A bivariate INAR(1) process with application

Transcription

1 A bivariate INAR1 process wit application Xanti Pedeli and Dimitris Karlis Department of Statistics Atens University of Economics and Business Abstract Te study of time series models for count data as become a topic of special interest during te last years. However, wile researc on univariate time series for counts now flouris, te literature on multivariate time series models for count data is notably more limited. In te present paper, a bivariate integer-valued autoregressive process of order 1 BINAR1 is introduced. Empasis is placed on model wit bivariate Poisson and bivariate negative binomial innovations. We discuss properties of te BINAR1 model and propose te metod of conditional maximum likeliood for te estimation of its unknown parameters. Issues of diagnostics and forecasting are considered and predictions are produced by means of te conditional forecast distribution. Estimation uncertainty is accommodated by taking advantage of te asymptotic normality of maximum likeliood estimators and constructing appropriate confidence intervals for te -step-aead conditional probability mass function. Te proposed model is applied to a bivariate data series concerning daytime and nigttime road accidents in te Neterlands. Keywords: BINAR; count data; Poisson; negative binomial; bivariate time series. 1 Introduction Multivariate count data occur in several different disciplines like epidemiology, marketing, criminology and engineering just to name a few. In many cases te data are observed across time leading to multivariate time series data as, for example, wen one studies te purcases of different products across time, or te occurrence of different diseases across time. Corresponding autor 1

2 In te literature tere are several models to fit univariate count time series models see Davis et al., A commonly used class of suc models consists of te so-called integer autoregressive time series models, introduced by McKenzie 1985 and Al-Os and Alzaid Te interested reader is referred to McKenzie 2003 and Jung and Tremayne 2006 for a brief but detailed review of suc models. Te literature on multivariate time series models for count data is less developed. Some interesting attempts ave been made during te last decade but most of tem do not arise in te context of INAR processes. Among te models tat ave been built in te aforementioned setting are tose of Latour 1997; Brännäs and Nordström 2000; Heinen and Rengifo 2007; Silva et al and Quoresi Te aim of tis paper is to introduce and examine in detail a bivariate integer-valued autoregressive model of order 1 BINAR1. To motivate te model consider te case of road traffic accidents. Accident analysis assumes tat even if te beavior of crases differs between day and nigt, bot types of accidents sare some common azards. Weater conditions, te road s quality and caracteristics and uman error, i.e. fallible perception, attention and/or memory, count between te factors tat introduce correlation. On te oter and, serial correlation between successive daily cras counts, i.e. autocorrelation, is reported as an important callenge for all accident models Brijs et al., Tus, appropriate time series models are needed to andle te presence of correlation bot between and witin te series of daytime and nigttime cras counts. Te remaining of te paper is structured as follows. A general specification of te BINAR1 process and alternative metods for te estimation of its unknown parameters are given in section 2. In sections 3 we concentrate on te special cases of bivariate Poisson and bivariate negative binomial innovations respectively. In section 4 we give a specification of te model residuals as a diagnostic tool wile issues of forecasting are discussed in section 5. An application to real data concerning daytime and nigttime road accidents follows in section 6. Some concluding remarks are presented in section 7. 2 Te BINAR1 Process 2.1 Model Let X and R be non-negative integer-valued random 2-vectors. Let A be a 2 2 diagonal matrix wit independent elements {α jj } j1,2. Te bivariate 2

3 integer-valued autoregressive process of order 1 can be defined as [ ] [ ] [ ] α1 0 X1,t 1 R1t X t A X t 1 + R t +, t Z α 2 X 2,t 1 R 2t were is te binomial tinning operator defined as α X X i1 Y i Y,were{Y i } X i1 is a sequence of iid Bernoulli random variables suc tat P Y i 1α 1 P Y i 0andα [0, 1] Steutel and van Harn, In te bivariate case, te A operation is a matricial operation wic acts as te usual matrix multiplication keeping in te same time te properties of te binomial tinning operation. One can see tat wit te above definition te jt element, j 1, 2isgivenbyX jt α j X j,t 1 + R jt. Te elements R t wic entered te system in te interval t 1,t] are usually called as innovations. Assuming independence between and witin te tinning operations and {R jt } an iid sequence wit mean λ j and variance σ 2 j υ j λ j, υ j > 0,j 1, 2, te unconditional first and second order moments based on second order stationarity conditions are: EX jt µ Xj λ j 1 α j 2.2 VarX jt σ 2 X j α j + υ j λ j 1 α 2 j 2.3 Cov X jt,x j,t+ γ Xj α j σ 2 X j ; 1, 2, Corr X jt,x j,t+ ρ Xj α j ; 1, 2, Obviously, te mean, variance and autocovariance functions can take only positive values, since λ j,σj 2 and α j are all positive. Depending on weter υ j > 1, υ j 0, 1, or υ j 1, te variance may be larger tan te mean overdispersion, smaller tan te mean underdispersion, or equal to te mean equidispersion respectively. Dependence between te two series tat comprise te BINAR1 process is introduced by allowing for dependence between R 1t and R 2t wile retaining all te previous assumptions fixed. Watever te underlying joint distribution of {R 1t,R 2t } is, it can be sown tat te covariance between te innovations of te two series at time t, totally determines te covariance between te current value of te one process and te innovations of te oter process at te same point in time t and vice versa see Appendix: 3

4 CovX 1t,R 2t CovR 1t,R 2t 2.6 As expected, te covariance between te sequences {X 1t } and {X 2t } at time t is also affected by te corresponding survival parts of te two processes. More specifically it can be sown tat, CovX 1,t+,X 2t α 1 1 α 1 α 2 CovR 1t,R 2t ; 0, 1,... and 2.7 α CorrX 1,t+,X 2t 1 1 α α2 2 1 α 1 α 2 CovR 1t,R 2t ; 0, 1,... α 1 + υ 1 α 2 + υ 2 λ 1 λ Covariances and correlations between X 1t and X 2,t+, 0, 1,...,canbe defined analogously. Note tat 2.7 presumes tat {X t } is a strictly stationary process, i.e. X1t X1,t+ tat te joint distribution of istesameastatof, for X 2t X 2,t+ all. Using te analytical representations [ ] [ ] [ ] X1t α1 0 X1,t 1 R1t X 2t 0 α 2 X 2,t 1 R 2t and [ ] [ ] [ ] X1,t+ α1 0 X1,t+ 1 R1,t X 2,t+ 0 α 2 X 2,t+ 1 R 2,t+ it is easy to see tat strict stationarity does indeed old for {X t } since te variables involved in te rigt-and sides of 2.9 and 2.10 ave identical distributions see also Latour, Estimation As already noted, te structure of te BINAR1 model implies tat te two innovation series {R 1t,R 2t } follow jointly a bivariate distribution. Let G R1,R 2 s 1,s 2 be te joint probability generating function jpgf of{r 1t,R 2t }. Ten, te jpgf of X t {X 1t,X 2t } is given by G Xt s G X1t,X 2t s 1,s 2 G X1,0 1 α1 t + α1s t 1 G X2,0 1 α2 t + α2s t 2 t 1 G R1,R 2 1 α1 i + α1s i 1, 1 α2 i + α2s i

5 wic reduces to G Xt s G X1t,X 2t s 1,s 2 G R1,R 2 1 α1+α i 1s i 1, 1 α2+α i 2s i Te moment generating function M Xt s G Xt e s can ten be used to obtain appropriate sample moments for te estimation of te unknown model parameters. However, wen te definition of a full density function is feasible, maximum-likeliood ML estimation is generally preferable. In te remaining of te present section we describe a general setting for ML estimation of te unknown parameters involved in te conditional mean and covariance functions. Te conditional density for te BINAR1 model can be expressed as te convolution of two binomials, namely X1,t 1 f 1 x 1 α x α 1 X 1,t 1 x f 2 x 2 x 1 X2,t 1 x 2 α x α 2 X 2,t 1 x 2, 2.14 and a bivariate distribution of te form f 3 k, s P R 1t k, R 2t s. Tus te conditional density becomes fx t x t 1, θ f 1 x 1t kf 2 x 2t sf 3 k, s 2.15 k s were θ is te vector of unknown parameters. Te conditional likeliood function is ten given by Lθ x T fx t x t 1, θ 2.16 t1 for some initial value x 0 and ence maximization provides wit te ML estimates. Numerical maximization is straigtforward wit standard statistical packages. 3 Parametric Cases In tis section we discuss two specific BINAR1 models. Te first one comes from te assumption tat te innovations of te two series follow jointly a bivariate Poisson distribution. Te second model assumes a bivariate negative 5

6 binomial distribution for te two innovation processes. Te two representations can be viewed as appropriate tools for modeling equidispersed and overdispersed bivariate time series respectively. Some additional specifications for negative correlation time series data are also briefly considered. 3.1 Te Poisson BINAR1 Process Model Let assume tat te joint probability mass function jpmf of te two innovation processes {R 1t,R 2t } is a bivariate Poisson distribution given by P R 1t x, R 2t y e λ 1+λ 2 φ λ 1 φ x λ 2 φ y x! y! s x y i i i φ i! 3.1 λ 1 φλ 2 φ were s minx, y, λ 1,λ 2 > 0andφ [0, minλ 1,λ 2. We will denote tis distribution as BPλ 1,λ 2,φ. Te bivariate Poisson distribution defined in 3.1 allows for dependence between te two random variables. Marginally eac random variable follows a Poisson distribution wit parameters λ 1 and λ 2 respectively. Parameter φ is te covariance between te two random variables. If φ 0 ten te two variables are independent and te bivariate Poisson distribution reduces to te product of two independent Poisson distributions. For a compreensive treatment of te bivariate Poisson distribution and its multivariate extensions te reader can refer to te books of Kocerlakota and Kocerlakota 1992 and Jonson et al Te above assumption leads to te equidispersion case, i.e. υ j 1,or equivalently assume tat R jt are iid Poisson sequences wit σ 2 j λ j, j 1, 2. Obviously, in tis case te covariance function 2.7 remains unaffected wile te correlation function 2.8 is simplified due to te simplification of te variances of te two processes. Hence, te Poisson BINAR1 model is caracterized by te vector of expectations µ Xt EX t wit elements µ Xjt λ j ; j 1, α j te variance-covariance matrix γ Xt wit diagonal elements CovX j,t+,x jt α j λ j ; 1 α j j 1, 2, 0, 1, and off-diagonal elements 6

7 CovX j,t+,x it α j φ ; j i, 0, 1, α 1 α 2 and te correlation matrix ρ Xt wit diagonal and off-diagonal elements equal to and CorrX j,t+,x jt α j ; j 1, 2, 0, 1, CorrX j,t+,x it α j 1 α1 1 α 2 φ 1 α 1 α 2 ; j i, 0, 1, λ 1 λ 2 respectively. Note also tat conditionally on te previous observations X t 1 {X 1,t 1,X 2,t 1 }, te vector of conditional means µ Xt t 1 EX t t 1 as elements µ Xjt t 1 α j X j,t 1 + λ j, j 1, 2. For 0, te conditional variance-covariance matrix γ Xt t 1 as diagonal and off-diagonal elements equal to CovX j,t+,x jt X j,t 1 α j 1 α j X j,t 1 + λ j and CovX j,t+,x it X j,t 1,X i,t 1 φ respectively, wile oterwise it is te zero matrix Estimation Te conditional density for te Poisson BINAR1 model can be obtained by substituting f 3 k, s e λ 1+λ 2 φ mink,s m0 λ 1 φ k m λ 2 φ s m φ m k m!s m!m! 3.7 in Ten we get fx t x t 1,α 1,α 2,λ 1,λ 2,φe λ 1+λ 2 φ x1,t 1 α x 1t k 1 1 α x 1t k 1 x 1,t 1 x 1t +k g 1 g 2 k0 s0 x2,t 1 x 2t s were g 1 minx 1t,x 1,t 1 andg 2 minx 2t,x 2,t 1. 7 mink,s m0 λ 1 φ k m λ 2 φ s m φ m k m!s m!m! α x 2t s 2 1 α 2 x 2,t 1 x 2t +s 3.8

8 3.2 A BINAR1 Process wit BVNB Innovations Model Assume tat te jpmf of te innovations {R 1t,R 2t } is a bivariate negative binomial distribution of te following form Marsall and Olkin, 1990; Boucer et al., 2008; Ceon et al., 2009: Γβ 1 + x + y P R 1t x, R 2t y Γβ 1 Γx +1Γy +1 x y λ 1 λ 1 + λ 2 + β 1 λ 2 λ 1 + λ 2 + β 1 β 1 λ 1 + λ 2 + β 1 β were λ 1,λ 2,β > 0. We will denote tis distribution as BVNBλ 1,λ 2,β. Note tat te marginal distribution of R jt is univariate negative binomial wit parameters β 1 and p j β 1 /λ j + β 1, j 1, 2 and tat te correlation between te two count variables R 1t and R 2t λ 1 λ 2 β Corrx, y λ 1 β1 + λ 2 β 3.10 must be positive. Tis assumption allows for more flexibility tan te Poisson BINAR1 model does, due to te involvement of te overdispersion parameter β in te model s specification. Recall tat in section 2.1, {R jt } was generally defined as an iid sequence wit mean λ j and variance σ 2 j υ j λ j, υ j > 0, j 1, 2. For te BVNB model, σ 2 j λ j 1 + βλ j implying tat υ j 1+βλ j, λ j,β > 0. Consequently υ j > 1 wic indicates te overdispersion case. However, te resulting model is not a BINAR model wit negative binomial marginals but a model tat effectively accounts for overdispersion. In specific, te statistical properties of te BINAR1 model wit BVNB innovations are encompassed in te vector of expectations µ Xt EX t wit elements µ Xjt λ j ; j 1, α j te variance-covariance matrix γ Xt wit diagonal and off-diagonal elements equal to CovX j,t+,x jt α j λ j 1 + βλ j + α j 1 α 2 j ; j 1, 2, 0, 1,

9 and CovX j,t+,x it α j βλ 1 λ 2 1 α 1 α 2 ; j i, 0, 1, respectively, and te correlation matrix ρ Xt wit diagonal elements CorrX j,t+,x jt αj ; j 1, 2, 0, 1, and off-diagonal elements αj β 1 α CorrX j,t+,x it 11 2 α2λ 2 1 λ 2 ; j i, 0, 1,... 1 α 1 α βλ 1 + α βλ 2 + α Conditionally on te previous observations X t 1 {X 1,t 1,X 2,t 1 },te vector of conditional means µ Xt t 1 EX t t 1 as elements µ Xjt t 1 α j X j,t 1 + λ j, j 1, 2. For 0, te conditional variance-covariance matrix γ Xt t 1 as diagonal and off-diagonal elements equal to CovX j,t+,x jt X j,t 1 α j 1 α j X j,t 1 + λ j 1 + βλ j and CovX j,t+,x it X j,t 1,X i,t 1 βλ 1 λ 2 respectively, wile oterwise it is te zero matrix Estimation For te BINAR1 model wit BVNB innovations it olds tat k s f 3 k, s Γβ 1 + k + s λ 1 λ 2 β 1 Γβ 1 k!s! λ 1 + λ 2 + β 1 λ 1 + λ 2 + β 1 λ 1 + λ 2 + β 1 Tus te conditional density 2.15 becomes β fx t x t 1,α 1,α 2,λ 1,λ 2,β g 1 g 2 Γβ 1 k s + k + s λ 1 λ 2 Γβ 1 k!s! λ k0 s0 1 + λ 2 + β 1 λ 1 + λ 2 + β 1 x1,t 1 α x 1t k 1 1 α x 1t k 1 x 1,t 1 x 1t +k x2,t 1 x 2t s were g 1 minx 1t,x 1,t 1 andg 2 minx 2t,x 2,t 1. 9 β 1 λ 1 + λ 2 + β 1 α x 2t s 2 1 α 2 x 2,t 1 x 2t +s β

10 3.3 Oter distributional coices As mentioned before, te coice of te joint distribution for R 1t and R 2t determines te properties of te underlying process. Wile te bivariate negative binomial provides overdispersion, it is interesting to note tat a selection of a distribution wit negative correlation can also produce negative correlation between te two series see 2.7. Te literature on bivariate count distributions wit negative correlation is limited. One of te reasons is tat negative correlation in bivariate counts occurs rater infrequently. However tere are suc models in te literature, as for example te bivariate Poisson-lognormal model of Aitcinson and Ho 1989 see also Cib and Winkelmann, 2001, te finite mixture model developed in Karlis and Meligkotsidou 2007 and models based on copulas see, e.g. Nikoloulopoulos and Karlis 2009 and te references terein. Finally, noted tat wile we used a certain bivariate negative binomial distribution, tere are certain oter alternatives in te literature wic could ave been used. We ave selected tis one mainly because of its relative simplicity. 4 Diagnostics In tis section we describe diagnostics for assessing te goodness of fit. Usually, in model fitting, tis is accomplised by means of residual analysis. However, due to te structural distinctiveness of INAR-type models, te classical definition of residuals as differences between te observed and fitted values, may prove to be inadequate as a diagnostic tool. We follow Freeland and McCabe 2004a by introducing a definition for residuals for count data tat distinguises between a set of residuals for te continuation process r 1t α X t 1 αx t 1 and anoter for te arrival component r 2t R t λ. In tis section we attempt to extend te ideas of Freeland and McCabe 2004a to te BINAR1 model. For eac one of te two series {X 1t,X 2t }, we define two sets of residuals; one for eac random component. So, for te continuation components we let r j 1t α j X j,t 1 α j X j,t 1 and for te arrival components we let r j 2t R jt λ j, j 1, 2. In order to arrive at a sensible and practical form of te above definitions, te unobservable quantities α j X j,t 1 and R jt sould be replaced wit E t [α j X j,t 1 ]ande t [R jt ] respectively, i.e. wit teir conditional expectations given te observed values of X jt and X j,t 1. PROPOSITION 1. Let E t [ ] denote te conditional expectation to te sigma field, I t σx j0,x j1,..., X jt, j 1, 2. For te BINAR1 model 10

11 wit bivariately distributed innovations te following equalities old: E t [α 1 X 1,t 1 ] α 1X 1,t 1 P x 1t 1 X 1,t 1 1,X 2,t 1 P x 1t X 1,t 1,X 2,t E t [α 2 X 2,t 1 ] α 2X 2,t 1 P x 2t 1 X 1,t 1,X 2,t P x 2t X 1,t 1,X 2,t 1 g 1 g 2 kf 1 x 1t kf 2 x 2t sf 3 k, s x E t [R 1t ] 2t k0 s0 4.3 P x 1t X 1,t 1,X 2,t 1 g 1 g 2 sf 1 x 1t kf 2 x 2t sf 3 k, s x E t [R 2t ] 1t k0 s0 4.4 P x 2t X 1,t 1,X 2,t 1 were te densities f 1 and f 2 are given in 2.13 and 2.14, f 3 k, s P R 1t k, R 2t s, g 1 minx 1t,x 1,t 1 andg 2 minx 2t,x 2,t 1. Using Proposition 1 we can now define te residuals as r j 1t r j 2t E t [r j 1t ]E t [α j X j,t 1 ] α j X j,t 1, and 4.5 E t [r j 2t ]E t [R jt ] λ j, for j 1, Regarding separately eac one of te two series tat comprise te BI- NAR1 model, it is noted tat adding te components of te two new sets of residuals gives te usual definition of residuals, i.e. r j 1t + r j 2t E t [α j X j,t 1 ] α j X j,t 1 + E t [R jt ] λ j E t [α j X j,t 1 + R jt ] α j X j,t 1 λ j X jt α j X j,t 1 λ j r j t. 4.7 Tus, te adequacy of eac component of te model may by assessed by plotting te aformentioned sets of residuals. 5 Forecasting Te usual way to produce forecasts in time series models is via te conditional forecast distribution. Freeland and McCabe 2004b establised te -stepaead conditional distribution of te Poisson INAR1 model, based on te remark of Al-Os and Alzaid 1987 tat 1 X t,x t d α X t + α i R t i,x t

12 were R t is a sequence of uncorrelated non-negative integer-valued random variables wit finite mean and variance. Te above result olds also for te marginal distribution of eac one of te two series X 1t,X 2t tat consist a BINAR1 model. As in te univariate case, αj X j,t X j,t, j 1, 2, as a binomial distribution wit parameters αj,x j,t. Moreover, te joint and marginal distributions of 1 αi 1 R 1,t i and 1 αi 2 R 2,t i are determined by te joint and marginal distributions { of X 1t and X 2t. Tis relation can be described 1 in terms of te jpgf of α1 i R 1,t i, 1 α2 i R 2,t i }. Denote by S j te quantity 1 αj i R 1,t j, j 1, 2. Ten, 1 G S1,S 2 s 1,s 2 G R1,R 2 1 α1 i + α1s i 1, 1 α2 i + α2s i Hence, te joint distribution of {X 1t,X 2t } given {X 1,t,X 2,t } is a convolution of two binomial distributions wit parameters α 1,X 1,t and α 2,X 2,t respectively, and a bivariate distribution wit jpgf of te form 5.2. Obviously, if 5.2 as not a closed-form expression, ten neiter te -step-aead forecast distribution can be specified in closed-form. However, it is straigtforward to evaluate it numerically. For te Poisson BINAR1 model, it can be proved tat [ 1 α G S1,S 2 s 1,s 2 exp 1 1 α 1 1 α + 1 α2 1 α 1 α 2 λ 1 s φs 1 1s α 2 1 α 2 ] λ 2 s wile te corresponding jpgf for te BINAR1 model wit BVNB innovations is given by 1 [ G S1,S 2 s 1,s 2 1 βλ1 α1s i 1 1 βλ 2 α2s i 2 1 ] β wic is not of a convenient form. Note owever tat irrespective of te jpgf of { 1 α1 i R 1,t i, 1 α i 2 R 2,t i }, closed-form expressions are available for teir conditional expectations and 12

13 variances conditional on X 1t,X 2t. More specifically, it can be proved tat 1 1 α E αj i j R j,t i λ j α j and Var 1 αj i R j,t i 1 α 2 j 1 α j υ 1 αj 2 j λ j + 1 α2 j 1 α j 1 αj 2 λ j 5.6 THEOREM 1. Tejpmf of {X 1,T +,X 2,T + } given {x 1T,x 2T } is given by f wit means, P X 1,T + x 1,X 2,T + x 2 x 1T,x 2T minx 1,x 1T minx 2,x 2T x1t x 1 k k0 x2t x 2 s 1 s0 α 2 x 2 s 1 α 2 x 2T x 2 +s α 1 x 1 k 1 α 1 x 1T x 1 +k 1 α1 i R 1,T + i k, α2 i R 2,T + i s x 1T,x 2T 1 α Ex j,t + x 2T,x 2T αj j x jt + ER jt ; j 1, 2, 1, 2,... 1 α j 5.7 and variances, 1 α 2 Varx j,t + x 1T,x 2T αj 1 αj j x jt + VarR 1 αj 2 jt 1 α j + 1 α2 j ER 1 α j 1 αj 2 jt ; j 1, 2, 1, 2, Te corresponding jpgf of {X 1,T +,X 2,T + } given {x 1T,x 2T } is of te form 13

14 G X1,T +,X 2,T + s 1,s 2 x 1T,x 2T 1 α 1+α 1s 1 X 1T 1 α 2+α 2s 2 X 2T G S1,S 2 s 1,s were G S1,S 2 s 1,s 2 is given in 5.2. Corollary 1. For te Poisson BINAR1 model, te jpgf and jpmf of {X 1,T +,X 2,T + } given {x 1T,x 2T } are given by G X1,T +,X 2,T + s 1,s 2 x 1T,x 2T 1 α1 + α1s 1 X 1T 1 α2 + α2s 2 X 2T { } 1 α exp 1 1 α λ 1 s α λ 2 s α2 φs 1 1s α 1 1 α 2 1 α 1 α 2 and P X 1,T + x 1,X 2,T + x 2 x 1T,x 2T minx 1,x 1T minx 2,x 2T x1t x 1 k k0 x2t s0 x 2 s { [ 1 α exp 1 1 α 1 mink,s m0 α 2 x 2 s 1 α 2 x 2T x 2 +s [ 1 α 1 1 α 1 λ 1 respectively, wit means, λ 1 + α 1 x 1 k 1 α 1 x 1T x 1 +k 1 α 2 1 α λ 2 1 α2 1 α 2 1 α 1 α 2 ] k m [ 1 α 2 1 α 2 λ 2 1 α 1 α 2 1 α 1 α 2 φ k m!s m!m! ]} φ 1 α 1 α 2 1 α 1 α 2 φ ] s m [ 1 α 1 α 2 1 α 1 α 2 φ ] m Ex j,t + x 1T,x 2T α j x jt + 1 α j 1 α j λ j ; j 1, 2, 1, 2, variances, Varx j,t + x 1T,x 2T αj 1 αj x jt + 1 α j λ j ; 1 α j j 1, 2, 1, 2, and covariance, 14

15 1 α Covx 1,T +,x 2,T + x 1T,x 2T 1 α2 1 α 1 α 2 φ ; 1, 2, Corollary 2. For te BINAR1 model wit BVNB innovations, te jpgf and te jpmf of {X 1,T +,X 2,T + } given {x 1T,x 2T } are given by and G X1,T +,X 2,T + s 1,s 2 x 1T,x 2T 1 α1 + α1s 1 X 1T 1 α2 + α2s 2 X 2T 1 [ 1 βλ1 α1s i 1 1 βλ 2 α2s i 2 1 ] β 1 f P X 1,T + x 1,X 2,T + x 2 x 1T,x 2T minx 1,x 1T minx 2,x 2T x1t x 1 k k0 x2t x 2 s 1 s0 α 2 x 2 s 1 α 2 x 2T x 2 +s α 1 x 1 k 1 α 1 x 1T x 1 +k 1 α1 i R 1,T + i k, α2 i R 2,T + i s x 1T,x 2T respectively, were f 1 α1 i R 1,T + i k, 1 can be numerically calculated. Te means and variances of tis process are given by, α i 2 R 2,T + i s x 1T,x 2T Ex j,t + x 1T,x 2T αj x jt + 1 α j λ j ; 1 α j j 1, 2, 1, 2, and 1 α 2 Varx j,t + x 1T,x 2T αj 1 αj j x jt βλ 1 αj 2 j λ j 1 α j + 1 α2 j λ 1 α j 1 αj 2 j ; j 1, 2, 1, 2,

16 wereas te covariance function is not of a closed-form. Te marginal probabilities P x 1 x 1T,x 2T andp x 2 x 1T,x 2T canbecalculated directly as, P x 1 x 1T,x 2T x 2 P x 1,x 2 x 1T,x 2T and P x 2 x 1T,x 2T x 1 P x 1,x 2 x 1T,x 2T respectively. Given te fact tat te vector of parameters θ is unknown, in practice we are only able to compute P x 1,x 2 x 1T,x 2T ; ˆθ were ˆθ are typically te maximum likeliood estimators introduced in section 2.2. Lack of knowledge about te true values of te model parameters and te need to estimate tem introduce uncertainty in te estimation of te -step-aead jpmf s. Estimation uncertainty, i.e. te error made in estimating tese probabilities, can be assessed by taking advantage of te asymptotic normality of ML estimators. Under standard regularity conditions, te ML estimator θ, denoted by ˆθ, is asymptotically normally distributed around te true parameter value, i.e. T ˆθ θ0 a N0, i 1, were i 1 is te inverse of te Fiser information matrix Bu and McCabe, Te δ-metod can ten be used for finding te asymptotic distribution of a random variable g ˆθ. An application of te δ-metod to g ˆθ P x x T ; ˆθ provides us wit a confidence interval for te probability associated wit any fixed value of x x 1,x 2 in te forecast distribution. Obviously, tese intervals may be truncated outside [0,1] Freeland and McCabe, 2004b. THEOREM 2 Freeland and McCabe, 2004b: Te quantity P x x T ; ˆθ as an asymptotically normal distribution wit mean P x x T ; θ 0 andvariance { } σx; 2 θ 0 T 1 P ˆθ i 1 P ˆθ 5.15 θθ0 θθ0 It is apparent tat analytical expressions for 5.15 are only available in cases were P x x T as a closed-form expression as is te case for te Poisson BINAR1 model see Appendix. Closing tis section, it is wort noting tat summarizing te forecast distribution by means of conditional expectations, wile ensures a minimum mean square error, it as drawbacks wit respect to data coerency since te integer-valued property of te time series is not taken into account. Freeland and McCabe 2004b suggest instead te use of te median of tis distribution wic always lies in te support of te series and is terefore coerent. Also, Pavlopoulos and Karlis 2008 propose a parametric bootstrap approac wic guarantees bot integer-valued predictions and prediction 16

17 intervals wit integer-valued ends. In our case, since te distributions are discrete, it is relatively easy to find te median to use as prediction instead of te mean, as te median will satisfy te discrete nature of te data. 6 Application Te data used in tis application refer to te joint modelling of daytime and nigttime road accidents in Scipol area, in te Neterlands for te year As nigttime accidents we refer to accidents appened between 10.00am-06.00pm, wile te rest were considered as daytime accidents. In accident analysis tose types of accidents are considered to ave different beavior. During nigttimes te traffic is of different nature e.g. more people travel for entertainment. On te oter and since bot types sare te same environment, like weater conditions, caracteristics of te road, tey are correlated. Te data are daily observations. Data from successive days are typically correlated as tey refer to similar conditions. Ignoring tis time series nature of te data can lead to incorrect inference see Brijs et al., Hence joint modelling of te two time series can be very useful. For suc data typically te autocorrelations are relatively small and tus AR1 models are enoug to capture te time dependence. Te data can be seen in figure 1. Te daytime and nigttime accidents ave mean values variances equal to and respectively implying ovedispersion. Te autocorrelation functions for bot series present a rater exponential decay wit a few exceptions. Te first order autocorrelation coefficient is 0.12 for daytime accidents and 0.13 for nigttime accidents. Finally te correlation between te two time series is revealing a sort of correlation between te series. In order to model te data we considered bot te bivariate Poisson INAR1 model and te INAR1 model wit BVNB innovations. Te results can be seen in Table 1. Comparing te log-likeliood one can see tat bot te time series context and te correlation between te series are needed. Te negative binomial BINAR1 model can also model te overdispersion and tus it provides te better fit. We ave also fitted a model wit bivariate Poisson lognormal innovations wic aving two more parameters offered very small improvement being muc more computationally demanding results are not presented ere. It is clear tat te time series models are better tan te models tat neglect tis. In addition te BVNB INAR1 model is muc better as it captures te overdispersion in te dataset togeter wit te correlation between te two series but also te autocorrelation witin eac series. Te standard 17

18 daytime number of accidents number of accidents days nigttime days Figure 1: Time series plots and acf plots for te series of daytime and nigttime accidents. 18

19 Table 1: Maximum Likeliood Estimates from fitting alternatively a BI- NAR1, two independent INAR1 models and a simple bivariate Poisson model. BINAR1 Independent INAR1 Biv. Poisson Neg.Bin BINAR1 Estimate SE Estimate SE Estimate SE Estimate SE ˆα ˆα ˆλ ˆλ ˆφ ˆβ Log-Lik AIC daytime accidents nigttime accidents errors of te estimates obtained by te two approaces standard errors are derived numerically from te Hessian sow tat fitting a BINAR1 model to te data generally improves te precision of te produced estimates. On te oter and it is apparent tat ignoring any form of te correlation eiter witin or between or te overdispersion leads to incorrect standard errors and ence incorrect inferences. Figures 2-4 and te related inference concern results obtained from te BINAR1 model wit BVNB innovations. Figure 2 includes te plots of te residuals of te two series. Since tese residuals ave not been standardized, te survival and arrival residuals add up to te Pearson residuals. Moreover, a large Pearson residual is comprised by a large survival and arrival residual, wile a small Pearson residual consists of a small survival and arrival residual. Te signs of survival and arrival residuals may also differ in some cases. However, tey still keep teir similarity in pattern. Anoter interesting point is te reflection of te model structure in te correlation between different pairs of residuals. More specifically, te sample correlations between te survival and arrival residuals of eac series are very ig: 0.72 for te daytime series and 0.67 for te nigttime series. Te arrival residuals of te two series are also significantly correlated at 0.16 depicting te structural assumption underlying te BINAR1 model tat te correlation between te two series as been introduced by using correlated innovation terms. Figure 3 sows te one-step-aead marginal predictive distributions P x 1,n+1 x 1n,x 2n andp x 2,n+1 x 1n,x 2n werex 1 corresponds to te daytime series and X 2 corresponds to te nigttime series. Te last observation was equal to 4 for te former series and equal to 1 for te latter series. As 19

20 one can see in Figure 3, bot distributions are skewed to te rigt wic is in accordance wit te sape of te negative binomial distribution. Te most probable one-step-aead predictive value is equal to 7 for te daytime series and equal to 1 for te nigttime series. Te larger dispersion of te series of daytime accidents compared wit te nigttime accidents series is also reflected in te plot of its predictive distribution. Figure 4 sows te observed values of te series of daytime and nigttime accidents wit te corresponding one-step-aead predictions. Te divergence between real data and forecasts is also portrayed. Te orizontal lines correspond to te observed mean values of te two series. Obviously, divergence is larger for observations tat lie far away from te mean. Tis seems to be expected since te one-step-aead predictions ave te same mean but are less dispersed tan te original series. Note also tat te correlation coefficient of te two series of forecasts is equal to te correlation coefficient of te real data series. daytime accidents Arrival Survival Pearson Observation Number nigttime accidents Arrival Survival Pearson Observation Number Figure 2: Non-standardized residuals of te daytime and nigttime accidents series. 20

21 prob prob daytime accidents nigttime accidents Figure 3: Te one-step-aead predictive distributions P x 1,T +1 x 1T,x 2T P x 2,T +1 x 1T,x 2T of te series of daytime and nigttime accidents respectively. Te last observed values n 365 are equal to 4 and 1 accordingly. 21

22 daytime accidents Data Forecasts divergence Observation Number nigttime accidents Data Forecasts divergence Observation Number Figure 4: Observed values of te series of daytime and nigttime accidents and te corresponding one-step-aead predictions.te orizontal lines correspond to te observed mean values of te two series. 22

23 7 Concluding Remarks Te main focus in tis paper is on bivariate time series for count data. Generally, te desired BINAR1 model can be constructed in two different ways: Te first approac prespecifies te form of te marginal distributions and subsequently identifies te required form of te distribution of te innovations in order for stationarity to old. In te second approac it is te coice of te form of te innovations distribution tat leads to te specification of te underlying marginal distributions. Te models proposed in tis paper ave been built following te last approac. In particular, we considered two different BINAR1 models, one wit bivariate Poisson innovations and anoter one wit bivariate negative binomial innovations. Te former specification as te useful property tat te joint distribution of te two series under consideration is also bivariate Poisson. In te latter case, we don t end up wit a bivariate negative binomial INAR1 process but we obtain a BINAR1 model tat effectively accounts for overdispersion. Deviations from te equidispersion restriction could alternatively accounted for by assuming anoter distribution for te innovations, e.g. mixed Poisson, or by te inclusion of appropriate regressors. Results on suc extensions will be reported elsewere. It is of course self-evident tat te proposed model is not a panacea. For example, wen significant correlation between te series under consideration is present at lags iger tan 1, fitting a BINAR1 model proves to be rater inadequate. Tus, extensions of te present model to iger orders would be a useful contribution to te improvement of its flexibility. Moreover, te structure of real-life data frequently implies need for te inclusion of bot autoregressive and moving average components wen for example seasonal patterns are observed in time series of counts. So, extending te bivariate INAR model to a bivariate INARMA model seems to be anoter interesting callenge. Finally, generalization of te proposed process to te multivariate case would provide a great opportunity for modelling more tan two time series of correlated count data. In tis case, te definition of a multivariate discrete distribution for te innovation process is needed. Te existing models ave certain limitations and tey do not lead to models wit well specified marginals. Hence inference can be difficult wit standard metods like maximum likeliood and some alternatives, like composite likeliood, sould be considered. 23

24 References J. Aitcinson and C. Ho. Te multivariate Poisson-log normal distribution. Biometrika, 75: , M.A. Al-Os and A.A. Alzaid. First-Order Integer-Valued Autoregressive Process. Journal of Time Series Analysis, 83: , J.P. Boucer, D. Micel, and G. Montserrat. Models of insurance claim counts wit time dependence based on generalization of poisson and negative binomial distributions. Variance, 21: , K. Brännäs and J. Nordström. A Bivariate Integer Valued Allocation Model for Guest Nigts in Hotels and Cottages. Umeå Economic Studies, 547, T. Brijs, D. Karlis, and G. Weets. Studying te effect of weater conditions on daily cras counts using a discrete time-series model. Accident Analysis and Prevention, 40: , R. Bu and B. McCabe. Model selection, estimation and forecasting in INARp models: A likeliood-based Markov Cain approac. International Journal of Forecasting, 24: , S. Ceon, S.H. Song, and B.C. Jung. Tests for independence in a bivariate negative binomial model. Journal of te Korean Statistical Society, 38: , S. Cib and R. Winkelmann. Markov Cain Monte Carlo Analysis of Correlated Count Data. Journal of Business & Economic Statistics, 194: , R.A. Davis, W.T. Dunsmuir, and Y. Wang. Modelling time series of count data, pages Ed. S. Gos, Marcel Dekker, R.K. Freeland and B.P.M. McCabe. Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis, 255: , 2004a. R.K. Freeland and B.P.M. McCabe. Forecasting discrete valued low count time series. International Journal of Forecasting, 20: , 2004b. A. Heinen and E. Rengifo. Multivariate autoregressive modeling of time series count data using copulas. Journal of Empirical Finance, 14: ,

25 N. Jonson, S. Kotz, and N. Balakrisnan. Multivariate Discrete Distributions. Wiley, New York, R.C. Jung and A.R. Tremayne. Binomial tinning models for integer time series. Statistical Modelling, 6:21 96, D. Karlis and L. Meligkotsidou. Finite multivariate Poisson mixtures wit applications. Journal of Statistical Planning and Inference, 137: , S. Kocerlakota and K. Kocerlakota. Bivariate Discrete Distributions, Statistics: textbooks and monograps, volume 132. Markel Dekker, New York, A. Latour. Te Multivariate GINARp Process. Advances in Applied Probability, 291: , A.W. Marsall and I. Olkin. Multivarirate distribution generated from mixtures of convolution and product families, pages Topics in Statistical Dependence, Block,Sampson y Sanits Eds, Institute of Matematical Statistics, E. McKenzie. Some Simple Models for Discrete Variate Time Series. Water Resources Bulletin, 214: , E. McKenzie. Discrete Variate Time Series, volume 21. Sanbag, D. and Rao, C. eds., Elsevier Science, A.K. Nikoloulopoulos and D. Karlis. Finite normal mixture copulas for multivariate discrete data modeling. Journal of Statistical Planning and Inference In Press, H. Pavlopoulos and D. Karlis. INAR1 modeling of overdispersed count series wit an environmental application. Environmetrics, 194: , A.M.M.S. Quoresi. Bivariate Time Series Modeling of Financial Count Data. Communications in Statistics - Teory and Metods, 35: , N. Silva, I. Pereira, and E.S. Siva. Estimation and forecasting in SUINAR1. REVSTAT - Statistical Journal, 63: , F.W. Steutel and K. van Harn. Discrete Analogues of Self - Decomposability and Stability. Te Annals of Probability, 75: ,

26 Appendix I. PROOF OF EQUATION 2.6. CovX 1t,R 2t EX 1t R 2t EX 1t ER 2t EX 1t R 2t µ 1 λ [ 2 ] E α1 i R 1,t i R 2t µ 1 λ 2 α1 i {ER 1,t i R 2t } µ 1 λ 2 ER 1t R 2t + α1 i {ER 1,t 1 R 2t } µ 1 λ 2 i1 CovR 1t,R 2t + CovR 1t,R 2t α1 i {λ 1 λ 2 } µ 1 λ 2 II. ESTIMATION UNCERTAINTY OF THE POISSON BINAR1 MODEL. For te Poisson BINAR1 model, we let ˆθ T ˆα 1, ˆα 2, ˆλ 1, ˆλ 2, ˆφ bete ML estimators of θ α 1,α 2,λ 1,λ 2,φ based on a sample of size T. Ten, 5.15 can be written as σx; 2 θ 0 T 1 P α 1 P + λ 2 θθ0 P P + 2 α 1 λ 1 P P + 2 α 2 λ 1 P P + 2 λ 1 λ 2 θθ0 2 i 1 4,4 + θθ0 θθ0 θθ0 2 i 1 1,1 + P φ i 1 1,3 +2 i 1 2,3 +2 i 1 3,4 +2 P α 2 θθ0 P θθ0 2 i 1 5,5 +2 P α 1 λ 2 P α 2 λ 2 P λ 1 φ P P 2 i 1 2,2 + θθ0 θθ0 θθ0 2 P λ 1 i 1 3,3 θθ0 P P α 1 α 2 i 1 1,2 θθ0 i 1 P P 1,4 +2 α 1 φ i 1 P P 2,4 +2 α 2 φ i 1 P P 3,5 +2 λ 2 φ θθ0 θθ0 θθ0 i 1 1,5 i 1 2,5 i 1 4,5 } 26

27 were i 1 k,j is te k, j-element of te matrix i 1, k, j 1, 2,..., 5and P α 1 { x 1 α1 1T P x 1 1,x 2 x 1T 1,x 2T α1 1 P x 1,x 2 x 1T,x 2T } λ 11 α1 1 1 α1 {P 1 α1 2 x 1,x 2 x 1T,x 2T P x 1 1,x 2 x 1T,x 2T } + α 2φ1 α 1 1 α α 1α 2 1 α 1 α 2 2 {P x 1,x 2 x 1T,x 2T P x 1 1,x 2 x 1T,x 2T P x 1,x 2 1 x 1T,x 2T +P x 1 1,x 2 1 x 1T,x 2T } P α 2 P φ { x 1 α2 2T P x 1,x 2 1 x 1T,x 2T 1 α2 1 P x 1,x 2 x 1T,x 2T } λ 21 α2 1 1 α2 {P 1 α2 2 x 1,x 2 x 1T,x 2T P x 1,x 2 1 x 1T,x 2T } + α 1φ1 α 1 1 α α 1α 2 1 α 1 α 2 2 {P x 1,x 2 x 1T,x 2T P x 1 1,x 2 x 1T,x 2T P x 1,x 2 1 x 1T,x 2T +P x 1 1,x 2 1 x 1T,x 2T } P λ 1 P λ 2 1 α 1 1 α 1 1 α 2 1 α 2 {P x 1,x 2 x 1T,x 2T +P x 1 1,x 2 x 1T,x 2T } {P x 1,x 2 x 1T,x 2T +P x 1,x 2 1 x 1T,x 2T } 1 α 1 α2 {P x 1,x 2 x 1T,x 2T P x 1 1,x 2 x 1T,x 2T 1 α 1 α 2 P x 1,x 2 1 x 1T,x 2T +P x 1 1,x 2 1 x 1T,x 2T } In order to obtain analytical expressions for te elements tat comprise te Fiser information matrix i 1 we follow te notation of Freeland and McCabe 2004b and denote by l θ te second derivatives of te log-likeliood of te Poisson BINAR1 model wit respect to θ [α 1,α 2,λ 1,λ 2,φ] : l θ l α1 α 1 lα1 α 2 lα1 λ 1 lα1 λ 2 lα1 φ l α2 α 2 lα2 λ 1 lα2 λ 2 lα2 φ l λ1 λ 1 lλ1 λ 2 lλ1 φ l λ2 λ 2 lλ2 φ l φφ 27

28 Troug ordinary algebra it can be sown tat l α1 α α 1 2 T { 2x1,t 1 P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 x 1,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 t1 + x 1,t 1x 1,t 1 1P x 1t 2,x 2t x 1,t 1 2,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 } 2 x1,t 1 P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 l α2 α α 2 2 T { 2x2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1 x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 t1 + x 2,t 1x 2,t 1 1P x 1t,x 2t 2 x 1,t 1,x 2,t 1 2 P x 1t,x 2t x 1,t 1,x 2,t 1 } 2 x2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1 P x 1t,x 2t x 1,t 1,x 2,t 1 l λ1 λ 1 l λ2 λ 2 T t1 T t1 { P x 1t 2,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 { P x 1t,x 2t 2 x 1,t 1,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 } 2 P x1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 } 2 P x1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t,x 2t x 1,t 1,x 2,t 1 28

29 l φφ T { 1 P x t1 1t,x 2t x 1,t 1,x 2,t 1 {2P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 2P x 1t 2,x 2t 1 x 1,t 1,x 2,t 1 2P x 1t 1,x 2t 2 x 1,t 1,x 2,t 1 +Px 1t 2,x 2t 2 x 1,t 1,x 2,t 1 + P x 1t 2,x 2t x 1,t 1,x 2,t 1 +Px 1t,x 2t 2 x 1,t 1,x 2,t 1 } + 1 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {2P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 + 2P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 2P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P 2 x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 P 2 x 1t 1,x 2t x 1,t 1,x 2,t 1 P 2 x 1t,x 2t 1 x 1,t 1,x 2,t 1 } } l α1 α 2 T { x 1,t 1 x 2,t 1 1 α t1 1 1 α 2 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1 1,x 2,t 1 1 } P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1} l α1 λ 1 T { x 1,t 1 1 α t1 1 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t 2,x 2t x 1,t 1 1,x 2,t 1 } P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 } l α2 λ 2 T { x 2,t 1 1 α t1 2 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 2 x 1,t 1,x 2,t 1 1 } P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1} 29

30 l α1 λ 2 T { x 1,t 1 1 α t1 1 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1 1,x 2,t 1 } P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 } l α2 λ 1 T { x 2,t 1 1 α t1 2 P 2 x 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 1 } P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1} l α1 φ T { { x1,t α t1 1 P x 1t,x 2t x 1,t 1,x 2,t 1 [P x 1t 2,x 2t 1 x 1,t 1 1,x 2,t 1 P x 1t 2,x 2t x 1,t 1 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1 1,x 2,t 1 ] P x 1t 1,x 2t x 1,t 1 1,x 2,t 1 [P x P 2 1t 1,x 2t 1 x 1,t 1,x 2,t 1 x 1t,x 2t x 1,t 1,x 2,t 1 }} P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 ] l α2 φ T { { x2,t α t1 2 P x 1t,x 2t x 1,t 1,x 2,t 1 [P x 1t 1,x 2t 2 x 1,t 1,x 2,t 1 1 P x 1t,x 2t 2 x 1,t 1,x 2,t 1 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 1] P x 1t,x 2t 1 x 1,t 1,x 2,t 1 1 [P x P 2 1t 1,x 2t 1 x 1,t 1,x 2,t 1 x 1t,x 2t x 1,t 1,x 2,t 1 }} P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t 1,x 2t x 1,t 1,x 2,t 1 ] l λ1 λ 2 T { 1 P 2 x t1 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 } P x 1t,x 2t 1 x 1,t 1,x 2,t 1 P x 1t 1,x 2t x 1,t 1,x 2,t 1 } 30

31 l λ1 φ T { 1 P x t1 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t 2,x 2t 1 x 1,t 1,x 2,t 1 P x 1t 2,x 2t x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 } P x 1t 1,x 2t x 1,t 1,x 2,t 1 {P x P 2 1t 1,x 2t 1 x 1,t 1,x 2,t 1 x 1t,x 2t x 1,t 1,x 2,t 1 } P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 } l λ2 φ T { 1 P x t1 1t,x 2t x 1,t 1,x 2,t 1 {P x 1t 1,x 2t 2 x 1,t 1,x 2,t 1 P x 1t,x 2t 2 x 1,t 1,x 2,t 1 P x 1t 1,x 2t 1 x 1,t 1,x 2,t 1 } P x 1t,x 2t 1 x 1,t 1,x 2,t 1 {P x P 2 1t 1,x 2t 1 x 1,t 1,x 2,t 1 x 1t,x 2t x 1,t 1,x 2,t 1 } P x 1t 1,x 2t x 1,t 1,x 2,t 1 P x 1t,x 2t 1 x 1,t 1,x 2,t 1 } Note tat, in contrast to te univariate case, te information as well as te scores of te Poisson BINAR1 model cannot be decomposed into quantities associated wit eac component of te model seperately. Tis barrier is just due to te model s structure, i.e. to its bivariate nature. Te Fiser information matrix i can ten be calculated as usual: [ ] 2 lθ ] i E θ E [ lθ θ θ 2 were lθ is te log-likeliood of te Poisson BINAR1 model. 31