Summary. This note reconsiders the nonnegative integer-valued bilinear processes introduced by Doukhan, Latour, and Oraichi (2006). Using a hidden Markov argument, we extend their result of the existence of a stationary solution for the INBL(1,0,1,1) process to the class of superdiagonal INBL(p; q; m; n) models. Our approach also yields improved parameter restrictions for several moment conditions compared to the ones in Doukhan, Latour, and Oraichi (2006).count data;integer-valued time series;bilinear model

Akker, R. van den

Drost, F.C.

Werker, B.J.M.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2007–47 
 
NOTE ON INTEGER-VALUED BILINEAR  
TIME SERIES MODELS 
 
By Feike C. Drost, Ramon van den Akker, Bas J.M. Werker 
 
 
 
June 2007 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Note on integer-valued bilinear time series models
Feike C. Drost ² Ramon van den Akker ² Bas J.M. Werker
Econometrics and Finance group, CentER, Tilburg University, The Netherlands
Summary. This note reconsiders the nonnegative integer-valued bilinear processes introduced
by Doukhan, Latour, and Oraichi (2006). Using a hidden Markov argument, we extend their
result of the existence of a stationary solution for the INBL(1,0,1,1) process to the class of su-
perdiagonal INBL(p;q;m;n) models. Our approach also yields improved parameter restrictions
for several moment conditions compared to the ones in Doukhan, Latour, and Oraichi (2006).
Keywords: count data, integer-valued time series, bilinear model
JEL: C22
1. Introduction
In many sciences one encounters nonnegative discrete valued time series, often as counts of
events or objects at consecutive points in time. Especially in economics and medicine many
interesting variables are (nonnegative) integer-valued. For example: the number of transac-
tions in IBM during each minute, the number of patients in a hospital at the end of the day,
the number of claims an insurance company receives during each day, the number of epilep-
tic seizures a patient su®ers each day, etcetera. Until the mid seventies modeling discrete
valued time series did not attract much attention since most traditional representations
of dependence become either impossible or impractical. The last two decades there were
many developments in the literature on integer-valued time series; see McKenzie (2003) for
a detailed review. In empirical work integer-valued moving average processes (Al-Osh and
Alzaid (1991)), integer-valued autoregressive processes (Al-Osh and Alzaid (1987), Al-Osh
and Alzaid (1990), and Du and Li (1991)), and generalized integer-valued autoregressive
processes (Latour (1998)) are the most common choices. Doukhan et al. (2006) introduced
the class of nonnegative INteger-valued BiLinear time series, which contains the three afore-
mentioned classes.
Recall the de¯nition of an INBL(p;q;m;n) process: let (X¡1;:::;X¡p_m;"¡1;:::;"¡q_n)
be the (nonnegative) integer-valued initial points generated by some probability distribution
º, and de¯ne Xt, t = 0;1;2;:::, recursively by,
Xt =
p X
i=1
®i ± Xt¡i +
q X
j=1
¯j ± "t¡j +
m X
k=1
n X
`=1
°k;` ± (Xt¡k"t¡`) + "t; (1)
where ("t)t¸0 is a collection of i.i.d. nonnegative integer-valued variables with distribution G
with mean ¹G 2 [0;1] and variance ¾2
G 2 [0;1], and where the Steutel-van Harn operators
´± applied to some nonnegative integer-valued random variable Z are de¯ned as
´ ± Z =
Z X
s=1
U(´)
s ;2 Drost, van den Akker, and Werker
where (U
(´)
s )s¸1 is a collection of i.i.d. nonnegative integer-valued variables with distribution
function F´ with mean ´.¤ In proofs we suppose, without loss of generality, p = q = m = n
since we may introduce additional lags in (1) with ´ = 0 or equivalently F´f0g = 1. The
class of GINAR(p) processes arises by taking q = m = n = 0. Specializing further to the
situation where the F´-distributions are Bernoulli, one gets the class of INAR(p) models.
Similarly INMA(q) processes are obtained when p = m = n = 0.
In this note we consider stationarity and existence of moments of superdiagonal
INBL(p;q;m;n) processes with °k;` = 0 if k < `. Using Markov chain techniques we ob-
tain the existence and uniqueness of a strictly stationary solution of (1) when
P
i ®i +
¹G
P
k;` °k;` < 1 (see Theorem 2.1). Theorem 2.2 provides su±cient parameter restrictions
ensuring the existence of (higher-order) moments under the stationary distribution. Along
completely di®erent lines our results generalize Sections 2 and 3 in Doukhan et al. (2006)
who restricted attention to the INBL(1;0;1;1) process. Moreover our parameter restric-
tions in Theorem 2.2 are less severe. Estimation of superdiagonal INBL processes can be
performed along the same lines as in Doukhan et al. (2006). The details are beyond the
scope of this note.
2. Results
First we discuss the existence of a stationary solution. The insightful proof of Theorem 2.1
in Doukhan et al. (2006) gives an explicit construction of approximating processes for the
INBL(1;0;1;1) process. It seems impossible to extend this approach to general superdiag-
onal INBL processes. As an alternative we prove our results by Markov chain techniques.
Let F ¤ G denote the convolution of the probability measures F and G and introduce
the process Z = (Zt)t¸0 by
Zt = (Xt¡1;:::;Xt¡p_m;"t¡1;:::;"t¡q_n): (2)
Notice that Z0 » º. It is easy to see that Z is a Markov chain. For t ¸ 0 and zt =
(xt¡1;:::;xt¡p_m;et¡1;:::;et¡q_n) we have
P(Zt+1 = zt+1 j Zt = zt)
= Gfeg
¡
¤
p
i=1(¤
xt¡i
s=1 F®i) ¤
q
j=1 (¤
et¡j
s=1 F¯j) ¤m
k=1 ¤n
`=1(¤
xt¡ket¡`
s=1 F°k;`)
¢
fz ¡ eg;
if zt+1 = (z;xt¡1;:::;xt+1¡p_m;e;et¡1;:::;et+1¡q_n) for some e;z, and with
P(Zt+1 = zt+1 j Zt = zt) = 0 otherwise. From (1) it is immediate that, for t ¸ 0,
E[Xt j Zt] = ¹G +
p X
i=1
®iXt¡i +
q X
j=1
¯j"t¡j +
m X
k=1
n X
`=1
°k;`Xt¡k"t¡` 2 [0;1]: (3)
The following proposition provides su±cient conditions for Z to be an irreducible aperiodic
Markov chain. The proof is immediate from the de¯nition of the state space.
¤Implicitly it is understood that distinct Steutel-van Harn operators are independent and also
that Steutel-van Harn operators applied to di®erent random variables are independent.Note on integer-valued bilinear time series models 3
Proposition 2.1 Let Z be the time series de¯ned in (2) linked to the INBL process X
de¯ned in (1) and assume, for all relevant i;j;k;`,
0 < Gf0g; 0 < F®if0g; 0 < F¯jf0g; 0 < F°k;`f0g: (4)
Then Z is an irreducible aperiodic Markov chain on the state space
S = fz j 9t ¸ 1 : P(Zt = z j Z0 = 0) > 0g:
Remark Condition (4) allows the process X to arrive at 0 and therefore (4) seems to be
harmless in real-life applications. At the cost of lengthy derivations, condition (4) might be
weakened but this is outside the scope of this short note.
Theorem 2.1 Let X be the superdiagonal INBL process as de¯ned in (1) with °k;` = 0 if
k < ` and assume (4), ¾2
G < 1,
Pq
j=1 ¯j < 1, and
p X
i=1
®i + ¹G
m X
k=1
n X
`=1
°k;` < 1: (5)
Then there exists a unique initial distribution º¤ such that X is strictly stationary under
Pº¤. Furthermore the ¯rst moment of X is ¯nite under Pº¤.
Proof. We prove that there exists a unique initial distribution º¤ such that Z is sta-
tionary under Pº¤, which yields the result. Since Z is irreducible and aperiodic on S (and
since, for countable Markov chains, ¯nite sets are petite) it su±ces to prove, see Theo-
rem 15.0.1 in Meyn and Tweedie (1994), that there exists a mapping V : z 7! [1;1) such
that the following Foster-Lyapunov drift criterium holds: there exists ± > 0 and a ¯nite set
such that for all Zt outside this ¯nite set we have
W(Zt) = (1 ¡ ±)V (Zt) ¡ E[V (Zt+1) j Zt] ¸ 0: (6)
Take, for notational simplicity, p = q = m = n. To verify (6) choose ± > 0 su±ciently small
such that X
i
®i + ¹G
X
k¸`
°k;` < (1 ¡ ±)n ¡ n± ¡
1
2
n(n + 1)¹G±;
and de¯ne V : z 7! [1;1) by
V (Zt) = 1 +
X
i
~ ®iXt¡i +
X
j
~ ¯j"t¡j +
X
k¸`
~ °k;`Xt¡k"t¡`;
where the nonnegative tilde parameters are recursively de¯ned from ~ ®n+1 = ~ ¯n+1 =
~ °n+1;` = 0 and
(1 ¡ ±)~ ®i = ~ ®i+1 + ¹G~ °i+1;1 + ®i + ±;
(1 ¡ ±)~ ¯j = ~ ¯j+1 + ¯j + ±;
(1 ¡ ±)~ °k;` = ~ °k+1;`+1 + °k;` + ±:
Note, by the choice of ±,
µ ´ ~ ®1 + ¹G~ °1;1 =
X
i
(®i + ±)=(1 ¡ ±)i + ¹G
X
k¸`
(°k;` + ±)=(1 ¡ ±)k < 1:4 Drost, van den Akker, and Werker
Hence we obtain, using (1), (3), and the independence of "t and Xt ¡ "t,
W(Zt) = ¡ ± ¡ ¹G(~ ®1 + ~ ¯1) ¡ (¹2
G + ¾2
G)~ °1;1
+
X
i
[(1 ¡ ±)~ ®i ¡ (~ ®1 + ¹G~ °1;1)®i ¡ ~ ®i+1 ¡ ¹G~ °i+1;1]Xt¡i
+
X
j
[(1 ¡ ±)~ ¯j ¡ (~ ®1 + ¹G~ °1;1)¯j ¡ ~ ¯j+1]"t¡j
+
X
k¸`
[(1 ¡ ±)~ °k;` ¡ (~ ®1 + ¹G~ °1;1)°k;` ¡ ~ °k+1;`+1]Xt¡k"t¡`
= ¡ ± ¡ ¹G(~ ®1 + ~ ¯1) ¡ (¹2
G + ¾2
G)~ °1;1 + (1 ¡ µ) £
2
4
X
i
(®i +
±
1 ¡ µ
)Xt¡i +
X
j
(¯j +
±
1 ¡ µ
)"t¡j +
X
k¸`
(°k;` +
±
1 ¡ µ
)Xt¡k"t¡`
3
5:
Conclude W(Zt) ¸ 0 outside a ¯nite set. Hence the drift condition (6) holds, which concludes
the proof of the stationarity. The existence of the ¯rst moment under º¤ is immediate from
Theorem 2.2 below. 2
Remark For p = m = n = 1 and q = 0, (5) reduces to the condition of Theorem 2.1 in
Doukhan et al. (2006).
The next theorem gives a su±cient condition for the existence of higher order moments
of Xt under Pº¤.
Theorem 2.2 Let X be the superdiagonal INBL process as de¯ned in (1) with °k;` = 0 if
k < `. Suppose, for some K 2 N, the existence of the K-th order moments of F®i;F¯j, and
F°k;`, and assume (4), EG"2K
0 < 1,
Pq
j=1 ¯j < 1, and
p X
i=1
®i +
¡
EG"K
0
¢1=K
m X
k=1
n X
`=1
°k;` < 1: (7)
Then, with º¤ the stationary distribution of Theorem 2.2,
Eº¤XK
0 < 1:
Proof. Recall from Proposition 2.1 that Z is an irreducible aperiodic Markov chain on
S and 0 2 S. Hence, since Z has stationary distribution º¤, we have Zt
d ¡! º¤, under P±0
as t ! 1. Thus the convergence also holds for marginals. By the Portmanteau theorem,
and nonnegativity of X, this implies for L 2 N,
Eº¤X
L¡1
0 · liminf
t!1
E±0X
L¡1
t · sup
0·t<1
E±0X
L¡1
t : (8)
To prove that the right hand side of (8) is bounded for L · K+1, we use induction. Clearly
the statement holds true for L = 1. To show the boundedness for L = 2 we make some
preliminary remarks. Denote, for notational convenience, E±0 by E0 and let, for P ¸ 1,
kZkP =
¡
E0jZjP¢1=P
denote the LP(P±0) norm. Recall k¢kP · k¢kQ for 1 · P · Q. HenceNote on integer-valued bilinear time series models 5
(7) implies that
P
i ®i +k"0kv
P
k;` °k;` < 1 for all 1 · v · K. Since, for t · s, Xt ¡"t and
"s are independent, we have, for P ¸ 1,
kXt"skP · k(Xt ¡ "t)"skP + k"t"skP · k"0kPkXtkP + k"2
0kP: (9)
Using (3), (9) with P = 1, the zero starting values, and (7), the boundedness for L = 2 is
obtained from,
E0Xt = ¹G +
X
i
®iE0Xt¡i +
X
j
¯j¹G +
X
k¸`
°k;`E0Xt¡k"t¡`
· ¹G + ¹G
X
j
¯j + (¹2
G + ¾2
G)
X
k¸`
°k;` +
X
i
®iE0Xt¡i + ¹G
X
k¸`
°k;`E0Xt¡k
·
2
4¹G + ¹G
X
j
¯j + (¹2
G + ¾2
G)
X
k¸`
°k;`
3
5
1 X
s=0
0
@
X
i
®i + ¹G
X
k¸`
°k;`
1
A
s
< 1:
To complete the induction, we assume that the right hand side of (8) is bounded for some
2 · L · K and we prove that this implies sup0·t<1 E±0XL
t < 1. We use the following
result (see, for example, Dharmadhikari et al. (1968)).
Lemma 2.1 If Z1;:::;Zn are i.i.d. variables with mean zero and a ¯nite k-th moment,
k ¸ 2, then we have the bound
E
¯
¯ ¯
¯
¯
n X
s=1
Zs
¯
¯ ¯
¯
¯
k
· Cknk=2EjZ1jk;
where the constant Ck > 0 only depends on k (and not on the distribution of Z1).
Using that the Steutel-van Harn operator ´ ± Z, conditional on Z, follows a ¤Z
s=1F´ distri-
bution, Lemma 2.1 implies the following inequality for L ¸ 2,
Ej´ ± Z ¡ ´ZjL = EE
£
j´ ± Z ¡ ´ZjL j Z
¤
· CLEZL=2EjU(´) ¡ ´jL · CL;´kZk
L=2
L¡1: (10)
We have for t ¸ 0, using (9),
kXtkL · kXt ¡
X
i
®iXt¡i ¡
X
j
¯j"t¡j ¡
X
k¸`
°k;`Xt¡k"t¡`kL
+ k
X
i
®iXt¡i +
X
j
¯j"t¡j +
X
k¸`
°k;`Xt¡k"t¡`kL
·
X
i
k®i ± Xt¡i ¡ ®iXt¡ikL +
X
j
k¯j ± "t¡j ¡ ¯j"t¡jkL
+
X
k¸`
k°k;` ± (Xt¡k"t¡`) ¡ °k;`Xt¡k"t¡`kL + k"tkL
+
X
i
®ikXt¡ikL +
X
j
¯jk"t¡jkL +
X
k¸`
°k;`(k"0kLkXt¡kkL + k"2
0kL)
· M +
X
i
®ikXt¡ikL + k"0kL
X
k¸`
°k;`kXt¡kkL;6 Drost, van den Akker, and Werker
where the appropriately chosen constant M < 1, not depending on t, can be obtained
from the assumed moment conditions and the induction hypothesis applied to (10) and (9).
Condition (7) completes the induction argument just as for the case L = 2,
kXtkL · M
1 X
s=0
0
@
X
i
®i + k"0kL
X
k¸`
°k;`
1
A
s
:
This completes the proof. 2
Remark For the INBL(1,0,1,1) process Doukhan et al. (2006) showed that the condition
kU(®)kK + k"0kKkU(°)kK < 1 su±ces to obtain the existence of the K-th moment of Xt
under Pº¤. Condition (7) is weaker since, for K ¸ 1, ´ = kU(´)k1 · kU(´)kK.
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A simple integer-valued bilinear time series model.

An integer-valued pth-order autoregressive structure (INAR(p)) process.

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Existence and stochastic structure of a non-negative integer-valued autoregressive process.

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Note on Integer-Valued Bilinear Time Series Models

Note on Integer-Valued Bilinear Time Series Models

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