ARTICLE IN PRESS

0167-7152/$ - se

doi:10.1016/j.sp

�Correspond4200-465 Porto

E-mail addr

Statistics & Probability Letters 76 (2006) 1655–1663

www.elsevier.com/locate/stapro

Asymptotic distribution of the Yule–Walker estimator forINARðpÞ processes

Isabel Silvaa,b,�, M. Eduarda Silvab,c

aDepartamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, PortugalbDepartamento de Matematica Aplicada, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687,

4169-007 Porto, PortugalcUI&D Matematica e Aplicac- oes, Universidade de Aveiro, Aveiro, Portugal

Received 6 December 2005; received in revised form 10 March 2006; accepted 3 April 2006

Available online 19 May 2006

Abstract

The INteger-valued AutoRegressive (INAR) processes were introduced in the literature by Al-Osh and Alzaid [1987.

First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275] and McKenzie [1988. Some

ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822–835] for modelling correlated

series of counts. These processes have been considered as the discrete counter part of AR processes, but their highly

nonlinear characteristics lead to some statistically challenging problems, namely in parameter estimation. Several

estimation procedures have been proposed in the literature, mainly for processes of first order. For some of these

estimators the asymptotic properties as well as finite sample properties have been obtained and studied. This paper

considers Yule–Walker parameter estimation for the pth-order integer-valued autoregressive, INARðpÞ, process. In

particular, the asymptotic distribution of the Yule–Walker estimator is obtained and it is shown that this estimator is

asymptotically normally distributed, unbiased and consistent.

r 2006 Elsevier B.V. All rights reserved.

Keywords: INAR process; Autocovariance distribution; Yule–Walker estimation; Delta method

1. Introduction

Recently, there has been a growing interest in modelling non-negative integer-valued time series and,specially, time series of counts. Several models have been proposed and, in particular, the INteger-valuedAutoRegressive (INAR) model has been the subject of study in several papers. The pth-order integer-valuedautoregressive (INARðpÞ) process is defined as follows (Latour, 1998). A discrete time non-negative integer-valued stochastic process, fX tg, is said to be an INARðpÞ process if it satisfies the following equation:

X t ¼ a1 � X t�1 þ a2 � X t�2 þ � � � þ ap � X t�p þ et, (1)

e front matter r 2006 Elsevier B.V. All rights reserved.

l.2006.04.008

ing author. Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias,

, Portugal.

esses: ims@fe.up.pt (I. Silva), mesilva@fc.up.pt (M.E. Silva).

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–16631656

where

(1)

fetg, designated the innovation process, is a sequence of independent and identically distributed (i.i.d) non-negative integer-valued random variables with E½et� ¼ me, Var½et� ¼ s2e , E½e

3t � ¼ ge and E½e4t � ¼ ke;

(2)

the symbol � represents the thinning operation (Steutel and Van Harn, 1979; Gauthier and Latour, 1994)defined by

ai � X t�i ¼XXt�i

j¼1

Y i;j ; i ¼ 1; . . . ; p,

where fY i;jg, designated the counting series, is a set of i.i.d. non-negative integer-valued random variablessuch that E½Y i;j � ¼ ai, Var½Y i;j � ¼ s2i , E½Y

3i;j � ¼ gi and E½Y 4

i;j � ¼ ki. All the counting series are assumedindependent of fetg;

(3)

0paio1, i ¼ 1; . . . ; p� 1, and 0oapo1.

Alternatively, an INARðpÞ process can be represented as a p-dimensional INARð1Þ process (Franke andSubba Rao, 1995). Accordingly, by using the vector thinning operation, defined as a random vector whose ithcomponent is given by

½A � X�i ¼Xp

j¼1

aij � X j ; i ¼ 1; . . . ; p,

where X ¼ ½X 1 � � �X p�T is a random vector, A is a p� p matrix with entries aij satisfying 0paijp1, for

i; j ¼ 1; . . . ; p, and the counting series of all aij � X j ; i; j ¼ 1; . . . ; p, are assumed independent, the INARðpÞprocess defined in (1) can be written as

Xt ¼ A � Xt�1 þWt, (2a)

X t ¼ HXt, (2b)

where H ¼ ½1 0 � � � 0�, Xt ¼ ½X t X t�1 � � � X t�pþ1�T, Wt ¼ ½et 0 � � � 0�

T, for fetg a sequence of i.i.d. randomvariables, with E½et� ¼ me, Var½et� ¼ s2e ; E½e

3t � ¼ ge, and E½e4t � ¼ keo1, and

A ¼

a1 a2 � � � ap�1 ap

1 0 � � � 0 0

0 1 � � � 0 0

..

. ... . .

. ... ..

.

0 0 � � � 1 0

26666664

37777775.

Furthermore, model (2a) can be expressed as

Xt¼dAk� Xt�k þ

Xk�1j¼0

Aj�Wt�j, (3)

where ¼dstands for equality in distribution. Then, Eqs. (2) can be written as

Xt¼dX1j¼0

Aj�Wt�j, (4a)

X t¼dHXt, (4b)

where H ¼ ½1 0 � � � 0�, provided the spectral radius of the matrix A is less than one, that is, rðAÞo1.The existence and stationarity condition for the INARðpÞ processes is that the roots of zp � a1zp�1 � � � � �

ap�1z� ap ¼ 0 lie inside the unit circle (Du and Li, 1991) or, equivalently, thatPp

j¼1 ajo1 (Latour, 1997,1998). Probabilistic characteristics of the INAR models, in terms of second- and third-order moments andcumulants, have been obtained by Silva and Oliveira (2004, 2005). In particular, it is found that the

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–1663 1657

autocovariance function, Rð�Þ, satisfies a set of Yule–Walker type difference equations, which can be written inscalar and vectorial form as

Rð0Þ ¼ V p þPpi¼1

aiRðiÞ;

RðkÞ ¼Ppi¼1

aiRði � kÞ;

8>>><>>>:

() Rp a ¼ ½�V p 0 � � � 0�T,

respectively, where

Rp ¼

Rð0Þ Rð1Þ � � � RðpÞ

Rð1Þ Rð0Þ � � � Rðp� 1Þ

..

. ... . .

. ...

RðpÞ Rðp� 1Þ � � � Rð0Þ

2666664

3777775; a ¼

�1

a1

..

.

ap

2666664

3777775,

and V p is defined by

Vp ¼ s2e þ mX

Xp

i¼1

s2i ,

where s2e ¼ Var½et�, mX ¼ E½X t� ¼ me=ð1�Pp

i¼1 aiÞ, and s2i is the variance of the counting series,Y j;i ðj ¼ 1; . . . ;X t�i; i ¼ 1; . . . ; pÞ, of the ith thinning operation, ai � X t�i.

Several estimation methods for the INARðpÞ model have been proposed in the literature both in the timedomain, namely, Yule–Walker and Conditional Least Squares (Du and Li, 1991; Latour, 1998), GeneralizedMethod of Moments (Brannas, 1994, 1995), Conditional Maximum Likelihood (Franke and Seligmann, 1993;Franke and Subba Rao, 1995) and in the frequency domain, the Whittle Criterion (Silva and Oliveira, 2004,2005). Methods based on higher-order statistics (moments and cumulants) have also been considered (Silva,2005). The small sample properties of these estimators have been studied empirically by Monte Carlo methods(Silva, 2005) but little is known about the asymptotic properties. It must be said that these depend on higher-order moments and cumulants which are very difficult to obtain.

Here the asymptotic distribution of the sample autocovariance for INARðpÞ processes is obtained and it isshown that the Yule–Walker estimator is asymptotically normally distributed, unbiased and consistent. Thisresult generalizes the work of Park and Oh (1997), who derived the asymptotic distribution of theYule–Walker estimator for an alternative parametrization of the Poisson INAR(1) process with binomialthinning operation.

In Section 2.1 the asymptotic properties of the sample mean and sample autocovariance matrix are obtainedand in Section 2.2 expressions for the asymptotic covariance matrix of the Yule–Walker estimator areprovided.

2. Yule–Walker estimation of the INARðpÞ model

Given a realization fX 1; . . . ;X Ng from an INARðpÞ process, the Yule–Walker estimator a of a is obtainedby solving the following system of linear equations:

Rp�1a ¼ rp 3

Rð0Þ Rð1Þ Rð2Þ � � � Rðp� 1Þ

Rð1Þ Rð0Þ Rð1Þ � � � Rðp� 2Þ

Rð2Þ Rð1Þ Rð0Þ � � � Rðp� 3Þ

..

. ... ..

. . .. ..

.

Rðp� 1Þ Rðp� 2Þ Rðp� 3Þ � � � Rð0Þ

266666664

377777775

a1

..

.

..

.

..

.

ap

26666666664

37777777775¼

Rð1Þ

Rð2Þ

Rð3Þ

..

.

RðpÞ

266666664

377777775,

with RðkÞ ¼ ð1=NÞPN�k

t¼1 ðX t � X ÞðX tþk � X Þ; k 2 Z, and X ¼ ð1=NÞPN

t¼1 X t.

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–16631658

The estimators for me and s2e are, respectively,

me ¼ X 1�Xp

i¼1

ai

!; s2e ¼ V p � X

Xp

i¼1

s2i ,

where

Vp ¼ Rð0Þ �Xp

i¼1

aiRðiÞ

and s2i is an estimator of the variance of the counting series for the ith thinning operation,ai � X t�i; i ¼ 1; . . . ; p. The estimation of s2i depends on the distribution of the counting series. For instance,in the case of the binomial thinning operation (when the counting series are Bernoulli distributed),s2i ¼ aið1� aiÞ, for i ¼ 1; . . . ; p.

Now, in order to obtain the asymptotic distribution of the Yule–Walker estimator, the asymptoticproperties of the sample covariance matrix are needed. These properties are obtained in 2.1 and the asymptoticdistribution of the Yule–Walker estimator follows in 2.2.

2.1. Asymptotic distribution of the autocovariance function

In the following theorem the asymptotic multivariate normality of the sample mean, X , and of the sampleautocovariance function, RðkÞ, is established.

The details of the proof are omitted since it follows closely Brockwell and Davis (1991, Chapter 7) byconsidering an auxiliary function

R�ðkÞ ¼1

N

XN

t¼1

ðX t � mX ÞðX tþk � mX Þ; k 2 N 0,

obtaining its asymptotic properties and proving that Rð�Þ has the same asymptotic properties as the auxiliaryfunction, R�ð�Þ.

Theorem 1. If fX tg is an INARðpÞ process with representation (4), whereP1

j¼0 jAjjo1, and if Rð�Þ is the

autocovariance function of fX tg, then for any non-negative integer h,

X

Rð0Þ

..

.

RðhÞ

2666664

3777775 is AN

mX

Rð0Þ

..

.

RðhÞ

2666664

3777775; N�1VR

0BBBBB@

1CCCCCA

where

VR ¼

½V 11�1�1 ½V 12�1�ðhþ1Þ

½V 12�T1�ðhþ1Þ ½V 22�ðhþ1Þ�ðhþ1Þ

" #

is an ðhþ 2Þ � ðhþ 2Þ matrix for any non-negative integer h, such that

V11 ¼ limN!1

NVarðX Þ

¼X1

h¼�1

RðhÞ, ð5aÞ

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–1663 1659

½V 12�kþ1 ¼ limN!1

NCovðX ;R�ðkÞÞ; k ¼ 0; 1; . . . ; h,

¼X1

h¼�1

CX ðh; k þ hÞ, ð5bÞ

½V 22�kþ1;jþ1 ¼ limN!1

NCovðR�ðkÞ;R�ðjÞÞ; k; j ¼ 0; 1; . . . ; h,

¼X1

h¼�1

RðhÞRðhþ j � kÞ þ Rðhþ jÞRðh� kÞ þ CY ðh; k; j þ hÞ, ð5cÞ

where CX ð�; �Þ is the third-order cumulant of X t and CY ð�; �; �Þ is the fourth-order cumulant of the process

Y t ¼ X t � mX (Silva, 2005).

Proof. Note that for 0pkph,

ffiffiffiffiffiNpðR�ðkÞ � RðkÞÞ ¼

1ffiffiffiffiffiNp

XN

t¼N�kþ1

ðX t � mX ÞðX tþk � mX Þ

þ1ffiffiffiffiffiNp �mX

XN�k

t¼1

X t � mX

XN�k

t¼1

X tþk � ðN � kÞm2X

þ XXN�k

t¼1

X t þ XXN�k

t¼1

X tþk � ðN � kÞX2

!

¼1ffiffiffiffiffiNp

XN

t¼N�kþ1

ðX t � mX ÞðX tþk � mX Þ

þNffiffiffiffiffiNp ðX � mX Þ

1

N

XN�k

t¼1

X t þ1

N

XN�k

t¼1

X tþk � 1�k

N

� �ðX þ mX Þ

!.

The first term, ð1=ffiffiffiffiffiNpÞPN

t¼N�kþ1 ðX t � mX ÞðX tþk � mX Þ, is opð1Þ, since

1ffiffiffiffiffiNp E

XN

t¼N�kþ1

ðX t � mX ÞðX tþk � mX Þ

����������

" #p

1ffiffiffiffiffiNp kRð0Þ �!

N!10.

The last term is also opð1Þ, sinceffiffiffiffiffiNpðX � mX Þ is Opð1Þ,

as N !1, becauseffiffiffiffiffiNpðX � mX Þ)

dY , where Y�Nð0;V 11Þ, and )

dstands for convergence in distribution.

Note that by the weak law of large numbers (Brockwell and Davis, 1991, p. 208),

1

N

XN�k

t¼1

X t þ1

N

XN�k

t¼1

X tþk � 1�k

N

� �ðX þ mX Þ

!!P

0.

This remark leads to the following resultffiffiffiffiffiNpðR�ðkÞ � RðkÞÞ is opð1Þ,

as N !1. Then the proposition follows from the fact that

X

R�ð0Þ

..

.

R�ðhÞ

2666664

3777775 is AN

mX

Rð0Þ

..

.

RðhÞ

2666664

3777775; N�1VR

0BBBBB@

1CCCCCA

and Proposition 6.3.3 of Brockwell and Davis (1991, p. 205). &

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–16631660

Note that this result is in agreement with the asymptotic distribution obtained by Park and Oh (1997) forthe case of Poisson INAR(1) models with binomial thinning operation.

2.2. Asymptotic distribution of the Yule– Walker estimator

In this section, the asymptotic properties of the sample covariance matrix is used to obtain the asymptoticdistribution of the Yule–Walker estimator. Hence, consider the p� p Toeplitz sample autocovariance matrix,Rp�1, and the vector of sample autocovariance function with p elements, rp. Let

VRr ¼vecðRp�1Þ

rp

" #(6)

be a vector with pðpþ 1Þ elements, where the vec operator is the column-stacking operator of a matrix(Neudecker, 1969), that is,

vecðAÞ ¼

A�1

A�2

..

.

A�q

2666664

3777775

is a pq� 1 vector where A�i is the ith column of the p� q matrix A.Note that for p ¼ 1,

VRr ¼ ½Rð0Þ Rð1Þ�T

and for p41, the ith element of VRr can be written as

VRrðiÞ ¼Rðjði � 1Þmod p� ½ði � 1Þ=p�jÞ if ipp2;

Rðjimod pjÞ if i4p2;

(

where ½a� represents the integer part of a 2 R.Then, using the previous results it is found that

VRr ¼vecðRp�1Þ

rp

" #is AN

vecðRp�1Þ

rp

" #;1

NRRr

!,

where RRr is the pðpþ 1Þ � pðpþ 1Þ covariance matrix of VRr,

RRrði; jÞ ¼ CovðVRrðiÞ; VRrðjÞÞ, (7)

defined as in (5c), for i; j ¼ 1; . . . ; pðpþ 1Þ.Now, let

DT ¼ �½Rð1ÞIp � � � RðpÞIp�ððR�1p�1Þ

T� R�1p�1Þ½Ip2 0p2�p� þ ½0p�p2 R�1p�1�, (8)

where In is the n� n identity matrix, 0n�m the n�m matrix of zeros and � the Kronecker product (Graham,1981) defined as follows. Let A and B be p� q and m� n matrices, respectively. Then A� B is a pm� qn

matrix given by

A� B ¼

a11B a12B � � � a1qB

a21B a22B � � � a2qB

..

. ... . .

. ...

ap1B ap2B � � � apqB

2666664

3777775.

The asymptotic distribution of the Yule–Walker estimator of an INARðpÞ process is given in the followingtheorem.

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–1663 1661

Theorem 2. Let fX tg be an INARðpÞ process, satisfying (1), and a the Yule– Walker estimator of a. Then

N1=2ða� aÞ is ANð0p;DTRRrDÞ,

where 0n is a vector of n zeros, RRr is given in (7) and DT is defined by (8).

Proof. The Delta method (van der Vaart, 1998) is used to demonstrate the result. Thus, let j be the functionfrom Dj � Rpðpþ1Þ into Rp defined by

jðXÞ ¼ ðunvecp�pþ1ðXÞC1Þ�1ðunvecp�pþ1ðXÞC2Þ,

where X is a vector with pðpþ 1Þ elements,

C1 ¼Ip

0Tp

" #; C2 ¼

0p

1

� �,

are a ðpþ 1Þ � p matrix and a vector with pþ 1 elements, respectively, 0p is a vector of p zeros and unvecn�m isthe inverse of the vec operator, that is, unvecp�qðXÞ, for a vector X of pq� 1 elements is defined as a p� q

matrix, such that unvecp�qðvecðAÞÞ ¼ A (Swami and Giannakis, 1994).Note that for VRr defined in (6)

jðVRrÞ ¼ R�1

p�1rp ¼ a.

Then, by the Delta method,

a is AN a;1

NDTRRrD

� �,

where

D ¼qjqX

� �jX¼VRr

¼qjqXT

� �T

jX¼VRr

is the pðpþ 1Þ � p derivative matrix of the function j, defined as follows. Let f : Rn ! Rm be a vector valuedfunction with vector variable. The n�m matrix derivative of f is defined as (Rao and Rao, 1998, p. 225)

qfqX

9qf

qXT

� �T

9

qf1qX1

� � �qf1qXn

..

. . .. ..

.

qfm

qX1� � �

qfm

qXn

26666664

37777775

T

,

in order to meet the needs of the Jacobian matrix.Note that jðXÞ can be written as the product of two matrices, jðXÞ ¼M�11 ðXÞM2ðXÞ. Then, by using the

product rule (Rao and Rao, 1998, p. 234), it is found that

qjqXT

� �T

¼qðM�11 M2Þ

qXT

� �T

¼ ðMT2 � IpÞ

qvecðM�11 Þ

qXTþ ðI1 �M�11 Þ

qvecðM2Þ

qXT

� �T

.

Now, using the matrix derivative of an inverse matrix,

qvecðM�11 Þ

qXT¼ �ððM�11 Þ

T�M�11 Þ

qvecðM1Þ

qXT,

and then, replacing M1 by unvecp�pþ1ðXÞC1, and using the properties of the vec and unvecn�m operators(Neudecker, 1969) and the derivative matrix rule of f : Rk ! R such that f ¼ AX, where A is a n� k constant

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–16631662

matrix (Rao and Rao, 1998, p. 233), it is found that

qvecðM�11 Þ

qXT¼ � ððunvecp�pþ1ðXÞC1Þ

�1ÞT� ðunvecp�pþ1ðXÞC1Þ

�1� � qvecðunvecp�pþ1ðXÞC1Þ

qXT

¼ � ððunvecp�pþ1ðXÞC1Þ�1ÞT� ðunvecp�pþ1ðXÞC1Þ

�1� � qðCT

1 � IpÞX

qXT

¼ � ððunvecp�pþ1ðXÞC1Þ�1ÞT� ðunvecp�pþ1ðXÞC1Þ

�1� �

ðCT1 � IpÞ.

In a similar manner

qvecðM2Þ

qXT¼

qvecðunvecp�pþ1ðXÞC2Þ

qXT¼

qðCT2 � IpÞX

qXT¼ ðCT

2 � IpÞ.

Then,

qjqXT

� �T

¼ �ððunvecp�pþ1ðXÞC2ÞT� IpÞðððunvecp�pþ1ðXÞC1Þ

�1ÞT� ðunvecp�pþ1ðXÞC1Þ

�1ÞðCT

1 � IpÞ�þ ðI1 � ðunvecp�pþ1ðXÞC1Þ

�1ÞðCT

2 � Ip�T.

Therefore,

DT ¼ � ðrTp � IpÞððR�1p�1Þ

T� R�1p�1ÞðC

T1 � IpÞ þ ðI1 � R�1p�1ÞðC

T2 � IpÞ

¼ � ½Rð1ÞIp � � � RðpÞIp�ððR�1p�1Þ

T� R�1p�1Þ½Ip2 0p2�p� þ ½0p�p2 R�1p�1�,

and the theorem follows. &

For the particular case of INAR(1) processes with Poisson, PðlÞ, innovations and binomial thinningoperation, it is found that R0 ¼ ½Rð0Þ� ¼ ½l=ð1� aÞ�,

RRr ¼VarðRð0ÞÞ CovðRð0Þ; Rð1ÞÞ

CovðRð0Þ; Rð1ÞÞ VarðRð1ÞÞ

" #

¼

lð2lð1þ a2Þ þ ð1� aÞð1þ aÞ2Þ

ð1� aÞ2ð1� a2Þ

2lað1þ 2l� a2Þ

ð1� aÞ2ð1� a2Þ

2lað1þ 2l� a2Þ

ð1� aÞ2ð1� a2Þ

lðlð1þ 4a2 � a4Þ þ að1� aÞð1þ aÞ2Þ

ð1� aÞ2ð1� a2Þ

266664

377775

and

DT ¼ � ½Rð1Þ�ð½1=Rð0Þ� � ½1=Rð0Þ�Þ½1 0� þ ½0 1=Rð0Þ�

¼ �Rð1Þ

Rð0Þ21

Rð0Þ

" #¼�að1� aÞ

l1� al

� �.

Hence, DTRRrD ¼ ð1� a2Þ þ ðað1� aÞ2Þ=l. Therefore, Theorem 2 states that

N1=2ða� aÞ is AN 0; ð1� a2Þ það1� aÞ2

l

� �,

which is in agreement with the asymptotic distribution obtained by Park and Oh (1997) for the a parameter.

References

Al-Osh, M.A., Alzaid, A.A., 1987. First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275.

Brannas, K., 1994. Estimation and testing in integer-valued AR(1) models. Umea Economic Studies, 335, Umea University.

Brannas, K., 1995. Explanatory variables in the AR(1) count data model. Umea Economic Studies, 381, Umea University.

Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods, second ed. Springer, New York.

ARTICLE IN PRESSI. Silva, M.E. Silva / Statistics & Probability Letters 76 (2006) 1655–1663 1663

Du, J.-G., Li, Y., 1991. The integer-valued autoregressive (INARðpÞ) model. J. Time Ser. Anal. 12, 129–142.

Franke, J., Seligmann, T., 1993. Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling

epileptic seizure counts. In: Subba Rao, T. (Ed.), Developments in Time Series Analysis. Chapman & Hall, London, pp. 310–330.

Franke, J., Subba Rao, T., 1995. Multivariate first order integer valued autoregressions. Technical Report, Mathematics Department,

UMIST.

Gauthier, G., Latour, A., 1994. Convergence forte des estimateurs des parametres d’un processus GENAR(p). Ann. Sci. Math. Quebec 18,

49–71.

Graham, A., 1981. Kronecker Products and Matrix Calculus: With Applications. Ellis Horwood, Chichester.

Latour, A., 1997. The multivariate GINARðpÞ process. Adv. Appl. Probab. 29, 228–248.

Latour, A., 1998. Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19,

439–455.

McKenzie, E., 1988. Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822–835.

Neudecker, H., 1969. Some theorems on matrix differentiation with special reference to Kronecker matrix product. J. Amer. Statist.

Assoc. 64, 953–963.

Park, Y., Oh, C.W., 1997. Some asymptotic properties in INAR(1) processes with Poisson marginals. Statist. Papers 38, 287–302.

Rao, C.R., Rao, M.B., 1998. Matrix Algebra and its Applications to Statistics and Econometrics. World Scientific, Singapore.

Silva, I., 2005. Contributions to the analysis of discrete-valued time series. Ph.D. Thesis, Universidade do Porto, Portugal.

Silva, M.E., Oliveira, V.L., 2004. Difference equations for the higher order moments and cumulants of the INAR(1) model. J. Time Ser.

Anal. 25, 317–333.

Silva, M.E., Oliveira, V.L., 2005. Difference equations for the higher order moments and cumulants of the INARðpÞ model. J. Time Ser.

Anal. 26, 17–36.

Steutel, F.W., Van Harn, K., 1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7, 893–899.

Swami, A., Giannakis, G., 1994. Multichannel ARMA processes. IEEE Trans. Signal Process. 42, 898–913.

van der Vaart, A.W., 1998. Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University

Press, Cambridge.