apresentacao_JOCLAD2014

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


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


MotivationCount time series

Discrete time non-negative integer-valued time series

Motivation JOCLAD 2014 3 / 25




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









operation defined for integer values Thinning operation









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) =



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)



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


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 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



Bivariate INteger-valued Moving Average models

Bivariate INMA(1) [Brnns and Nordstrm, 2000]: Aggregation of INAR(1) models

BINMA models JOCLAD 2014 6 / 25




X1,t = 1,t +1,1 1,t1 + +1,q1 1,tq1X2,t = 2,t +2,1 2,t1 + +2,q2 2,tq2





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(1,1) models

X1,t = 1,t +1,1 1,t1X2,t = 2,t +2,1 2,t1



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




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



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




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




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 ()




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)]




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




MM and GMM for Poisson BINMA(1, 1) models


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 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))]


Generalized Method of Moments: q = 9 > p = 5Parameter Estimation JOCLAD 2014 12 / 25


Parameter EstimationSimulation Study

Illustrate the small sample properties of MM and GMM estimators

Compare their behaviour

Analyse the choice/number of moment conditions



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)



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)





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)




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


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


0.5

0

0.5

1

Bias 2


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



Results for Negative Binomial BINMA(1, 1) models


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)





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)





0

0.2

0.4

0.6

0.8

Bias 1,1


0

0.5

Bias 2,1


1

0.5

0

0.5

Bias 1


0.4

0.2

0

0.2

0.4

0.6

Bias 2


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



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



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



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)



How many and which moment conditions to choose?

Example: Poisson BINMA(1, 1) models

Available: 7 conditions


Method of Moments: 5 conditions

2 means, 2 auto-covariances (lag 1) and 1 cross-covariance (lag 0)

Generalized Method of Moments: 7 conditions


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)





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)




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


0.5

0

0.5

1

1.5

Bias 2


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



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


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


References

Al-Osh, M.A. and Alzaid, A.A. (1988).

Integer-valued moving average (INMA) process.

Statistical Papers, Vol. 29, pp 281-300.

Brnns, K. and J. Nordstrm (2000).

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.

Journal of the Korean Statistical Society, Vol. 38(2), pp. 185U190.

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).

Discrete analogues of self-decomposability and stability.

The Annals of Probability, Vol. 7, pp. 893-899.

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 WorkReferences

Documents

apresentacao_JOCLAD2014