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GENERALIZED INTEGER-VALUED AUTOREGRESSIONKurt Brännäs a & Jörgen Hellström aa Department of Economics, Umeå University, Umeå, SE-90187, SwedenVersion of record first published: 06 Feb 2007.

To cite this article: Kurt Brännäs & Jörgen Hellström (2001): GENERALIZED INTEGER-VALUED AUTOREGRESSION,Econometric Reviews, 20:4, 425-443

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GENERALIZED INTEGER-VALUEDAUTOREGRESSION

Kurt Brannas* and Jorgen Hellstrom{

Department of Economics, Umea University, SE-90187 Umea, Sweden

ABSTRACT

The integer-valued AR(1) model is generalized to encompass some of the

more likely features of economic time series of count data. The generalizations

come at the price of loosing exact distributional properties. For most

specifications the first and second order both conditional and unconditional

moments can be obtained. Hence estimation, testing and forecasting are

feasible and can be based on least squares or GMM techniques. An illustration

based on the number of plants within an industrial sector is considered.

Key Words: Characterization; Dependence; Time series model; Estimation;

Forecasting; Entry and exit

JEL Classification: C12, C13, C22, C25, C51.

1. INTRODUCTION

Applied micro-economic interest in count data models has been steadily

increasing in recent years, and introductory treatises can now be found in

econometric textbooks (Greene, 1997) or in specialized monographs (e.g.,

Cameron and Trivedi, 1998; Winkelmann, 1997). While many applied studies

are based on cross-sectional data some studies are based on panel data. Time series

characteristics are then introduced by correlated unobserved heterogeneity (the

Zeger (1988) approach) and more seldomly by an explicit lag structure in the

endogenous count variable. In this paper we focus on a model with an explicit

lag structure which should be of interest also for economic time series at

ECONOMETRIC REVIEWS, 20(4), 425–443 (2001)

425

Copyright # 2001 by Marcel Dekker, Inc. www.dekker.com

*E-mail: kurt.brannas@econ.umu.se{E-mail: jorgen.hellstrom@econ.umu.se

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semi-aggregate levels. The considered model class is the integer-valued auto-

regression (INAR).

The INAR model is one useful model for non-negative sequences of

dependent count variables. The first order INAR [INAR(1)] model is particularly

attractive partly thanks to its interpretational appeal. It explains the present number

of, say, individuals in some situation as the sum of those that remain (or survive)

from the previous period, and those that enter (or are born) in the intervening

period. The INAR(1) model was introduced by McKenzie (1985) and has been

elaborated on in subsequent papers by McKenzie (e.g., 1988), Al-Osh and Alzaid

(e.g., 1987) and others. Al-Osh and Alzaid (1987) considered Yule-Walker,

conditional least squares and maximum likelihood estimation in the Poisson

case. Brannas (1994, 1995) considered estimation by generalized method of

moments for Poisson and generalized Poisson models and the inclusion of

explanatory variables. Empirical economic applications of the model are still

few, though it has been used in studies of, e.g., the entry and exit of plants

(Berglund and Brannas, 1996). The related integer-valued moving average model

was recently studied and employed for a financial application by Brannas and Hall

(1998).

In this paper we relax some of the independence assumptions underlying the

basic INAR(1) model to make the model more readily available for economic

applications. In the treatments of the original model distributional properties have

been stressed, while we look for more flexibility by relaxing assumptions and by

only considering the first and second order moments of the model. The main focus

in the paper is on the INAR(1) model, but extensions to general INAR( p) as well

as multivariate INAR(1) are also briefly considered. Beyond model properties we

also focus on aspects of estimation, testing and forecasting.

In Section 2 the basic model is introduced. The implications of relaxing

some of the basic assumptions are presented in Section 3. To give some intuition to

the modelling the discussion is in terms of the entry and exit decisions of firms.

Section 4 covers estimation and testing, while forecasting is dealt with in Section 5.

For two of the extended model specifications we provide Monte Carlo evidence of

estimator and test statistic performance for time series of finite length in Section 6.

An illustration based on the number of Swedish mechanical paper and pulp mills is

presented in Section 7. Some concluding remarks close the paper.

2. BASIC MODEL

The paper is concerned with extensions to the integer-valued autoregressive

model of order one [INAR(1)] given by

yt ¼ a � ytÿ1 þ et; ð1Þ

where yt is a non-negative integer-valued random variable and t is the time index.

The scalar multiplication of the Gaussian AR model is in the integer-valued case

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replaced by the binomial thinning operator, defined as a � y ¼Py

i¼1 ui; where fuig

is a sequence of independent and identically distributed 0–1 random variables

(Steutel and van Harn, 1979). The fuig sequence is independent of ytÿ1 and et; and

Prðui ¼ 1Þ ¼ 1ÿ Prðui ¼ 0Þ ¼ a; a 2 ½0; 1�: Further, ytÿ1 is assumed independent

of et: The fetg sequence of non-negative, integer-valued random variables has mean

l, finite variance d, and Covðet; esÞ ¼ 0; for all t 6¼ s: This et is usually assumed to

possess some specified distribution. To be able to generalize the basic model we,

however, abstain from making a full distributional assumption. Instead, we only

offer results for the first and second order moments, since useful results in terms,

e.g., of the probability generating function are difficult to obtain for the general-

izations that we consider.

Under the assumptions of the basic model the thinning operator has the

properties Eða� yjyÞ ¼ ay; Eða� yÞ ¼ aEðyÞ; V ða� yjyÞ ¼ að1ÿaÞy; and

V ða� yÞ ¼ a2V ðyÞþað1ÿaÞEðyÞ:The first and second order conditional and unconditional moments for the

base case INAR(1) model are then

Eð ytjytÿ1Þ ¼ aytÿ1 þ l

Eð ytÞ ¼ l=ð1ÿ aÞ

V ð ytjytÿ1Þ ¼ að1ÿ aÞytÿ1 þ d

V ð ytÞ ¼ ½að1ÿ aÞEð ytÿ1Þ þ d�=ð1ÿ a2Þ:

We note that the model embodies a conditional heteroskedasticity effect (cf. Engle,

1982). Since this does not exactly match a conventional ARCH model effect we

use the label INARCH for the property. The INARCH effect is larger the larger is

ytÿ1: The autocovariance function at lag k, gk ¼ akV ð ytÿkÞ; and the autocorrelation

function is rk ¼ ak : Obviously, both functions are positive.

As an example of an integer-valued process, the number of firms in a region

at a certain time ( yt) is the number of firms surviving from the previous period

ða � ytÿ1Þ plus the entering new firms ðetÞ: Since one would expect the survival of

an individual firm to depend on the survival of other firms we incorporate this

feature into the model. There is also reason to believe that survival may depend on

the number of existing firms, and that the entry process may be correlated with the

survival mechanism. These and other model extensions are considered in the next

section.

3. GENERALIZED MODELS

The generalization of the model proceeds by considering the first and second

order moments (including the autocorrelation function) for different and weaker

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assumptions about the basic model. We retain the original basic model structure

in (1) but relax assumptions according to EðuiujÞ 6¼ EðuiÞEðujÞ; for i 6¼ j;EðuietÞ 6¼ EðuiÞEðetÞ; and Eð ytÿ1uiÞ 6¼ Eð ytÿ1ÞEðuiÞ: Further, dependence

within the et process, INAR( p), Threshold INAR(1), time dependent entry and

exit as well as multivariate models are also considered. We prefer in most cases to

consider one extension at a time so that effects are more transparent. The number

of firms in a specific region serves as a working example throughout the paper. As

we wish to focus on model properties we abstain from attempts to discuss in detail

the role of economic determinants to various bits of the models. Economic

variables could be included by letting parameters be functions of economic

variables. Other illustrative discussions based on any ‘birth-death’ type of

phenomenon could equally well be used.

3.1. Dependence Between Exit Decisions

It is reasonable to question the assumption that the individual firms survive or exit

independently. They all operate in the same macroeconomic milieu and would, it

appears, be affected in much the same way. To account for this, we modify the

model by letting EðuiujÞ ¼ ys 6¼ EðuiÞEðujÞ ¼ a2; for i 6¼ j: At this stage we

maintain that fuig is independent of the past stock ytÿ1 and of the number of

entrants et:The correlation between survival indicators ui and uj is

ks ¼ Corrðui; ujÞ ¼ ðys ÿ a2Þ=að1ÿ aÞ; i 6¼ j:

For the basic model of the previous section ys ¼ a2 so that ks ¼ 0: When ys < a2

there is a negative correlation between exit decisions and when ys > a2 there is

positive correlation.

It is straightforward to show that both the conditional and unconditional first

order moments remain unchanged from the basic model. Note that this holds under

even less restrictive dependence specifications. The dependence has an effect only

on the second or higher order moments. We obtain the conditional and uncondi-

tional variances as

V ð ytjytÿ1Þ ¼ ðaÿ ysÞytÿ1 þ ðys ÿ a2Þy2tÿ1 þ d

V ð ytÞ ¼ ½ðaÿ ysÞEð ytÿ1Þ þ ðys ÿ a2ÞEð y2tÿ1Þ þ d�=ð1ÿ a2Þ:

Note that the INARCH effect is in this case larger than in the basic model when

ys > a2 (positively correlated ui and uj) for ytÿ1 > 1; while for ytÿ1 ¼ 0 or

ytÿ1 ¼ 1 there is no INARCH effect and the conditional variance is then constant

as it is in the base case model. The lag one autocorrelation coefficient can be

shown to remain unchanged from the basic model, i.e. r1 ¼ a:

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3.2. Dependence Among Entrants

To allow for dependence between entry decisions, we may write the model

on the form yt ¼ a � ytÿ1 þ b � zt; where b is the probability of entry among zt

potential entrants. Let b � zt ¼Pzt

i¼1 vi; where vi represents the independent 0–1

decision to start a firm ðvi ¼ 1Þ or not ðvi ¼ 0Þ: The zt is assumed to be

independent of the fvig sequence, and zt and fvig are assumed to be independent

of a � ytÿ1: Let EðvivjÞ ¼ yp; for i 6¼ j; then it follows that the first order moments

are unchanged, but with l now corresponding to bEðztÞ: The correlation between

entry decisions is kp ¼ ðyp ÿ b2Þ=ðbð1ÿ bÞÞ:

For the second order moments we get

V ð ytjytÿ1Þ ¼ að1ÿ aÞytÿ1 þ ðbÿ ypÞEðztÞ þ ypEðz2t Þ ÿ b2

E2ðztÞ

V ð ytÞ ¼ ½að1ÿ aÞEð ytÿ1Þ þ ðbÿ ypÞEðztÞ þ ypEðz2t Þ ÿ b2

E2ðztÞ�=ð1ÿ a2Þ:

The lag one autocorrelation is r1 ¼ a:

3.3. Dependence Between Entry and Exit Mechanisms

It may be reasonable, e.g., for management and technology reasons, to expect

some dependence between the entry and exit mechanisms. We consider the case

where exit and entry are correlated in the following sense, EðuietÞ ¼ ye 6¼

EðuiÞEðetÞ; for any i.1 We obtain

ke ¼ Corrðui; etÞ ¼ ðye ÿ alÞ=½ðaÿ a2Þd�1=2:

Since the model in (1) is additive we immediately see that the first order

moments remain unchanged. The situation is more involved when it comes to

the second order moments, since we now have a covariance term in the variance

expression:

V ð ytÞ ¼ V ða � ytÿ1Þ þ V ðetÞ þ 2Covða � ytÿ1; etÞ:

We have that Covða � ytÿ1; etÞ ¼ ðye ÿ alÞEð ytÿ1Þ: Note that under independence

ye ¼ al; so that the covariance term is equal to zero. The conditional and

unconditional variances are given by

V ð ytjytÿ1Þ ¼ ½að1ÿ aÞ þ 2ðye ÿ alÞ� ytÿ1 þ d

V ð ytÞ ¼ f½að1ÿ aÞ þ 2ðye ÿ alÞ�Eð ytÿ1Þ þ dg=ð1ÿ a2Þ

1We could also consider a more basic dependence with the number of entrants. Let, as in Section 3.2,

et ¼Pzt

i¼1 vi; where zt may represent the number of potential entrepreneurs and vi represents the

independent 0–1 decision to start a firm ðvi ¼ 1Þ or not ðvi ¼ 0Þ: Let EðviÞ ¼ b and EðuiviÞ ¼ c:From this follows that EðuietÞ ¼ cEðztÞ ¼ ye:

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and the lag one autocorrelation coefficient is again r1 ¼ a: The variance expres-

sions exceed those of the basic model only when ye > al:

3.4. Stock Dependent Survival Mechanism

In a market one may expect dependence between the exit mechanism and the

number of operating firms. This would imply that Eðuijytÿ1Þ ¼Prðui ¼ 1jytÿ1Þ ¼ ay;t

is a function of ytÿ1 and hence varying over time. This dependence makes it difficult

to obtain general and explicit expressions for the unconditional moments except for

under some very simplified and then probably artificial specifications. Since

Eðay;t � yÞ ¼ Ey½Py

i¼1 EðuijyÞ� ¼ Eyðay;tyÞ we get the moments

Eð ytjytÿ1Þ ¼ ay;tytÿ1 þ l

Eð ytÞ ¼ Eðay;tytÿ1Þ þ l

V ð ytjytÿ1Þ ¼ ay;tð1ÿ ay;tÞytÿ1 þ d

V ð ytÞ ¼ V ðay;tytÿ1Þ þ E½ay;tð1ÿ ay;tÞytÿ1� þ d:

Not surprisingly there is no explicit expression for r1. The ay;t probability may, for

instance, be modelled by a logistic distribution function.

In a related way we could also introduce stock dependent entry behavior, e.g.,

through the conditional representation Eð ytj ytÿ1Þ ¼ ay;tytÿ1 þ ly;t; where ly;t is a

function of ytÿ1 and then time-varying.

3.5. INAR( p)

While the INAR(1) models have features that makes for easy interpretations

higher order autoregressions are not equally easy to interpret. They may be viewed

as duals to INMA(q) models for which Al-Osh and Alzaid (1988) and Brannas

and Hall (1998) have offered some interpretations. Let the INAR( p) process be

defined as

yt ¼ a1 � ytÿ1 þ � � � þ ap � ytÿp þ et

with ai 2 ½0; 1�; i ¼ 1; . . . ; pÿ 1; and ap 2 ð0; 1�: To simplify we present results for

the p ¼ 2 case, but generalize previous model suggestions to account for dependence

between exit decisions and dependence between entry and exit decisions. We get

Eð ytjy1; . . . ; ytÿ1Þ ¼ a1ytÿ1 þ a2 ytÿ2 þ l

Eð ytÞ ¼ l=ð1ÿ a1 ÿ a2Þ

V ð ytjy1; . . . ; ytÿ1Þ ¼ ½a1 ÿ ys þ 2ðye1 ÿ a1lÞ� ytÿ1 þ ðys ÿ a21Þy

2tÿ1

þ ½a2 ÿ ys þ 2ðye2 ÿ a2lÞ� ytÿ2 þ ðys ÿ a22Þy

2tÿ2

þ 2ðy12s ÿ a1a2Þ ytÿ1ytÿ2 þ d:

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Here, ykls ¼ Eðui;tÿkuj;tÿlÞ is a measure of survival dependence at times t ÿ k and

t ÿ l; with ys for k ¼ l ¼ 0; fej ¼ EðetuijÞ is the dependence measure between the

entry mechanism at time t and the survival decision at time t ÿ j; j ¼ 1; 2: The

variance can be obtained from the given moments.

3.6. A Bivariate Model

We consider a bivariate process and note that results are easily generalized to

a general multivariate context. We write yit ¼ ai � yi;tÿ1 þ ei;t and allow for

a general dependence structure. Specifically, we let fkls ¼ EðuikujlÞ; i; j ¼ 1; 2;

reflect the dependence between survival=exit decisions in equations k and l,

fkle ¼ EðuikeltÞ; for k, l¼ 1, 2, reflects the dependence between entry and exit

mechanisms, and f ¼ Eðe1te2sÞ; for t ¼ s; and f ¼ 0 otherwise. Given this setup

we find no changes from previous results for the first order moments. For the

second order moments we get among other results that

V ð yitÞ ¼ f½ai ÿfiis þ 2ðfii

e ÿ liaiÞ�Eð yi;tÿ1Þ

þ ðfiis ÿ a2

i ÞEð y2i;tÿ1Þ þ dig=ð1ÿ a2

i Þ; i ¼ 1;2

Covð y1t; y2tÞ ¼ f12s Eð y1;tÿ1y2;tÿ1Þ ÿ a1a2Eð y1;tÿ1ÞEð y2;tÿ1Þ

þ ðf12e ÿ a1l2ÞEð y1;tÿ1Þ þ ðf

21e ÿ a2l1ÞEð y2;tÿ1Þ þ ðfÿ l1l2Þ:

Note that these expressions simplify under stronger assumptions. In parti-

cular, given the basic independence assumptions on single equations as well as

between equations the covariance between y1t and y2t reduces to zero. Other results

such as the full cross-covariance function can be obtained, but is not given.

Empirical applications of related multivariate models have been reported by

Blundell et al. (1999), Berglund and Brannas (1999) and Brannas and Brannas

(1998).

3.7. Time Dependent Entry and Exit

Following Brannas (1995), we may introduce explanatory variables through

the parameters of the model, keeping in mind that the restrictions at 2 ½0; 1� and

lt � 0 should be respected. Two convenient and in other situations widely adopted

specifications are the logistic distribution function, i.e. at ¼ 1=½1þ expðxt bÞ�, and

the exponential function, i.e. lt ¼ expðztpÞ: The explanatory variable vectors xt

and zt are treated as fixed and measured at the beginning of the period starting

at time t ÿ 1: The b and p are the corresponding vectors of parameters. Note the

relationship between this specification and the stock dependent model of

Section 3.4.

The full time varying model is written

yt ¼ at � ytÿ1 þ et:

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We have the following unconditional moment relations

EðytÞ ¼ atEðytÿ1Þ þ lt

V ðytÞ ¼ a2t V ðytÿ1Þ þ atð1ÿ atÞEðytÿ1Þ þ dt

gk;t ¼Ykÿ1

i¼0

atÿi

" #V ðytÿkÞ; k ¼ 1; 2; . . .

rk;t ¼ gk;t=½V ðytÞV ðytÿkÞ�1=2¼

Ykÿ1

i¼0

atÿi

" #V ðytÿkÞ

V ðytÞ

� �1=2

; k ¼ 1; 2; . . . ;

where gk;t and rk;t are the autocovariance and the autocorrelation functions at time

t and lag k. Hence, both functions are time dependent. With variances approxi-

mately equal we expect that rk;t > rkþ1;t; for any t and k ¼ 1; 2; . . . : The

conditional moments are of the form given for the basic model, albeit with time

dependent parameters.

3.8. Threshold INAR(1) Models

We consider two types of threshold models, in one the switching between

regimes is governed by a random and non-observable process, while in the other

switching occurs with respect to a threshold level and the past stock ytÿ1:For the first case, let the previous stock ytÿ1 be split randomly into two parts

wtÿ1 and ytÿ1 ÿ wtÿ1; with corresponding survival probabilities a1 and a2: We

assume that wtÿ1 cannot be observed and that wtÿ1 given ytÿ1 follows a binomial

distribution such that Eðwtÿ1jytÿ1Þ ¼ pytÿ1 and V ðwtÿ1jytÿ1Þ ¼ pð1 ÿ pÞytÿ1 ¼

p �ppytÿ1: With this we write the model as

yt ¼ ða1 � wtÿ1Þ þ ða2 � ðytÿ1 ÿ wtÿ1ÞÞ þ et:

Assume that the assumptions about the basic model are otherwise satisfied and that

wtÿ1 is independent of et: After some tedious but straightforward algebra we can

prove the following results:

Eðytjytÿ1Þ ¼ ½a1pþ a2 �pp�ytÿ1 þ l ¼ ~aaytÿ1 þ l

EðytÞ ¼ l=ð1ÿ ~aaÞ

V ðytjytÿ1Þ ¼ ½a1pð1ÿ a1pÞ þ a2 �ppð1ÿ a2 �ppÞ�ytÿ1 þ d

V ðytÞ ¼ ½1ÿ ða21p

2 þ a22 �pp2Þ�

ÿ1f½a1pð1ÿ a1pÞ þ a2 �ppð1ÿ a2 �ppÞ�Eðytÿ1Þ þ dg

r1 ¼ ~aa:

Bockenholt (1999) studies a mixture Poisson INAR(1) model in which mixing is

related to the et-part of the model.

For the second type of threshold INAR(1) model we assume that there are

two mean functions governed by the lagged level ytÿ1 of the process:

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yt ¼a1 � ytÿ1 þ e1t; ytÿ1 � y0

a2 � ytÿ1 þ e2t; ytÿ1 > y0

�:

This model reduces to an INAR(1) if a1 ¼ a2 and e1t ¼ e2t and can have the low

order moments of INAR(1) if a1 ¼ a2 and the low order moments of the eit are

equivalent. The conditional first order moment is

Eðytjytÿ1Þ ¼a1ytÿ1 þ l1; ytÿ1 � y0

a2ytÿ1 þ l2; ytÿ1 > y0

�:

In this case the unconditional and conditional variances are formidable and not

very illuminating.

3.9. Remarks

As could be anticipated, changes in the dependence structure of the basic

INAR model will generally not change the first order moments, but will change

higher order moments. There are a number of other generalizations that could have

been considered. For instance, we could let the thinning operations be dependent

over time in the INAR(1). There will be no effect of this on moments of the type

considered here, but there may be effects on other moments. Obviously, we could

also consider combinations of the studied extensions and let some of the

dependence parameters be functions of time varying economic variables.

We find that the INARCH effect as well as the variance properties vary

substantially with model type, and for this reason we argue that empirical

discrimination between the model types should be possible.

Note that as dependence is introduced obtaining distributional properties,

e.g., using probability generating functions, will become exceedingly complicated

for most specifications and hence too complicated for empirical use.

4. ESTIMATION

Maximum likelihood (ML) estimation of a and l in the basic INAR(1) model

is more complicated than ML estimation in the Gaussian AR(1). This is due to

more complicated distributional forms that complicate numerical calculations.

Estimation with conditional least squares (CLS), conditional ML as well as exact

ML and the Yule-Walker estimators were studied by Al-Osh and Alzaid (1987) in

the Poisson case. In a recent study Park and Oh (1997) show asymptotic normality

for the Yule-Walker type estimator for a slightly different parameterization and

also show that the Yule-Walker asymptotically is more efficient than the CLS

estimator. Generalized method of moments (GMM) estimation was considered by

Brannas (1994) for the Poisson and the generalized Poisson model. In this section

we will consider estimation of the generalized INAR(1) models of Section 3.

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Method of moments or Yule-Walker based on unconditional moments, CLS

(weighted and unweighted) and GMM are the considered estimators.

4.1. Yule-Walker Estimation

It is simple to get estimates of a, l and d in the basic INAR(1) model. The

method of moments is related to the Yule-Walker estimator and yields

aa ¼ r1; ll ¼ ð1ÿ r1Þ�yy and dd ¼ ð1ÿ r21Þs

2 ÿ r1ð1ÿ r1Þ�yy; where r1 is the sample

autocorrelation coefficient at lag one, �yy is the sample mean, and s2 is the sample

variance.

Obtaining estimates for the generalized models is not straightforward, since

we then have, at least, four unknown parameters, but only have ready access to the

mean, variance and autocorrelations. Since we have alternative estimators there is

no strong reason for pursuing a search for additional unconditional moments of

higher orders to make this approach feasible.

4.2. Conditional Least Squares Estimators

Weighted or unweighted conditional least squares (WCLS or CLS) estima-

tors are simple to use and have been found to perform well for univariate models

and short time series (Brannas, 1995). The conditional mean or the one-step-ahead

prediction error can be used to obtain the estimates. The conditional mean is for

most of the specifications considered in Section 3:

Eðytjytÿ1Þ ¼ aytÿ1 þ l;

where a and l are the unknown parameters to be estimated. The CLS estimators of

a and l minimize the criterion function

Q ¼XT

t¼2

½yt ÿ aytÿ1 ÿ l�2:

For both the basic and generalized models V ðytjytÿ1Þ vary with both a and las well as other parameters. Depending on which model type is considered we may

apply OLS to estimate any remaining parameters (y and d) from the empirical

conditional variance expression

ee2t ¼ gðy; d; aa; ll; ytÿ1Þ þ xt;

where eet is the residual from the CLS estimation phase and xt is a disturbance

term.

The WCLS estimator of a and l minimize a criterion function in which the

conditional variance is taken as given (i.e. evaluated at estimates)

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QW ¼XT

t¼2

ðyt ÿ aytÿ1 ÿ lÞ2

gðyy; dd; aa; ll; ytÿ1Þ:

In this case the estimators of a and l have the shape of the CLS estimator

expressions, but where each sum contains a term 1=gð:; ytÿ1Þ and T ÿ 1 is replaced

byPT

t¼2 1=gð:; ytÿ1Þ: For both the CLS and WCLS estimators, estimated covar-

iance matrices for parameters based on the Gauss-Newton algorithm was studied

by Brannas (1995). He also considered Eicker-White type covariance matrices for

the CLS estimator and found that the WCLS estimator has better bias and mean

square error (MSE) properties and that its associated test statistics had the best

power properties.

For the stock dependence specification as well as when explanatory variables

are present estimation by the Gauss-Newton algorithm with or without weighting

is straightforward (cf. Brannas, 1995).

4.3. Generalized Method of Moments

In this subsection we consider GMM (Hansen, 1982) estimation. Two

approaches to GMM estimation can be considered. One approach employs

unconditional moment restrictions and the other, considered here, is based

on conditional moment restrictions (e.g., Newey, 1985; Tauchen, 1986). Note

that estimation based on unconditional moment restrictions is related to the

Yule-Walker estimator.

The conditional GMM estimator can be seen as an extension of the CLS

estimator and minimizes the quadratic form

q ¼ mðcÞ0WWÿ1mðcÞ;

where mðcÞ is a vector of moment restrictions and c is the vector of unknown

parameters. The estimator is consistent and asymptotically normal subject to mild

regularity conditions (e.g., Davidson and MacKinnon, 1993, ch.17) for any

symmetric and positive definite weight matrix WW: The estimator is efficient

when WW is the asymptotic covariance matrix of mðcÞ: To obtain WW; q can in a

first stage be minimized using, for instance, the identity matrix I for WW: In a

second stage the consistent estimates cc from stage one can be used to form WW

based on the consistent Newey and West (1987) estimator.

To estimate a and l the empirical moment restrictions Tÿ1PT

t¼2 et ¼ 0;for l, and Tÿ1

PTt¼2 ytÿ1et ¼ 0; for a, with corresponding theoretical moments

EðetÞ ¼ 0 and Eðytÿ1etÞ ¼ 0 can be used. Here, et ¼ yt ÿ aytÿ1 ÿ l is the one-

step-ahead forecast error. The restrictions match the normal equations of the CLS

estimator. For d and y (depending on which extension of the base case is

considered) the GMM estimators may, e.g., be formed by letting the moment

restrictions be the difference between sample and theoretical moments

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ðT ÿ 1Þÿ1PTt¼2 e2

t ÿ V ðytjytÿ1Þ ¼ 0 and ðT ÿ 1Þÿ1PTt¼2 ½e

2t ÿ V ðytjytÿ1Þ� ytÿ1

¼ 0:When the numbers of unknown parameters and moment restrictions are equal

the estimated asymptotic covariance matrix of the GMM estimator is

CovðccÞ ¼ Tÿ1½GG0WWÿ1GG�ÿ1;

where the GG matrix with rows G j¼ @mj=@c

0 and WW are both evaluated at cc:

4.4. Remarks on Specification Testing

For some of the discussed INAR specifications the changes arise in the first

order moment, while for others tests need to focus on second order moments. The

former problem is of a more conventional type and standard procedures can be

expected to perform relatively well.

To test if we have dependence between the survival probability and the

existing stock as well as on a vector of explanatory variables we may specify

a functional form for ay,t, e.g., of the logistic distribution function type:

ay;t ¼ 1=ð1þ expðxtbþ yytÿ1ÞÞ:

A Wald test for y ¼ 0 is therefore an immediate test of stock dependence, while a

test for b ¼ 0 tests for the presence of explanatory variables. A joint test of y ¼ 0

and b ¼ 0 is a test for constant survival probability. The Wald tests can be based

on CLS, WCLS or GMM estimators.

For the other specifications, testing must be based on the second order

moments of the form V ðytjytÿ1Þ ¼ gðy; d; a; l; ytÿ1Þ: The GMM estimator

provides a unified framework for doing this and testing can be related to LR,

LM or Wald testing ideas. Testing the hypothesis of no dependence in the exit

mechanism corresponds to testing RðcÞ ¼ ys ÿ a2 ¼ 0: The simple Wald test in

the GMM framework, W ¼ RðccÞ0½hðccÞ0WWÿ1hðccÞ�ÿ1RðccÞwith hðccÞ ¼ @RðcÞ=@c0;is distributed w2ð1Þ:

Note that when CLS or WCLS techniques are used, it will not be possible to

account for the covariances between estimators of parameters contained in the first

and second order moments, so that the resulting test statistics are at most

approximative.

5. FORECASTING

We consider the forecasting of future values yTþh of the INAR(1) process

given past observations up through time T. Since the first order moments of the

basic and extended models in most cases are the same this leaves their forecasts

unaltered. By repeated substitution we can write the future values of the process as

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yTþh ¼ ah � yT þXh

i¼1

ahÿi � eTþi; h ¼ 1; 2; :::: :

Then the h-step-ahead forecast is obtained as

yyTþhjT ¼ EðyTþhjy1; . . . ; yT Þ ¼ ahyT þ lð1þ aþ . . .þ ahÿ1Þ

¼ ah yT ÿl

1ÿ a

� �þ

l1ÿ a

;

where the equality ð1þ aþ . . .þ ahÿ1Þ ¼ ð1ÿ ahÞ=ð1ÿ aÞ has been used. The

term in brackets measures the deviation of the process from the mean of the

process. As h goes to infinity and with a< 1, the first part of the expression goes to

zero and hence the forecast approaches the mean of the process. As a! 1 the

forecast approaches yT , which is to be expected on comparison with a random

walk model. Brannas (1995) gives corresponding results for the time-varying

parameter model.

The forecast error is eTþh ¼ yTþh ÿ yyTþhjT ; so that the forecast is unbiased.

The one-step-ahead forecast error variance is in the basic model

V ðeTþ1Þ ¼ að1ÿ aÞEðyTþ1Þ þ d ¼ alþ dð1ÿ a2ÞV ðyT Þ:

It can be shown that the forecast error variances are affected by the generalizations

of Section 3 only through changes in the variance term V(yT). The h-step-ahead

forecast error variance is in the basic model

V ðeTþhÞ ¼ að1ÿ aÞEðyT Þ þ a2ð1ÿ ahÞV ðyT Þ þ d ¼ ð1ÿ a2hÞV ðyT Þ;

where the error variance increases with the forecast horizon, h, for 0 < a < 1: For

the extended models we can show that the forecast error variance h steps ahead

will again change only due to the changes in the variance term V(yT).

6. FINITE SAMPLE PROPERTIES

To give an indication of the small sample performance of estimators and tests

we conduct two Monte Carlo experiments. One is for the case of correlated

survival=exit decisions (cf. Section 3.1) and the other for the stock dependent case

(cf. Sections 3.4 and 4.4).

The factors to be varied in the first experiment are a ¼ 0:5; 0:7; 0:9;l ¼ 5; 10; and to obtain a positive correlation ks, we set ys ¼

a2 þ ðiÿ 1Þ � 0:02; i ¼ 1; . . . ; 5: Positively correlated binary data are generated

by specializing a result of Lunn and Davies (1998).2 The time series length is set at

2To obtain a constant correlation between binary random variables we modify the algorithm of Lunn

and Davies (1998, Section 2.1), such that Ui ¼ ð1ÿWiÞVi þWiZ; i ¼ 1; . . . ; ytÿ1: In Ui all

variables are independently Bernoulli distributed with probabilities a for Vi and Z, and jk1=2s j for

Wi. This gives a constant correlation ks � 0 between any pair Ui, Uj.

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T ¼ 50; 100; 200; and the distribution for et is throughout Poisson. For the second

experiment we specify ay;t ¼ 1=½1þ expðÿ2:2þ yytÿ1Þ� with y ¼ 0ð0:02Þ0:14; so

that for y ¼ 0 ay;t ¼ 0:9 and smaller for larger y. The other parts of the model are

as in the basic model, i.e. with ys ¼ a2 so that ks ¼ 0: The other values are set as

in the first experiment. In each cell 1000 replications are generated, and to avoid

start-up transients a first set of 150 observations is dropped in each replication.

For the first experiment the CLS estimator is based on the explicit expres-

sions of Section 4.2 and a LS estimator is used for the conditional variance

expression with a and l set at CLS estimates. The obtained estimates are used to

initialize a GMM estimator with W¼ I. In the second experiment a Gauss-Newton

algorithm is used for the nonlinear CLS estimator.

6.1. Results

Starting with the dependence-between-exits case the bias results for the CLS

estimators of a and ys are illustrated in Figure 1. The bias is smaller the larger is T

and there is a weak tendency for increasing bias for the ys parameter with larger

correlation ks. The biases (and MSEs) are practically the same for the GMM

estimator based on the four moment restrictions mentioned in Section 4.3. This

similarity is anticipated in view of the Ahn and Schmidt (1995) asymptotic

argument. With moment conditions of the present sequential nature no efficiency

gain can be expected for the parameters contained in the conditional mean

function. For the other parameters, i.e. l and d we find biases that are quite

large for large a and ys for both estimators. One explanation is that these instances

correspond to much larger variances, V(yt), of the series. For the largest a and ys

the variance is 136.3, while for a ¼ 0:5 and ys ¼ a2 the variance of the series is

only 10.

Figure 1. Biases of CLS estimators of a (right) and ys (left) against ks for the dependence between

exits model with true a ¼ 0:9 and l ¼ 5: Solid line and white symbol ðT ¼ 50Þ; dot-dashed line and

grey symbol (T¼ 100) and dotted line and black symbol (T¼ 200).

438 BRANNAS AND HELLSTROM

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The power properties of the GMM based Wald test of H0 : ys ¼ a2; i.e. of

zero-correlation between exit decisions, are illustrated in Figure 2. The test is more

powerful for a ¼ 0:5 than for a ¼ 0:9 and moreover the size properties are better

in the former case. Again this is likely to be due to the very different variances in

the two cases.

For the second experiment with stock dependence we give the bias and MSE

properties for the CLS estimator of y in Figure 3. Both measures appear smaller

for larger T and the bias is smaller for larger y. A larger y corresponds to a smaller

ay,t (and smaller V(yt)), so that there appears to be a general bias improvement as agets smaller. In terms of the power of a Wald test of y ¼ 0, using a White-type of

covariance matrix estimator, we find sizes that increase with sample size and are

Figure 2. Power functions for Wald test statistic of the hypothesis ys¼ a2 plotted against ks for

T¼ 50, 100, 200 at true values l ¼ d ¼ 5; a ¼ 0:5 (left) and a ¼ 0:9 (right). Lines and symbols are

defined in Figure 1.

Figure 3. Biases (left) and MSEs (right) for the CLS estimator of y in the stock dependence case

with l ¼ 5. Lines and symbols are defined in Figure 1.

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significantly too large for T¼ 200. Numerically, the Gauss-Newton algorithm

diverged in a large number of replications for small y, i.e. when ay,t is large.

7. ILLUSTRATION

Consider as an illustration (cf. Brannas, 1995) the number of Swedish

mechanical paper and pulp mills 1921–1981, Figure 4. This industrial production

technology is obviously on its way out and new production capacity is created in

plants of a more recent technology and larger scale. From this follows that 1ÿ amay reflect exits that are entries in other production technologies. Table 1 gives

parameter estimates for a simple model with industrial gross profit margin and

GNP used as explanatory variables. The fit of the model is exhibited in Figure 4

and is quite good. Note that the fit for a model without explanatory variables is

almost equally good (R2¼ 0.95 instead of 0.96). Testing for stock dependence

Figure 4. The number of Swedish mechanical paper and pulp mills (solid line) and fitted values

(dashed line), and estimated conditional variances (right) for base case model (dotted line), models

for exit dependence (dash-dotted line) and dependence between entry and exit (solid line).

Table 1. Estimation Results (CLS With S.E. in Parentheses) and Variable Definitions

Variable Survival Probability Mean Entry

Gross Profit Margin ÿ0.055 ÿ0.038(1950–72¼ 100) (0.009) (0.006)GNP – ÿ0.001(1900¼ 100) (0.000)Constant 3.605 5.051

(0.673) (0.607)

440 BRANNAS AND HELLSTROM

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indicates a nonsignificant effect. For these two reasons we only present additional

results for models without explanatory variables.

We present CLS and GMM estimates for all parameters in the first and

second order moment specifications for dependent exits and dependent entry and

exit in Table 2. The former specification suggests that d is much larger than l, and

that ys > a2: It follows that there is a positive correlation of ks ¼ 0:09 between the

survival=exit decisions. For the specification with dependence between entry and

exit decision d is also larger than l. Also, since ye > al there is a positive

correlation of ke ¼ 0:54 between the entry and exit decisions. The hypothesis of

no correlation was tested using a Wald test based on GMM estimates. The

correlation between exits was not significant, while the entry=exit correlation

was significant at the 0.05 level. The parameters a and ys are throughout precisely

estimated, while l and in particular d are imprecisely estimated.

Figure 4 also reports graphs for conditional variances for the basic as well as

for the two models with dependence. The latter two have rather similar paths,

while the former is quite flat. The differences arise from the weights given to ytÿ1

in the conditional variances. Note also the closeness (except for the level) between

the right and left panels of Figure 4.

8. CONCLUSIONS

The paper has demonstrated that several empirically motivated extensions to

the basic integer-valued autoregressive model can be made while maintaining that

models be easy to interpret as well as be estimable. Full characterizations of the

models could generally be obtained in terms of the first and second order

conditional and unconditional moments.

Full distributional properties could not be obtained on explicit forms, so that

empirically maximum likelihood estimation is not feasible, and instead least

Table 2. Estimation Results by CLS and GMM for Different Dependence Structures (S.E. in

Parentheses)

Specification a l d ys ye ks ke

CLS-estimatesBasic model 0.958

(0.028)0.233

(0.698)6.668

(3.832)Dependent exits (Section 3.1) 0.958

(0.028)0.233

(0.698)4.640

(3.167)0.921

(0.003)0.09

Dependent entry=exit (Section 3.3) 0.958(0.028)

0.233(0.698)

1.340(2.823)

0.348(0.197)

0.54

GMM-estimatesDependent exits (Section 3.1) 0.958

(0.001)0.233

(0.212)3.740

(5.527)0.922

(0.002)0.12

Dependent entry=exit (Section 3.3) 0.958(0.001)

0.233(0.212)

1.340(7.551)

0.348(0.183)

0.54

GENERALIZED INTEGER-VALUED AUTOREGRESSION 441

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squares and generalized method of moments (GMM) estimators are employed.

These estimators and the corresponding tests performed reasonably well in a small

scale Monte Carlo experiment. No doubt there is room for further improvements,

at least, in terms of GMM estimation, since additional as well as alternative

moment restrictions may be preferable to the ones evaluated here.

An interesting aspect of the model class is its conditional heteroskedasticity

property, which we label INARCH. It is through the conditional variance that the

various model extensions come through most clearly. In the reported Monte Carlo

results as well as in the empirical illustration the dependence parameters were quite

precisely estimated and the corresponding Wald test statistic based on GMM had

reasonable properties.

ACKNOWLEDGMENTS

The financial support from The Swedish Research Council for the Huma-

nities and Social Sciences is acknowledged. Thomas Aronsson, Colin Cameron,

Xavier de Luna and two anonymous referees are thanked for their comments on

previous versions of the paper. A previous version of the paper has been presented

at the Umea and Uppsala universities and at the 1998 EC2 Conference.

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