A Limit Theorem for Mildly Explosive
Autoregression with Stable Errors
Alexander Aue
1 2
and Lajos Horváth 1
Abstract: We discuss the limiting behavior of the serial correlation coefficient in mildly
explosive autoregression, where the error sequence is in the domain of attraction of an
α–stable law, α ∈ (0, 2]. Therein, the autoregressive coefficient ρ = ρn > 1 is assumed to
satisfy the condition ρn → 1 such that n(ρn − 1) → ∞ as n → ∞. In contrast to the vast
majority of existing literature in the area, no specific form of ρ is required. We show that
the serial correlation coefficient converges in distribution to a ratio of two independent
stable random variables.
AMS 2000 Subject Classification: Primary 62M10, Secondary 91B84.
Keywords and Phrases: near–integrated time series; mildly explosive autoregressive
time series; serial correlation coefficient; strictly stable random variables
1
Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, UT 84112–
0090, USA, emails: aue@math.utah.edu and horvath@math.utah.edu
2
Corresponding author. Phone: +1–801–581–5231, fax: +1–801–581–4148
Partially supported by NATO grant PST.EAP.CLG 980599 and NSF–OTKA grant INT–0223262
1
Introduction
Consider the first–order autoregressive AR(1) process {Xk } defined by the equations
Xk = ρXk−1 + εk ,
−∞ < k < ∞,
(1.1)
where ρ is a deterministic parameter and {εk } some noise sequence. These processes are
well understood if dealing with independent, identically distributed errors having a finite
variance. Exploiting its recursive structure, it can be shown that the defining equations
(1.1) allow for a unique causal (i.e. future–independent), strictly stationary solution if
and only if |ρ| < 1. It should be noted however, that, in case |ρ| > 1, strictly stationary
solutions exist which are future–dependent. Detailed expositions on this topic and on more
basic properties of AR(1) processes and its generalizations may be found in Brockwell and
Davis (1991, Chapter 3).
Assuming for simplicity that the AR(1) process is initialized with X0 = 0, statistical
inference for the unkonwn parameter ρ can be carried out, e.g., by the least squares
estimator
!
ρ̂ =
n
X
k=1
2
Xk−1
−1 n
X
k=1
Xk−1 Xk ,
n ≥ 2,
(1.2)
given observations
√ of X1 , . . . , Xn . Its limiting behavior has been examined thoroughly.
If |ρ| < 1, then n(ρ̂ − ρ) converges in distribution to a normal law with mean 0 and
variance 1 − ρ2 . The situation turns out to be more complicated for explosive AR(1)
processes. Under the additional assumption of a Gaussian noise sequence {εk }, White
(1958) showed that in case of a fixed |ρ| > 1 the limit distribution of ρ̂ − ρ is standard
Cauchy if normalized with ρn /(1−ρ2 ). In general, however, the limit distribution depends
on the distribution of the noise, as was pointed out by Anderson (1959), and hence no
central limit theorem applies on the explosive side.
One of the major concerns in econometrics, which has been discussed for several decades
now is to validate or reject the so–called random walk hypothesis: To what extent can
econometric time series such as (logarithms of changes in) stock–market prices be modeled
by AR(1) processes with parameter ρ = 1, that is by random walks? Evidence has, for
instance, been found in the contributions of Fama (1965) and, more recently, Nelson and
Plosser (1982). Commonly, the testing problem is tackled using derivatives of the popular
test statistics introduced by Dickey and Fuller (1979, 1981), which can, for example, be
based on the least squares estimator in (1.2). However, these tests feature a low power
due to the lack of difference of the test procedures under the random walk hypothesis and
the possible alternatives |ρ| < 1, ρ > 1 or ρ 6= 1.
Moreover, it has been found that in some financial applications [see Phillips (1988) and
the references therein] the parameter ρ tends to 1 with increasing sample size. To accomodate this observation, ρ = ρn is allowed to depend on n, the number of observations, such
that ρn → 1 as n → ∞. The process is then referred to as near–integrated. Depending on
whether ρn < 1 or ρn > 1, it is called near–stationary or mildly–explosive. Most of the existing literature is devoted to the so–called local to unity case ρn = 1 + c/n. For instance,
2
Chan and Wei (1987, 1988) studied both the near–stationary and the mildly–explosive
case and determined the limiting behavior of the estimator ρ̂n , which is defined in (2.6).
Lately, Phillips and Magdalinos (2005) [see also Giraitis and Phillips (2005)] investigated
the general parameter case in the near–integrated setting assuming that ρn → 1 with a
rate slower than 1/n [so–called moderate deviations from unity].
All papers cited in the previous paragraph are working under the assumption of a
finite variance along with independent, identically distributed or weakly dependent errors.
However, extensive empirical research conducted shows that this assumption seems to
be violated for quite a variety of time series. Thus, Mandelbrot (1963, 1969) already
argued for what he called the infinite variance hypothesis and suggested to model noise
sequences by stable random variables. While, in general, there is still controversy whether
or not stable variables can reasonably describe financial data, we would like to mention
the monograph by Mittnik and Rachev (2000) who happen to concur with Mandelbrot’s
approach.
Consequently, we will focus on giving a corresponding limit theory for the serial correlation coefficient in the mildly explosive setting with moderate deviations from a unit
root if the innovations {εk } are heavy–tailed, that is, in the domain of attraction of an
α–stable law, where α ∈ (0, 2]. Pioneering work has been carried out by Chan and Tran
(1989) in the random walk case ρ = 1, and by Chan (1990) who studied the local to unity
case both on the stationary and explosive side.
To start with, we will introduce the model and basic assumptions, and state the result
in Section 2, which also contains those properties of strictly α–stable random variables,
which are important in the discourse. The proof is relegated to Section 3.
2
Main Result
We study the following sequence of first–order autoregressive AR(1) models. Let
Xk (n) = ρn Xk−1 (n) + εk ,
k = 1, . . . , n,
(2.1)
where, for simplicity, the initial values X0 (n) = ε0 are constant. To take into account
the infinite variance hypothesis, the innovation sequence {εk } is assumed to consist of
independent, identically distributed random variables being in the domain of attraction
of a strictly α–stable law.
A random variable ξα is called strictly α–stable, α ∈ (0, 2], if its characteristic function
has the form
φα (t) =









exp(−t2 /2)
if α = 2,
exp(−|t|α [1 − iβ sgn(t) tan( 21 α)])
if α ∈ (0, 1) ∪ (1, 2), β ∈ [−1, 1],
exp(−|t|)
(2.2)
if α = 1.
The definition includes the normal law (α = 2) and the Cauchy law (α = 1). There are
no explicit expressions for densities or distribution functions of random variables being
3
in the domain of attraction of an α–stable law, so there exist various indirect ways of
defining them. For convenience [they are easier applicable in the proof], we choose the
characteristic function version.
The class of stable distributions has become more and more important as well in theory
as in applications. Stable laws appear in a natural way in areas such as radio engineering,
electronics, biology and economics [see Zolotarev (1986, Chapter 1)]. For other extensive expositions on α–stable random variables and processes confer Samorodnitsky and
Taqqu (1994), and Bingham et al. (1987).
The assumption that {εk } is in the domain of attraction of a strictly α–stable law
implicitely contains the limit result [cf. Ibragimov and Linnik (1971, Theorem 2.1.1)]
m
X
1
D
εi −→ ξα
1/α
m L(m) i=1
(m → ∞),
(2.3)
which bears some resemblence with the classical central limit theorem, but is obviously
more general: If α ∈ (0, 2), in which case variances do not exist, then L(x) > 0, x > 0, is
a slowly varying function at infinity, that is,
L(cx)
=1
x→∞ L(x)
lim
for all c > 0.
(2.4)
q
If α = 2, then we choose L(x) = Eε20 > 0 and condition (2.3) becomes the central limit
theorem with ξ2 being a standard normal random variable. Moreover, (2.2) yields that
Eε1 = 0 for α ∈ (1, 2].
It is assumed that the model parameter ρn satisfies
ρn → 1 and n(ρn − 1) → ∞
(n → ∞).
(2.5)
Since the latter condition implies that, for large n, ρn > 1 is farther away from unity than
O(1/n), this setting is referred to as mildly explosive with moderate deviations from a
unit root. Denote by
ρ̂n =
n
X
k=1
2
Xk−1
(n)
!−1
n
X
Xk−1 (n)Xk (n),
k=1
n ≥ 1,
(2.6)
the least squares estimator for ρn . Then, the serial correlation coefficient ρ̂n − ρn has,
under a suitable normalization, a limit which consists of a fraction of two independent
strictly stable random variables.
Theorem 2.1 Let {Xk (n)} follow (2.1) and let conditions (2.3) and (2.5) be satisfied.
Then
ρnn
D ξα
(ρ̂n − ρn ) −→
(n → ∞),
2 log ρn
ζα
where ρ̂n is defined in (2.6) and ξα and ζα denote independent strictly α–stable random
variables.
4
If α = 2, that is, if the innovations {εk } satisfy the central limit theorem, Theorem
2.1 has already been proved in Phillips and Magdalinos (2005). Here, ξ2 and ζ2 are
independent standard normally distributed, and thus the limit ξ2 /ζ2 becomes a standard
Cauchy random variable. The result is also in accordance with the asymptotics of ρ̂ − ρ
in the fixed parameter case ρ > 1 [see Mijnheer (2002)]. However, the results of Chan
(1990) indicate that the asymptotics in the local to unity case, where the limit is given as
function of integrals of α–stable processes, are qualitatively different from Theorem 2.1
and closer related to the unit root case ρ = 1 treated in Chan and Tran (1989).
3
Proof
First, we collect some facts which are used throughout the proof. Let
cn = log ρn ,
n ≥ 1.
(3.1)
The mean–value theorem and condition (2.5) imply that ncn → ∞ as n → ∞.
Let L(x) be a slowly varying function at infinity. Then,
L(n)
= O (ncn )δ
L(1/cn )
for all δ > 0,
(3.2)
and for any δ > 0 and T > 0 there is a constant C > 0 such that for all n ≥ 1
L(un)
≤ Cuδ
L(n)
for all T ≤ u < ∞.
(3.3)
Both relations follow from Corollary 3.1 in Csörgő and Horváth (1993, p. 420).
A process {Wα (t) : t ≥ 0} is called strictly α–stable if its increments are strictly α–
stable random variables. The assumption of {εk } being in the domain of attraction of a
strictly α–stable law yields the functional version of the central limit theorem (2.3) for all
α ∈ (0, 2]. By Gikhman and Skorohod (1969, pp. 479–481) [see also Petrov (1975, Section
IV.2)], the finite–dimensional distributions of the partial sum process converge to those
of a strictly α–stable process and the partial sum process is tight. Consequently, we have
that, for any T > 0,
⌊nt⌋
X D[0,T ]
1
εi −→ Wα (t)
n1/α L(n) i=1
(n → ∞),
(3.4)
D[0,T ]
where −→ denotes weak convergence in the Skorohod space D[0, T ].
Furthermore, as another consequence of (2.3),
n
X
i=1
ε2i = OP n2/α L2 (n)
5
(n → ∞).
(3.5)
If α = 2, then Eε21 < ∞, so (3.5) follows from the weak law of large numbers. If α ∈ (0, 2),
then (2.3) holds if and only if
P {|ε1| > x} = x−α K(x),
where K(x) is a slowly varying function at infinity. The function
L(x) satisfies the relation
√
1/α
2
−α/2
limy→∞ K(y L(y)) = 1. Clearly, P {ε1 > x} = x
K( x), so {ε2k } is in the domain of
attraction of a strictly α/2–stable law and
n
X
1
D
ε2i −→ ξα/2
2/α
2
n L (n) i=1
(n → ∞),
q
since limy→∞ K( y 2/α L2 (y)) = 1.
We will use the following convention concerning the integral sign: When −∞ < a <
b < ∞ and l is a left–continuous and r is a right–continuous function then
Z
b
a
r dl =
Z
[a,b)
r dl
and
Z
b
a
l dr =
Z
(a,b]
l dr,
whenever these integrals make sense as Lebesgue–Stieltjes integrals. In this case the usual
integration by parts formula
Z
b
a
r dl +
Z
b
a
l dr = l(b)r(b) − l(a)r(a)
(3.6)
is valid. If l or r are not finite at one of the endpoints or at least one of the endpoints
themselves are not finite, then the corresponding integrals are meant as improper integrals
[see Hewitt and Stromberg (1969, p. 419) and Csörgő et al. (1986, p. 87)].
Next, we will outline the proof steps. It is clear from (2.1) and (2.6) that
ρ̂n − ρn =
n
X
2
Xk−1
(n)
k=1
!−1
n
X
Xk−1 (n)εk ,
k=1
n ≥ 1.
(3.7)
Hence, we need to derive the asymptotics of both sums on the right–hand side. To do so,
a representation of Xk (n) in terms of the innovations εj , where j ≤ k, will be applied.
Since by assumption ε0 = X0 (n),
Xk (n) =
k
X
i=0
ρin εk−i =
k
X
ρnk−i εi ,
k = 1, . . . , n,
(3.8)
i=0
exploiting the recursive structure of model (2.1). Lemma 3.1 deals with the partial sums
2
of the Xk−1(n)εk , and the partial sums of the Xk−1
are handled in Lemma 3.2. Finally,
their joint convergence is established in Lemma 3.3. Theorem 2.1 will follow readily from
these auxiliary results.
6
Let T > 0 and denote by ⌊·⌋ the integer part. Then, using (3.8),
⌊T /cn ⌋
n
X
X
Xk−1 (n)εk =
k=1
n
X
Xk−1 (n)εk +
k=1
k=⌊T /cn ⌋+1
ρ−i
n εi
i=⌊T /cn ⌋+1
⌊T /cn ⌋
n
X
+
k
X
ρnk−1 εk
ρnk−1 εk
ρ−i
n εi
X
i=0
k=⌊T /cn ⌋+1
= RT,1 (n) + RT,2 (n) + RT,3 (n).
(3.9)
It will be shown that RT,3 (n) is the leading term, while RT,1 (n) and RT,2 (n) do not
contribute to the limit distribution. For α ∈ (0, 2] and x > 0 set Nα (x) = x1/α L(x).
Lemma 3.1 Let the assumptions of Theorem 2.1 be satisfied.
(i) For every T > 0 and ε > 0,
n
o
−2
lim P ρ−n
n Nα (1/cn )|RT,1 (n)| ≥ ε = 0,
n→∞
where RT,1 (n) is defined in (3.9).
(ii) For every ε > 0,
n
o
−2
lim lim sup P ρ−n
n Nα (1/cn )|RT,2 (n)| ≥ ε = 0,
T →∞
n→∞
where RT,2 (n) is defined in (3.9).
Proof. Set M = ⌊T /cn ⌋. It follows from (3.1) that M → ∞ as n → ∞.
(i) We recall that
RT,1 (n) =
M
X
M
X
Xk−1 (n)εk =
k=1
ρnk−1 εk
k−1
X
ρ−i
n εi .
i=0
k=1
It is easy to see that
M
X
k=1
ρ2k
n

k
X

ρ−i
n εi
i=0
!2
−
k−1
X
ρ−i
n εi
i=0
!2 
M
X
=

−k
ρ2k
n ρn εk
k=1
= 2
M
X
k=1
ρkn εk
"
k−1
X
2
k−1
X
ρ−i
n εi
+
ρ−k
n εk
i=0
ρ−i
n εi +
i=0
M
X
ε2k .
k=1
On the other hand, by Abel’s summation, we have
M
X
k=1


ρ2k
n
= ρ2M
n
k
X
ρ−i
n εi
i=0
M
X
i=0
!2
ρ−i
n εi
−
!2
k−1
X
ρ−i
n εi
i=0
− ρ2n ε20 +
!2 

M
−1 X
7
k=1
2k+2
ρ2k
n − ρn
k
X
i=0
ρ−i
n εi
!2
.
#
Hence, on combining the previous equations, we arrive at

M
1  2M X
RT,1 (n) =
ρn
ρ−i
n εi
2ρn
i=0
M
−1 X
ρ2k
n
+
k=1
−
!2
− ρ2n ε20
ρ2k+2
n
k
X
ρ−i
n εi
i=0
!2
−
M
X
k=1

ε2k  .
(3.10)
We will work on the right–hand side of equation (3.10) term by term.
Set S(x) = ε0 + . . . + ε⌊x⌋ . Note that S(x) is a right–continuous function and, hence,
using integration by parts along (3.6) implies
M
X
ρ−i
n εi
Z
= ε0 +
0
i=0
= cn
Z
=
Z
M
M
0
cn M
0
exp(−cn x)dS(x)
S(x) exp(−cn x)dx + ρ−M
n S(M)
S (x/cn ) exp(−x)dx + ρ−M
n S(M).
By (3.4) and the definition of M, it holds that
D
−1
ρ−M
n Nα (1/cn )S(M) −→ exp(−T )Wα (T )
(n → ∞)
(3.11)
and
Nα−1 (1/cn )
Z
0
cn M
D
S(x/cn ) exp(−x)dx −→
Z
T
0
Wα (x)dx
(n → ∞),
(3.12)
so we conclude that
M
X
i=0
ρ−i
n εi = OP (Nα (1/cn ))
(n → ∞).
Hence, using ncn → ∞, we obtain
−n −2
ρ2M
Nα (1/cn )
n
M
X
ρ−i
n εi
i=0
!2
= OP ρ−n
= OP (exp(−ncn )) = oP (1)
n
as n → ∞, showing that the first term on the right–hand side of (3.10) is negligible.
As a consequence of (2.3), integration by parts [see (3.6)] in combination with standard
estimates give for the third term in equation (3.10)
M −1 X
−2
ρ2k
ρ−n
N
(1/c
)
n n
n
α
k=1
!2 ρ−i
n εi
i=0
k
X
2k+2
− ρn
8
≤
≤
ρn−n
ρn−n
1+
1+
= OP ρ−n
n
= oP (1)
ρ2M
n
ρ2M
n
Nα−1 (1/cn )
!2
k
X
−i max ρn εi
1≤k≤M i=0
Nα−1 (1/cn )
"Z
cn M
0
|S (x/cn )| exp(−x)dx + sup
0≤t≤M
|ρ−t
n S(t)|
#!2
for any fixed T > 0 as n → ∞. To obtain the rate OP (ρ−n
n ), apply (3.12) and
D
−1
Nα−1 (1/cn ) sup |ρ−t
n S(t)| ≤ Nα (1/cn ) sup |S(t/cn )| −→ sup |Wα (t)|,
0≤t≤M
0≤t≤T
0≤t≤T
which follows from the fact that, for large n, ρ−t
n ≤ 1 and from (3.4).
Finally, (3.5) yields,
M
X
−2
ρ−n
n Nα (1/cn )
k=1
ε2k = OP ρ−n
= oP (1)
n
for any fixed T > 0 as n → ∞. Since ρ2n ε20 = O(1) a.s., the proof of (i) is complete.
(ii) Note that, similarly as in part (i),
RT,2 (n) =

n
X
1 

2ρn k=M +1

k
 X
ρ2k

n
i=M +1
2

k−1
X
 −
ρ−i
n εi
i=M +1
2 
 
ρ−i
−
n εi

n
X
k=M +1
ε2k  .

(3.13)
Starting with the second sum on the right–hand side, we get from (3.2) and (3.5)
−2
ρ−n
n Nα (1/cn )
n
X
k=1
ε2k = OP (1)(ncn )2/α+2δ exp(−ncn ) = oP (1)
(3.14)
for any fixed T > 0 as n → ∞, where δ > 0 can be chosen arbitrarily.
It remains to examine the first sum in (3.13). Observe that
n
X
k=M +1

k
 X
ρ2k

n
i=M +1


= ρ2n
n
n
X
i=M +1

+1) 
= ρ2(M
n
2

 −
ρ−i
n εi
2
 +
ρ−i
n εi
n
X
i=M +1
k−1
X
i=M +1
n−1
X
k=M +1
2
 +
ρ−i
n εi
2 
 
ρ−i

n εi
(3.15)

2k+2 
ρ2k
n − ρn
n−1
X
k=M +1
k
X
i=M +1

2k+2
ρ2k

n − ρn
9

2

ρ−i
n εi
k
X
i=M +1
2

 −
ρ−i
n εi
n
X
i=M +1
2 
 
ρ−i
.
n εi
It is clear that

−2
2M +2 
ρ−n
n Nα (1/cn )ρn
n
X
i=M +1
2

 = O(1)ρ−n N −1 (1/cn )
ρ−i
n εi
n
α
n
X
i=M +1
2
 .
ρ−i
n εi
(3.16)
Now, integration by parts and a subsequent change of variables yield
n
X
−i ρn εi i=M +1
≤ ρ−n
n |S(n)| +
ncn
Z
T
|S(x/cn )| exp(−x)dx.
The central limit theorem for strictly α–stable random variables in (2.3) implies
Nα (n) |S(n)|
Nα (1/cn ) Nα (n)
ρ−2n
n
!2
2/α+2δ
= OP (1)ρ−2n
= oP (1)
n (ncn )
for all δ > 0 according to (3.2). Also, with 0 < ν < min{α, 1}, an application of Markov’s
inequality leads to
(
Z
Nα−1 (1/cn )
P
= P
≤
Z
√
ncn
T
2
|S(x/cn )| exp(−x)dx
ν
ncn
|S(x/cn )| exp(−x)dx
T
εNα (1/cn )
−ν
E
Z
ncn
T
≥
≥ε
√
)
εNα (1/cn )
ν
|S(x/cn )| exp(−x)dx
ν .
(3.17)
By Minkowski’s inequality for integrals [see Kaczor and Nowak (2003, p. 49)] we have
E
Z
ncn
T
ν
|S(x/cn )| exp(−x)dx
≤
Z
ncn
T
E|S(x/cn )|ν exp(−νx)dx.
Using Theorem 6.1 of de Acosta and Giné (1979) and (3.3), we arrive at
Nα−ν (1/cn )
≤ C
≤ C
Z
Z
T
ncn
ZT∞
T
ncn
E|S(x/cn )|ν exp(−νx)dx
(3.18)
Nαν (x/cn )Nα−ν (1/cn ) exp(−νx)dx
xν/α+δ exp(−νx)dx
with any δ > 0 and some constant C > 0. Thus, we have proved




lim lim sup P Nα−2 (1/cn ) 
n→∞

T →∞
10
n
X
i=M +1
2



 ≥ ε = 0.
ρ−i
n εi


(3.19)
Now, for the second term in (3.15) we write
n−1
X
k=M +1
= −
= −
2k+2
ρ2k
n − ρn
n−1
X
k=M +1
n−1
X
k=M +1
= −2
+
n
X

k
 X

k=M +1

n
X
i=k+1

n
X
2k+2 
ρ2k
n − ρn
ρ−j
n εj
n−1
X
k=M +1
i=k+1
i=k+1
ρn−i εi
i=M +1

ρ−i
n εi



2k+2 
ρ2k
n − ρn
n
X
i=k+1
2 
 
ρ−i

n εi
k
X
ρ−j
n εj +
n
X
j=M +1
j=M +1
n
X
n
X
 2
ρ−i
n εi
2k+2 
ρ2k
n − ρn
First, integration by parts gives
n
X
n
X


 −
ρ−i
n εi
2k+2 
ρ2k
n − ρn
j=M +1
n−1
X
i=M +1
2
j=M +1
n
X
i=k+1
2
ρ−j
n εj −

ρ−j
n εj
j=k+1




ρ−j
n εj

ρ−i
n εi
 .
ρ−i
n εi
(3.20)
= S(n) exp(−ncn ) − S(k) exp(−kcn ) + cn
Z
k
n
S(x) exp(−xcn )dx.
Therein we conclude from (3.2),
−2
2n
|S(n)|ρ−n
ρ−n
n
n Nα (1/cn )ρn
2
= OP exp(−ncn )(ncn )2/α+2δ = oP (1)
for all δ > 0 as n → ∞. Also, (3.4) implies that
Z 1
X
1 n−1
D
−2
2
N (n)S (k) −→
Wα2 (t)dt
n i=1 α
0
(n → ∞).
Moreover, 1 − ρ2n = (1 + ρn )(1 − ρn ) = O(1)cn by Taylor expansion. Hence, relation (3.2)
gives
−2
ρ−n
n Nα (1/cn )
n−1
X
k=M +1
2k
ρn
−2k 2
ρn
S (k)
− ρ2k+2
n
2
= ρ−n
n (1 − ρn )
n−1
X
= O(1)ρ−n
n ncn
X
Nα2 (n) 1 n−1
N −2 (n)S 2 (k)
Nα2 (1/cn ) n k=1 α
Nα−2 (1/cn )S 2 (k)
k=1
1+2/α+2δ
= OP (1)ρ−n
n (ncn )
= oP (1)
11
for all δ > 0 as n → ∞. Finally, (3.4) and (3.2) yield
−2
c2n ρ−n
n Nα (1/cn )
n−1
X
2
ρ2k
n (ρn
− 1)
k=M +1
n−1
X
Z
≤
−2
2c3n ρ−n
n Nα (1/cn )
≤
−2
2c3n ρ−n
n Nα (1/cn )n
≤
−2
2
3
2c3n ρ−n
n Nα (1/cn )Nα (n)n
k=M +1
3+2/α+2δ
= OP ρ−n
n (ncn )
= oP (1)
|S(x)| exp(−xcn )dx
k
2
n
|S(x)| exp((k − x)cn )dx
k
|S(x)|dx
0
2
n
2
n
Z
Z
Z
1
0
2
Nα−2 (n)S(nx)dx
for all δ > 0 as n → ∞. Integration by parts gives
n
X
−j ρ
ε
n j
j=M +1
≤ |S(n)| exp(−ncn ) + cn
Z
n
M
|S(x)| exp(−xcn )dx.
On applying (2.5) and (3.2) we obtain
Nα−1 (1/cn ) exp(−ncn )|S(n)| = OP (1)(ncn )1/α+δ exp(−ncn ) = oP (1)
for all δ > 0 as n → ∞. Let 0 < ν < min{1, α}. As in (3.18), Markov’s inequality,
Minkowski’s inequality for integrals [cf. Kaczor and Nowak (2003, p. 49)], and Theorem
6.1 of de Acosta and Giné (1979) imply
P
Z
cn Nα−1 (1/cn ) = P
n
M
S(x) exp(−xcn )dx
ν
cn Nα−1 (1/cn )
≤ (εNα (1/cn ))
−ν
= (εNα (1/cn ))
−ν
≤ (εNα (1/cn ))−ν
≤ Cε−ν
Z
∞
T
Z
M
E cn
E
Z
Z
n
Z
n
M
ncn
M cn
ncn
M cn
≥ε
ν
S(x) exp(−xcn )dx
≥ εν
ν
|S(x)| exp(−xcn )dx
ν
|S(x/cn )| exp(−x)dx
E|S(x/cn )|ν exp(−xν)dx
xδ+ν/α exp(−xν)dx
12
(3.21)
for any δ > 0 and some constant C, where we also applied relation (3.3). Hence
lim lim sup P
T →∞
n→∞



n−1
X −2
ρ2k
ρ−n
n
n Nα (1/cn ) 

k=M +1
− ρ2(k+1)
n
Next, we show that
lim lim sup P
T →∞
n→∞
n
n−1
X
X −n −2
−j
ρ2k
ρ
N
(1/c
)
ρ
ε
n j
n
n
α
n

j=M +1
k=M +1

−i 

×
ρn εi i=k+1
Since we proved that
lim lim sup P
T →∞
n→∞
(3.23) is established if we verify
Towards this end, write
k=M +1

2k+2 
ρ2k
n − ρn
= (1 − ρ2n )
n
X
n
X
i=k+1
ρ−i
n εi
i=M +2
≥ε


n
X
= (1 −
ρ2n )
2k+2 
ρ2k
n − ρn


ρ−i
n εi
i−1
X

k=M +1
i=k+1
=
= 0. (3.22)


= 0.
(3.23)
≥ε


= 0,


 = OP (1).
ρ−i
n εi
n−1
X
n
X
n
X
ρ−i
n εi
ρn2k−i εi
k=M +1 i=k+1
ρ2k
= (1 − ρ2n )
n
+1)
ρ2(M
n
− ρ2(k+1)
n
≥ε




X
n
−i −1
ρn εi
N (1/cn )  α
i=M +1
k=M +1
n
X


n−1
X
−1
ρ−n
n Nα (1/cn )
n
X



n−1
X
2 
 ρ−i
ε
i
n
i=k+1

i=M +2
n
X
ρ−i
n εi
i=M +2
2(M +1)
ρ2i
n − ρn
ρ2n − 1
−
n
X
ρin εi .
i=M +2
We have already shown that
X
n
ρ−i
ε
Nα−1 (1/cn ) n i
i=M +2
= OP (1).
(3.24)
Since {εk } is a sequence of independent, identically distributed random variables we get
n
X
−1
i ρ−n
N
(1/c
)
ρ
ε
n n
α
n i
i=M +2
D
=
13
n−M
X
Nα−1 (1/cn ) i=0
ρ−i
n εi ,
and, thus, we have by (3.24) that
n
X
i −1
ρ
ε
ρ−n
N
(1/c
)
n n i
n
α
i=M +2
= OP (1).
This completes the proof of (3.23). Putting together (3.13)–(3.15), (3.19), (3.20) and
(3.22), we obtain part (ii) of Lemma 3.1.
2
So, it suffices to investigate the term RT,3 (n) to obtain the limit distribution. The
second part of the proof deals with the partial sums of the Xk2 (n). Let again T > 0.
Then, we obtain that
n
X
Xk2 (n)
n
X
=
k=1
k
X
ρ2k
n
i=0
k=1
⌊T /cn ⌋
X
=
ρ−i
n εi
k
X
ρ2k
n
ρ−i
n εi
i=0
k=1
n
X
+
!2
k=⌊T /cn ⌋+1

!2
n
X
+
k=⌊T /cn ⌋+1
2
⌊T /cn ⌋

ρ2k
n
X
i=0


ρ2k
n
k
X
i=⌊T /cn ⌋+1
2

ρ−i
n εi

ρ−i
n εi
= ST,1 (n) + ST,2 (n) + ST,3 (n).
(3.25)
The next lemma identifies ST,3 (n) as leading term by showing that ST,1 (n) and ST,2 (n)
are asymptotically small.
Lemma 3.2 Let the assumptions of Theorem 2.1 be satisfied.
(i) For every T > 0 and ε > 0,
n
o
lim P ρn−2n cn Nα−2 (1/cn )ST,1 (n) ≥ ε = 0
n→∞
where ST,1 (n) is defined in (3.25).
(ii) For every ε > 0,
n
o
lim lim sup P ρn−2n cn Nα−2 (1/cn )ST,2 (n) ≥ ε = 0,
T →∞
n→∞
where ST,2 (n) is defined in (3.25).
Proof. Recall that M = ⌊T /cn ⌋ and S(x) = ε0 + . . . + ε⌊x⌋ .
(i) Note that
k
X
i=0
ρ−i
n εi
!2
=
ρ−k
n S(k)
≤
2ρn−2k S 2 (k)
+ cn
+
Z
k
0
2c2n
14
!2
S(x) exp(−xcn )dx
Z
o
k
!2
|S(x)| exp(−xcn )dx
.
Since by (3.4)
T
Z
0
Nα−1 (1/cn )S(u/cn )
2
D
du −→
we get for the first term of the right–hand side
−2
ρ−2n
n cn Nα (1/cn )
M
X
−2k 2
ρ2k
n ρn S (k)
T
Z
0
Wα2 (u)du,
=
−2
ρ−2n
n cn Nα (1/cn )
=
ρ−2n
n
k=1
Z
T
0
Z
(3.26)
M
0
S 2 (x)dx
Nα−1 (1/cn )S(u/cn )
2
du
= oP (1)
for any fixed T > 0 as n → ∞. By (3.12), we obtain for the second term,
3
−2
ρ−2n
n cn Nα (1/cn )
M
X
Z
ρ2k
n
k
0
k=1
!2
|S(x)| exp(−xcn )dx
−2
2M
3
≤ ρ−2n
n cn Nα (1/cn )Mρn
=
)
OP (1)T ρ−2(n−M
n
Z
0
T
M
Z
0
!2
|S(x)| exp(−xcn )dx
Nα−1 (1/cn )|S(u/cn )| exp(−u)du
!2
= oP (1)
for any fixed T > 0 as n → ∞. This proofs part (i) of the lemma.
(ii) Arguments similar to those used in part (i) of the proof give the estimate
ST,2 (n) ≤ 2
n
X
k=M +1

ρ−2k S 2 (k) + ρ−2M S 2 (M) + c2
ρ2k
n
n
n
n
Z
k
M
!2 
S(x) exp(−xcn )dx
.
We will proceed termwise again. At first, note that by (3.4) and the definition of M,
n
1 X
D
N −2 (n)S 2 (k) −→
n k=M +1 α
Z
0
1
Wα2 (t)dt
(n → ∞).
Therefore, we obtain the following asymptotics for the first term:
−2
ρ−2n
n cn Nα (1/cn )
n
X
(ncn )1+2/α
S (k) ≤
ρ2n
n
k=M +1
2
L(n)
L(1/cn )
!2
1+δ+2α
= OP (1)ρ−2n
n (ncn )
= oP (1)
15
n
1X
N −2 (n)S 2 (k)
n k=1 α
for any δ > 0 as n → ∞, where we have used (3.2). Next observe that, since ρ−M
=
n
exp(−2T )(1 + o(1)),
n
X
−2
ρ−2n
n cn Nα (1/cn )
2
ρ2k
n exp(−2T )S (M)
k=M +1
ρ2n+1 − ρM
n
O(1)ρn−2n cn n
ρn − 1
=
L(T /cn )
L(1/cn )
!2
T 2/α exp(−2T )
S 2 (M)
Nα2 (M)
by (2.3) and (3.2). It is obvious that by (2.4)
ρ2n+1 − ρM
n
ρn−2n cn n
ρn − 1
Moreover,
L(M)
L(1/cn )
S 2 (M) D
−→ Wα2 (1)
Nα2 (M)
!2
= O(1).
(n → ∞).
Thus, we conclude
lim lim sup P
T →∞
n→∞
for all ε > 0. Finally,


−2
ρ−2n
n cn Nα (1/cn )

k=M +1
−2
ρ−2n
n cn Nα (1/cn )
n
X
ρ2k
cn
n
k=M +1
= ρ−2n
n cn
n
X
n
X
ρ2k
n
k=M +1
Z
n
M
Z
n
M
) 2
ρ2(k−M
S (M) ≥ ε
n


=0

2
|S(x)| exp(−xcn )dx
Nα−1 (1/cn )|S(x)|d exp(−xcn )
2
,
where the parts involving ρn are clearly bounded. Ultimately, choose 0 < ν < 21 min{1, α}.
Repeating the arguments used in (3.21) we obtain that
E
"
Nα−1 (1/cn )
Z
n
M
|S(x)|d exp(−xcn )
2 #ν
≤
Z
∞
T
x2ν/α+δ e−2νx dx,
so by Markov’s inequality we have that
lim lim sup P
T →∞
n→∞



−2
ρ−2n
n cn Nα (1/cn )
n
X
cn
ρ2k
n
k=M +1
Z
n
M
2
|S(x)| exp(−xcn )dx
for all ε > 0. Hence, the proof of part (ii) of Lemma 3.2 is complete.
≥ε



=0
2
Lemma 3.1 tells us that it suffices to derive the limit distribution of the remaining term
RT,3 (n). Checking the remaining summation ranges in the corresponding double sum in
(3.9) shows us that RT,3 (n) is a product of two independent factors, whose limit can be
calculated separately. We also note that the square of one of the terms in RT,3 (n) is the
only random part in ST,3 (n).
16
Lemma 3.3 Let the assumptions of Theorem 2.1 be satisfied. Then

M
X
N −1 (1/cn )
α
D
−→
Z
−n −1
ρ−i
n εi , ρn Nα (1/cn )
i=0
T
0
n
X
j=M +1
Z
exp(−x)dWα,1 (x),
∞
0

ρnj−1 εj 
!
Wα,2 (x) exp(−x)dx
(n → ∞)
for any fixed T > 0, where {Wα,1 (x) : x ≥ 0} and {Wα,2 (x) : x ≥ 0} denote two independent
strictly α–stable processes.
Proof. The proof is given in two steps, each of them investigating one of the components
of the vector of interest. The joint behavior is obtained on combining both results.
(i) Observe that
Nα−1 (1/cn )
M
X
ρ−i
n εi
=
Z
Nα−1 (1/cn )
T
0
i=0
!
exp(−x)dS(x/cn ) + ε0 .
Hence, integration by parts and (2.3) give the convergence in distribution result for the
first coordinate, since Nα−1 (1/cn ) is clearly asymptotically negligible.
(ii) Note that
n
X
ρ−n
n
ρnj−1 εj = ρ−2
n
j=M +1
n
X
D
εj = ρ−2
ρj−n+1
n
n
n−M
X
ρ−i
n εi .
i=1
j=M +1
We have shown that, for any T ∗ > 0,
M
X
∗
Nα−1 (1/cn )
ρ−i
n εi
i=1
D
−→
Z
T∗
0
exp(−x)dWα (x),
where M ∗ = ⌊T ∗ /cn ⌋. Next, we write
n−M
X
−1
−i N (1/cn )
ρn εi α
i=M ∗ +1
=
Z
n−M
M∗
Nα−1 (1/cn ) exp(−xcn )dS(x)
≤ Nα−1 (1/cn ) exp(−M ∗ cn )|S(M ∗ )| + Nα−1 (1/cn ) exp(−(n − M)cn )|S(n − M)|
+Nα−1 (1/cn )
Z
∞
M ∗ cn
|S(x/cn )| exp(−x)dx.
Condition (2.3) implies that
n
o
lim
lim sup P Nα−1 (1/cn ) exp(−M ∗ cn )S(M ∗ ) ≥ ε = 0.
∗
T →∞
n→∞
Similarly to (3.18), Theorem 6.1 of de Acosta and Giné (1979) yields, for all T > 0,
n
o
lim P Nα−1 (1/cn ) exp(−(n − M)cn )|S(n − M)| ≥ ε = 0.
n→∞
17
We have proved in (3.21) that
lim lim sup P
T ∗ →∞
n→∞
Z
∞
T∗
Nα−1 (1/cn )|S(x/cn )| exp(−x)dx
≥ ε = 0.
Let 0 < ν < min{1, α}. Applications of Minkowski’s inequality and the self–similarity of
Wα yield
Z
E ∞
T∗
ν
exp(−x)dWα (x) ≤ E exp(−T ∗ )Wα (T ∗ ) +
≤ 2
ν
≤ 2
ν
∗
∗
ν
exp(−νT )E|Wα (T )| +
∗
exp(−νT )T
∗ν/α
+
Z
∞
T∗
Z
∞
T∗
ν/α
x
∞
Z
T∗
ν
ν
Wα (x) exp(−x)dx
E|Wα (x)| exp(−νx)dx
exp(−νx)dx .
So, by Markov’s inequality we have
lim P
T ∗ →∞
∞
Z
T∗
Therefore, the proof is complete.
exp(−x)dWα (x)
≥ ε = 0.
(3.27)
2
Lemma 3.4 Let the assumptions of Theorem 2.1 be satisfied. Then
−2
ρ−n
n Nα (1/cn )
n
X
Xk−1 (n)εk ,
ρn−2n cn Nα−2 (1/cn )
k=1
D
−→
Z
0
∞
n
X
Xk2 (n)
k=1
exp(−x)dWα,1 (x)
Z
∞
0
1
exp(−x)dWα,2 (x),
2
!
Z
0
∞
exp(−x)dWα,1 (x)
as n → ∞.
Proof. Recalling that M = ⌊T /cn ⌋, it is easy to see that
n
X
ρn−2n cn
ρ2k
n
k=M +1
=
+1)
ρ2(n+1)
− ρ2(M
n
n
−2n
ρn cn
ρ2n − 1
−→
1
2
as n → ∞. Let ε, δ > 0. By Lemmas 3.1 and 3.2 there are n0 and T0 such that
P
(
P
Also,
n
X
−2
ρ−n
N
(1/c
)
Xk−1 (n)εk
n
n
α
k=1
(
n
X
ρn−2n cn Nα−2 (1/cn ) P
Z
∞
T
Xk2 (n)
k=1
− RT,3 (n)
− ST,3 (n)
exp(−x)dWα,1 (x)
18
)
≥ ε ≤ δ,
)
≥ ε ≤ δ.
≥ε ≤δ
2 !
for all T ≥ T0 and n ≥ n0 . Since Lemma 3.3 holds true for any T > 0, the proof is
complete.
2
We are now in a position to derive the limit distribution of the serial correlation coefficient.
Proof of Theorem
2.1. Let α ∈ (0, 2]. We show that for any strictly α–stable process
R∞
Wα (t) the integral 0 exp(−x)dWα (x) is also a strictly α–stable random variable multiplied with α−1/α . The claim is trivial if α = 2, so we assume α ∈ (0, 2). Let ξ1 , ξ2, . . .
be an independent sequence of strictly α–stable random variables. Then, by (3.11) and
(3.12),
Z ∞
n
α1/α X
D
−k
1/α
Zn =
exp(−x)dWα (x)
ρ ξk −→ α
(1/cn )1/α k=1 n
0
(n → ∞).
The characteristic function of the ξi is φα (t) given in (2.2). Therefore, if α 6= 1, the
characteristic function of Zn satisfies
α
E exp(itZn ) = exp −|t| αcn
n
X
ρ−αk
[1
n
k=1
− iβ
!
sgn(ρ−k
n t) tan(α/2)]
−→ φα (t)
as n → ∞, since clearly sgn(ρ−k
n t) = sgn(t) and
αcn
n
X
k=1
ρ−αk
n
= αcn
1 − ρn−α(n+1)
1 − ρ−α
n
!
−→ 1.
[Note that, by Taylor expansion, 1 − ρ−α
n = αcn (1 + o(1)).] A similar argument applies
also in case α = 1.
In Lemmas 3.1 and 3.2 we have identified the leading terms of the two partial sums
determining the difference ρ̂n − ρn in (3.7). According to Lemma 3.4, these leading terms
jointly converge towards stochastic integrals with respect to (independent) strictly α–
stable processes. Hence, the assertion follows from the previous paragraph.
2
Acknowledgement. The authors would like to thank P.C.B. Phillips and two anonymous referees for a careful reading of the manuscript and helpful advice that improved
the presentation.
References
[1] Anderson, T.W. (1959). On asymptotic distributions of estimates of parameters of
stochastic difference equations. The Annals of Mathematical Statistics 30, 676–687.
[2] Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge, U.K.
[3] Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods (2nd ed.).
Springer–Verlag, Berlin.
19
[4] Chan, N.H. (1990). Inference for near–integrated time series with infinite variance.
Journal of the American Statistical Association 85, 1069–1074.
[5] Chan, N.H., and Tran, L.T. (1989). On the first–order autoregressive process with
infinite variance. Econometric Theory 5, 354–362.
[6] Chan, N.H., and Wei, C.Z. (1987). Asymptotic inference for nearly nonstationary
AR(1) processes. The Annals of Statistics 15, 1050–1063.
[7] Chan, N.H., and Wei, C.Z. (1988). Limiting distributions of least squares estimates
of unstable autoregressive processes. The Annals of Statistics 16, 367–401.
[8] Csörgő, M., and Horváth, L. (1993). Weighted Approximations in Probability and
Statistics. Wiley, Chichester.
[9] Csörgő, M., Csörgő, S., Horváth, L., and Mason, D.M. (1986). Normal and stable
convergence of integral functions of the empirical distribution function. The Annals
of Probability 14, 86–118.
[10] De Acosta, A., and Giné, E. (1979). Convergence of moments and related functionals
in the general central limit theorem in Banach spaces. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 48, 213–231.
[11] Dickey, D.A., and Fuller, W.A. (1979). Distributions of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association
74, 427–431.
[12] Dickey, D.A., and Fuller, W.A. (1981). Likelihood ratio statistics for autoregressive
time series with a unit root. Econometrica 49, 1057–1072.
[13] Fama, E.F. (1965). The bevavior of stock–market prices. The Journal of Business
38, 34–105.
[14] Gikhman, I.I., and Skorohod, A.V. (1969). Introduction to the Theory of Random
Processes. Saunders, Philadelphia.
[15] Giraitis, L., and Phillips, P.C.B. (2005). Uniform limit theory for stationary autoregression. Journal of Time Series Analysis, forthcoming.
[16] Hewitt, E., and Stromberg, K. (1969). Real and Abstract Analysis. Springer–Verlag,
New York.
[17] Ibragimov, I.A., and Linnik, Y.V. (1971). Independent and Stationary Sequences of
Random Variables. Wolters–Noordhoff, Groningen.
[18] Kazcor, W.J., and Nowak, M.T. (2003). Problems in Mathematical Analysis III:
Integration. The American Mathematical Society, Providence, RI.
20
[19] Mandelbrot, B. (1963). The variation of certain speculative prices. The Journal of
Business 36, 394–419.
[20] Mandelbrot, B. (1969). Long–run linearity, locally Gaussian process, H–spectra and
infinite variances. International Economic Review 10 82–111.
[21] Mijnheer, J. (2002). Asymptotic inference for AR(1) processes with (nonnormal)
stable innovations. V. The explosive case. Journal of Mathematical Sciences 111,
3854–3856.
[22] Mittnik, S., and Ratchev, S. (2000). Stable Paretian Models in Finance. Wiley, New
York.
[23] Nelson, C.R., and Plosser, C. (1982). Trends and random walks in macroeconomic
time series: some evidence and implications. Journal of Monetary Economics 10,
139–162.
[24] Petrov, V.V. (1975). Sums of Independent Random Variables. Springer–Verlag, New
York.
[25] Phillips, P.C.B. (1988). Regression theory for near–integrated time series. Econometrica 56, 1021–1043.
[26] Phillips, P.C.B., and Magdalinos, T. (2005). Limit theory for moderate deviations
from a unit root. Journal of Econometrics, forthcoming.
[27] Samorodnitsky, G., and Taqqu, M. (1994). Stable Non–Gaussian Random Processes.
Chapman & Hall, New York.
[28] White, J.S. (1958). The limiting distribution of the serial correlation coefficient in
the explosive case. The Annals of Mathematical Statistics 29, 1188–1197.
[29] Zolotarev, V.M. (1986). One–Dimensional Stable Distributions. Translations of
Mathematical Monographs, Vol. 65. The American Mathematical Society, Providence, RI.
21