PARAMETER ESTIMATION FOR PERIODICALLY STATIONARY
TIME SERIES
By Paul L. Anderson and Mark M. Meerschaert
Albion College and University of Otago
First Version received December 2002
Abstract. The innovations algorithm can be used to obtain parameter estimates for
periodically stationary time series models. In this paper, we compute the asymptotic
distribution for these estimates in the case, where the innovations have a finite fourth
moment. These asymptotic results are useful to determine which model parameters are
significant. In the process, we also develop asymptotics for the Yule–Walker estimates.
Keywords. Time series; periodically stationary; innovations algorithm.
AMS 1991 subject classifications. Primary 62M10, 62E20; secondary 60E07, 60F05.
1.
INTRODUCTION
A stochastic process Xt is called periodically stationary (in the wide sense) if
lt ¼ EXt and ct(h) ¼ EXtXtþh for h ¼ 0, ±1, ±2, . . . are all periodic functions
of time t with the same period m 1. If m ¼ 1, then the process is stationary.
Periodically stationary processes manifest themselves in such fields as
economics, hydrology and geophysics, where the observed time series are
characterized by seasonal variations in both the mean and covariance
structure. An important class of stochastic models for describing periodically
stationary time series are the periodic ARMA models, in which the model
parameters are allowed to vary with the season. Periodic ARMA models are
developed in Adams and Goodwin (1995), Anderson and Vecchia (1993),
Anderson and Meerschaert (1997, 1998), Anderson et al. (1999), Jones and
Brelsford (1967), Lund and Basawa (2000), Pagano (1978), Salas et al. (1985),
Tjostheim and Paulsen (1982), Troutman (1979), Vecchia and Ballerini (1991)
and Ula (1993).
Anderson et al. (1999) develop the innovations algorithm for periodic ARMA
model parameters. In this paper, we provide the asymptotic estimates necessary to
determine which of these estimates are statistically different from zero, under the
classical assumption that the innovations have finite fourth moment. Brockwell
and Davis (1988), discuss asymptotics of the innovations algorithm for stationary
time series, using results of Berk (1974) and Bhansali (1978). Our results reduce to
theirs, when the period m ¼ 1. Since our technical approach extends that of
Brockwell and Davis (1988), we also need to develop periodically stationary
0143-9782/05/04 489–518
JOURNAL OF TIME SERIES ANALYSIS Vol. 26, No. 4
Ó 2005 Blackwell Publishing Ltd., 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main
Street, Malden, MA 02148, USA.
490
P. L. ANDERSON AND M. M. MEERSCHAERT
analogues of results in Berk (1974) and Bhansali (1978). In particular, we obtain
asymptotics for the Yule–Walker estimates of a periodically stationary process.
Although the innovations estimates are more useful in practice, the asymptotics of
the Yule–Walker estimates are also of some independent interest.
2.
THE INNOVATIONS ALGORITHM FOR PARMA PROCESSES
The periodic ARMA process fX~t g with period m [denoted by PARMAm(p, q)] has
representation
Xt p
X
/t ðjÞXtj ¼ et j¼1
q
X
ht ðjÞetj ;
ð1Þ
j¼1
where Xt ¼ X~t lt and fetg is a sequence of random variables with mean zero
and SD rt such that fr1
t et g is i.i.d. The autoregressive (AR) parameters /t(j),
the moving average parameters ht(j), and the residual SDs rt are all periodic
functions of t with the same period m 1. In this paper, we will make the
classical assumption Ee4t < 1, which leads to normal asymptotics for the
parameter estimates. We also assume that the model admits a causal
representation
Xt ¼
1
X
wt ðjÞetj ;
ð2Þ
j¼0
where wt(0) ¼ 1 and
condition
P1
j¼0
jwt ðjÞj < 1 for all t, and satisfies an invertibility
et ¼
1
X
pt ðjÞXtj ;
ð3Þ
j¼0
P
where pt(0) ¼ 1 and 1
j¼0 jpt ðjÞj < 1 for all t.
ðiÞ
Let X^iþk ¼ PHk;i Xiþk denote the one-step predictors, where Hk,i ¼
spfXi, . . ., Xiþk1g, k 1, and PHk,i is the orthogonal projection onto this
space, which minimizes the mean squared error (MSE)
ðiÞ
ðiÞ
vk;i ¼ kXiþk X^iþk k2 ¼ EðXiþk X^iþk Þ2 :
Then
ðiÞ
ðiÞ
ðiÞ
X^iþk ¼ /k;1 Xiþk1 þ þ /k;k Xi ;
ðiÞ
ðiÞ
k 1;
ð4Þ
ðiÞ
where the vector of coefficients /k ¼ ð/k;1 ; . . . ; /k;k Þ0 solves the prediction
equation
ðiÞ
ðiÞ
Ck;i /k ¼ ck
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ð5Þ
491
PARAMETER ESTIMATION
ðiÞ
with ck ¼ ðciþk1 ð1Þ; ciþk2 ð2Þ; . . . ; ci ðkÞÞ0 and
Ck;i ¼ ciþk‘ ð‘ mÞ
ð6Þ
‘;m¼1;...;k
is the covariance matrix of (Xiþk1, . . . , Xi)0 for each i ¼ 0, . . . , m 1. Let
^ci ð‘Þ ¼ N 1
N 1
X
ð7Þ
Xjmþi Xjmþiþ‘
j¼0
denote the (uncentered) sample autocovariance, where Xt ¼ X~t lt . If we replace
the autocovariances in the prediction equation (5) with their corresponding
ðiÞ
ðiÞ
sample autocovariances, we obtain the estimator /^k;j of /k;j .
Because the process is nonstationary, the Durbin–Watson algorithm for
ðiÞ
computing /^k;j does not apply. However, the innovations algorithm still applies to
a nonstationary process. Writing
ðiÞ
X^iþk ¼
k
X
ðiÞ
ðiÞ
hk;j ðXiþkj X^iþkj Þ
ð8Þ
j¼1
ðiÞ
yields the one-step predictors in terms of the innovations Xiþkj X^iþkj .
Proposition 4.1 of Lund and Basawa (2000), shows that if r2i > 0 for i ¼
0, . . . , m 1, then for a causal PARMAm(p, q) process the covariance matrix Ck,i is
nonsingular for every k 1 and each i. Anderson et al. (1999) shows that if
EXt ¼ 0 and Ck,i is nonsingular for each k 1, then the one-step predictors X^iþk ,
k 0, and their mean-square errors (MSEs)vk,i, k 1, are given by
v0;i ¼ ci ð0Þ;
‘1
X
ðiÞ
ðiÞ
ðiÞ
h‘;‘j hk;kj vj;i ;
hk;k‘ ¼ ðv‘;i Þ1 ciþ‘ ðk ‘Þ ð9Þ
j¼0
vk;i ¼ ciþk ð0Þ k1
X
ðiÞ
ðhk;kj Þ2 vj;i ;
j¼0
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
where (9) is solved in the order v0,i, h1;1 , v1,i, h2;2 , h2;1 , v2,i, h3;3 , h3;2 , h3;1 , v3,i, . . . The
results in Anderson et al. (1999) show that
ðhikiÞ
hk;j
! wi ðjÞ;
vk;hiki ! r2i ;
ðhikiÞ
/k;j
ð10Þ
! pi ðjÞ
as k ! 1 for all i, j, where
j m½j=m
if j ¼ 0; 1; . . .,
hji ¼
m þ j m½j=m þ 1 if j ¼ 1; 2; . . .
and [Æ] is the greatest integer function.
Ó Blackwell Publishing Ltd 2005
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P. L. ANDERSON AND M. M. MEERSCHAERT
If we replace the autocovariances in (9) with the corresponding sample
ðiÞ
autocovariances (7), we obtain the innovations estimates h^k;‘ and ^vk;i . Similarly,
replacing the autocovariances in (5) with the corresponding sample
ðiÞ
autocovariances yields the Yule–Walker estimators /^k;‘ . The consistency of
these estimators was also established in Anderson et al. (1999). Suppose that fXtg
is the mean zero PARMA process with period m given by (1) and that Eðe4t Þ < 1.
Assume that the spectral density matrix f(k) of the equivalent vector ARMA
process is such that mz0 z z0 f(k)z Mz0 z, p k p, for some m and M such
that 0 < m M < 1 and for all z in Rm. If k is chosen as a function of the
sample size N, so that k2/N ! 0 as N ! 1 and k ! 1, then the results in
Anderson et al. (1999) also show that
ðhikiÞ P
^
hk;j
! wi ðjÞ;
P
^vk;hiki ! r2i ;
ð11Þ
P
^ ðhikiÞ !
pi ðjÞ
/
k;j
for all i, j. This yields a practical method for estimating the model parameters, in
the classical case of finite fourth moments. The results of Section 3, can then be
used to determine which of these model parameters are statistically significantly
different from zero.
3.
ASYMPTOTIC RESULTS
We compute the asymptotic distribution for the innovations estimates of the
parameters in a periodically stationary time series (2) with period m 1. In the
process, we also obtain the asymptotic distribution of the Yule–Walker estimates.
For any periodically stationary time series, we can construct an equivalent
(stationary) vector moving average process in the following way: Let Zt ¼
(etm, . . . , e(tþ1)m1)0 and Yt ¼ (Xtm, . . . , X(tþ1)m1)0 , so that
Yt ¼
1
X
Wj Ztj ;
ð12Þ
j¼1
where Wj is the m m matrix with i‘ entry wi(tm þ i ‘), and we number the rows
and columns 0, 1, . . . , m 1 for ease of notation. Also, let N(m, C) denote a
Gaussian random vector with mean m and covariance matrix C, and let )
indicate convergence in distribution. Our first result gives the asymptotics of the
Yule–Walker estimates. A similar result was obtained by Lewis and Reinsel (1985)
for vector AR models, however the prediction problem here is different. For
example, suppose that (2) represents monthly data with m ¼ 12. For a periodically
stationary model, the prediction equations (4) use observations for earlier months
in the same year. For the equivalent vector moving average model, the prediction
equations use only observations from past years.
Ó Blackwell Publishing Ltd 2005
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PARAMETER ESTIMATION
Theorem 1. Suppose that the periodically stationary moving average (2) is
causal, invertible, Ee4t < 1, and that for some 0 < g G < 1 we have
gz0 z z0 f(k)z Gz0 z for all p k p, and all z in Rm, where f(k) is the
spectral density matrix of the equivalent vector moving average process (12). If k ¼
k(N) ! 1 as N ! 1 with k3/N ! 0 and
1
X
N 1=2
jp‘ ðk þ jÞj ! 0 for ‘ ¼ 0; 1; . . . ; m 1
ð13Þ
j¼1
then for any fixed positive integer D
hiki
N 1=2 pi ðuÞ þ /^k;u : 1 u D; i ¼ 0; . . . ; m 1 ) N ð0; KÞ;
ð14Þ
where
K ¼ diagðr20 Kð0Þ ; r21 Kð1Þ ; . . . ; r2m1 Kðm1Þ Þ
ð15Þ
with
ðKðiÞ Þu;v ¼
m1
X
pimþs ðsÞpimþs ðs þ jv ujÞr2
imþs
ð16Þ
s¼0
and m ¼ min(u, v), 1 u, v D.
In Theorem 1, note that K(i) is a D D matrix and the Dm-dimensional vector
given in (14) is ordered
h0ki
h0ki
hm1ki
N 1=2 ðp0 ð1Þþ /^k;1 ;...;p0 ðDÞþ /^k;D ;...;pm1 ð1Þ þ /^k;1
hm1ki 0
;...;pm1 ðDÞ þ /^k;D
Þ:
Next we present our main result, giving asymptotics for innovations estimates of a
periodically stationary time series.
Theorem 2. Suppose that the periodically stationary moving average (2) is
causal, invertible, Ee4t < 1, and that for some 0 < g G < 1 we have
gz0 z z0 f(k)z Gz0 z for all p k p, and all z in Rm, where f(k) is the
spectral density matrix of the equivalent vector moving average process (12). If k ¼
k(N) ! 1 as N ! 1 with k3/N ! 0 and
N 1=2
1
X
jp‘ ðk þ jÞj ! 0
for ‘ ¼ 0; 1; . . . ; m 1
ð17Þ
j¼1
then
ðhikiÞ
N 1=2 ðh^k;u
wi ðuÞ : u ¼ 1; . . . ; D; i ¼ 0; . . . ; m 1Þ ) N ð0; V Þ;
ð18Þ
V ¼ A diagðr20 Dð0Þ ; . . . ; r2m1 Dðm1Þ ÞA0 ;
ð19Þ
where
Ó Blackwell Publishing Ltd 2005
494
P. L. ANDERSON AND M. M. MEERSCHAERT
A¼
D1
X
En P½DmnðDþ1Þ ;
ð20Þ
n¼0
Dn
n
n
zfflfflfflffl}|fflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflffl}|fflfflfflffl{
En ¼ diagð0; . . . ; 0; w0 ðnÞ; . . . ; w0 ðnÞ ; 0; . . . ; 0;
w1 ðnÞ; . . . ; w1 ðnÞ ; . . . ; 0; . . . ; 0; wm1 ðnÞ; . . . ; wm1 ðnÞ Þ;
|fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
n
Dn
Dn
2
2
DðiÞ ¼ diagðr2
i1 ; ri2 ; . . . ; riD Þ
and P an orthogonal Dm Dm cyclic permutation matrix,
1
0
0 1 0 0 0
B0 0 1 0 0C
C
B
.. C
B. . . .
P ¼ B .. .. .. ..
C:
.
C
B
@0 0 0 0 1A
1
0 0
ð21Þ
0
ð22Þ
ð23Þ
0
Note that P0 is the Dm Dm identity matrix and P‘ (P0 )‘. Matrix
multiplication yields Corollary 1.
Corollary 1.
Regarding Theorem 2, in particular, we have that
ðhikiÞ
wi ðuÞÞ ) N ð0; r2
N 1=2 ðh^k;u
iu
u1
X
r2in w2i ðnÞÞ:
ð24Þ
n¼0
Remark. Corollary 1 also holds the asymptotic result for the second-order
stationary process, where the period is just m ¼ 1. In this case r2i ¼ r2 so (24)
becomes
N 1=2 ðh^k;u wðuÞÞ ) N ð0;
u1
X
w2 ðnÞÞ;
n¼0
which agrees with Theorem 2.1 in Brockwell and Davis (1988).
4.
PROOFS
The proof of Theorem 1 requires some preliminaries. First we show that (K(i))u,v is
the limit of the u, v entry in the inverse covariance matrix (6).
Lemma 1. Suppose that the periodically stationary moving average (2) is causal,
invertible, and that Ee4t < 1. If Ck,i is given by (6) and (A)u,v denotes the u, v entry
of the matrix A, then for any i ¼ 0, 1, . . ., v 1 we have
Ó Blackwell Publishing Ltd 2005
495
PARAMETER ESTIMATION
ðiÞ
ðC1
k;hiki Þu;v ! ðK Þu;v ¼
m1
X
pimþs ðsÞpimþs ðs þ jv ujÞr2
imþs
ð25Þ
s¼0
as k ! 1, where m ¼ min(u, v).
Proof.
The prediction equation (5) yield
ðiÞ
ðiÞ
ciþ‘ ðk ‘Þ /k;1 ciþ‘ ðk 1 ‘Þ /k;k ciþ‘ ð‘Þ ¼ 0
ð26Þ
ðiÞ
for 0 ‘ k1 and since vk;i ¼ hXiþk ; Xiþk X^iþk i we also have
ðiÞ
ðiÞ
vk;i ¼ ciþk ð0Þ /k;1 ciþk ð1Þ /k;k ciþk ðkÞ:
ð27Þ
Define
0
1
B
B0
B
F ¼B
B ...
B
@
0
/k1;1
ðiÞ
/k1;2
ðiÞ
1
/k2;1
..
.
ðiÞ
0
0
ðiÞ
/k1;k1
1
C
ðiÞ
/k2;k2 C
C
C
..
C
.
C
ðiÞ
/1;1 A
1
and use the fact that cr(s) ¼ crþs(s), so that (Ck,i)j,‘ ¼ ciþk‘(‘j), to compute
that for 1 j < ‘ and 2 ‘ k the (j, ‘) element of the matrix FCk,i is
ðiÞ
ðiÞ
ciþk‘ ð‘ jÞ /kj;1 ciþk‘ ð‘ j 1Þ /kj;kj ciþk‘ ð‘ kÞ:
Substitute n0 ¼ k j and ‘0 ¼ k ‘ to obtain
ðiÞ
ðiÞ
ðF Ck;i Þj;‘ ¼ ciþ‘0 ðn0 ‘0 Þ /n0 ;1 ciþ‘0 ðn0 ‘0 1Þ /n0 ;n0 ciþ‘0 ð‘0 Þ ¼ 0
in view of (26), so that FCk,i is lower triangular. Also (27) yields (FCk,i)j,j ¼ vkj,i
for 1 j k. Since the transpose F0 is also lower triangular, the matrix FCk,iF 0 is
still lower triangular, with (j, j) element vkj,i. But FCk,iF 0 is a symmetric matrix,
1
1
so it is diagonal. In fact, FCk,iF 0 ¼ H1, where H ¼ diagðv1
k1;i ; vk2;i ; . . . ; v0;i Þ.
1
0
Then Ck;i ¼ F HF , and a computation shows that
ðC1
k;i Þu;v ¼
minðu;vÞ
X
ðiÞ
ðiÞ
/k‘;u‘ /k‘;v‘ v1
k‘;i ;
ð28Þ
‘¼1
ðiÞ
where the number of summands is fixed and finite for all k, and we set /k;0 1
to simplify the notation. Substitute hi ki for i in (28) and apply (10) using
hi ki ¼ h(i ‘)(k ‘)i to obtain
vk‘;hiki ! r2i‘ ;
ðhikiÞ
/k‘;u‘ ! pi‘ ðu ‘Þ;
ðhikiÞ
/k‘;v‘
ð29Þ
! pi‘ ðv ‘Þ
Ó Blackwell Publishing Ltd 2005
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P. L. ANDERSON AND M. M. MEERSCHAERT
as k ! 1, so that
ðC1
k;hiki Þu;v !
minðu;vÞ
X
pi‘ ðu ‘Þpi‘ ðv ‘Þr2
i‘
‘¼1
as k ! 1. Substitute m ¼ min(u, v) and commute the first two terms if m ¼ v to
finish the proof of Lemma 1.
u
In order to prove Theorem 1, it will be convenient to use a slightly different
ðiÞ
ðiÞ
estimator fk;j for the coefficients /k;j of the one-step predictor (4), obtained by
minimizing
wk;i ¼ ðN kÞ1
NX
k1
ðiÞ
ðiÞ
ðXjmþiþk fk;1 Xjmþiþk1 fk;k Xjmþi Þ2
ð30Þ
j¼0
ðiÞ
for i ¼ 0, . . . , v 1. For i fixed, take partials in (30) with respect to fk;‘ for each
‘ ¼ 1, . . . , k to obtain
NX
k1
ðiÞ
ðiÞ
ðXjmþiþk fk;1 Xjmþiþk1 fk;k Xjmþi ÞXjmþiþk‘ ¼ 0
j¼0
and rearrange to get
ðiÞ ðiÞ
ðiÞ ðiÞ
ðiÞ
fk;1^sk1;k‘ þ þ fk;k ^s0;k‘ ¼ ^sk;k‘
ð31Þ
for each ‘ ¼ 1, . . . , k, where
1
^sðiÞ
m;n ¼ ðN kÞ
NX
k1
Xjmþiþm Xjmþiþn :
j¼0
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
Now define ^rk ¼ ð^sk;k1 ; . . . ; ^sk;0 Þ0 and fk ¼ ðfk;1 ; . . . ; fk;k Þ0 and let
0
1
ðiÞ
ðiÞ
^sk1;k1 ^s0;k1
B
C
.. C
..
^ ðiÞ ¼ B
R
k
@
. A
.
ðiÞ
ðiÞ
^sk1;0 ^s0;0
so that (31) becomes
^ ðiÞ f ðiÞ ¼ ^rðiÞ
R
k k
k
ð32Þ
analogous to the prediction equations (5).
Theorem 1 and Lemma 1 depend on modulo m arithmetic, which requires our
hi ki-notation. Since Lemmas 2–8 do not have this dependence, we proceed
with the less cumbersome i-notation.
ðiÞ
ðiÞ
Lemma 2. Let pk ¼ ðpiþk ð1Þ; . . . ; piþk ðkÞÞ0 and Xj ðkÞ ¼ ðXðjkÞmþiþk1 ; . . . ;
XðjkÞmþi Þ0 . Then for all i ¼ 0, . . ., m 1 and k 1 we have
Ó Blackwell Publishing Ltd 2005
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PARAMETER ESTIMATION
ðiÞ
ðiÞ
^ ðiÞ Þ1
pk þ fk ¼ ðR
k
N 1
1 X
ðiÞ
X ðkÞeðjkÞmþiþk;k ;
N k j¼k j
ð33Þ
where et,k ¼ Xt þ pt(1)Xt1 þ þ pt(k)Xtk.
Note that
Proof.
^ ðiÞ ¼
R
k
N 1
1 X
ðiÞ
ðiÞ
X ðkÞXj ðkÞ0
N k j¼k j
ðiÞ
and ^rk ¼
N1
1 X
ðiÞ
X ðkÞXðjkÞmþiþk
N k j¼k j
and apply (32) to obtain
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
^ Þ1^r
pk þ fk ¼ pk þ ðR
k
hk
i
ðiÞ 1 ^ ðiÞ ðiÞ
ðiÞ
^
¼ ðRk Þ Rk pk þ ^rk
^ ðiÞ Þ1
¼ ðR
k
ðiÞ
^ Þ1
¼ ðR
k
N 1
1 X
ðiÞ
ðiÞ
ðiÞ
ðiÞ
X ðkÞXj ðkÞ0 pk þ Xj ðkÞXðjkÞmþiþk
N k j¼k j
N 1
h
i
1 X
ðiÞ
ðiÞ
ðiÞ
Xj ðkÞ Xj ðkÞ0 pk þ XðjkÞmþiþk ;
N k j¼k
u
which is equivalent to (33).
Lemma 3.
For all i ¼ 0, . . . , m 1 and k 1 we have
ðiÞ 0 ðiÞ
wk;i ¼ ^riþk ð0Þ fk ^rk ;
where ^ri ð0Þ ¼ ðN kÞ
Proof.
j¼k
ð34Þ
2
XðjkÞmþi
.
The right-hand side of (34) equals
^riþk ð0Þ ¼
1 PN 1
N 1
1 X
ðiÞ 0 ðiÞ
fk Xj ðkÞXðjkÞmþiþk
N k j¼k
N 1h
i
1 X
ðiÞ
ðiÞ
XðjkÞmþiþk fk;1 XðjkÞmþiþk1 fk;k XðjkÞmþi XðjkÞmþiþk ;
N k j¼k
^ 2 , where we let Y ¼ (Xiþk, Xmþiþk,. . .,
while wk;i ¼ ðN kÞ1 kY Yk
0
X(Nk1)mþiþk) , Xt ¼ (Xiþkt, Xmþiþkt, . . . , X(Nk1)mþiþkt)0 for t ¼ 1, . . . , k
^ ¼ f ðiÞ X1 þ þ f ðiÞ Xk . Since hY;
^ Y Yi
^ ¼ 0, we also have
and Y
k;1
k;k
^ Y Yi
^ ¼ hY Y
^ þ Y;
^ Y Yi
^ ¼ hY; Y Yi
^
ðN kÞwk;i ¼ hY Y;
N 1h
i
X
ðiÞ
ðiÞ
XðjkÞmþiþk fk;1 XðjkÞmþiþk1 fk;k XðjkÞmþi XðjkÞmþiþk
¼
j¼k
and (34) follows easily.
u
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P. L. ANDERSON AND M. M. MEERSCHAERT
Lemma 4.
For all i ¼ 0, . . . , m 1 and k 1 we have
ðiÞ
ðiÞ
ðiÞ
wk;i r2iþk ¼ ð^riþk ð0Þ ciþk ð0ÞÞ ðfk þ pk Þ0 ck
ðiÞ 0
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
þ pk ð^rk ck Þ ðfk þ pk Þ0 ð^rk ck Þ
1
X
piþk ðjÞciþkj ðjÞ;
ð35Þ
j¼kþ1
ðiÞ
where ck ¼ ðciþk1 ð1Þ; ciþk2 ð2Þ; . . . ; ci ðkÞÞ0 .
Proof.
From (34) we have
ðiÞ 0 ðiÞ
wk;i r2iþk ¼ ^riþk ð0Þ fk ^rk r2iþk
ðiÞ
ðiÞ
ðiÞ
¼ ð^riþk ð0Þ ciþk ð0ÞÞ ðfk þ pk Þ0 ck
ðiÞ 0
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
þ pk ð^rk ck Þ ðfk þ pk Þ0 ð^rk ck Þ
ðiÞ 0 ðiÞ
þ ciþk ð0Þ þ pk ck r2iþk
since the remaining terms cancel. Define X^t ¼ Xt et so that X^iþk ¼ Xiþk eiþk ¼
piþk ð1ÞXiþk1 piþk ð2ÞXiþk2 and r2iþk ¼ Eðe2iþk Þ ¼ E½ðXiþk X^iþk Þ2 . Since
E½X^iþk ðXiþk X^iþk Þ ¼ 0 it follows that r2iþk ¼ E½Xiþk ðXiþk X^iþk Þ ¼ ciþk ð0Þþ
piþk ð1Þciþk1 ð1Þ þ piþk ð2Þciþk2 ð2Þ þ and (35) follows easily.
u
Lemma 5. Let ci(‘), di(‘) for ‘ ¼ 0, 1, 2, . . . be arbitrary sequences of real
numbers, let
1
1
X
X
ci ðkÞetmþik and vtmþj ¼
dj ðmÞetmþjm
utmþi ¼
and set
k¼0
Ci2 ¼
m¼0
1
X
ci ðkÞ2
k¼0
for 0 i, j < m. Then
2
E4
M
X
and
D2j ¼
1
X
dj ðmÞ2
m¼0
!2 3
utmþi vtmþj 5 4M 2 C 2 D2 g;
ð36Þ
t¼1
where C 2 ¼ maxðCi2 Þ, D2 ¼ maxðD2j Þ, g ¼ max(gt), and gt ¼ Eðe4t Þ.
Proof. Since r1
t et are i.i.d., g ¼ max(gt: 1 < t < 1) ¼ max(gt: 0 i < m) < 1. The left-hand side of (36) consists of M2 terms of the form
Eðui vj u‘mþi v‘mþj Þ ¼
1 X
1 X
1 X
1
X
ci ðkÞdj ðmÞci ðrÞdj ðsÞEðeik ejm e‘mþir e‘mþjs Þ: ð37Þ
k¼0 m¼0 r¼0 s¼0
The terms in (37) with i k ¼ j m ¼ ‘m þ i r ¼ ‘m þ j s sum to
Ó Blackwell Publishing Ltd 2005
499
PARAMETER ESTIMATION
1
X
ci ðkÞdj ðk þ j iÞci ðk þ ‘mÞdj ðk þ ‘m þ j iÞEðe4ik Þ
k¼0
g
g
1
X
k¼0
1
X
ci ðkÞdj ðk þ j iÞci ðk þ ‘mÞdj ðk þ ‘m þ j iÞ
jci ðkÞdj ðkÞj2
k¼0
g
1
X
ci ðkÞ2
k¼0
1
X
dj ðmÞ2
m¼0
gC 2 D2
P
using the
factP that
for an, bn > 0 we always have ð an bn Þ2 P
P
maxðan Þ2 b2n a2n b2n . The terms in (37) with i k ¼ j m 6¼ ‘m þ
i r ¼ ‘m þ j s sum to
1 X
1
X
ci ðkÞdj ðk þ j iÞci ðrÞdj ðr þ j iÞEðe2ik e2‘mþir Þ
k¼0 r¼0
g
g
g
1 X
1
X
jci ðkÞdj ðk þ j iÞci ðrÞdj ðr þ j iÞj
k¼0 r¼0
1
X
jci ðkÞdj ðkÞj
k¼0
1
X
1
X
jci ðrÞdj ðrÞj
r¼0
ci ðkÞ2
k¼0
1
X
dj ðmÞ2
m¼0
gC 2 D2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
since Eðe2ik e2‘mþir Þ ¼ Eðe2ik ÞEðe2ir Þ Eðe4ik ÞEðe4ir Þ ¼ gik gir g by the
Schwarz inequality. Similarly, the terms in (37) with i k ¼ ‘m þ i r 6¼
j m ¼ ‘m þ j s sum to
1 X
1
X
ci ðkÞdj ðmÞci ð‘m þ kÞdj ð‘m þ mÞEðe2ik e2jm Þ gC 2 D2
k¼0 m¼0
and the terms in (37) with i k ¼ ‘m þ j s 6¼ j m ¼ ‘m þ i r sum to
1 X
1
X
ci ðkÞdj ðmÞci ð‘m þ m þ i jÞdj ð‘m þ k þ j iÞEðe2ik e2jm Þ gC 2 D2 ;
k¼0 m¼0
while the remaining terms are zero. Then E(uivju‘mþiv‘mþj) < 4gC2D2 and (36)
follows immediately.
u
Ó Blackwell Publishing Ltd 2005
500
P. L. ANDERSON AND M. M. MEERSCHAERT
Lemma 6. For et,k as in Lemma 2 and utmþi as in Lemma 5 we have
2
!2 3
N 1
1
X
X
E4
uðtkÞmþi ðeðtkÞmþ‘;k eðtkÞmþ‘ Þ 5 4gðN kÞ2 C 2 B max
p‘ ðk þ jÞ2 ;
‘
t¼k
where C2, g are as in Lemma 5 and B ¼
Proof.
P P
m1
1
i¼0
‘¼0
jwi ð‘Þj
2
j¼1
ð38Þ
.
Write
1
X
etmþ‘ etmþ‘;k ¼
¼
¼
p‘ ðmÞXtmþ‘m
m¼kþ1
1
X
p‘ ðmÞ
m¼kþ1
1
X
1
X
w‘m ðrÞetmþ‘mr
r¼0
d‘;k ðk þ jÞetmþ‘kj ;
j¼1
where
d‘;k ðk þ jÞ ¼
j
X
p‘ ðk þ sÞw‘ðkþsÞ ðj sÞ:
s¼1
Since fXtg is causal and invertible,
1
X
d‘;k ðk þ jÞ ¼
j¼1
¼
¼
j
1 X
X
p‘ ðk þ sÞw‘ðkþsÞ ðj sÞ
j¼1 s¼1
1
X
p‘ ðk þ sÞ
s¼1
1
X
1
X
w‘ðkþsÞ ðj sÞ
j¼s
p‘ ðk þ sÞ
s¼1
1
X
w‘ðkþsÞ ðrÞ
r¼0
P
2
is finite, and hence we also have 1
j¼1 d‘;k ðk þ jÞ < 1. Now apply Lemma 5 with
vtmþ‘ ¼ etmþ‘,k etmþ‘ to see that
2
2
!2 3
!2 3
NX
k1
N 1
X
E4
utmþi ðetmþ‘;k etmþ‘ Þ 5 ¼ E4
uðtkÞmþi ðeðtkÞmþ‘;k eðtkÞmþ‘ Þ 5
t¼0
t¼k
4ðN kÞ2 C 2 D2‘;k g
4ðN kÞ2 C 2 D2k g;
where D2k ¼ maxðD2‘;k : 0 ‘ < mÞ and D2‘;k ¼
Ó Blackwell Publishing Ltd 2005
P1
j¼1 d‘;k ðk
þ jÞ2 . Next compute
501
PARAMETER ESTIMATION
D2‘;k
¼
j
1
X
X
j¼1
¼
¼
j
X
p‘ ðk þ rÞw‘ðkþrÞ ðj rÞ
r¼1
j1
1
X
X
j¼1
!
!
p‘ ðk þ sÞw‘ðkþsÞ ðj sÞ
s¼1
!
j1
X
p‘ ðk þ j pÞw‘ðkþjpÞ ðpÞ
p¼0
!
p‘ ðk þ j qÞw‘ðkþjqÞ ðqÞ
q¼0
1 X
1
X
1
X
w‘ðkþjpÞ ðpÞw‘ðkþjqÞ ðqÞp‘ ðk þ j pÞp‘ ðk þ j qÞ
p¼0 q¼0 j¼maxðp;qÞþ1
m1 X
1 X
1
X
jwi ðpÞwiþqp ðqÞj
i¼0 p¼0 q¼0
1
X
jp‘ ðk þ j pÞp‘ ðk þ j qÞj;
j¼maxðp;qÞþ1
where (without loss of generality we suppose p q)
1
X
jp‘ ðk þ j pÞp‘ ðk þ j qÞj ¼
1
X
jp‘ ðk þ jÞp‘ ðk þ j þ p qÞj
j¼1
j¼maxðp;qÞþ1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uX
1
X
u1
p‘ ðk þ jÞ2
p‘ ðk þ j þ p qÞ2
t
j¼1
1
X
j¼1
p‘ ðk þ jÞ2
j¼1
and
m1 X
1 X
1
X
jwi ðpÞwiþqp ðqÞj i¼0 p¼0 q¼0
Then D2‘;k B
P1
j¼1
m1 X
1
X
!2
jwi ð‘Þj
¼ B:
i¼0 ‘¼0
p‘ ðk þ jÞ2 and (38) follows easily.
u
ðiÞ
Lemma 7. For et,k and Xj ðkÞ as in Lemma 2 we have for some real constant
A > 0 that
"
2 #
N 1
1
X
X
ðiÞ
1
E Xj ðkÞðeðjkÞmþiþk;k eðjkÞmþiþk Þ
p‘ ðk þ jÞ2 :
ðN kÞ
Ak max
‘
j¼k
j¼1
ð39Þ
Proof.
Rewrite the left-hand side of (39) in the form
2
!2 3
k1
N 1
X
X
2
ðN kÞ
E4
XðtkÞmþiþs ðeðtkÞmþiþk;k eðtkÞmþiþk Þ 5
s¼0
t¼k
and apply Lemma 6, k times with u(tk)mþi ¼ X(tk)mþiþs for each s ¼ 0, . . . , k 1
to obtain the upper bound of (39) with A ¼ 4gC2B.
u
Ó Blackwell Publishing Ltd 2005
502
P. L. ANDERSON AND M. M. MEERSCHAERT
Lemma 8. Suppose that the periodically stationary moving average (2) is causal,
invertible, Ee4t < 1, and that for some 0 < g G < 1 we have gz0 z z0 f(k)z Gz0 z for all p k p, and all z in Rm, where f(k) is the spectral
density matrix of the equivalent vector moving average process (12). If k ¼
k(N) ! 1 as N ! 1 with k3/N ! 0 and (13) holds then
ðiÞ
ðiÞ
ðN kÞ1=2 bðkÞ0 ðpk þ fk Þ ðN kÞ1=2 bðkÞ0 C1
k;i
N 1
X
P
ðiÞ
Xj ðkÞeðjkÞmþiþk ! 0 ð40Þ
j¼k
ðiÞ
for any b(k) ¼ (bk1, . . . , bkk)0 such that kb(k)k2 remains bounded, where fk is from
ðiÞ
(32) and Xj ðkÞ is from Lemma 4.2.
Using (33) the left-hand side of (40) can be written as
"
#
N 1
N
1
X
X
ðiÞ 1
ðiÞ
ðiÞ
1=2
0
1
^
bðkÞ ðR Þ
Xj ðkÞeðjkÞmþiþk;k C
Xj ðkÞeðjkÞmþiþk ¼ I1 þ I2 ;
ðN kÞ
Proof.
k;i
k
j¼k
where
j¼k
"
I1 ¼ ðN kÞ
1=2
#
N 1
X
ðiÞ
ðiÞ
1
^ Þ C1
bðkÞ ðR
Xj ðkÞeðjkÞmþiþk;k ;
k;i
k
0
j¼k
"
I2 ¼ ðN kÞ
1=2
bðkÞ
0
C1
k;i
N
1
X
ðiÞ
Xj ðkÞðeðjkÞmþiþk;k
#
eðjkÞmþiþk Þ ;
j¼k
so that
jI1 j ðN kÞ
1=2
NX
1
ðiÞ 1
ðiÞ
1
^
kbðkÞk kðRk Þ Ck;i k Xj ðkÞeðjkÞmþiþk;k ¼ J1 J2 J3 ;
j¼k
where J1 ¼ kb(k)k is bounded by assumption,
N 1
ðN kÞ1=2 1 X ðiÞ
Xj ðkÞeðjkÞmþiþk;k J3 ¼
1=2
N k j¼k
k
^ ðiÞ Þ1 C1 k ! 0 in probability by an argument similar to
and J2 ¼ k 1=2 kðR
k;i
k
Theorem 1 in Anderson et al. (1999). In fact, if we let
(
)
1 1
2
X X
jwi ðm1 Þjjwj ðm2 Þj < 1; 0 i; j < m ;
ð41Þ
M ¼ max g 3
m1 ¼0 m2 ¼0
^ ðiÞ Ck;i k then ðN kÞVarð^sðiÞ
where g ¼ Eðe4t Þ and if we let Qk;i ¼ kR
m;n Þ M in
k
general and hence Eðk 1=2 Q2k;i Þ k 3 M=ðN kÞ ! 0 as k ! 1 since k3/N ! 0 as
N ! 1, and the remainder of the argument is exactly the same as Theorem 3.1 in
Anderson et al. (1999). Using the inequality Var(X þ Y) 2(Var(X) þ Var(Y))
we obtain
Ó Blackwell Publishing Ltd 2005
503
PARAMETER ESTIMATION
2
2 3
1 X
N 1
N
k
ðiÞ
EðJ32 Þ ¼
Xj ðkÞeðjkÞmþiþk;k 5 2ðT1 þ T2 Þ;
E4
N k j¼k
k
where
2
N k 4
T1 ¼
E N
k
2
N k 4
T2 ¼
E N
k
2 3
N 1
1 X
ðiÞ
Xj ðkÞeðjkÞmþiþk 5
k j¼k
2 3
N 1
1 X
ðiÞ
Xj ðkÞðeðjkÞmþiþk;k eðjkÞmþiþk Þ 5:
k j¼k
Lemma 7 implies that
T2 AðN kÞ max
‘
1
X
p‘ ðk þ jÞ2
j¼1
and
2
!2 3
k1
N 1
X
X
T1 ¼ k 1 ðN kÞ1
E4
XðjkÞmþiþt eðjkÞmþiþk 5
t¼0
1
1
¼ k ðN kÞ
k1
X
"
E
t¼0
¼ k 1 ðN kÞ1
j¼k
N 1
X
!
XðjkÞmþiþt eðjkÞmþiþk
j¼k
k1 X
N 1 X
N 1
X
N 1
X
!#
XðrkÞmþiþt eðrkÞmþiþk
r¼k
EðXðjkÞmþiþt XðrkÞmþiþt eðjkÞmþiþk eðrkÞmþiþk Þ
t¼0 j¼k r¼k
¼ k 1 ðN kÞ1
k1 X
N 1
X
2
EðXðjkÞmþiþt
ÞEðe2ðjkÞmþiþk Þ
t¼0 r¼k
¼ k 1
k1
X
ciþt ð0Þ ðN kÞ1
t¼0
N 1
X
r2iþk
r¼k
so that T1 D ¼ maxi ðci ð0ÞÞ maxi ðr2i Þ. Hence
EðJ32 Þ 2D þ 2AðN kÞ max
‘
P1
1
X
p‘ ðk þ jÞ2 ;
j¼1
2
where ðN kÞ max‘ j¼1 p‘ ðk þ jÞ ! 0 in view of (13), so that J3 is bounded in
probability. Since J2 ! 0 in probability, it follows that I1 ! 0 in probability.
ðiÞ
Apply Lemma 6 with ‘ ¼ i þ k and uðtkÞmþi ¼ bðkÞ0 C1
k;i Xt ðkÞ to see that
EðI22 Þ 4gðN kÞC 2 B max
‘
1
X
p‘ ðk þ jÞ2 ;
j¼1
Ó Blackwell Publishing Ltd 2005
504
P. L. ANDERSON AND M. M. MEERSCHAERT
P1
where
C 2 ¼ maxi k¼0 ci ðkÞ2 comes from the representation u(tk)mþi ¼
P
2
kci(k)e(tk)mþik so that C is finite if and only if Var(u(tk)mþi) < 1. Since
ðiÞ
ðiÞ
ðiÞ
ðiÞ
0
0
Ck;i ¼ E½Xt ðkÞXt ðkÞ we have VarðuðtkÞmþi Þ ¼ bðkÞ0 C1
k;i E½Xt ðkÞXt ðkÞ 0 1
1
Ck;i bðkÞ ¼ bðkÞ Ck;i bðkÞ and Theorem A.1 in Anderson et al. (1999) shows that
1
2
kC1
k;i k ð2pgÞ , when the assumed spectral bounds hold, so C < 1. Then it
follows from (13) that I2 ! 0 in probability, completing the proof of Lemma 8.u
P
Proof of Theorem 1. Let Xtm ¼ m
j¼0 wt ðjÞetj and define eu(k) to be the k
dimensional vector with 1 in the uth place and zero entries elsewhere. Let
ðhikiÞ
ðmÞ
tNm;k ðuÞ ¼ ðN kÞ1=2 eu ðkÞ0 ðCk;hiki Þ1
N 1
X
ðhikiÞ
Xjm
ðkÞeðjkÞmþhikiþk
j¼k
so that
ðhikiÞ
tNm;k ðuÞ ¼ ðN kÞ1=2
N 1
X
ðhikiÞ
;
ð42Þ
ðkÞeðjkÞmþhikiþk ;
ð43Þ
wuj;k
j¼k
where
ðhikiÞ
wuj;k
ðhikiÞ
Xjm
ðmÞ
ðhikiÞ
¼ eu ðkÞ0 ðCk;hiki Þ1 Xjm
ðkÞ ¼ ðXðjkÞmþhikiþk1;m ; . . . ; XðjkÞmþhiki;m Þ0 and
ðmÞ
ðhikiÞ
Ck;hiki ¼ E½Xjm
ðhikiÞ
ðkÞXjm
For each 0 r < k, X(jk)mþhikiþr,m is
(e(jk)mþhikiþrs : 0 s m). Hence we write
ðhikiÞ
wuj;k
a
ðkÞ0 :
linear
ð44Þ
combination
¼ LðeðjkÞmþhikim ; . . . ; eðjkÞmþhikiþk1 ÞeðjkÞmþhikiþk ;
of
ð45Þ
where L(e1, . . . , eq) denotes a linear combination of e1, . . . , eq. Note that for i, u, k
ðhikiÞ
fixed, wuj;k are identically distributed for all j since the first two terms in (43) are
nonrandom and do not depend on j, while the last two terms are identically
distributed for all j by definition, using the fact that Xt is periodically
strictly
ðhikiÞ
ðhikiÞ ðhi0 kiÞ
stationary. Also, wuj;k
are uncorrelated since E½wuj;k wvj0 ;k ¼ 0 unless
(j k)m þ hi ki þ k ¼ (j0 k)m þ hi0 ki þ k, which requires i ¼ i0 mod m.
ðhikiÞ ðhikiÞ
Since 0 i < m, this implies that i ¼ i0 . Then E½wuj;k wvj0;k ¼ 0 if j 6¼ j0 and
otherwise
ðhikiÞ
E½wuj;k
ðhikiÞ
wvj;k
ðmÞ
¼ r2i eu ðkÞ0 ðCk;hiki Þ1 ev ðkÞ
from (44). It follows immediately from (42) that the covariance matrix of the
vector
ðhikiÞ
tNm;k ¼ ðtNm;k ðuÞ : 1 u D; 0 hi ki m 1Þ0
ðm1Þ
2 ð1Þ
2
is Cm ¼ diagðr20 Cð0Þ
Þ, where
m ; r1 Cm ; . . . ; rm1 Cm
Ó Blackwell Publishing Ltd 2005
505
PARAMETER ESTIMATION
ðmÞ
0
1
ðCðiÞ
m Þu;v ¼ eu ðkÞ ðCk;hiki Þ ev ðkÞ
and 1 u, v D. Note that
ðh0kiÞ
ðh0kiÞ
ðhm1kiÞ
tNm;k ¼ ðtNm;k ð1Þ; . . . ; tNm;k ðDÞ; . . . ; tNm;k
ðhm1kiÞ
ð1Þ; . . . ; tNm;k
ðDÞÞ0 :
We also have for any k 2 RDm that Var(k0 tNm,k) ¼ k0 Cmk. Apply Lemma 1 to the
periodically stationary process Xtm to see that
ðmÞ
eu ðkÞ0 ðCk;hiki Þ1 ev ðkÞ ! ðKðiÞ
m Þu;v
as N ! 1, where
ðKðiÞ
m Þu;v ¼
minðu;vÞ1
X
piminðu;vÞþs;m ðsÞpiminðu;vÞþs;m ðs þ jv ujÞr2
iminðu;vÞþs
s¼0
and et ¼
P1
j¼0
pt;m ðjÞXtj;m (with pt,m(0) ¼ 1). Since Cm ! Km as k ! 1, then
lim Varðk0 tNm;k Þ ¼ k0 Km k;
N !1
where
ðm1Þ
2 ð1Þ
2
Km ¼ diagðr20 Kð0Þ
Þ:
m ; r1 Km ; . . . ; rm1 Km
Next, we want to use the Lindeberg–Lyapounov central limit theorem to show
that
k0 tNm;k ) N ð0; k0 Km kÞ:
Towards this end, define the vector
ðh0kiÞ
wj;k ¼ k0 ðw1j;k
ðh0kiÞ
; . . . ; wDj;k
ðhm1kiÞ
; . . . ; w1j;k
ðhm1kiÞ 0
; . . . ; wDj;k
Þ
for any k 2 RDm so that
k0 tNm;k ¼ ðN kÞ1=2
N
1
X
wj;k :
j¼k
Then, Var(wj,k) ¼ Var(k0 tNm,k) ¼ k0 Cmk. Also, fwj,k : j ¼ k, . . . , N 1g are mean
zero, identically distributed and uncorrelated. Now, let K ¼ [(k þ m)/m] þ 2,
where again [Æ] is the greatest integer function and let N1 be an integer such that
N1/(N K) ! 0 and K/N1 ! 0 (the largest integer less than ((N K)K)1/2
suffices). Let M1 be the greatest integer less than or equal to (N K)/N1, so that
M1N1/N ! 1 and M1K/N ! 0. Define the random variables
Ó Blackwell Publishing Ltd 2005
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P. L. ANDERSON AND M. M. MEERSCHAERT
1=2
z1N ;k ¼ ðwk;k þ þ wkþN1 K1;k Þ=N1 ;
1=2
z2N ;k ¼ ðwkþN1 ;k þ þ wkþ2N1 K1;k Þ=N1
..
.
1=2
zM1 N ;k ¼ ðwkþðM1 1ÞN1 ;k þ þ wkþM1 N1 K1;k Þ=N1 :
i.i.d. with mean zero and
The choice of K ensures that z1N,k, . . . , zM1N, k are
1=2 PM1
variance s2M1 ¼ ððN1 KÞ=N1 Þk0 Cm k. Hence, M1
j¼1 zjN ;k is also mean zero
with variance s2M1 , where s2M1 ! s2 as M1 ! 1 and s2 ¼ k0 Kmk. The Lindeberg–
Lyapounov central limit theorem (see, e.g. Billingsley, 1968, Thm 7.3) implies
that
1=2
M1
M1
X
zjN ;k ) N ð0; s2 Þ
ð46Þ
j¼1
if we can show that
s4
M1
M1
X
1=2
E ðM1 zjN ;k Þ4 ! 0
j¼1
as N ! 1. Letting N~1 ¼ N1 K, we have
Eðz4jN ;k Þ ¼ N12 N~1 Eðw4j;k Þ þ ðN~1 2 N~1 ÞEðw2j;k w2‘;k Þ
N12 N~1 Eðw4j;k Þ þ ðN~1 2 N~1 ÞEðw4j;k Þ
Eðw4j;k Þ;
where Eðw4j;k Þ < 1. Hence,
s4
M1
M1
X
1=2
EððM1
1
4
zjN ;k Þ4 Þ s4
M1 M1 Eðwj;k Þ
j¼1
s4 M11 Eðw4j;k Þ
!0
as N ! 1 (hence N1 ! 1 and M1 ! 1). Thus, (46) holds. Also,
1=2
M1
M1
X
j¼1
Ó Blackwell Publishing Ltd 2005
zjN ;k ðN kÞ1=2
N 1
X
j¼k
wj;k
507
PARAMETER ESTIMATION
has limiting variance 0, since its variance is no more than
ðM1 þ 1ÞKðVarðwt;k ÞÞ ðM1 þ 1ÞKðVarðwt;k ÞÞ
;
N k
N
which approaches 0 as N ! 1. We can therefore conclude that
k0 tNm;k ) N ð0; s2 Þ
recalling that s2 ¼ k0 Kmk. Now, an application of the Cramer–Wold device yields
tNm;k ) N ð0; Km Þ:
ð47Þ
Next we want to show that Km ! K as m ! 1. This is equivalent to showing
ðiÞ
that KðiÞ
as m ! 1. Recall that wt(0) ¼ 1 and write
m ! K
1
X
Xt ¼
wt ð‘Þet‘
‘¼0
¼
1
X
wt ð‘Þ
1
r
X
X
r¼1
so that pt ðrÞ ¼ Pr
‘¼1
Xtm ¼
pt‘ ðjÞXt‘j
j¼0
‘¼0
¼ et þ
1
X
!
wt ð‘Þpt‘ ðr ‘Þ Xtr
‘¼1
wt ð‘Þpt‘ ðr ‘Þ for r 1. Similarly
m
X
wt ð‘Þ
pt‘;m ðjÞXt‘j;m
j¼0
‘¼0
¼ et þ
1
X
1
X
minðr;mÞ
X
r¼1
‘¼1
!
wt ð‘Þpt‘;m ðr ‘Þ Xtr;m
Pminðr;mÞ
so that pt;m ðrÞ ¼ ‘¼1
wt ð‘Þpt‘;m ðr ‘Þ for r 1. Apply these two formulae recursively to see that pt,m(r) ¼ pt(r) for 1 r m. Then we actually have
ðiÞ
KðiÞ
for all m sufficiently large.
m ¼ K
Now we want to show that
ðhikiÞ
lim lim sup P ðjtN ;k
m!1
N !1
ðhikiÞ
ðuÞ tNm;k ðuÞj > dÞ ¼ 0
ð48Þ
for any d > 0, where
ðhikiÞ
tN ;k
ðuÞ ¼ ðN kÞ1=2 eu ðkÞ0 C1
k;hiki
N 1
X
ðhikiÞ
Xj
ðkÞeðjkÞmþhikiþk :
j¼k
Apply the Chebyshev inequality to see that the probability in (48) is bounded
above by
"
#
N 1
X
ðhikiÞ
ðhikiÞ
1=2
d2 VarðtN ;k ðuÞ tNm;k ðuÞÞ ¼ d2 Var ðN kÞ
Aj eðjkÞmþhikiþk ;
j¼k
Ó Blackwell Publishing Ltd 2005
508
P. L. ANDERSON AND M. M. MEERSCHAERT
ðhikiÞ
ðmÞ
ðhikiÞ
where Aj ¼ eu ðkÞ0 C1
ðkÞ eu ðkÞ0 ðCk;hiki Þ1 Xjm
k;hiki Xj
mands are uncorrelated, we have
ðhikiÞ
VarðtN ;k
ðhikiÞ
ðuÞ tNm;k ðuÞÞ ¼ ðN kÞ1
ðkÞ. Since the sum-
N 1 h
i
X
E A2j e2ðjkÞmþhikiþk ;
j¼k
¼ r2i ðN kÞ1
N 1
X
EðA2j Þ
j¼k
¼ r2i EðA2j Þ
because each Aj has the same distribution. Write EðA2j Þ ¼ VarðI1 þ I2 Þ, where
h
i
ðhikiÞ
ðhikiÞ
ðkÞ Xjm
ðkÞ ;
I1 ¼ eu ðkÞ0 C1
k;hiki Xj
h
i
ðmÞ
ðhikiÞ
1
ðC
Þ
ðkÞ;
I2 ¼ eu ðkÞ0 C1
Xjm
k;hiki
k;hiki
so that VarðI1 Þ ¼ eu ðkÞ0 C1
k;hiki Rk;hiki Ck;hiki eu ðkÞ; where Rk,hiki is the covariance
matrix of the random vector (Xtmþhiki þ k1Xtmþhiki þ k1, m, . . . , Xtmþhiki
Xtmþhiki, m)0 . Since keu(k)k ¼ 1, Theorem A.1 in Anderson et al. (1999) implies
that Var(I1) (G/g)kRk,hikik. In order to show that kRk,hikik!0 as m ! 1, we
consider the spectral density matrix fd (k) of the vector
P moving average Wt ¼
(Xtmþm1Xtmþm1,m, . . . , XtmXtm,m)0 . If we let Yt ¼ 1
‘¼0 W‘ Zt‘ , where Yt ¼
(Xtmþm1, . . . , Xtm)0 , Zt ¼ (etmþm1, . . . , etm)0 , and (W‘)ij ¼ wm1i(‘m i þ j) and
0
Y
then
Wt ¼ Yt Ytm.
Since
Xt Xtm ¼
Ptm ¼ (Xtmþm1,m, . . . , Xtm,m) ,
j>m wt(j)etj, the moving average
1
X
Wt ¼
~ ‘ Zt‘ ;
W
‘¼dmþ2
m e1
where dxe is the smallest integer greater than or equal to x and
~ ‘Þ ¼ w
except that we zero out the entries with
ðW
m1i ð‘m i þ jÞ
ij
‘m i þ j m. Then the spectral density matrix
0
1 0
10
1
1
X
1 @ X
~ ‘ eik‘ AR@
~ ‘ eik‘ A ;
fd ðkÞ ¼
W
W
2p
mþ2
mþ2
‘¼d
m
e1
‘¼d
m
e1
where R is the covariance matrix of Zt. As in the proof of Theorem A.1
in Anderson et al. (1999), we now define Cw(h) ¼ Cov(Wt, Wtþh), W ¼
(Wn1, . . . , W0)0 and
C ¼ CovðW ; W Þ ¼ ½Cw ði jÞni;j¼0 ¼ CovðXnm1 Xnm1;m ; . . . ; X0 X0;m Þ0 :
For fixed i and k, let n ¼ [(k þ hi ki)/m] þ 1. Then Rk,hiki is a submatrix of
C ¼ Rnm,0. Fix an arbitrary vector y ¼ (y0, . . . , yn1)0 in Rnm, whose jth entry
yj 2 Rm. Then
Ó Blackwell Publishing Ltd 2005
509
PARAMETER ESTIMATION
y 0 Cy ¼
n1 X
n1
X
yj0 Cw ðj kÞyk
j¼0 k¼0
¼
n1 X
n1
X
yj0
Z
j¼0 k¼0
¼
Z p X
n1
p
Z
eikðjkÞ fd ðkÞdk yk
p
eikj yj0
0
fd ðkÞ
j¼0
p
dm
p
p
X
n1
e
ikk
yk dk
k¼0
X
m1
yj0 yj dk ¼ 2pdm y 0 y
j¼0
so that kCk 2pdm, where dm is the largest modulus of the elements of fd(k). But
dm ! 0 in view of the causality condition (2), hence kRk,hikik kCk 2pdm !
0, which implies that Var(I1) ! 0 as well.
ðmÞ
ðmÞ
1 ðmÞ
1
1
Next write VarðI2 Þ ¼ eu ðkÞ0 ½C1
k;hiki ðCk;hiki Þ Ck;hiki ½Ck;hiki ðCk;hiki Þ ðmÞ
eu ðkÞ and recall that Ck;hiki is the covariance matrix of (Xtmþhiki þ k1,m, . . . ,
Xtmþhiki,m)0 for each t. Then as in the preceeding paragraph write
Ytm ¼ ðXtmþm1;m ; . . . ; Xtm;m Þ ¼
½ðm1Þ=mþ1
X
‘ Zt‘ ;
W
‘¼0
‘ Þ ¼ wm1i ð‘m i þ jÞ except that we zero out the entries with
where ðW
ij
‘mi þ j > m, and let fm(k) denote the spectral density matrix of Ytm. Let
B(Zt) ¼ Zt1 denote the backward shift operator, and write Yt ¼ W(B)Zt
1
ik
0 ik
and similarly Ytm ¼ WðBÞZ
) and fm ðkÞ ¼
t . Then f(k) ¼ (2p) W(e )RW (e
1 ik
0
ik
ðe Þ, whereqasffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2pÞ Wðe ÞRW
before
R
is
the
covariance
matrix
of Zt. Using
ffi
P 2
the Frobenius norm kAkF ¼
a
,
we
have
i;j ij
ik ÞRW
0 ðeik Þk
kf ðkÞ fm ðkÞkF ¼ ð4p2 Þ1 kWðeik ÞRW0 ðeik Þ Wðe
F
ik ÞRW0 ðeik Þk
ð4p2 Þ1 kWðeik ÞRW0 ðeik Þ Wðe
F
ik ÞRW0 ðeik Þ Wðe
ik ÞRW
0 ðeik Þk
þ ð4p2 Þ1 kWðe
F
0 ik
2 1
ik
ik
ð4p Þ kWðe Þ Wðe Þk kRk kW ðe Þk
F
F
F
2 1
ik Þk kRk kW0 ðeik Þ W
0 ðeik Þk
þ ð4p Þ kWðe
F
F
F
0 ðeik Þk ! 0 as m ! 1 in view of the causality condition (2)
where kW0 ðeik Þ W
F
and the remaining norms are bounded independent of m, so that
|z0 [f(k) fm(k)]z| d(m)z0 z for any z 2 Rm, where d(m) ! 0 as m ! 1. Now
Theorem A.1 in Anderson et al. (1999) yields
z0 ðg dðmÞÞz z0 f ðkÞz z0 dðmÞz z0 fm ðkÞz z0 f ðkÞz þ z0 dðmÞz z0 ðG þ dðmÞÞz:
Let C(h) ¼ Cov(Yt, Ytþh), Y ¼ (Yn1, . . . , Y0)0 and Cnm;0 ¼ CovðY ; Y Þ ¼
0
½Cði jÞn1
i;j¼0 , the covariance matrix of (Xnm1, . . . , X0) . For fixed i and k, let
Ó Blackwell Publishing Ltd 2005
510
P. L. ANDERSON AND M. M. MEERSCHAERT
n ¼ [(k þ hi ki)/m] þ 1 as before. Similarly, let Cm(h) ¼ Cov(Ytm, Ytþh,m),
ðmÞ
Ym ¼ (Yn1,m, . . . , Y0m)0 and Cnm;0 ¼ CovðYm ; Ym Þ, the covariance matrix of
ðmÞ
0
(Xnm1,m, . . . , X0m) . Since Ck,hiki is a submatrix of Cnm,0 and Ck;hiki is a
ðmÞ
submatrix of Cnm;0 , it follows that
ðmÞ
ðmÞ
kCk;hiki Ck;hiki k kCnm;0 Cnm;0 k
ðmÞ
¼ sup jy 0 ðCnm;0 Cnm;0 Þyj
kyk¼1
X
n1 X
n1
0
¼ sup yj ðCm ðj ‘Þ Cðj ‘ÞÞy‘ kyk¼1 j¼0 ‘¼0
X
n1 X
n1 Z p
0
ikðj‘Þ
¼ sup yj
e
ðfm ðkÞ f ðkÞÞdk y‘ kyk¼1 j¼0 ‘¼0
p
Z
!0
! n1
n1
p X
X
ikj
ik‘
¼ sup e yj ðfm ðkÞ f ðkÞÞ
e y‘ dk
kyk¼1 p j¼0
‘¼0
is bounded above by 2pd(m), hence
ðmÞ
ðmÞ
ðmÞ
1
1
1
kC1
k;hiki ðCk;hiki Þ k ¼ kCk;hiki ðCk;hiki Ck;hiki ÞðCk;hiki Þ k
ðmÞ
ðmÞ
1
kC1
k;hiki k kCk;hiki Ck;hiki k kðCk;hiki Þ k
1
1
2pdðmÞ :
2pg
2pðg dðmÞÞ
Then
VarðI2 Þ 1
1
2pdðmÞ 2pg
2pðg dðmÞÞ
2
2pðG þ dðmÞÞ ! 0
as m ! 1, and so Var(I1 þ I2) ! 0 as well, which proves (48). Together with
ðiÞ
(47), the fact that KðiÞ
as m ! 1, and Theorem 4.2 in Billingsley (1968),
m ! K
this proves that
tN ;k ) N ð0; KÞ
ð49Þ
as N ! 1, where
ðhikiÞ
tN;k ¼ ðtN;k
ðuÞ : 1 u D; 0 hi ki m 1Þ0 :
ðhikiÞ
ðhikiÞ
ðhikiÞ
Applying Lemma 8 yields ðN kÞ1=2 eu ðkÞ0 ðpk
þ fk
Þ tN ;k ðuÞ ! 0 in
ðhikiÞ
ðhikiÞ
ðhikiÞ
probability. Note that pk
¼ ðpi ð1Þ; . . . ; pi ðkÞÞ0 and fk
¼ ðfk;1 ; . . . ;
ðhikiÞ
ðhikiÞ
0 ðhikiÞ
fk;k
Þ.
Combining
statement
(49)
with
eu ðkÞ ðpk
þ fk
Þ¼
ðhikiÞ
pi ðuÞ þ fk;u
and the fact that (N k)/N ! 1, implies that
Ó Blackwell Publishing Ltd 2005
511
PARAMETER ESTIMATION
ðhikiÞ
N 1=2 ðpi ðuÞ þ fk;u
ðhikiÞ
Þ tN;k
P
ðuÞ ! 0
ð50Þ
ðhikiÞ
hiki
is defined as in (30). Then (14) holds with /^k;u
as N ! 1, where fk;u
hiki
by fk;u .
Finally, we must show that
ðhikiÞ
N 1=2 ðfk;u
replaced
ðhikiÞ P
/^k;u Þ ! 0:
ð51Þ
Write
ðhikiÞ
N 1=2 kfk;u
ðhikiÞ
ðhikiÞ
^ k;hiki Þ1^r
k ¼ N 1=2 kðR
k
/^k;u
ðhikiÞ
^ 1 c^
C
k;hiki k
k
^ 1 k k^rðhikiÞ k
^ k;hiki Þ1 C
N 1=2 kðR
k;hiki
k
^ 1 k k^rðhikiÞ c^ðhikiÞ k
þ N 1=2 kC
k;hiki
k
k
ðhikiÞ
and recall that ðN kÞVarð^sk;j Þ M, where M is given by (41), so that
ðhikiÞ
N 1=2 k^rk
k is bounded in probability. Also, recall from the proof of Lemma 8
^ 1 C1 k ! 0 in probability, so the same is true with i replaced with
that k 1=2 kR
k;i
k;i
1
^ 1
hiki. A very similar argument yields k 1=2 kC
k;hiki Ck;hiki k ! 0 in probability,
1
1=2 ^ 1
^
so that k kRk;hiki Ck;hiki k ! 0 in probability as well. Next observe that
^ 1 k kC
^ 1
kC
C1 k þ kC1 k, where kC1 k is uniformly bounded
k;hiki
k;hiki
k;hiki
k;hiki
k;hiki
ðhikiÞ
by Theorem A.1 of Anderson et al. (1999). Since ^sk;j
and c^i ðj kÞ differ by
only k terms and a factor (N k)/N, it is easy to check that
ðhikiÞ
ðhikiÞ
ðhikiÞ
N 1=2 k^rk
c^k
k ! 0 in probability. It follows that N 1=2 ðfk;u ðhikiÞ
/^k;u Þ ! 0 in probability, which completes the proof of Theorem 1.
u
Proof of Theorem 2.
it follows that
ðiÞ
From the two representations of X^iþk given by (4) and (8),
ðhikiÞ
hk;j
¼
j
X
ðhikiÞ ðhikiÞ
hk‘;j‘
ð52Þ
/k;‘
‘¼1
ðhikiÞ
for j ¼ 1, . . . , k if we define hkj;0 ¼ 1 and replace i with hi ki. Equation (52)
can be modified and written as
0
ðhikiÞ
hk;1
1
0
ðhikiÞ
/k;1
1
B ðhikiÞ C
B ðhikiÞ C
B hk;2 C
B /k;2 C
C
C
B
B
B ðhikiÞ C
ðhikiÞ B ðhikiÞ C
B hk;3 C ¼ Rk
B /k;3 C;
C
C
B
B
B .. C
B .. C
@ . A
@ . A
ðhikiÞ
ðhikiÞ
hk;D
/k;D
ð53Þ
where
Ó Blackwell Publishing Ltd 2005
512
P. L. ANDERSON AND M. M. MEERSCHAERT
0
ðhikiÞ
Rk
1
ðhikiÞ
hk1;1
ðhikiÞ
hk1;2
B
B
B
¼B
B
B
@
..
.
ðhikiÞ
hk1;D1
0
1
ðhikiÞ
hk2;1
..
.
ðhikiÞ
hk2;D2
0
0
0
0
1
0
..
.
ðhikiÞ
hkDþ1;1
1
0
0C
C
C
0C
C
.. C
.A
1
ð54Þ
ðiÞ
ðiÞ
for fixed lag D. From the definitions of h^k;u and /^k;u , we also have
1
0 ðhikiÞ 1
ðhikiÞ
h^k;1
/^k;1
B ^ðhikiÞ C
B ^ðhikiÞ C
B hk;2 C
B /k;2 C
C
C
B
B
B ^ðhikiÞ C ^ ðhikiÞ B ^ðhikiÞ C
B hk;3 C ¼ Rk
B /k;3 C;
C
C
B
B
B .. C
B .. C
@ . A
@ . A
ðhikiÞ
ðhikiÞ
h^
/^
0
k;D
where
know
ðhikiÞ
h^
k‘;u
ð55Þ
k;D
^ ðhikiÞ is defined as in (54) with h^ðhikiÞ replacing hðhikiÞ . From (11) we
R
k
k;u
k;u
ðhikiÞ P
that h^k;u ! wi ðuÞ, hence for fixed ‘ with k0 ¼ k ‘, we have
ðhi‘k 0 iÞ P
¼ h^ 0
! w ðuÞ.
i‘
k ;u
Thus,
P
^ ðhikiÞ !
R
RðiÞ ;
k
ð56Þ
where
0
B
B
RðiÞ ¼ B
@
1
wi1 ð1Þ
..
.
wi1 ðD 1Þ
0
1
..
.
wi2 ðD 2Þ 0
0
..
.
1
0
0C
C
.. C:
.A
ð57Þ
wiDþ1 ð1Þ 1
We have
^ ðhikiÞ ð/^ðhikiÞ /ðhikiÞ Þ þ ðR
^ ðhikiÞ RðhikiÞ Þ/ðhikiÞ ;
h^ðhikiÞ hðhikiÞ ¼ R
k
k
k
ð58Þ
ðhikiÞ
ðhikiÞ
ðhikiÞ
ðhikiÞ
where hðhikiÞ ¼ ðhk;1 ; . . . ; hk;D Þ0 , /ðhikiÞ ¼ ð/k;1 ; . . . ; /k;D Þ0 , and h^ðhikiÞ
(hiki)
(hiki)
ðhikiÞ
and /^
are the respective estimators of h
and /
. Note that
ðhikiÞ
^
ðR
k
ðhikiÞ
Rk
^ ðhikiÞ R
^ ðhikiÞ Þ/ðhikiÞ
Þ/ðhikiÞ ¼ ðR
k
k
^ ðhikiÞ RðhikiÞ Þ/ðhikiÞ
þ ðR
k
k
ðhikiÞ
þ ðRk
where
Ó Blackwell Publishing Ltd 2005
ðhikiÞ
Rk
Þ/ðhikiÞ ;
ð59Þ
513
PARAMETER ESTIMATION
0
ðhikiÞ
Rk
1
B hðhi1kiÞ
B k;1
B ðhi1kiÞ
h
¼B
B k;2
..
B
@
.
ðhi1kiÞ
hk;D1
0
1
ðhi2kiÞ
hk;1
0
0
0
0
1
0
..
..
.
ðhi2kiÞ
hk;D2
1
.
ðhiDþ1kiÞ
hk;1
0
0C
C
C
0C
C
.. C
.A
1
ð60Þ
^ ðhikiÞ is the corresponding matrix obtained by replacing hðhikiÞ with h^ðhikiÞ
and R
k
k;u
k;u
for every season i and lag u. We next need to show that
ðhikiÞ
ðhikiÞ
^ ðhikiÞ R
^ ðhikiÞ ¼ oP ðN 1=2 Þ. This is equivalent
Rk
Rk
¼ oðN 1=2 Þ and R
k
k
to showing that
ðhi‘kiÞ
N 1=2 ðhk;u
ðhikiÞ
hk‘;u Þ ! 0
ð61Þ
ðhi‘kiÞ
ðhikiÞ P
N 1=2 ðh^k;u
h^k‘;u Þ ! 0
ð62Þ
and
for ‘ ¼ 1, . . . , D 1 and u ¼ 1, . . . , D. Using estimates from the proof of
Anderson et al. (1999, Cor. 2.2.4) and condition (13) of Theorem 1, it is not hard
ðhikiÞ
to show that N 1=2 ð/k;u
þ pi ðuÞÞ ! 0 as N ! 1 for any u ¼ 1, . . . , k. This leads
to
ðhi‘kiÞ
N 1=2 ð/k;u
þ pi‘ ðuÞÞ ! 0
ð63Þ
ðhi‘kiÞ
by replacing i with i ‘ for fixed ‘. Letting ak ¼ N 1=2 ð/k;u
þ pi‘ ðuÞÞ and
ðhikiÞ
bk ¼ N 1=2 ð/k‘;u þ pi‘ ðuÞÞ we see that bk ¼ ak‘. Since ak ! 0 then bk ! 0 as
k ! 1. Hence,
ðhikiÞ
N 1=2 ð/k‘;u þ pi‘ ðuÞÞ ! 0
ð64Þ
as k ! 1. Subtracting (64) from (63) yields
ðhi‘kiÞ
N 1=2 ð/k;u
ðhikiÞ
/k‘;u Þ ! 0;
ð65Þ
which holds for ‘ ¼ 1, . . . , u 1 and u ¼ 1, . . . , k. Since
ðhi‘kiÞ
hk;1
ðhikiÞ
ðhi‘kiÞ
hk‘;1 ¼ /k;1
ðhikiÞ
/k‘;1
we have (61) with u ¼ 1. The cases u ¼ 2, . . . , D follow iteratively using (10), (53),
and (65). Thus, (61) is established. To prove (62), we need Lemma 9.
u
Lemma 9.
For all ‘ ¼ 1, . . . , u 1 and u ¼ 1, . . . , k, we have
ðhi‘kiÞ
N 1=2 ð/^k;u
ðhikiÞ
P
/^k‘;u Þ ! 0:
ð66Þ
Ó Blackwell Publishing Ltd 2005
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P. L. ANDERSON AND M. M. MEERSCHAERT
Proof.
where
ðhi‘kiÞ
Starting from (42), we need to show that tNm;k
ðhi‘kiÞ
tNm;k
ðuÞ ¼ ðN kÞ1=2
N 1
X
ðhikiÞ
P
ðuÞ tNm;k‘ ðuÞ ! 0,
ðhi‘kiÞ
wuj;k
j¼k
ðhikiÞ
tNm;k‘ ðuÞ ¼ ðN k þ ‘Þ1=2
N 1
X
ðhikiÞ
wuj;k‘
j¼k‘
ðhi‘kiÞ
wuj;k
¼ eu ðkÞ
0
ðmÞ
ðhi‘kiÞ
ðCk;hi‘ki Þ1 Xjm
ðkÞeðjkÞmþhi‘kiþk
ðhikiÞ
ðmÞ
ðhikiÞ
wuj;k‘ ¼ eu ðk ‘Þ0 ðCk‘;hiki Þ1 Xjm
ðk ‘Þeðjkþ‘Þmþhikiþk‘
and
ðhi‘kiÞ
Xjm
ðhikiÞ
Xjm
ðkÞ ¼ ðXðjkÞmþhi‘kiþr;m : r ¼ 0; . . . ; k 1Þ0 ;
ðk ‘Þ ¼ ðXðjkþ‘Þmþhikiþr;m : r ¼ 0; . . . ; k ‘ 1Þ0 ;
ðmÞ
ðhi‘kiÞ
ðmÞ
ðkÞ, and Ck‘;hiki is the covariance
Ck;hi‘ki is the covariance matrix of Xjm
ðhikiÞ
ðhi‘kiÞ ðhikiÞ
matrix of Xjm
ðk ‘Þ. Note that Eðwuj;k
wuj0 ;k‘ Þ ¼ 0 unless
j0 ¼ j ‘ þ ðhi ‘ ki þ ‘ hi kiÞ=m;
ð67Þ
which is always an integer. For each j, there is at most one j0 that satisfies (67) for
j0 2 fk ‘, . . . , N1g. If such a j0 exists then
ðhi‘kiÞ
Eðwuj;k
ðhikiÞ
ðmÞ
ðmÞ
wuj0 ;k‘ Þ ¼ r2i‘ eu ðkÞ0 ðCk;hi‘ki Þ1 CðCk‘;hiki Þ1 eu ðk ‘Þ;
ðhi‘kiÞ
ðhikiÞ
where C ¼ EðXjm
ðkÞXj0 m ðk ‘Þ0 Þ. Note that the (k ‘)-dimensional
ðhikiÞ
vector Xj0 m ðk ‘Þ is just the first (k‘) of the k entries of the vector
ðhi‘kiÞ
ðmÞ
Xjm
ðkÞ. Hence, the matrix C is just Ck;hi‘ki with the last ‘ columns deleted.
ðmÞ
1
Then ðCk;hi‘ki Þ C ¼ Iðk; ‘Þ, which is the k k identity matrix with the last ‘
columns deleted. But then for any fixed u, for all k large, we have
ðmÞ
eu ðkÞ0 ðCk;hi‘ki Þ1 C ¼ eu ðk ‘Þ0 . Then
ðhi‘kiÞ
Eðwuj;k
ðhikiÞ
ðmÞ
wuj0 ;k‘ Þ ¼ r2i‘ eu ðk ‘Þ0 ðCk‘;hiki Þ1 eu ðk ‘Þ
ðmÞ
¼ r2i‘ ðCk‘;hiki Þ1
uu :
Consequently,
ðhi‘kiÞ
Varðwuj;k
ðhikiÞ
ðhi‘kiÞ
wuj0 ;k‘ Þ ¼ E½ðwuj;k
ðhikiÞ
wuj0 ;k‘ Þ2 ðmÞ
ðmÞ
1
2
¼ r2i‘ ðCk;hi‘ki Þ1
uu þ ri‘ ðCk‘;hiki Þuu
ðmÞ
2r2i‘ ðCk‘;hiki Þ1
uu
ðmÞ
ðmÞ
1
¼ r2i‘ ½ðCk;hi‘ki Þ1
uu ðCk‘;hiki Þuu !0
Ó Blackwell Publishing Ltd 2005
515
PARAMETER ESTIMATION
by Lemma 1 applied to the finite moving average. Thus,
ðhi‘kiÞ
VarðtNm;k
ðhikiÞ
ðhi‘kiÞ
ðuÞ tNm;k‘ ðuÞÞ ðN kÞ1 Varðwuj;k
ðhikiÞ
wuj0 ;k‘ ÞðN kÞ
!0
since for each j ¼ k, . . . , N 1 there is at most one j0 satisfying (67) along with
ðhi‘kiÞ
j0 ¼ k ‘, . . . , N 1. Then Chebyshev’s inequality shows that tNm;k
ðuÞ
ðhikiÞ
P
tNm;k‘ ðuÞ ! 0.
P
ðhikiÞ
tN ;k‘ ðuÞ ! 0.
Since
ðhikiÞ
ðhikiÞ
tNm;k ðuÞ tN ;k
P
ðuÞ ! 0
we
also
get
ðhikiÞ
tN ;k
ðuÞ
Now the Lemma follows easily using (50) along with (51).
Now, since
ðhi‘kiÞ
ðhikiÞ
ðhi‘kiÞ
ðhikiÞ
h^k;1
h^k‘;1 ¼ /^k;1
/^k‘;1
we have (62) with u ¼ 1. The cases u ¼ 2, . . . , D follow iteratively using (55),
^ /.
^ Thus, (62) is established. From (58)
Lemma 9 and (52) with h, / replaced by h;
to (62), it follows that
^ ðhikiÞ ð/^ðhikiÞ /ðhikiÞ Þ
h^ðhikiÞ hðhikiÞ ¼ R
k
^ ðhikiÞ RðhikiÞ Þ/ðhikiÞ þ oP ðN 1=2 Þ:
þ ðR
k
k
ð68Þ
To accommodate the derivation of the asymptotic distribution of h^ðiÞ hðiÞ , we
need to rewrite (68). Define
ðh0kiÞ
ðh0kiÞ
ðh0kiÞ
ðh0kiÞ
h^ h ¼ðh^k;1
hk;1 ; . . . ; h^k;D
hk;D ; . . . ;
ðhm1kiÞ
ðhm1kiÞ
ðhm1kiÞ
ðhm1kiÞ 0
h^k;1
hk;1
; . . . ; h^k;D
hk;D
Þ
ð69Þ
and
ðh0kiÞ
/ ¼ð/k;1
ðh0kiÞ
; . . . ; /k;D
ðhm1kiÞ
; . . . ; /k;1
ðhm1kiÞ 0
; . . . ; /k;D
Þ:
Using (69) we can rewrite (68) as
^ k ð/^ /Þ þ ðR
^ R Þ/ þ oP ðN 1=2 Þ;
h^ h ¼ R
k
k
ð70Þ
where /^ is the estimator of / and
ðh0kiÞ
Rk ¼ diagðRk
ðh1kiÞ
; Rk
ðhm1kiÞ
; . . . ; Rk
Þ
and
ðh0kiÞ
Rk ¼ diagðRk
ðh1kiÞ
; Rk
ðhm1kiÞ
; . . . ; Rk
Þ
noting that both Rk and Rk are Dm Dm matrices. The estimators of Rk and Rk are
^ k and R
^ . Now write ðR
^ R Þ/ ¼ Ck ðh^ hÞ, where
respectively R
k
k
k
Ck ¼
D1
X
Bn;k P½DmnðDþ1Þ
ð71Þ
n¼1
Ó Blackwell Publishing Ltd 2005
516
P. L. ANDERSON AND M. M. MEERSCHAERT
and
Dn
Bn;k
n
n
zfflfflfflffl}|fflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
zfflfflfflffl}|fflfflfflffl{
ðh0kiÞ
ðh0kiÞ
¼diagð0; . . . ; 0 ; /k;n ; . . . ; /k;n
; 0; . . . ; 0;
ðh1kiÞ
ðh1kiÞ
ðhm1kiÞ
ðhm1kiÞ
/k;n ; . . . ; /k;n
; . . . ; 0; . . . ; 0; /k;n
; . . . ; /k;n
Þ
|fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
n
Dn
Dn
with P the orthogonal Dm Dm cyclic permutation matrix (23). Thus, we write
equation (70) as
^ k ð/^ /Þ þ Ck ðh^ hÞ þ oP ðN 1=2 Þ:
h^ h ¼ R
ð72Þ
Then,
^ k ð/^ /Þ þ oP ðN 1=2 Þ
ðI Ck Þðh^ hÞ ¼ R
so that
^ k ð/^ /Þ þ oP ðN 1=2 Þ:
h^ h ¼ ðI Ck Þ1 R
ðhikiÞ
Let C ¼ limk!1Ck so that C is Ck with /k;u
R ¼ limk!1Rk, where
ð73Þ
replaced with pi(u). Also, let
R ¼ diagðRð0Þ ; . . . ; Rðm1Þ Þ
P
^k !
and R(i) as defined in (57). Equation (56) shows that R
R and then Theorem 1
along with equation (73) yield
N 1=2 ðh^ hÞ ) N ð0; V Þ;
where
V ¼ ðI CÞ1 RKR0 ½ðI CÞ1 0
ð74Þ
and K is as in (15). Let
0
1
pi1 ð1Þ
..
.
B
B
S ðiÞ ¼ B
@
0
1
..
.
pi1 ðD 1Þ pi2 ðD 2Þ
0
0
..
.
1
0
0C
C
.. C:
.A
ð75Þ
piDþ1 ð1Þ 1
It can be shown that
0
2
2
ðiÞ
KðiÞ ¼ S ðiÞ diagðr2
i1 ; ri2 ; . . . ; riD ÞS :
P
From the equation, wi ðuÞ ¼ u‘¼1 pi ð‘Þwi‘ ðu ‘Þ, it follows that R(i)S(i) ¼
IDD, the D D identity matrix. Therefore,
Ó Blackwell Publishing Ltd 2005
517
PARAMETER ESTIMATION
0
0
0
2
2
ðiÞ ðiÞ
RðiÞ KðiÞ RðiÞ ¼ RðiÞ S ðiÞ diagðr2
i1 ; ri2 ; . . . ; riD ÞS R
2
2
¼ diagðr2
i1 ; ri2 ; . . . ; riD Þ
and it immediately follows that
RKR0 ¼ diagðr20 Dð0Þ ; . . . ; r2m1 Dðm1Þ Þ;
2
2
where DðiÞ ¼ diagðr2
i1 ; ri2 ; . . . ; riD Þ. Thus, eqn (74) becomes
V ¼ ðI CÞ1 diagðr20 Dð0Þ ; . . . ; r2m1 Dðm1Þ Þ½ðI CÞ1 0 :
Also, from the relation wi ðuÞ ¼
Pu
‘¼1
ðI CÞ1 ¼
ð76Þ
pi ð‘Þwi‘ ðu ‘Þ, it can be shown that
D1
X
En P½DmnðDþ1Þ ;
n¼0
where En is defined in (21). Using estimates from the proof of Corollary 2.2.3
in Anderson et al. (1999) along with condition (17) it is not hard to show that
N1/2(w h) ! 0. Then it follows that
N 1=2 ðh^ wÞ ) N ð0; V Þ;
where w ¼ (w0(1), . . . , w0(D), . . . , wm1(1), . . . , wm1(D))0 . We have proved the
theorem.
u
ACKNOWLEDGEMENTS
Mark M. Meerschaert is partially supported by NSF Grants DES-9980484,
DMS-0139927, DMS-0417869 and by the Marsden Foundation in New Zealand.
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