behaviour of the sample ACVF and ACF of the stationary
AR(1) process with ARCH(1) considered in the previous sections. The point process results of
the section 3 will be crucial.
Dene the sample ACVF of (Xt ) by
nXh
n;X (h) = n1 Xt Xt+h ; h = 0; 1; ::: ;
t=1
15
and the corresponding sample ACF by
n;X (h) = n;X (h)=n;X (0) ; h = 0; 1; ::: :
The sample ACVF and ACF for (jXt j) and (Xt2) are given in the same way. Moreover, we write
X (h) = E (X0Xh ) ; jX jl (h) = E (jX0jl jXh jl ) ;
and
X (h) = X (h)=X (0) ; jX jl (h) = jX jl (h)=jX jl (0) ; l = 1; 2 h = 0; 1; :::
if these quantities exist. If this is the case straightforward calculations yield
X (h) = h X (0) = h =(1 2 E ("2 ))
and
X (h) = ( + E ("
2
2
2
))h
X 2 (0) + E ("
2
)X (0)
h
X
1
j =0
(2 + E ("2 ))j ; h 0 ;
where
X (0) = 2X (0)(22E ("2) + E ("4 ))=(1 4 62 E ("2 ) 2 E ("4)) :
2
Mean-corrected versions for the sample ACVF and ACF can also be investigated. However one
can show (with the same approach as in the proof of Theorem 4.1) that the limits stay the same
(see also Remark 3.6 of Davis and Mikosch (1998)).
In order to state our results we have to introduce several mappings. Let > 0, xt =
m+1
(m)
(x(0)
n f0g and dene the mappings
t ; :::; xt ) 2 R
Th;k; : M ! R
by
P
P
n = 1 n1
T 1; 1; 1
;
P1 t=1 t xt P1 t=1 (th)fjx(kt) j>g
Th;k; t=1 nt xt = t=1 nt xt xt 1fjxt j>g ; h; k 0 ;
(0)
(0)
where nt 2 N0 for any t 1. Since the set fx 2 Rm+1 n f0g j jx(h) j > g is bounded for any
h = 0; :::; m the mappings are a.s. continuous with respect to the limit point processes N X, N jXj
and N X . Consequently, by the continuous mapping theorem, we have in particular
2
XX
X
1fjPiQij j>g
T 1; 1; (NnX) = 1fjXt j>g !d T 1; 1; (N X) =
1 1
n
t=1
(0)
i=1 j =1
16
(0)
(4.1)
and for any h; k 0
X X 2 (h) (k)
X
Pi Qij Qij 1fjPi Qij j>g : (4.2)
Th;k; (NnX) = Xt(h) Xt(k) 1fjXt j>g !d Th;k; (N X) =
1 1
n
(0)
t=1
i=1 j =1
(0)
Note that, with obvious modications, (4.1) and (4.2) hold also for NnjXj and N jXj respectively
NnX and N X . The following theorem collects the weak limit results of the sample ACVF and
ACF of (Xt), (jXt j) and (Xt ) depending on the tail index > 0. The weak limits turn to be
innite variance stable random vectors. However, they are only functionals of point processes
and have no explicit representation. Therefore, the results are only of qualitative nature and
explicit asymptotic condence bounds for the sample ACVFs and ACFs can't be constructed.
2
2
Theorem 4.1 Assume (Xt) is the stationary AR(1) process with ARCH(1) errors satsifying
the conditions of Theorem 2.1 with E ("2) = 1. Let > 0 be the tail index in (2.6) and (an) be
the sequence satisfying (3:3) for k = 0. Then the following statements hold:
(1) (a) If 2 (0; 2), then
nan 2n;X (h) h=0;:::;m !d (VhX )h=0;:::;m ;
d (V X =V X )
(n;X (h))h=1;:::;m !
h 0 h=1;:::;m ;
and
nan 2n;jX j (h) h=0;:::;m !d (VhjX j )h=0;:::;m ;
n;jX j (h) h=1;:::;m !d (VhjX j =V0jX j )h=1;:::;m ;
where the vectors (V0X ; :::; VmX ) and (V0jX j ; :::; VmjX j ) are jointly =2-stable in Rm+1 with point
process representation
VhX
and
VhjX j =
1 X
1
X
i=1 j =1
=
1 X
1
X
i=1 j =1
(h)
Pi2 Q(0)
ij Qij
(h)
Pi2jQ(0)
ij jjQij j ; h = 0; :::; m ; respectively .
(b) If 2 (0; 4), then
nan 4n;X (h) h=0;:::;m !d (VhX )h=0;:::;m ;
n;X (h) h=1;:::;m !d (VhX =V0X )h=1;:::;m ;
2
2
2
2
17
2
where (V0X ; :::; VmX ) is jointly =4-stable in Rm+1 with point process representation
2
2
VhX 2
=
1 X
1
X
i=1 j =1
(h) 2
Pi4 (Q(0)
ij Qij ) ; h = 0; :::; m :
respectively,
(2) (a) If 2 (2; 4) and E ("4) < 1, then
nan 2(n;X (h) X (h)) h=0;:::;m !d (VhX )h=0;:::;m ;
nan 2(n;X (h) X (h)) h=1;:::;m !d X1(0)(VhX X (h)V0X )h=1;:::;m
and
nan 2(n;jX j (h) jX j (h)) h=0;:::;m !d (VhjX j )h=0;:::;m ;
nan 2(n;jX j (h) jX j (h)) h=1;:::;m !d jX1j (0)(VhjX j jX j (h)V0jX j )h=1;:::;m ;
where the vectors (V0X ; :::; VmX ) and (V0jX j ; :::; VmjX j ) are jointly =2-stable in Rm+1 with
; V X = Ve X + V X ; m 1 ;
m
m
m
V0X = Ve0X 1 (2 + )
1
1
and
p
V0jX j = V0X ; VmjX j = VemjX j + E (j + "j)VmjX j1 ; m 1 :
Furthermore, (Ve0X ; :::; VemX ) and (Ve0jX j ; :::; VemjX j ) are the distributional limits of
T1;1; (N X) (2 + )T0;0; (N X); T0;h; (N X) T0;h
and
T1;1; (N jXj)
X
1; (N )
h=1;:::;m
nan 4(n;X
2
nan 4 (n;X (h) X (h)) h=0;:::;m !d (VhX )h=0;:::;m ;
d 1 (0)(V X (h)V X )
(h) X (h)) h=1;:::;m !
X
h=1;:::;m ;
0
h
X
2
2
2
2
2
2
2
where (V0X ; :::; VmX ) is jointly =4-stable in Rm+1 with
2
2
; V X = Ve X + ( + )V X ; m 1 ;
m
m
m
V0X = Ve0X 1 (4 + 62 + 2 E ("4))
2
p
( + )T0;0; (N jXj); T0;h; (N jXj) E (j + "j)T0;h 1; (N jXj) h=1;:::;m ;
2
respectively, as ! 0.
(b) If 2 (4; 8) and E ("8 ) < 1, then
2
2
1
18
2
2
2
1
and (Ve0X ; :::; VemX ) is the distributional limit of
2
2
T1;1; (N X ) (4 + 62 + 2 E ("4))T0;0; (N X ); T0;h; (N X ) (2 + )T0;h
2
2
as ! 0.
(3) (a) If 2 (4; 1), then
n1=2(n;X (h) X (h))
n1=2(n;X (h) X (h))
and
2
d (GX )
!
h h
;:::;m
h=0
d
1
! X (0)(GXh
h=1;:::;m
=0
X)
;
1; (N
h=1;:::;m
2
;:::;m ;
X (h)GX0 )h=1;:::;m
n1=2(n;jX j(h) jX j (h)) h=0;:::;m !d (GjhX j)h=0;:::;m ;
1=2
n (n;X (h) X (h)) h=1;:::;m !d X1 (0)(GjhX j X (h)Gj0X j)h=1;:::;m ;
where the limits are multivariate Gaussian with mean zero.
(b) If 2 (8; 1), then
=
n (n;X (h) X (h)) h
=
1 2
2
2
n1 2(n;X (h) X (h))
2
2
d (GX
!
h )h ;:::;m ;
;:::;m
!d X (0)(GXh X (h)GX )h
;:::;m
2
=0
=0
h=1
1
2
2
2
2
0
=1
;:::;m ;
where the limits are multivariate Gaussian with mean zero.
Remark 4.2 (a) Theorem 4.1 is a generalization of results for the ARCH(1) process (see Davis
and Mikosch (1998)). They use a dierent approach which does not extend to the general case
because of the autoregressive part of (Xt ).
(b) The assumption 2 := E ("2) = 1 in the theorem is not a restriction. In cases where the
second moment is dierent from one consider the process (Xbt) dened by the stochastic recurrence
equation
q
Xbt = Xbt 1 + =2 + Xbt2 1 "t = ; t 2 N ;
where the notation is the same as for the process (Xt) in (2.1). Note that (Xbt) = (Xt=2 ). Since
the assumptions in the theorem do not dependent on the parameter the results hold for (Xbt) and
hence they also hold for (Xt ) replacing the limits (VhX ; VhjX j ;VhX )h=0;:::;m by 4 (VhX ; VhjX j ;VhX )h=0;:::;m
and (GXh ; GjhX j;GXh )h=0;:::;m by 4 (GXh ; GjhX j;GXh )h=0;:::;m, respectively.
2
2
2
19
2
(c) Note that the description of the distributional limits in part (2) of Theorem 4.1 is dierent
than in Theorem 3.5 of Davis and Mikosch (1998). In the latter theorem the condition
lim
lim sup var an
#0
n!1
2
nXh
t=1
!
Xt Xt+h 1fjXtXt hjang = 0
+
2
is required. However, this condition is very strong and does not seem to be in general fullled when
(Xt ) is correlated (see e.g. Theorem 1.1 of Rio (1993) for a possible justication) . Therefore, we
choose another way and establish the convergence in distribution of the sample ACVF directly
from the point process convergence in Theorem 3.3.
Proof. Statements (1a) and (1b) are immediate consequences of Theorem 3.5(1) of Davis
and Mikosch (1998). Note that all conditions in this theorem are fullled because of Proposition 3.1 and Theorem 3.3. Statements (3a) and (3b) for the sample ACVFs follows from standard
limit theorems for strongly mixing sequences (see e.g. Theorem 3.2.1 of Zhengyan and Chuanrong (1996)). The limit behavior for the ACFs can be shown in the same way as e.g. in Davis
and Mikosch (1998), p.2062 .
It remains to show (2a) and (2b). We restrict ourselves to the case (jXtj) and only establish joint
convergence of (n;jX j(0); n;jX j (1)). All other cases can be treated similar or even easier. Recall
that (Xet ) =d (jXt j), where the process (Xet) is dened in (2.8). Thus it is sucient to study the
sample ACVF of the process (Xet).
We start by rewriting n;Xe (0) using the recurrence structure of (Xet )
nan n;Xe (0) Xe (0) = an
2
= (2 + )an 2
n X
e
2
n X
e
t=1
Xt2+1 E (Xe 2)
Xt2 E (Xe 2)
t
q
n X
e
2Xt + Xet "t + ( + Xet )("t
+ an
=1
2
2
t=1
We conclude that for any > 0 ,
n
X
1 (2 + ) nan 2 n;Xe (0) Xe (0)
= an 2
t=1
+1
( + Xet2 )("2t+1 1)1fXetang
+ 2an 2
q
n
X
e
t=1
Xt + Xet2 "t+11fXetang
20
2
2
+1
1) :
+ an
2
n q
X
e
Xt + Xet "t + ( + Xet )("t
t=1
2
2
+1
2
+1
1) 1fXet>ang + oP (1)
=: (I1) + (I2) + (I3) + oP (1) :
We show rst that (I1) and (I2) converge in probability to zero. Note that the summands in (I1)
are uncorrolated. Therefore,
var(I1) = an 4
an
4
n
X
t=1
var ( + Xet2 ) 1fjXetjang ("2t+1 1)
n X
t=1
E ( + Xet2 )2 1fjXetjang E ("2t+1 1)2
const ; as n ! 1 ;
! 0 ; as # 0 ;
4
where the asymptotic equivalence comes from Karamatas theorem on regular variation and the
tail behavior of the stationary distribution of (Xet). Note that the condition E ("4 ) < 1 is crucial.
Analogously, one can show that
lim
lim var(I2) = 0 :
#0 n!1
Now we consider (I3). (2.8) yields
(I3) = an 2
n
X
e
t=1
Xt2+11fXet>ang (2 + )an 2
n
X
e
t=1
Xt21fXet>ang an 2
=d T1;1; (NnjXj) (2 + )T0;0; (NnjXj) an 2 T 1; 1; (NnjXj)
!d T1;1; (N jXj) (2 + )T0;0; (N jXj) ;
n
X
t=1
1fXet>ang
(4.3)
where the limit has expectation zero. Finally, following the same arguments as in Davis and
Hsing (1995), pp. 897-898, the right hand side in (4.3) converges in distribution to a =2-stable
random variable, as ! 0 .
Now consider n;Xe (1). We proceed as above and write
nan 2 n;Xe (1) Xe (1) = an 2
= an 2
n 1
X
t=1
n
X
Xet Xet
1
t=1
+1
E (Xe0Xe1)
f"t (Xet) E (f"t (Xet)) + an 2
+1
+1
=: (J1) + (J2) ;
21
n X
e
1
t=1
p
p
Xt2j + "t+1j E (Xe 2)E (j + "j)
p
p
where fz (y) = jyj jy + + y2 z j jy + yz j for any y; z 2 R. First, we show that (J1)
converges in probability to zero. Observe for that purpose that
var jan 2
n
X
1
t=1
var an
= an 4
f"t (Xet ) E (f"t (Xet))j
+1
2
n
X
t=1
n X
n
X
p
t=1 s=1
!
+1
jf"t (Xet )j
!
(4.4)
+1
cov(jf"t (Xet )j; jf"s (Xes )j) :
+1
+1
Now note that jfz (y)j jyj jz j for any y; z 2 R. Therefore and since > 2 there exists a
> 0 such that
p
E (jf"(Xe )j2+) E (j"j2+)E (jXe j2+) < 1 :
(4.5)
Because of (4.5) and the geometric strong mixing property of (Xet) all assumptions of Lemma 1.2.5
of Zhengyan and Chuanrong (1996) are satised and we can bound (4.4) by
const an 4 n
n
X
s=1
(2=(2+))s
(4.6)
which converges to zero as n ! 1 since < 4. Next we rewrite (J2) and get
p
(J2) = E (j + "j)nan 2 n;Xe (0) Xe (0)
+ an 2
+ an 2
n
X
e
1
t=1
nX1
t=1
p
p
p
p
Xt2 1fXetang j + "t+1 j E (j + "j)
Xet2 1fXet>ang j + "t+1 j E (j + "j)
= (K1) + (K2) + (K3) :
d T (N jXj) (2 + )T (N jXj). Moreover, using the same arguments as
By (4.3), (K1) !
1;1;
0;0;
before one can show that lim#0 limn!1 var(K2) = 0. Hence (K2 ) = oP (1). It remains to consider
(K3). We begin with the decomposition
(K3) = an 2
n
X
e
1
t=1
q
p
Xt 1fXet>ang jXet + Xet "t+1j jXet + + Xet2"t+1 j
n
n
X
X
p
e
e
+ an
Xt Xt 1fXet>ang an
Xet 1fXet>angE (j + "j) :
1
2
t=1
1
2
+1
2
t=1
Proceeding the same way as in (4.4)-(4.6) the rst term converges in probability to zero. Thus,
p
(K3) =d oP (1) + T0;1; (NnjXj) E (j + "j)T0;0; (NnjXj)
p
!d T ; ; (N jXj) E (j + "j)T ; ; (N jXj) ;
00
01
22
where the limit has zero mean and converges again to a =2-stable random variable as # 0.
Since for the distributional convergence only the point process convergence and the continuous
mapping theorem has been used, it is immediate that the same kind of argument yields the joint
convergence of the sample autocovariances to a =2-stable limit as described in the statement.
Finally, the asymptotic behavior of the sample ACF can be shown in the same way as in Davis
and Mikosch (1998), p.2062 .
2
Acknowledgement
I am very grateful to Thomas Mikosch for useful comments and advice. Furthermore, I thank
Claudia Kluppelberg for making me aware to this problem.
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23
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24
1.0
-0.5
0.0
0.5
1.0
0.5
0.0
-0.5
5
10
15
20
0
5
10
15
20
0
5
10
15
20
-0.5
0.0
0.5
1.0
0
Figure 2: ACFs of the AR(1) process with ARCH(1) errors with standard normal distributed innovations ("t )
and parameters = 0:2; = 1 and = 0:4 (top, left), = 0:4, = 1 and = 0:6 (top, right) and = 0:8,
= 1, = 0:6 (bottom). In the rst case = 5:49, in the second = 2:87 and in the last = 1:35. The dotted
lines indicate the 5% and 95%-quantiles of the empirical distributions of the sample ACFs at dierent lags. The
underlying simulated sample paths have length 1000. The condence bands were derived from 1000 independent
simulations of the sample ACFs at these lags. The plots conrm the dierent limit behavior of the sample ACFs
as described in this article.
25