THE NON-STATIONARY BIVARIATE INAR UNDER AN UNCONSTRAINED INTER CORRELATION STRUCTURE
YUVRAJSUNECHER?
Department of Accounting, Finance and Economics, University of Technology Mauritius, Mauritius Email: [email protected]
NAUSHADMAMODEKHAN
Department of Economics and Statistics, University of Mauritius, Mauritius Email: [email protected]
VANDNAJOWAHEER
Department of Mathematics, University of Mauritius, Mauritius Email: [email protected]
SUMMARY
It is commonly observed in medical and financial studies that large volume of time series of count data are collected for several variates. The modelling of such time series and the estimation of parameters under such processes are rather challenging since these high dimensional time series are influenced by time-varying covariates that eventually render the data non-stationary. This paper considers the modelling of a bivariate integer-valued autoregressive (BINAR(1)) process where the innovation terms are distributed under non- stationary Poisson moments. Since the full and conditional likelihood approaches are cum- bersome in this situation, a Generalized Quasi-likelihood (GQL) approach is proposed to estimate the regression effects while the serial and time-dependent cross correlation effects are handled by method of moments. This new technique is assessed over several simula- tion experiments and the results demonstrate that GQL yields consistent estimates and is computationally stable since few non-convergent simulations are reported.
Keywords and phrases:Bivariate, Moving Average, Poisson, GQL, Non-Stationarity AMS Classification:65C60, 62J12, 62H12, 62J20, 62J10
?Corresponding author
c Institute of Statistical Research and Training (ISRT), University of Dhaka, Dhaka 1000, Bangladesh.
1 Introduction
Most of the univariate time series models that have been developed so far are based on the observation- driven (OD) integer-valued autoregressive (INAR(1)) process (See McKenzie, 1985; Al Osh and Alzaid, 1987; Kim and Park, 2008). In these models, it is relatively parsimonious to formulate the conditional likelihood estimating equations and used the iterative Newton-Raphson or Fisher-scoring algorithm to obtain the parameter estimates of the regression and serial correlation parameters. How- ever, in some important practical settings, the time series of counts are collected for two variables where apart from the serial correlation within each series, there exists as well a cross-correlation be- tween the two series and this lead to a bivariate INAR(1) (BINAR(1)) time series set-up. In addition, these time series of counts are subject to some time-independent and time-varying covariates that induce non-stationary correlations. Thus, the modelling of the BINAR(1) time series is challenging.
In literature, Pedeli and Karlis (2009, 2011) developed the first BINAR(1) time series of counts (CBINAR1M1) which was a simple extension of two INAR(1) processes where the cross correla- tion between the two series was induced by the innovation terms of the two series. The innovation series were assumed to follow the bivariate Poisson or some bivariate Generalized Poisson distri- butions. Later, Pedeli and Karlis (2013a) extended the CBINAR1M1 to a full BINAR(1) process (FBINAR1M1) by considering a more complex cross correlation structure where apart from the bivariately distributed innovation terms, the cross correlation was also induced by relating the obser- vation from the first variate with observation from the second variate via the binomial thinning oper- ator. However, both CBINAR1M1 and FBINAR1M1 were developed under stationary distributional or time-independent covariate assumptions. Recently, Mamode Khan et al. (2016b) introduced the first constrained BINAR(1) (CBINAR1M2) Poisson time series model under non-stationary set-up.
In their model, the mean and variance parameters of the bivariate innovation series were functions of time-dependent covariates, but so far, there is no full BINAR(1) process for non-stationary bivariate time series of counts such as the number of stock transactions of two companies operating in the same line of business, the number of daytime and nighttime accidents in an area, the number of day and night larceny acts and many others.
As for the inference procedures, Pedeli and Karlis (2009, 2011, 2013a) developed CMLE and method of moment approaches to estimate the mean and correlation parameters respectively in CBI- NAR1M1 and FBINAR1M1 models. Through simulation studies, CMLE was shown to provide esti- mates with lower bias than the moment based approach. Based on these results, Mamode Khan et al.
(2016b) developed the CMLE under time-dependent covariates for the non-stationary CBINAR1M2 and remarked that its log-information Hessian matrix was computationally intensive. As an alter- native, Mamode Khan et al. (2016b) proposed a generalized quasi-likelihood approach (GQL) that depends only on the notion of the marginal means, variances and joint covariances. This GQL equa- tion was then solved iteratively using Newton-Raphson to obtain estimates of the mean parameters while a robust method of moment was implemented to estimate the non-stationary serial and cross correlations based on the findings of Sutradhar and Das (1999) and Jowaheer and Sutradhar (2005).
Besides, through simulation studies, GQL yielded equally efficient estimates as CMLE but far more efficient estimates than GMM (Qu and Lindsay, 2003) and GEE (Liang and Zeger, 1986). Thus, in this paper, we propose to develop a GQL approach for the full non-stationary Poisson BINAR(1)
process (FBINAR1M2) as well as CMLE and GMM adaptive equations. The performance of these methodologies are tested through a simulation study.
The outline of the paper is as follows: In Section 2, we introduce the FBINAR1M2 Poisson time series model and the marginal moments and joint covariances are derived under non-stationary distributional assumptions. In Section 3, we develop CMLE, GMM adaptive and GQL estimating functions to estimate the regression, serial and cross correlation parameters under the non-stationary FBINAR1M2 model which have not yet been explored in time series literature so far. This paper also compares the performance of CMLE, GMM adaptive and GQL on their basis of their statistical and computational efficiencies through a simulation study. The paper is concluded in Section 5.
2 The Non-Stationary Poisson BINAR(1) Model
Following Mamode Khan et al. (2016b), we extend the CBINAR1M2 to account for the cross- correlation between the innovation series and the cross-correlation between the current observation at timetwith the other variate previous lagged observations such that the FBINAR1M2 model is written as:
Yt[1]=ρ1∗Yt−1[1] +α1,t−1∗Yt−1[2] +d[1]t (2.1) Yt[2]=ρ2∗Yt−1[2] +α2,t−1∗Yt−1[1] +d[2]t , (2.2) where0 < α1,t−1,α2,t−1, ρ1, ρ2 < 1 andρk represent the serial correlation andαk,t−1are the cross-correlation that needs to be estimated. Yt−1[k] andd[k]t are independent. As for the thinning operation (Steutel and Van Harn, 1979),θ∗Yt−1[k] =PY
[k]
t−1
j=1 bj(θ)andbj(θ)is a binary variable with prob[bj(θ) = 1] =θandprob[bj(θ) = 0] = 1−θsuch that
θ∗Yt−1[k]|Yt−1[k], θ∼Binomial(Yt−1[k], θ)
In the above model, theYt[k] are subject to ap×1vector of covariatesxt = [xt1, xt2, . . . , xtp] andµ[k]t = exp(xTtβ[k]),whereβ[k]is thep×1vector of regression coefficients for thekthseries.
From Equations (2.1) and (2.2),Yt−1[k] ∼ Poisson(µ[k]t−1). d[1]t andd[2]t follow the bivariate Poisson distribution (BivPoisson) (Kocherlakota and Kocherlakota, 2001) such that
P d[1]t =u, d[2]t =v
=e− λ[1]t +λ[2]t −φt
(λ[1]t −φt)u(λ[2]t −φt)v u!v!
×
s
X
i=0
u i
v i
φiti!h
λ[1]t −φt
λ[2]t −φt
i−i
(2.3)
and following equation (2.3),d[k]t ∼Poisson(λ[k]t ), whereλ[1]t = (µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1), λ[2]t = (µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1),s= min(u, v),λ[1]t , λ[2]t >0, andφt∈[0,min(λ[1]t , λ[2]t )].
φtdenotes the covariance between the two error terms, such that
Cov d[1]t , d[2]
t0
=
φt it=t0, 0 ift6=t0. and alternatively
φt=ρ12,t
q λ[1]t
q λ[2]t ,
whereρ12,t denotes the bivariate cross-correlation between the innovation seriesd[1]t andd[2]t and 0≤ρ12,t≤1. From the above assumption, it can be proved that
E(Yt[k]) =V ar(Yt[k]) =µ[k]t Similarly, the lag-hcorrelation for thekthseries is
Corr Yt[k], Yt+h[k]
=ρhk q
µ[k]t q
µ[k]t+h
(2.4)
As for the joint covarianceσt,12=Cov(Yt[1], Yt[2]), from Equations (2.1) and (2.2)
E(Yt[1]Yt[2]) =ρ1ρ2E[Yt−1[1]Yt−1[2]] +α1,t−1α2,t−1E[Yt−1[1]Yt−1[2]] +α2,t−1ρ1µ[1]t−1+α2,t−1ρ1(µ[1]t−1)2 +ρ1µ[1]t−1µ[2]t −ρ1ρ2µ[1]t−1µ[2]t−1−ρ1α2,t−1(µ[1]t−1)2+α1,t−1ρ2µ[2]t−1
+α1,t−1ρ2(µ[2]t−1)2+α1,t−1µ[2]t−1µ[2]t −α1,t−1ρ2(µ[2]t−1)2−α1,t−1α2,t−1µ[1]t−1µ[2]t−1 +ρ2µ[1]t µ[2]t−1−ρ2ρ1µ[1]t−1µ[2]t−1−α1,t−1ρ2(µ[2]t−1)2+α2,t−1µ[1]t−1µ[1]t
−ρ1α2,t−1(µ[1]t−1)2−α1,t−1α2,t−1µ[1]t−1µ[2]t−1+E(d[1]t d[2]t ) and by simplifyingσt,12, a first-order difference equation is obtained, where
σt,12= (ρ1ρ2+α1,t−1α2,t−1)Cov Yt−1[1], Yt−1[2]
+ρ12,t
q λ[1]t
q λ[2]t +ρ1α2,t−1µ[1]t−1+α1,t−1ρ2µ[2]t−1
For the other off-diagonal covariance,
E(Yt[1]Yt+1[2]) =ρ2Cov Yt[1], Yt[2]
+α2,tCov Yt[1], Yt[1]
+µ[1]t µ[2]t+1 (2.5) E(Yt[1]Yt+2[2]) =ρ2Cov Yt[1], Yt+1[2]
+α2,t+1Cov Yt[1], Yt+1[1]
+µ[1]t µ[2]t+2 (2.6) (See detailed derivation for Equations (2.5) and (2.6) in Appendix A).
E(Yt[1]Yt+h[2]) =ρ2Cov Yt[1], Yt+h−1[2]
+α2,t+h−1Cov Yt[1], Yt+h−1[1]
+µ[1]t µ[2]t+h (2.7)
Using Equations(2.5) – (2.7), we have Cov Yt[1], Yt+h[2]
=ρ2Cov Yt[1], Yt+h−1[2]
+α2,t+h−1Cov Yt[1], Yt+h−1[1]
=ρh2h
ρ1ρ2+α1,t−1α2,t−1
Cov Yt−1[1], Yt−1[2]
+ρ12,t
q λ[1]t
q λ[2]t
+ρ1α2,t−1µ[1]t−1+α1,t−1ρ2µ[2]t−1i +µ[1]t
h−1
X
j=0
ρj2ρh−j−11 α2,t+h−1−j
and similarly,
Cov Yt+h[1], Yt[2]
=ρh1h
ρ1ρ2+α1,t−1α2,t−1
Cov Yt−1[1], Yt−1[2]
+ρ12,t
q λ[1]t
q λ[2]t
+ρ1α2,t−1µ[1]t−1+α1,t−1ρ2µ[2]t−1i +µ[2]t
h−1
X
j=0
ρj1ρh−j−12 α1,t+h−1−j.
Under time-independent covariates assumption,σt,12is reduced to
σ12= ρ1α2,1µ[1]1 +α1,1ρ2µ[2]1 +ρ12
q λ[1]1
q λ[2]1 (1−ρ1ρ2−α1,1α2,1)
Cov Yt[1], Yt+h[2]
=ρ2Cov Yt[1], Yt+h−1[2]
+α2,1Cov Yt[1], Yt+h−1[1]
=ρh2σ12+α2,1µ[1]1
h−1
X
j=0
ρj2ρh−j−11
Cov Yt+h[1], Yt[2]
=ρ1Cov Yt+h−1[1] , Yt[2]
+α1,1Cov Yt+h−1[1] Yt[1]
=ρh1σ12+α1,1µ[2]1
h−1
X
j=0
ρj1ρh−j−12
whereα1,1andα2,1are the stationary cross-dependence terms in Equations (2.1) and (2.2). Note that these stationary cross-covariance expressions have the same formulation as Pedeli and Karlis (2013a).
3 Estimation of Parameters
This section reviews the estimation methodologies employed in estimating the regression, serial- and cross-correlation parameters. For the CBINAR1M1 and FBINAR1M1, Pedeli and Karlis (2009, 2011, 2013a) developed the CMLE approach and method of moments approach but as mentioned, these two models were designed under stationary assumptions and yet the CMLE was computa- tionally intensive. Mamode Khan et al. (2016b) recently developed the GQL estimation technique for the non-stationary CBINAR1M2 where through simulation studies, it was noted that CMLE and GQL yielded equally efficient estimates asymptotically while far more efficient estimates than
GMM. Besides, GQL is more parsimonious than CMLE since it depends only on the notion of the marginal and joint moments. However, CMLE, GMM and GQL have not yet been developed for the FBINAR1M2 model. In the subsections that follow, these estimation approaches are derived in details and their merits and drawbacks are also highlighted.
3.1 CMLE
The conditional density for the FBINAR1M2 model is expressed as (Pedeli and Karlis, 2013a):
f1(k) =
k
X
j1=0
yt−1[1]
j1
y[2]t−1 k−j1
ρj11(1−ρ1)y[1]t−1−j1αk−j1,t−11(1−α1,t−1)y[2]t−1−k+j1
f2(s) =
s
X
j2=0
yt−1[2]
j2
yt−1[1]
s−j2
ρj22(1−ρ2)y[2]t−1−j2αs−j2,t−12(1−α2,t−1)yt−1[1] −s+j2
and a bivariate distribution of the innovation terms of the formf3(u, v) =P d[1]t =u, d[2]t =v . Then, the conditional density is written as (Pedeli and Karlis, 2013a)
f y[1]t , y[2]t
y[1]t−1, yt−1[2] , θ
=
g1
X
k=0 g2
X
s=0
f1(k)f2(s)f3 yt[1]−k, y[2]t −s
=
g1
X
k=0 g2
X
s=0
hXk
j1=0
y[1]t−1 j1
y[2]t−1 k−j1
ρj11(1−ρ1)y[1]t−1−j1αk−j1,t−11(1−α1,t−1)y[2]t−1−k+j1i
× s
X
j2=0
yt−1[2]
j2
yt−1[1]
s−j2
ρj22(1−ρ2)y[2]t−1−j2α2,t−1s−j2 (1−α2,t−1)y[1]t−1−s+j2
×e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)
×
min(k,s)
X
m=0
h
µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt
yt[1]−k−m
× µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt
yt[2]−s−m
φmt i
/[(yt[1]−k−m)!(yt[2]−s−m)!m!
whereθ = ρk, αk,t−1, φt, β[k]0
is the vector of unknown parameters,g1 = min(yt[1], yt−1[1] )and g2= min(y[2]t , y[2]t−1).
The conditional likelihood function is then given by
L(θ|y) =
T
Y
t=1
f yt[k]|y[k]t−1, θ
for some initial values ofθand by maximizing the conditional likelihood function logL(θ|y) = log
T
X
t=1
f yt[k]|yt−1[k] , θ
the following expressions are obtained:
Differentiating with respect toρ1, we have
∂logL(θ|y)
∂ρ1
=
T
X
t=1
" g1
X
k=0 g2
X
s=0
hf2(s)f3 yt[1]−k, y[2]t −s∂f1(k)
∂ρ1
+f1(k)f2(s) ∂
∂ρ1f3(yt[1]−k, yt[2]−s)ihXg1
k=0 g2
X
s=0
f(yt[k]|yt−1[k] , θ)i−1# , (3.1) where
∂f1(k)
∂ρ1
=
k
X
j1=0
yt−1[1]
j1
y[2]t−1 k−j1
αk−j1,t−11(1−α1,t−1)y[2]t−1−k+j1
×h
j1ρj11−1(1−ρ1)yt−1[1] −j1−ρj11(yt−1[1] −j1)(1−ρ1)yt−1[1] −j1−1i and
∂
∂ρ1
f3(y[1]t −k, yt[2]−s)
=
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (y[1]t −k−m)!(y[2]t −s−m)!m!
×h
µ[1]t−1(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)yt[2]−s−mφmt
−µ[1]t−1(y[1]t −k−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m−1
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)yt[2]−s−mφmt i and the double derivative with respect toρ1is
T
X
t=1
∂2logL(θ|y)]
∂ρ21
=
T
X
t=1
" Pg1
k=0
Pg2
s=0f2f3∂2f1
∂ρ21 +f2∂f1
∂ρ1
∂f3
∂ρ1 +f1f2∂2f3
∂ρ21 +f2∂f1
∂ρ1
∂f3
∂ρ1
Pg1 k=0
Pg2
s=0f(yt[k]|y[k]t−1, θ)
− Pg1
k=0
Pg2
s=0 f2f3∂f1
∂ρ1 +f1f2∂f3
∂ρ1
2
Pg1 k=0
Pg2 s=0
f(y[k]t |yt−1[k] , θ) 2
#
, (3.2)
where
∂2f1(k)
∂ρ21 =
k
X
j1=0
y[1]t−1 j1
yt−1[2]
k−j1
αk−j1,t−11(1−α1,t−1)yt−1[2] −k+j1
×h
j1(j1−1)ρj11−2(1−ρ1)y[1]t−1−j1−j1ρj11−1(yt−1[1] −j1)(1−ρ1)y[1]t−1−j1−1
−j1ρj11−1(yt−1[1] −j1)(1−ρ1)y[1]t−1−j1−1
+ρj11(yt−1[1] −j1)(y[1]t−1−j1−1)(1−ρ1)y[1]t−1−j1−2i
and
∂2
∂ρ21f3(y[1]t −k, yt[2]−s) =
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (y[1]t −k−m)!(y[2]t −s−m)!m!
×[−µ2t−1[1](µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
+µ2t−1[1](y[1]t −k−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m−1
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
−µ2t−1[1](y[1]t −k−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m−1
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt +µ2t−1[1](y[1]t −k−m)(y[1]t −k−m−1)
×(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m−2
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt ].
The above derivation confirms that the double derivative for CMLE is computationally intensive under non-stationary assumptions. Differentiating with respect toρ2, we have
T
X
t=1
∂logL(θ|y)
∂ρ2
=
T
X
t=1
g1
X
k=0 g2
X
s=0
f yt[k]|yt−1[k] , θ −1 g1
X
k=0 g2
X
s=0
f1(k)f3(y[1]t −k, yt[2]−s)∂f2(s)
∂ρ2
+f1(k)f2(s) ∂
∂ρ2
f3(y[1]t −k, yt[2]−s)
,
where
∂f2(s)
∂ρ2
=
s
X
j2=0
yt−1[2]
j2
yt−1[1]
s−j2
αs−j2,t−12(1−α2,t−1)y[1]t−1−s+j2 h
j2ρj22−1(1−ρ2)y[2]t−1−j2−ρj22(y[2]t−1−j2)(1−ρ2)y[2]t−1−j2−1i
and
∂
∂ρ2f3(y[1]t −k, yt[2]−s) =
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (yt[1]−k−m)!(y[2]t −s−m)!m!
×[µ[2]t−1(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)yt[2]−s−mφmt
−µ[2]t−1(y[2]t −s−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)yt[1]−k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)yt[2]−s−m−1φmt ].
Differentiating with respect toα1,t−1, we have
T
X
t=1
∂logL(θ|y)
∂α1,t−1 =
T
X
t=1
g1 X
k=0 g2
X
s=0
f(y[k]t |y[k]t−1, θ) −1 g1
X
k=0 g2
X
s=0
f2(s)f3(y[1]t −k, yt[2]−s)∂f1(k)
∂α1,t−1
+f1(k)f2(s) ∂
∂α1,t−1f3(yt[1]−k, y[2]t −s)
, where
∂f1(k)
∂α1,t−1 =
k
X
j1=0
y[1]t−1 j1
yt−1[2]
k−j1
ρj11(1−ρ1)y[1]t−1−j1h
(k−j1)αk−j1,t−11−1(1−α1,t−1)yt−1[2] −k+j1
−αk−j1,t−11(y[2]t−1−k+j1)(1−α1,t−1)y[2]t−1−k+j1−1i and
∂
∂α1,t−1f3(y[1]t −k, yt[2]−s)
=
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (y[1]t −k−m)!(y[2]t −s−m)!m!
×
µ[2]t−1(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)yt[1]−k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
−µ[2]t−1(yt[1]−k−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)yt[1]−k−m−1
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
. Differentiating with respect toα2,t−1, we have
T
X
t=1
∂logL(θ|y)
∂α2,t−1 =
T
X
t=1
g1
X
k=0 g2
X
s=0
f(yt[k]|yt−1[k] , θ) −1 g1
X
k=0 g2
X
s=0
f1(k)f3(yt[1]−k, y[2]t −s)∂f2(s)
∂α2,t−1
+f1(k)f2(s) ∂
∂α2,t−1f3(yt[1]−k, y[2]t −s)
,
where
∂f2(s)
∂α2,t−1
=
k
X
j2=0
y[2]t−1 j2
y[1]t−1 s−j2
ρj22(1−ρ2)y[2]t−1−j2h
(s−j2)αs−j2,t−12−1(1−α2,t−1)y[1]t−1−s+j2
−α2,t−1s−j2 (y[1]t−1−s+j2)(1−α2,t−1)y[1]t−1−s+j2−1i
and
∂
∂α2,t−1f3(yt[1]−k, y[2]t −s)
=
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (yt[1]−k−m)!(yt[2]−s−m)!m!
×h
µ[1]t−1(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)yt[1]−k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
−µ[1]t−1(yt[2]−s−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−m−1φmt i .
Differentiating with respect toφt, we have
T
X
t=1
∂logL(θ|y)
∂φt =
T
X
t=1
Pg1
k=o
Pg2
s=0f1(k)f2(s)∂φ∂
tf3(y[1]t −k, yt[2]−s) Pg1
k=0
Pg2
s=0f(y[k]t |yt−1[k] , θ)
,
where
∂
∂φt
f3(y[1]t −k, yt[2]−s) =
min(k,s)
X
m=0
e−(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1+µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt) (y[1]t −k−m)!(y[2]t −s−m)!m!
×h
(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt +m(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφm−1t
−(y[1]t −k−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m−1
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−mφmt
−(y[2]t −s−m)(µ[1]t −ρ1µ[1]t−1−α1,t−1µ[2]t−1−φt)y[1]t −k−m
×(µ[2]t −ρ2µ[2]t−1−α2,t−1µ[1]t−1−φt)y[2]t −s−m−1φmt i .