ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE
MODELS
By Alexander Aue, Lajos Horváth and Josef Steinebach
University of Utah and Universität zu Köln
First Version received September 2004
Abstract. We propose the quasi-maximum likelihood method to estimate the
parameters of an RCA(1) process, i.e. a random coefficient autoregressive time series of
order 1. The strong consistency and the asymptotic normality of the estimators are derived
under optimal conditions.
Keywords. Random coefficient autoregressive time series; parameter estimation;
quasi-maximum likelihood; consistency; asymptotic normality.
AMS 2000 subject classification. Primary 62M10; Secondary 62F10, 62F12.
1.
INTRODUCTION
During the last 30 years, there has been an increasing interest in nonlinear time
series models. One of the first examples is the random coefficient model
introduced and studied by Nicholls and Quinn (1982). Denote by N the set of
positive integers and by Z the set of all integers. Then, the random coefficient
autoregressive model of order 1, abbreviated RCA(1), is given by the equations
Xk ¼ ðu þ bk ÞXk1 þ ek ;
k 2 Z;
ð1Þ
where u is a real parameter. Throughout this paper, we assume that
fðbk ; ek Þ : k 2 Zg are independent and identically distributed pairs of
random variables on some probability space ðX; F ; P Þ:
ð2Þ
Time series fXk : k 2 Zg satisfying (1) have been used in the context of
random perturbations of dynamical systems and they have found a variety of
applications, for example, in finance and biology (cf. Nicholls and Quinn, 1982;
and Tong, 1990).
In this paper, we are interested in the quasi-maximum likelihood estimation of
the unknown parameters of an RCA(1) process. The procedure was introduced by
Quinn and Nicholls (1981) for more general RCA processes (see Chapter 4 of
Nicholls and Quinn, 1982), who established strong consistency and a central limit
theorem. However, their results require additional restrictive assumptions such as
the existence of second-order moments of the random variables in (2) (see
0143-9782/06/01 61–76
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 1
Ó 2006 Blackwell Publishing Ltd., 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main
Street, Malden, MA 02148, USA.
62
A. AUE, L. HORVÁTH AND J. STEINEBACH
Nicholls and Quinn, 1982, p. 70) as well as the boundedness of the second
moments of the Xk itself (see Assumption (v) in Quinn and Nicholls, 1981). In
contrast to this, we shall impose natural (and minimal) conditions solely on the
noise sequences fbkg and fekg. Further studies along these lines were conducted
by Schick (1996), who dealt with estimating the deterministic coefficient u in (1).
His method could, in principle, be extended to obtain similar results for the joint
estimation of all parameters.
The paper is organized as follows. In Section 2, we discuss some basic properties
of RCA(1) time series such as necessary and sufficient conditions for the existence
and uniqueness of a strictly stationary solution of (1) as well as the finiteness of
moments. Our main results are given in Section 3, where the strong consistency and
the asymptotic normality of the quasi-maximum likelihood estimator is obtained
under optimal assumptions. Section 4 contains the proofs of the latter results.
2.
THE STRUCTURE OF RCA(1) TIME SERIES
The process fXk : k 2 Zg satisfies a stochastic recurrence equation, so the
necessary and sufficient condition for the existence and the uniqueness of the
solution of (1) can be derived from Brandt (1986) and Bougerol and Picard (1992).
Related pioneering studies on bilinear equations, which may also be applied to
RCA(1) processes, were conducted by Quinn (1980, 1982). Andél (1976) studied
conditions for the weak stationarity of fXk : k 2 Zg without assuming (2). First,
we study the properties of
X ¼
1
X
ei
i¼0
i1
Y
ðu þ bj Þ:
ð3Þ
j¼0
(We use the convention P; ¼ 1.) Let logþx ¼ maxflog x, 0g be the positive part
of the logarithm.
Lemma 1.
We assume that (2) holds,
E logþ je0 j < 1
and
E logþ ju þ b0 j < 1:
ð4Þ
(i) If
1 E log ju þ b0 j < 0;
ð5Þ
then X defined in (3) converges absolutely with probability 1 (a.s.).
(ii) Conversely, if Pfe0 ¼ 0g < 1 and Pf|X| < 1g ¼ 1, then (5) holds.
Proof. (i) First we assume that (5) holds. By the strong law of large numbers,
there is a random variable i0 such that
Ó Blackwell Publishing Ltd 2006
ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
log ju þ b1 j þ log ju þ b2 j þ þ log ju þ bi j 1
ic;
2
if i i0 ;
63
ð6Þ
when 1 < c ¼ E log |u þ b0| < 0. Using (6), we have
jX j i0
X
jei j
i¼0
i1
Y
ju þ bj j þ
i0
Y
jei jeic=2 :
ð7Þ
i¼i0 þ1
k¼0
j¼0
1
X
ju þ bk j
Lemma 2.2 of Berkes et al. (2003) yields that
(
)
1
X
ic=2
jei je
< 1 ¼ 1;
P
i¼0
completing the first part of the lemma, when 1 < E log |u þ b0| < 0.
If E log|u þ b0| ¼ 1, then (6) holds for any c < 0 (cf. Chow and Teicher,
1997, pp. 125–126), so the argument in (7) can be repeated.
(ii) Since Pf|e0| ¼ 0g < 1, there is a constant a > 0 such that
Pf|e0| ag > 0. Let the events Ak be defined as
(
!
)
k 1
Y
ju þ bi j 2 ½a; 1Þ ½1; 1Þ ; k 2 N:
Ak ¼ x : jek j;
i¼0
Next we introduce an increasing sequence of r-algebras given by F0 ¼ f;, Xg
and Fk ¼ r((ei, bi), k i 0). Clearly, Ak 2 Fk. Applying (2), we get
(
)
k 1
X
log ju þ bi j 0 :
P fAk jF k1 g ¼ P fje0 j agI
i¼0
Hence
1
X
P fAk jF k1 g ¼ P fje0 j ag
k¼1
1
X
(
I
k¼1
k1
X
)
log ju þ bi j 0 :
ð8Þ
i¼0
Next we show that
1
X
k¼1
(
I
k1
X
)
log ju þ bi j 0
¼1
a.s.
ð9Þ
i¼0
If E log|u þ b0| > 0, then (9) follows from the strong law of large numbers.
If E log|u þ b0| ¼ 0, then the Chung–Fuchs law (cf. Chow and Teicher, 1997,
p. 154) yields (9). Corollary 3.2 of Durrett (1996, p. 240) with (8) and (9) gives
PfAk infinitely ofteng ¼ 1, and therefore
(
)
k1
Y
P lim ek
ðu þ bj Þ ¼ 0 ¼ 0:
k!1
j¼0
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64
A. AUE, L. HORVÁTH AND J. STEINEBACH
h
This contradicts the finiteness of X, completing the proof of Lemma 1.
Lemma 2. We assume that (2) and (5) hold, E|b0| < 1 and E|e0| < 1 with
some > 0. Then there is a d > 0 such that E|X|d < 1.
Proof. Let M(t) ¼ E|u þ b0|t, 0 t . We note that M(0) ¼ 1 and, by (5),
M (0þ) < 0. Hence M(t) is decreasing in a right neighbourhood of 0, so there is a
d > 0 such that M(d) < 1. We can assume, without loss of generality, that
0 < d 1. Since (a þ b)d ad þ bd for a, b 0 by concavity,
0
d
jX j 1
X
jei j
i1
Y
i¼0
!d
ju þ bj j
j¼0
1
X
i¼0
jei jd
i1
Y
ju þ bj jd :
j¼0
Using (2) and M(d) < 1, we conclude
EjX jd Eje0 jd
1
i1
1
X
X
Y
E
ju þ bj jd ¼ Eje0 jd
M i ðdÞ < 1:
i¼0
j¼0
i¼0
h
This completes the proof.
Lemma 3 provides a condition for the existence of E|X|m, where m 1,
generalizing Lemma 2.
Lemma 3. We assume that (2) and (5) hold and, for some m 1,
E|e0|m < 1, E|b0|m < 1, and E|u þ b0|m < 1. Then E|X|m < 1.
Proof. Using (2) and then the Minkowski inequality (cf. Chow and Teicher,
1997, p. 110), we get
ðEjX jm Þ1=m ðEje0 jm Þ1=m
1
X
ðEju þ b0 jm Þi=m < 1;
i¼0
h
completing the proof.
Eb20
2
¼ x and
Remark 1. If e0 and b0 are independent, Eb0 ¼ Ee0 ¼ 0,
Ee20 < 1, then EX2 < 1 if and only if u2 þ x2 < 1. This result was derived
from Andél (1976) by Nicholls and Quinn (1982, p. 31).
Theorem 1 gives the necessary and sufficient condition for the existence of a
unique solution of (1). The solution of (1) has the same distribution as X and we
see in Theorem 1 that (1) has a unique (strictly) stationary solution if and only if X
exists. We say that fXk : k 2 Zg is a nonanticipative solution of (1), if Xj is
independent of f(bk, ek) : k > jg for all j 2 Z.
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ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
Theorem 1. We assume that (1) and (4) hold.
(i) If (5) is satisfied, then
Xk ¼
1
X
eki
i¼0
i1
Y
ðu þ bkj Þ
j¼0
converges absolutely a.s. and the process fXk : k 2 Zg is the unique,
stationary nonanticipative solution of (1).
(ii) If
P fc1 b0 þ c2 e0 ¼ c3 g < 1 for any real c1 ; c2 ; c3 with ðc1 ; c2 Þ 6¼ ð0; 0Þ ð10Þ
and (1) has a nonanticipative solution, then (5) holds.
Proof. The first part of the theorem is an immediate consequence of Lemma 1
and the results of Brandt (1986). Condition (10) yields that Xk is irreducible in the
sense of Bougerol and Picard (1992), so their Theorem 2.5 implies the second part.
Confer also the discussion in Quinn (1982), in which a reference is made to Quinn
(1980).
h
3.
ESTIMATION OF THE PARAMETERS
We assume that
x2
covðb0 ; e0 Þ ¼
0
Eðb0 ; e0 Þ ¼ ð0; 0Þ and
0
;
r2
x2 ; r2 > 0:
ð11Þ
In this section, we are interested in the estimation of the parameter vector h ¼
(u, x2, r2) based on the quasi-maximum likelihood method. Observe that, for
k 2 Z,
EðXk jGk1 Þ ¼ uXk1
and
2
varðXk jGk1 Þ ¼ x2 Xk1
þ r2 ;
where Gk1 ¼ rððbi ; ei Þ : i k 1Þ:
So, if the vectors (bk, ek) were bivariate normal, then the (conditional)
likelihood would be given by
!
1=2
n Y
1
ðXi sXi1 Þ2
exp ;
Ln ðuÞ ¼
2 þ yÞ
2 þ yÞ
2ðxXi1
2pðxXi1
i¼1
where u ¼ (s, x, y). The quasi-maximum likelihood estimator ^hn is defined by
sup Ln ðuÞ ¼ Ln ð^hn Þ
ð12Þ
u2C
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66
A. AUE, L. HORVÁTH AND J. STEINEBACH
with some suitably chosen C R3. It is more convenient to work with the loglikelihood function
‘n ðuÞ ¼
n
1X
gi ðuÞ;
n i¼1
gi ðuÞ ¼
ðXi sXi1 Þ2
2
þ yÞ:
þ logðxXi1
2 þy
xXi1
Now, (12) can be written as
inf ‘n ðuÞ ¼ ‘n ð^hn Þ:
u2C
We assume that
1
1
C ¼ ðs; x; yÞ : s0 s s0 ; x x0 ; y y0
x0
y0
with some
s0 > 0; x0 > 1
and
ð13Þ
y0 > 1:
Even though the likelihood function was derived under the assumption of
normality, the consistency and the asymptotic normality of ^hn will be derived
without this distributional assumption. The proof of the consistency follows the
method developed by Pfanzagl (1969).
The first main result of this section is the strong consistency of the quasimaximum likelihood estimator ^
hn .
Theorem 2.
If (2), (4), (5), (10), (11) and (13) hold,
P fðu þ b0 Þe0 ¼ 0g < 1
ð14Þ
h 2 C;
ð15Þ
and
then
^
hn ! h
Remark 2.
equivalently,
a:s:
ðn ! 1Þ:
ð16Þ
If b0 and e0 are independent, then (14) is satisfied, or
P fðu þ b0 Þe0 6¼ 0g > 0:
This can easily be checked from the facts that
P fb0 > 0g > 0;
P fb0 < 0g > 0;
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P fe0 > 0g > 0;
P fe0 < 0g > 0;
ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
67
which imply, for example, in case of u 0, that
0 < P fu þ b0 > 0gP fe0 > 0g P fðu þ b0 Þe0 > 0g:
A similar argument applies in case of u < 0.
Remark 3. We imposed only (5) on the parameters which is the necessary and
sufficient condition for the existence of a unique strictly stationary version of the
RCA(1) process. Nicholls and Quinn (1982) proved that u2 þ x2 < 1 is
equivalent to the existence of a weakly stationary solution, if the sequences
fbk : k 2 Zg and fek : k 2 Zg are independent and (11) is satisfied. In the
presence of (2), this solution is also strictly stationary.
Remark 4. We assumed that x2 > 0 and r2 > 0. One of these assumptions
can be dropped. Assuming x2 0 and choosing
C ¼ fðs; x; yÞ : s0 s s0 ; 0 x x0 ; 1=y0 y y0 g;
Theorem 2 remains true. If x2 ¼ 0, then fXk : k 2 Zg is an AR(1) process, and
Theorem 2 remains true, if we also assume that EX02 < 1: If r2 0 is allowed,
then we can choose
Cfðs; x; yÞ : s0 s s0 ; 1=x0 x x0 ; 0 y y0 g;
and again Theorem 2 remains valid.
Next we consider the asymptotic normality of n1=2 ð^hn hÞ. Let
Yi1 ðu; j; cÞ ¼
j
Xi1
2
ðxXi1 þ
yÞc
;
j ¼ 0; 1; . . . ; 2c;
c 2 N:
ð17Þ
The proof will depend on the properties of
@
@
@
0
gi ðuÞ; gi ðuÞ; gi ðuÞ ;
gi ðuÞ ¼
@s
@x
@y
where
@
gi ðuÞ ¼ 2ðu s þ bi ÞYi1 ðu; 2; 1Þ 2ei Yi1 ðu; 1; 1Þ;
@s
@
gi ðuÞ ¼ ðu s þ bi Þ2 Yi1 ðu; 4; 2Þ 2ðu s þ bi Þei Yi1 ðu; 3; 2Þ
@x
e2i Yi1 ðu; 2; 2Þ þ Yi1 ðu; 2; 1Þ
ð18Þ
ð19Þ
@
gi ðuÞ ¼ ðu s þ bi Þ2 Yi1 ðu; 2; 2Þ 2ðu s þ bi Þei Yi1 ðu; 1; 2Þ
@y
e2i Yi1 ðu; 0; 2Þ þ Yi1 ðu; 0; 1Þ
ð20Þ
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A. AUE, L. HORVÁTH AND J. STEINEBACH
Let
A ¼ Eðg01 ðhÞÞT g01 ðhÞ;
ð21Þ
where xT denotes the transpose of x.
Our next result shows that the quasi-maximum likelihood estimator is
asymptotically normal. Let N(h, A) denote a normal random vector with mean
h and covariance matrix A, and let int(C) denote the interior of C. Set
2
3
2að2; 1Þ
0
0
H ¼ H ðhÞ ¼ 4
0
að4; 2Þ að2; 2Þ 5
ð22Þ
0
að2; 2Þ að0; 2Þ
with
aðj; cÞ ¼ E
Theorem 3.
X0j
< 1;
þ r 2 Þc
j ¼ 0; 1; . . . ; 2c;
ðx2 X02
c 2 N:
ð23Þ
If (2), (4), (5), (11), (13), (14), h 2 int(C) hold,
Eb40 < 1;
Ee40 < 1
ð24Þ
and
P fc1 b0 þ e0 2 fc2 ; c3 gg < 1 for any real c1 ; c2 ; c3 with c1 6¼ 0;
ð25Þ
then
D
h hÞ ! Nð0; H 1 AH Þ;
n1=2 ð^
where A and H are nonsingular matrices defined in (21) and (22), respectively.
Remark 5. If b0 and e0 are independent, condition (25) is satisfied.
Assume that, otherwise,
P fc1 b0 þ e0 2 fc2 ; c3 gg ¼ 1
for some c1 ; c2 ; c3 with c1 6¼ 0:
ð26Þ
Note that c2 6¼ c3, e.g. c2 < c3, and then c2 < 0 < c3, since E(c1b0 þ e0) ¼ 0,
but Eðc1 b0 þ e0 Þ2 ¼ c21 x2 þ r2 > 0: Now, from (26),
Z
P fc1 b0 þ e0 2 fc2 ; c3 gje0 ¼ vgdPe0 ðvÞ ¼ 1;
i.e.
P b0 2
c2 v c3 v
;
c1
c1
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¼1
for Pe0 almost-all v:
ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
69
Hence, b0 has a two-point distribution, e.g.
P fb0 ¼ u1 g > 0;
P fb0 ¼ u2 g > 0
for some u1 < 0 < u2 :
P fe0 ¼ v2 g > 0
for some v1 < 0 < v2 :
Similarly,
P fe0 ¼ v1 g > 0;
This, however, implies
P fc1 b0 þ e0 ¼ c1 u1 þ v1 g > 0;
P fc1 b0 þ e0 ¼ c1 u1 þ v2 g > 0;
P fc1 b0 þ e0 ¼ c1 u2 þ v2 g > 0;
with c1u1 þ v1 < c1u1 þ v2 < c1u2 þ v2, if c1 > 0, which contradicts (26). A
similar argument applies in case of c1 < 0.
4.
PROOFS OF THE MAIN RESULTS
We start with two technical lemmas.
Lemma 4.
If (2), (4), (5), (11), and (13) hold, then
E inf g1 ðuÞ > 1;
u2C
and Eg1(u) is continuous on C.
Proof.
It follows from Lemma 2 that
2
X
1 E sup j logðxX02 þ yÞj Elog 0 þ
þ Ej logðx0 X02 þ y0 Þj þ j log y0 j < 1:
x0 y0 u2C
So, we have
ðX1 sX0 Þ2
E sup logðxX02 þ yÞ > 1;
u2C xX 2 þ y
u2C
0
E inf g1 ðuÞ E inf
u2C
since ðX1 sX0 Þ2 ðxX02 þ yÞ1 0 for all u 2 C. Recalling (17), easy calculations
show that
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A. AUE, L. HORVÁTH AND J. STEINEBACH
gi ðuÞ ¼ ðu s þ bi Þ2 Yi1 ðu; 2; 1Þ þ e2i Yi1 ðu; 0; 1Þ
2
þ yÞ:
þ 2ðu s þ bi Þei Yi1 ðu; 1; 1Þ þ logðxXi1
Clearly, for all u 2 C and j ¼ 0, 1, 2,
E
jXi1 jj
< 1:
2 þy
xXi1
ð27Þ
Hence, by (11) and (23), we have
Eg1 ðuÞ ¼ ðu sÞ2 þ x2 að2; 1Þ þ r2 að0; 1Þ E log Y0 ðu; 0; 1Þ:
It is immediate that Eg1(u) is continuous on u 2 C.
Lemma 5.
h
If (2), (4), (5), (10), (11), (13) and (14) hold, then
Eg1 ðuÞ > Eg1 ðhÞ
Proof.
ð28Þ
for all
u 6¼ h; u 2 C:
By (28), we have
Eg1 ðuÞ ¼ ðu sÞ2 E
2 2
X02
x X 0 þ r2
þ
Eh
þ E logðx2 X02 þ r2 Þ;
xX02 þ y
xX02 þ y
where h(x) ¼ x log x, x > 0. Elementary arguments give that h(x) > h(1) ¼ 1
for all real x 6¼ 1. Hence Eg1(u) Eg1(h) and Eg1(u) ¼ Eg1(h) if and only if
2 2
x X0 þ r2
¼ 1 ¼ 1:
ð29Þ
s ¼ u and P
xX02 þ y
If x2 X02 þ r2 ¼ xX02 þ y a.s., then also ðx2 xÞX02 ¼ y r2 a.s. Now, if x2 6¼ x
or y 6¼ r2, then P fX02 ¼ cg ¼ 1 with some real c. By (1), we obtain
X12 ¼ ðu þ b1 Þ2 X02 þ e21 þ 2ðu þ b1 Þe1 X0
a.s.
ð30Þ
Using the stationarity of fXk : k 2 Zg, we also have P fX12 ¼ cg ¼ 1, so (30)
gives
c ¼ ðu þ b1 Þ2 c þ e21 þ 2ðu þ b1 Þe1 X0
a.s.
ð31Þ
Note that, in view of (1), (2) and (11), EX1 ¼ E(u þ b1)E(X0) þ Ee1 ¼uE(X0),
with |u| < 1, and hence, by stationarity, EXk ¼ 0 for all k 2 Z. On taking
conditional expectations with respect to (b1, e1) in (31), we have, almost surely,
c ¼ ðu þ b1 Þ2 c þ e21 . Plugging the latter relation back into (31) and noting that
Ó Blackwell Publishing Ltd 2006
ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
71
pffiffiffi
jX0 j ¼ c 6¼ 0, we must have (u þ b1)e1 ¼ 0 with probability 1, which contradicts
assumption (14).
Hence (29) implies that s ¼ u, x ¼ x2 and y ¼ r2.
h
Proof of Theorem 2. Our proof follows the general method developed by
Pfanzagl (1969). First, we observe that, by (15),
inf ‘n ðuÞ ‘n ðhÞ;
u2C
and therefore
lim sup inf ‘n ðuÞ lim sup ‘n ðhÞ
u2C
n!1
a.s.
n!1
The sequence fgi(h) : i 2 Zg is a stationary and ergodic sequence with
E|g1(h)| < 1. So, by the ergodic theorem, we have
lim ‘n ðhÞ ¼ Eg1 ðhÞ
n!1
a.s.;
resulting in
lim sup inf ‘n ðuÞ Eg1 ðhÞ
n!1
u2C
ð32Þ
a.s.
For each positive integer n, ‘n (u) is continuous on the compact set C, and
therefore ^
hn 2 C. Now, by (32), we have
hn Þ Eg1 ðhÞ
lim sup ‘n ð^
ð33Þ
a.s.
n!1
Let C C be a compact set such that the distance between C and h is positive.
The set C can easily be constructed. For example, let U be an open ball around h
with a small enough radius. Clearly, C ¼ CnU would satisfy the required
assumptions on C. Since g1(u) is continuous on C (and therefore on C), for any
r < Eg1(u), u 2 C, there exists an open ball U(u) around u such that
r < E inf g1 ðtÞ
t2U ðuÞ
(cf. Pfanzagl, 1969, p. 271). The sets U(u), u 2 C, are an open covering of C, and
therefore they contain a finite covering, say, U(u1), U(u2), . . . , U(uk) of C. By the
ergodic theorem, for any 1 j k, we have
n
1X
inf gi ðuÞ ¼ E inf g1 ðuÞ > r
n!1 n
u2U ðuj Þ
u2U ðuj Þ
i¼1
lim
a.s.
ð34Þ
Observing that
inf ‘n ðuÞ min
u2C
n
1X
inf gi ðuÞ;
1jk n
u2U ð uj Þ
i¼1
inf ‘n ðuÞ min
1jk u2U ðuj Þ
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A. AUE, L. HORVÁTH AND J. STEINEBACH
we have, by (34),
lim inf inf ‘n ðuÞ min lim inf
n!1 u2C
1jk n!1
n
1X
inf gi ðuÞ > r;
n i¼1 u2U ðuj Þ
except on an event Br with P(Br) ¼ 0 for any rational r satisfying
r < inf Eg1 ðuÞ:
ð35Þ
u2C
Let
B¼
[
fBr : r is rational and satisfies (35)g:
Then P(B) ¼ 0 and, on the complement of B, we have
lim inf inf ‘n ðuÞ > r
n!1 u2C
for all r rational satisfying (35). Thus,
lim inf inf ‘n ðuÞ inf Eg1 ðuÞ
n!1 u2C
u2C
ð36Þ
a.s.
According to Lemma 4, Eg1(u) is continuous and, by Lemma 5, h 2j C is the
unique minimum of Eg1(u), so (36) yields
lim inf inf ‘n uÞ > Eg1 ðhÞ a.s.
ð37Þ
n!1 u2C
Let U be an open ball around h with a small enough radius and U ¼ U \ C. If
^
hn 2j U infinitely often, there is a random subsequence nk such that, with C ¼
CnU , we have
hnk Þ lim sup ‘n ð^hn Þ
lim inf inf ‘n ðuÞ lim inf ‘nk ð^
n!1 u2C
k!1
a.s.
ð38Þ
n!1
However, (38) contradicts (33) and (37). So, for any U as above, there is a random
variable n0 such that ^
hn 2 U for all n n0. This completes the proof.
h
The proof of Theorem 3, i.e. of the asymptotic normality of n1=2 ð^hn hÞ, will
also be divided into several lemmas.
Lemma 6. We assume that (2), (4), (5), (11), (13), (15), (24) and (25) hold. Then
Eg0 1ðhÞ ¼ 0 and the matrix A from (21) exists and is nonsingular.
Proof.
From (2), (11), (18)–(20) and (23), it follows that
@
@
@
gi ðhÞ ¼ gi ðhÞ ¼ gi ðhÞ ¼ 0:
@s
@x
@y
It is clear that A is non-negative definite and finite, so it is enough to prove that
A is nonsingular. Let us assume that A is singular. Then
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ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
c1
@
@
@
g1 ðhÞ þ c2 g1 ðhÞ þ c3 g1 ðhÞ ¼ 0 a.s.
@s
@x
@y
73
ð39Þ
with some constants c1, c2 and c3, not all zero. Using the formulas for the partial
derivatives of g1(h), (39) holds if and only if
ðc3 þ c2 X02 Þ
ðb1 X0 þ e1 Þ2
ðx2 X02 þ r2 Þ
2c1 X0
2
b1 X0 þ e 1
c3 þ c2 X02
þ
¼0
x2 X02 þ r2 x2 X02 þ r2
a.s.
ð40Þ
First observe that c2 6¼ 0 or c3 6¼ 0. Otherwise, c1 6¼ 0 and X0(b1X0 þ e1) ¼ 0
a.s., which implies
ðb1 X0 þ e1 ÞIfX0 6¼0g ¼ 0
a.s.
On taking the conditional expectation with respect to (b1, e1), we have
b1 EðX0 IfX0 6¼0g Þ þ e1 P fX0 6¼ 0g ¼ e1 P fX0 6¼ 0g ¼ 0:
Since PfX0 6¼ 0g > 0, e1 ¼ 0 a.s., which contradicts our assumptions.
So, c2 6¼ 0 or c3 6¼ 0, which results in P fc3 þ c2 X02 6¼ 0g ¼ 1, because otherwise
P fX02 ¼ cg ¼ 1 for some c, a contradiction.
Now, from (40),
P fb1 X0 þ e1 2 fC2 ; C3 gg ¼ 1;
where C2 ¼ C2(X0), C3 ¼ C3(X0). This implies
EP fb1 X0 þ e1 2 fC2 ðX0 Þ; C3 ðX0 ÞgjX0 g ¼ 1;
i.e. since X0 and (b1, e1) are independent,
Z
P fb1 x þ e1 2 fC2 ðxÞ; C3 ðxÞggdPX0 ðxÞ ¼ 1;
resulting in
P fb1 x þ e1 2 fC2 ðxÞ; C3 ðxÞgg ¼ 1 for PX0 almost-all x:
In view of PfX0 6¼ 0g > 0, the latter equality contradicts our assumption (25),
which completes the proof.
h
Our next result shows that n1=2 ‘0n ðhÞ converges in distribution to a
nondegenerate three-dimensional normal random vector.
Lemma 7.
Then
We assume that (2), (4), (5), (11), (13), (15), (24) and (25) hold.
D
n1=2 ‘0n ðhÞ ! Nðh; AÞ;
where A is defined in (21).
Ó Blackwell Publishing Ltd 2006
74
A. AUE, L. HORVÁTH AND J. STEINEBACH
Proof.
We note that
n1=2 ‘0n ðhÞ ¼ n1=2
n
X
g0i ðhÞ;
i¼1
Eg0i ðhÞ
0; covðg0i ðhÞÞ
g0i ðhÞ; g0j ðhÞ
¼
¼ A, and
are uncorrelated if i 6¼ j.
where
Moreover, by (2) and (11), we have
!
k
k 1
X
X
0
gi ðhÞGk1 ¼
g0i ðhÞ:
E
i¼1
i¼1
Thus, Theorem 23.1 of Billingsley (1968, p. 206) and an application of the
Cramér–Wold device (Billingsley, 1968, pp. 48–49) yield Lemma 7.
h
Set H ðuÞ ¼ Eg001 ðuÞ. By (11), similarly to (27), it is obvious that H(u) is finite for
all u 2 C.
Let |Æ| denote the maximum norm of a matrix.
Lemma 8.
Then
We assume that (2), (4), (5), (11), (13), (14), (15) and (24) hold.
sup j‘00n ðuÞ H ðuÞj ! 0
ð41Þ
a:s:
u2C
Moreover, H(u) is continuous on C and H ¼ H(h) as in (22) is nonsingular.
Proof.
The ergodic theorem yields
lim sup sup j‘ð3Þ
n ðuÞj lim sup
n!1
u2C
n!1
n
1X
ð3Þ
ð3Þ
sup jg ðuÞj ¼ E sup jg1 ðuÞj < 1
n i¼1 u2C i
u2C
a.s.
Hence, the mean value theorem applied to the coordinates of ‘00n ðuÞ yields (41).
It is immediate that H(u) is continuous on C. If we show that
að4; 2Þ að2; 2Þ
H ¼
að2; 2Þ að0; 2Þ
is nonsingular, the proof of Lemma 8 will be complete. We note that H ¼ EgTg,
where
X02
1
g¼
;
:
x2 X02 þ r2 x2 X02 þ r2
If H is singular, then there are c1 and c2, (c1, c2) 6¼ (0, 0), such that
X2
1
P c1 2 2 0 2 þ c2 2 2
¼
0
¼ 1:
x X0 þ r
x X 0 þ r2
Since (42) implies that c1 6¼ 0, we have
Ó Blackwell Publishing Ltd 2006
ð42Þ
ESTIMATION IN RANDOM COEFFICIENT AUTOREGRESSIVE MODELS
P fX02 ¼ c2 =c1 g ¼ 1;
75
ð43Þ
which is the same contradiction as in the proof of Lemma 6.
h
Now we are ready to prove the asymptotic normality of n1=2 ð^hn hÞ.
Proof of Theorem 3. We proved in Lemma 8 that H1 exists. Since ‘n(u) is
continuously differentiable on int(C), by Theorem 2 we have that ‘0n ð^hn Þ ¼ 0 and
therefore ‘0n ð^
hn Þ ‘0n ðhÞ ¼ ‘0n ðhÞ. Applying the mean value theorem to the
coordinates of ‘0n ðuÞ, by Lemma 8 and Theorem 2 we get
ð^
hn hÞðH þ oð1ÞÞ ¼ ‘0n ðhÞ
The result now follows from Lemma 7.
a.s.
h
ACKNOWLEDGEMENTS
A. Aue, L. Horváth and J. Steinebach were partially supported by NATO grant
PST. EAP.CLG 980599; A. Aue and L. Horváth were partially supported by NSF
grant INT-0223262.
NOTE
Corresponding author: J. Steinebach, Mathematisches Institut, Universität zu
Köln, Weyertal 86-90, D-50931 Köln, Germany. E-mail: jost@math.uni-koeln.de
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