Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Estimating bivariate integer-valued moving average
models with the generalized method of moments
Isabel Silva1 and Cristina Torres2 and Maria Eduarda Silva3
1 Faculdade de Engenharia da Universidade do Porto2 ISCAP and Universidade do Porto
3 CIDMA and Faculdade de Economia da Universidade do Porto
JOCLAD 2014
JOCLAD 2014 1 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Outline
Motivation
Bivariate INteger-valued Moving Average, BINMA(q1,q2), models
Poisson BINMA(1, 1) Negative Binomial BINMA(1, 1)
Parameter Estimation
Method of Moments (MM) Generalized Method of Moments (GMM) MM and GMM for Poisson and Negative Binomial BINMA(1, 1) models Simulation Study
Ongoing and Future work
References
Outline JOCLAD 2014 2 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationCount time series
Discrete time non-negative integer-valued time series
Motivation JOCLAD 2014 3 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationCount time series
Discrete time non-negative integer-valued time series
Traditional representations of dependence are either impossible or impractical:
low counts, asymmetric distributions, excess zeros, overdispersion, . . .
Motivation JOCLAD 2014 3 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationCount time series
Discrete time non-negative integer-valued time series
Traditional representations of dependence are either impossible or impractical:
low counts, asymmetric distributions, excess zeros, overdispersion, . . .
INteger-valued AutoRegressive Moving Average (INARMA) Models
Multiplication in standard ARMA models for time series replaced by a suitable
operation defined for integer values
Motivation JOCLAD 2014 3 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationCount time series
Discrete time non-negative integer-valued time series
Traditional representations of dependence are either impossible or impractical:
low counts, asymmetric distributions, excess zeros, overdispersion, . . .
INteger-valued AutoRegressive Moving Average (INARMA) Models
Multiplication in standard ARMA models for time series replaced by a suitable
operation defined for integer values Thinning operation
Motivation JOCLAD 2014 3 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationCount time series
Discrete time non-negative integer-valued time series
Traditional representations of dependence are either impossible or impractical:
low counts, asymmetric distributions, excess zeros, overdispersion, . . .
INteger-valued AutoRegressive Moving Average (INARMA) Models
Multiplication in standard ARMA models for time series replaced by a suitable
operation defined for integer values Thinning operation
Binomial thinning operation [Steutel and Van Harn, 1979]
Y: non-negative integer-valued random variable (r.v.), [0,1]
Y = Yj=1 Bj
{Bj} N0 (counting series): sequence of independent and identically distributed(i.i.d.) r.v., independent of Y : Pr(Bj = 1) = 1Pr(Bj = 0) =
Motivation JOCLAD 2014 3 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationINARMA(p,q) processes
Xt =
AR part
1 Xt1 + +p Xtp+t +1 t t1 + +q t tq
MA part
, t Z
i,j 0, i = 1, . . . ,p1; j = 1, . . . ,q1 and p,q > 0, such that pi=1 i < 1{t} N0 : i.i.d. discrete r.v. (arrival or innovation process)
Motivation JOCLAD 2014 4 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
MotivationINARMA(p,q) processes
Xt =
AR part
1 Xt1 + +p Xtp+t +1 t t1 + +q t tq
MA part
, t Z
i,j 0, i = 1, . . . ,p1; j = 1, . . . ,q1 and p,q > 0, such that pi=1 i < 1{t} N0 : i.i.d. discrete r.v. (arrival or innovation process)
Multivariate time series of count
Counts of several events observed over time and the counts are correlated
Few models for multivariate count data:
Dynamic models for multivariate count data need to account both for serial and
cross-section correlation
The generalization of the discrete distribution to multivariate context is not
straightforwardMotivation JOCLAD 2014 4 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Bivariate Poisson Distribution [Kocherlakota and Kocherlakota, 1992; Johnson et al., 1997]
Xi Po(i), i = 0,1,2X = X1 +X0Y = X2 +X0
}
(X,Y) BPo(1,2,0)
Joint probability function:
Pr[X = x,Y = y] = e(1+2+0)min(x,y)
i=0
xi1 yi2
i0
(x i)!(y i)!i!
Motivation JOCLAD 2014 5 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Bivariate Poisson Distribution [Kocherlakota and Kocherlakota, 1992; Johnson et al., 1997]
Xi Po(i), i = 0,1,2X = X1 +X0Y = X2 +X0
}
(X,Y) BPo(1,2,0)
Joint probability function:
Pr[X = x,Y = y] = e(1+2+0)min(x,y)
i=0
xi1 yi2
i0
(x i)!(y i)!i!
Bivariate Negative Binomial Distribution [Marshall and Olkin, 1990; Cheon et al., 2009]
X Po(1),Y Po(2), where Gamma(1,1)(X,Y) BNB(1,2,)Joint probability function:
Pr[X = x,Y = y] =
(1+x+y)(1)(x+1)(y+1)
(1
1+2+1
)x( 21+2+1
)y( 11+2+1
)1
Motivation JOCLAD 2014 5 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Bivariate INteger-valued Moving Average models
Bivariate INMA(1) [Brnns and Nordstrm, 2000]: Aggregation of INAR(1) models
BINMA models JOCLAD 2014 6 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Bivariate INteger-valued Moving Average models
Bivariate INMA(1) [Brnns and Nordstrm, 2000]: Aggregation of INAR(1) models
X1,t = 1,t +1,1 1,t1 + +1,q1 1,tq1X2,t = 2,t +2,1 2,t1 + +2,q2 2,tq2
BINMA models JOCLAD 2014 6 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Bivariate INteger-valued Moving Average models
Bivariate INMA(1) [Brnns and Nordstrm, 2000]: Aggregation of INAR(1) models
X1,t = 1,t +1,1 1,t1 + +1,q1 1,tq1X2,t = 2,t +2,1 2,t1 + +2,q2 2,tq2
BIINMA(q1,q2) model with independent binomial thinning operations
arrivals following a discrete distribution [Quoreshi, 2006] arrivals following a bivariate Poisson and Negative Binomial distributions [Torres et al.]
BINMA(q1,q2) model with dependent binomial thinning operations
arrivals following a discrete distribution [Torres et al) arrivals following a bivariate Poisson and Negative Binomial distributions [Torres et al.]
Same dependence structure as in Al-Osh and Alzaid (1988)
BINMA models JOCLAD 2014 6 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
BINMA(1,1) models
X1,t = 1,t +1,1 1,t1X2,t = 2,t +2,1 2,t1
BINMA models JOCLAD 2014 7 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
BINMA(1,1) models
X1,t = 1,t +1,1 1,t1X2,t = 2,t +2,1 2,t1
Characterization of first- and second-order:
Poisson BINMA(1, 1) Neg. Bin. BINMA(1, 1)
Moment (j = 1,2) t BPo(1,2,) t BNB(1,2,)E[Xj,t] (j +)(1+j,1) j(1+j,1)
Var(Xj,t) (j +)(1+j,1) j(1+j,1)+ 2j (1+ 2j,1)Xj(1) = Cov(Xj,t1,Xj,t) (j +)j,1 jj,1(1+j)
X1,X2(0) = Cov(X1,t,X2,t) (1+1,12,1) 12(1+1,12,1)X1,X2(1) = Cov(X1,t,X2,t1) 1,1 121,1X2,X1(1) = Cov(X1,t1,X2,t) 2,1 122,1
BINMA models JOCLAD 2014 7 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
Method of Moments
Moment conditions:E[m(Xt,)] = 0
{Xt : t = 1, ...T} : observed sample : unknown q1 parameter vector with true value 0 m(Xj,t,) : continuous p1 vector function of E[m(Xt,)] exist and be finite for all t and
Parameter Estimation JOCLAD 2014 8 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
Method of Moments
Moment conditions:E[m(Xt,)] = 0
{Xt : t = 1, ...T} : observed sample : unknown q1 parameter vector with true value 0 m(Xj,t,) : continuous p1 vector function of E[m(Xt,)] exist and be finite for all t and
The MM estimator T solves the analogous sample moment conditions
mT() = T1T
t=1
m(Xt,) = 0
Parameter Estimation JOCLAD 2014 8 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
Generalized Method of Moment estimator [Hansen, 1982]
The GMM estimator minimizes a quadratic form
QT() = mT()WTmT()
where WT any symmetric and positive definite weight matrix
WT = (Cov(mT()))1
Parameter Estimation JOCLAD 2014 9 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
Generalized Method of Moment estimator [Hansen, 1982]
The GMM estimator minimizes a quadratic form
QT() = mT()WTmT()
where WT any symmetric and positive definite weight matrix
WT = (Cov(mT()))1
Under some assumptions about the structure of m(Xt,), Xt and the parameterspace, the GMM estimator T is
Weakly Consistent Asymptotically Normal(
MT( T)WT VT WT MT( T)) 12 (
MT( T)WT MT( T))
T(
T 0)
dN(0, Iq
)
where VT = TVar[mT( 0)] and MT() =mT ()
Parameter Estimation JOCLAD 2014 9 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
MM and GMM for Poisson BINMA(1, 1) models{
Xj,1, ...,Xj,T , j = 1,2}
: sample of Poisson BINMA(1, 1)
= (1,1,1,2,1,2,)
m(Xj,t,) =
X1,t (1 +)(1+1,1)X2,t (2 +)(1+2,1)X1,t1X1,t [(1 +)1,1 +(1 +)2(1+1,1)2]X2,t1X2,t [(2 +)2,1 +(2 +)2(1+2,1)2]X1,tX2,t1 [1,1 +(1 +)(1+1,1)(2 +)(1+2,1)]X1,tX2,t [(1+1,12,1)+(1 +)(1+1,1)(2 +)(1+2,1)]X2,tX1,t1 [2,1 +(1 +)(1+1,1)(2 +)(1+2,1)]
Parameter Estimation JOCLAD 2014 10 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
MM and GMM for Poisson BINMA(1, 1) models
Method of Moments: p = q = 5
j,1 =j(1)
1 j(1), j =
xj
(1+ j,1) , = 1,2(0)
1+ 1,12,1, j = 1,2
xj is the sample mean of{
Xj,t , t = 1, ...,T}
for j = 1,2 j(1) is the sample autocorrelation in lag 1, for j = 1,2 1,2(0) is the sample cross-covariance in lag 0
Parameter Estimation JOCLAD 2014 11 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter Estimation
MM and GMM for Poisson BINMA(1, 1) models
Method of Moments: p = q = 5
j,1 =j(1)
1 j(1), j =
xj
(1+ j,1) , = 1,2(0)
1+ 1,12,1, j = 1,2
xj is the sample mean of{
Xj,t , t = 1, ...,T}
for j = 1,2 j(1) is the sample autocorrelation in lag 1, for j = 1,2 1,2(0) is the sample cross-covariance in lag 0
Generalized Method of Moments: q = 7 > p = 5
T = argmin
(mT()WTmT()
)
WT = (Cov(mT()))1
Parameter Estimation JOCLAD 2014 11 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter EstimationMM and GMM for Negative Binomial BINMA(1, 1) models{
Xj,1, ...,Xj,T , j = 1,2}
: sample of Negative Binomial BINMA(1, 1)
= (1,1,1,2,1,2,)
m(Xj,t,) =
X1,t 1(1+1,1)X2,t 2(1+2,1)X21,t [1(1+1,1)+ 21 (1+ 21,1)+ 21 (1+1,1)2]X22,t [2(1+2,1)+ 22 (1+ 22,1)+ 22 (1+2,1)2]X1,t1X1,t [11,1(1+1)+ 21 (1+1,1)2]X2,t1X2,t [22,1(1+2)+ 22 (1+2,1)2]X1,tX2,t [12((1+1,12,1)+(1+1,1)(1+2,1))]X1,tX2,t1 [12(1,1 +(1+1,1)(1+2,1))]X2,tX1,t1 [12(2,1 +(1+1,1)(1+2,1))]
Method of Moments: p = q = 5
Generalized Method of Moments: q = 9 > p = 5Parameter Estimation JOCLAD 2014 12 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter EstimationSimulation Study
Illustrate the small sample properties of MM and GMM estimators
Compare their behaviour
Analyse the choice/number of moment conditions
Parameter Estimation JOCLAD 2014 13 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Parameter EstimationSimulation Study
Illustrate the small sample properties of MM and GMM estimators
Compare their behaviour
Analyse the choice/number of moment conditions
1000 realizations of Poisson and Negative Binomial BINMA(1, 1) models
T = 200, 500 and 1000 observations
{(0.1,1,0.1,1,0.5),(0.1,1,0.1,1,1),(0.1,1,0.5,3,0.5),(0.1,1,0.5,3,1),(0.1,3,0.5,1,0.5),(0.1,3,0.5,1,1),(0.1,3,0.9,1,0.5),(0.1,3,0.9,1,1),
(0.5,1,0.5,1,0.5),(0.5,1,0.5,1,1)}MM and GMM
Initial values for GMM: MM
BN BINMA(1, 1): Initial values for MM: (0.5,1,0.5,1,1)
Parameter Estimation JOCLAD 2014 13 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Poisson BINMA(1, 1) models
Sample mean bias and sample standard deviation (in brackets)
T = 200 T = 500 T = 1000
0(i) MM GMM MM GMM MM GMM1,1 = 0.1 0.015 0.035 0.000 0.001 -0.001 -0.008
(0.075) (0.119) (0.049) (0.073) (0.038) (0.056)
1 = 3 0.475 -0.047 0.522 0.038 0.521 0.071(0.318) (0.429) (0.205) (0.289) (0.157) (0.240)
2,1 = 0.9 -0.097 -0.185 -0.050 -0.101 -0.025 -0.047(0.122) (0.251) (0.089) (0.181) (0.067) (0.143)
2 = 1 0.630 0.322 0.571 0.165 0.542 0.097(0.219) (0.525) (0.141) (0.324) (0.103) (0.222)
= 1 -0.758 -0.051 -0.756 -0.027 -0.755 -0.029(0.062) (0.281) (0.040) (0.186) (0.028) (0.132)
Parameter Estimation JOCLAD 2014 14 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Poisson BINMA(1, 1) models
Sample mean bias and sample standard deviation (in brackets)
T = 200 T = 500 T = 1000
0(i) MM GMM MM GMM MM GMM1,1 = 0.1 0.020 0.042 0.002 0.010 -0.001 0.002
(0.077) (0.130) (0.052) (0.080) (0.039) (0.061)
1 = 3 0.113 -0.074 0.159 -0.005 0.167 0.007(0.288) (0.396) (0.201) (0.272) (0.146) (0.207)
2,1 = 0.5 -0.011 -0.011 -0.005 -0.005 0.001 0.004(0.136) (0.216) (0.086) (0.152) (0.061) (0.109)
2 = 1 0.178 0.056 0.169 0.030 0.163 0.010(0.204) (0.307) (0.127) (0.209) (0.094) (0.156)
= 0.5 -0.331 -0.021 -0.330 -0.010 -0.330 -0.005(0.067) (0.204) (0.043) (0.135) (0.032) (0.098)
Parameter Estimation JOCLAD 2014 15 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Poisson BINMA(1, 1) models
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_10000.2
0
0.2
0.4
0.6
0.8
Bias 1,1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_10000.5
0.3
0.1
0.1
0.3
0.5
Bias 2,1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
1
0.5
0
0.5
1
Bias 1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
0.5
0
0.5
1
Bias 2
MM_500 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
0.4
0.2
0
0.2
0.4
0.6
0.8
Bias
Poisson BINMA(1,1), 1,1
=0.1; 2,2
=0.5; 1=3;
2=1; =0.5
Parameter Estimation JOCLAD 2014 16 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Negative Binomial BINMA(1, 1) models
Sample mean bias and sample standard deviation (in brackets)
T = 200 T = 500 T = 1000
0(i) MM GMM MM GMM MM GMM1,1 = 0.5 -0.024 0.006 -0.008 0.004 -0.007 0.001
(0.117) (0.165) (0.079) (0.097) (0.057) (0.060)
1 = 1 0.033 -0.022 0.016 -0.007 0.007 -0.007(0.135) (0.150) (0.085) (0.091) (0.058) (0.062)
2,1 = 0.5 -0.025 0.000 0.003 -0.008 -0.001 -0.006(0.343) (0.169) (0.310) (0.100) (0.267) (0.066)
2 = 1 0.065 -0.016 0.039 0.001 0.029 -0.001(0.269) (0.150) (0.226) (0.095) (0.181) (0.063)
= 1 -0.072 -0.076 -0.050 -0.041 -0.035 -0.023(0.262) (0.247) (0.181) (0.151) (0.136) (0.110)
Parameter Estimation JOCLAD 2014 17 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Negative Binomial BINMA(1, 1) models
Sample mean bias and sample standard deviation (in brackets)
T = 200 T = 500 T = 1000
0(i) MM GMM MM GMM MM GMM1,1 = 0.1 0.012 0.014 -0.004 0.001 -0.001 0.002
(0.077) (0.083) (0.051) (0.050) (0.045) (0.038)
1 = 3 -0.015 -0.061 0.017 -0.017 0.006 -0.015(0.260) (0.287) (0.187) (0.199) (0.140) (0.143)
2,1 = 0.5 0.020 0.010 -0.017 -0.013 -0.007 -0.004(0.353) (0.186) (0.316) (0.107) (0.284) (0.077)
2 = 1 0.024 -0.012 0.049 0.006 0.036 0.002(0.271) (0.156) (0.227) (0.097) (0.194) (0.068)
= 0.5 -0.003 -0.024 -0.012 -0.014 -0.010 -0.009(0.117) (0.108) (0.081) (0.069) (0.059) (0.051)
Parameter Estimation JOCLAD 2014 18 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Negative Binomial BINMA(1, 1) models
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
0
0.2
0.4
0.6
0.8
Bias 1,1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_10000.5
0
0.5
Bias 2,1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
1
0.5
0
0.5
Bias 1
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
0.4
0.2
0
0.2
0.4
0.6
Bias 2
MM_200 GMM_200 MM_500 GMM_500 MM_1000 GMM_1000
0.2
0
0.2
0.4
Bias
Negative Binomial BINMA(1,1),1,1
=0.1; 2,2
=0.5; 1=3;
2=1; =0.5
Parameter Estimation JOCLAD 2014 19 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Discussion
Poisson BINMA(1, 1) models
MM sample standard deviation < GMM sample standard deviation
MM sample bias > GMM sample bias
For and j : GMM is better than MM
For j,1 : MM GMM
Parameter Estimation JOCLAD 2014 20 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Discussion
Poisson BINMA(1, 1) models
MM sample standard deviation < GMM sample standard deviation
MM sample bias > GMM sample bias
For and j : GMM is better than MM
For j,1 : MM GMM
Negative Binomial BINMA(1, 1) models
GMM is better than MM in terms of sample bias and standard deviation
For 2,1 and 2 : MM has a poor performance
Parameter Estimation JOCLAD 2014 20 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
How many and which moment conditions to choose?
Example: Poisson BINMA(1, 1) models
Available: 7 conditions
2 means, 2 auto-covariances (lag 1) and 3 cross-covariances (lags -1, 0 and 1)
Parameter Estimation JOCLAD 2014 21 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
How many and which moment conditions to choose?
Example: Poisson BINMA(1, 1) models
Available: 7 conditions
2 means, 2 auto-covariances (lag 1) and 3 cross-covariances (lags -1, 0 and 1)
Method of Moments: 5 conditions
2 means, 2 auto-covariances (lag 1) and 1 cross-covariance (lag 0)
Generalized Method of Moments: 7 conditions
2 means, 2 auto-covariances (lag 1) and 3 cross-covariances (lags -1, 0 and 1)
Generalized Method of Moments: 6 conditions (A)
2 means, 2 auto-covariances (lag 1) and 2 cross-covariances (lags 0 and 1)
Generalized Method of Moments: 6 conditions (B)
2 means, 2 auto-covariances (lag 1) and 2 cross-covariances (lags -1 and 0)
Parameter Estimation JOCLAD 2014 21 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Poisson BINMA(1, 1) models
Sample mean bias and sample standard deviation (in brackets)
T = 500 MM GMM
0(i) 5 cond. 7 cond. 6 cond.(A) 6 cond.(B)1,1 = 0.1 0.000 0.001 0.011 -0.019
(0.049) (0.073) (0.081) (0.071)
1 = 3 0.522 0.038 0.013 0.117(0.205) (0.289) (0.280) (0.316)
2,1 = 0.9 -0.050 -0.101 -0.061 -0.084(0.089) (0.181) (0.138) (0.181)
2 = 1 0.571 0.165 0.110 0.138(0.141) (0.324) (0.269) (0.305)
= 1 -0.756 -0.027 -0.032 -0.019(0.040) (0.186) (0.200) (0.176)
Parameter Estimation JOCLAD 2014 22 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Results for Poisson BINMA(1, 1) models
MM GMM GMM_6A GMM_6B0.1
0
0.1
0.2
0.3
Bias 1,1
MM GMM GMM_6A GMM_6B
0.6
0.4
0.2
0
Bias 2,1
MM GMM GMM_6A GMM_6B1
0.5
0
0.5
1
Bias 1
MM GMM GMM_6A GMM_6B
0.5
0
0.5
1
1.5
Bias 2
MM GMM GMM_6A GMM_6B
0.5
0
0.5
Bias
Poisson BINMA(1,1), 500 obs.; 1,1
=0.1; 2,2
=0.9; 1=3;
2=1; =1
Parameter Estimation JOCLAD 2014 23 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Ongoing and Future Work
Decide how many and which moment conditions to choose
Estimation of the parameters of the models
Generating Probability Function Characteristic Function
Specify other structures of dependence between the several thinning operations
Application to real data
Ongoing and Future Work JOCLAD 2014 24 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
Ongoing and Future Work
Decide how many and which moment conditions to choose
Estimation of the parameters of the models
Generating Probability Function Characteristic Function
Specify other structures of dependence between the several thinning operations
Application to real data
Developing tools for the analysis of multivariate time series of counts: inference,
diagnostic, forecasting
Great care is needed when developing extensions, otherwise model specification
and inference becomes very demanding mathematically and computationally
Ongoing and Future Work JOCLAD 2014 24 / 25
Isabel Silva & Cristina Torres & Maria Eduarda Silva Estimating BINMA models with GMM
References
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Integer-valued moving average (INMA) process.
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A Bivariate Integer Valued Allocation Model for Guest Nights in Hotels and
Cottages.
Ume Economics Studies 547. Ume University, Sweden.
Cheon, S., S. H. Song and B. C. Jung (2009).
Tests for independence in a bivariate negative binomial model.
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Hansen, Lars Peter (1982).
Large sample properties of generalized method of moments estimators.
Econometrica: Journal of the Econometric Society, Vol. 50, pp. 1029-1054.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997).
Discrete Multivariate Distributions. Wiley, New York.
Kocherlakota, S. and K. Kocherlakota (1992).
Bivariate discrete distributions. Markel Dekker, New York.
Marshall, A. W. and I. Olkin (1990).
Multivariate distributions generated from mixtures of convolution and
product families.
Lecture Notes-Monograph Series, Vol. 16, pp. 371U393.
Mtys, Lszl (1999).
Generalized Method of Moments Estimation. Cambridge University Press.
Quoreshi, S. (2006).
Bivariate Time Series Modelling of Financial Count Data.
Ume Economics Studies, Vol. 35, pp. 1343-1358.
Torres, C., Silva, I. and Silva, M. E. (2012).
Modelos bivariados de mdias mveis de valor inteiro.
XX Congresso Anual da Sociedade Portuguesa de Estatstica, 27-29 de
Setembro, Porto, Portugal.
Steutel, F.W. and K. Van Harn (1979).
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References JOCLAD 2014 25 / 25
OutlineMotivationBINMA modelsPoisson and NB BINMA(1, 1)Parameter EstimationMMGMMMM and GMM for Poisson BINMA(1, 1) modelsMM and GMM for NB BINMA(1, 1) modelsSimulation StudyOngoing and Future WorkReferencesRecommended