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I. Silva and M.E. Silva Parameter estimation for INAR processes based on HOS

Parameter estimation for INAR processes

based on High-Order Statistics

Isabel Silva1 Maria Eduarda Silva2,3

1

Centro das Construes (CEC) e Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto2Grupo de Matemtica e Informtica, Faculdade de Economia da Universidade do Porto

3Unidade de Investigao Matemtica e Aplicaes (UIMA), Universidade de Aveiro

COMPSTAT 2008

August, 2008 1 / 1


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Outline

Introduction

High-Order Statistics (HOS)

INteger-valued AutoRegressive (INAR) processes

Least square estimation using HOS

Monte Carlo results and application to real data

Final remarks

August, 2008 2 / 1


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Moments and cumulants of order higher than two

Introduction August, 2008 3 / 1


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Lack of Gaussianity and/or non-linearity


I Sil d M E Sil P i i f INAR b d HOS


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Lack of Gaussianity and/or non-linearity

Notation:

{Xt} : kth-order stationary stochastic process

X(s1, . . . , sk1) : kth-order joint moment ofXt,Xt+s1 . . . , Xt+sk1 , (s1, . . . , sk1 R)

X(s1, . . . , sk1) = E[XtXt+s1 . . .Xt+sk1 ]

X = E[Xt]


I Sil a and M E Sil a Parameter e timation for INAR proce e ba ed on HOS


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INteger-valued AutoRegressive processes

INAR(p) [Latour, 1998]

Xt = 1 Xt1 +2 Xt2 + +p Xtp + et


I Silva and M E Silva Parameter estimation for INAR processes based on HOS


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Xt = 1 Xt1 +2 Xt2 + +p Xtp + et

0 i < 1, i = 1, . . . ,p1, and 0 < p < 1, such that pk=1k < 1

thinning operation [Steutel and Van Harn, 1979; Gauthier and Latour, 1994)]

i Xti =Xti

j=1Yi,j, for i = 1, . . . ,p,

{Yi,j} (counting series): set of i.i.d. non-negative integer-valued r.v. with

E[Yi,j] = i, Var[Yi,j] = 2i

and E[Y3i,j

] = i

{et} (innovation process): sequence of i.i.d. non-negative integer-valued r.v.

(independent of{Yi,j}) with E[et] = e, Var[et] = 2e and E[e3t] = e


I Silva and M E Silva Parameter estimation for INAR processes based on HOS


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Xt = 1 Xt1 +2 Xt2 + +p Xtp + et

0 i < 1, i = 1, . . . ,p1, and 0 < p < 1, such that pk=1k < 1

thinning operation [Steutel and Van Harn, 1979; Gauthier and Latour, 1994)]

i Xti =Xti

j=1Yi,j, for i = 1, . . . ,p,

{Yi,j} (counting series): set of i.i.d. non-negative integer-valued r.v. with

E[Yi,j] = i, Var[Yi,j] = 2i

and E[Y3i,j

] = i

{et} (innovation process): sequence of i.i.d. non-negative integer-valued r.v.

(independent of{Yi,j}) with E[et] = e, Var[et] = 2e and E[e3t] = e

Usually: Poisson INAR(p) process with binomial thinning operation




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Third-order moments of INAR processes

[Silva and Oliveira (2004, 2005) and Silva (2005)]

X(0, 0) =

p

i=1

p

j=1

p

k=1ijkX(ij, i k) + 3

p

i=1

p

j=1ji2X(ij) + 3Xe

p

i=1i2

+ e

+ 3X(2e +e

2)p

i=1

i + 3ep

i=1

p

j=1

ijX(ij) +Xp

i=1

(i 3ii2 3i )

X(0, k) =p

i=1iX(0, k i) +eX(0), k> 0

X(k, k) =p

i=1

p

j=1

ij X(k i, kj) +p

i=1

i2X(k i) + 2eX(k)X(e

2 e2), k> 0

X(k, m) =p

i=1

iX(k, m i) +eX(k), m > k> 0

Least square estimation using HOS August, 2008 5 / 1



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. S va a d . . S va a a ete est at o o N p ocesses based o OS

Third-order moments of INAR processes

[Silva and Oliveira (2004, 2005) and Silva (2005)]

X(0, 0) =

p

i=1

p

j=1

p

k=1ijkX(ij, i k) + 3

p

i=1

p

j=1ji2

X(ij) + 3Xe

p

i=1i2

+ e

+ 3X(2e +e

2)p

i=1

i + 3ep

i=1

p

j=1

ijX(ij) +Xp

i=1

(i 3ii2 3i )

X(0, k) =p

i=1iX(0, k i) +eX(0), k> 0

X(k, k) =p

i=1

p

j=1

ij X(k i, kj) +p

i=1

i2X(k i) + 2eX(k)X(e

2 e2), k> 0

X(k, m) =p

i=1

iX(k, m i) +eX(k), m > k> 0

INAR processes have a non-linear structure

1st

and 2nd

order moments are not sufficient to describe dependence structureLeast square estimation using HOS August, 2008 5 / 1



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p


{x1,x2, . . . ,xn} : realization of a non-negative integer-valued stationary stochastic

process with third-order moments (0, k), k> 0




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p


{x1,x2, . . . ,xn} : realization of a non-negative integer-valued stationary stochastic

process with third-order moments (0, k), k> 0Approximating model: INAR(p) with parameters 1, ,p,e,2e and

third-order moments X(0, k), k> 0, which can be represented in the following

matrix form:

3,X = M3,X+eX(0)1p

X(0, 1)

X(0, 2)

...

X(0,p)

=

X(0, 0) X(1, 1) . . . X(p1,p1)

X(0, 1) X(0, 0) . . . X(p2,p2)

......

. . ....

X(0,p1) X(0,p2) . . . X(0, 0)

1

2

...

p

+eX(0)

1

1

...

1

X(0) =p

i=1

iX(i) +eX + Vp, with Vp = e2 +X

p

i=1

i2




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Defining H = [M3,X X(0)1p] and = [ 1 p e ]T

3,X = H




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3,X = H

may be estimated by least squares, ie, minimizing the squared error between 3,Xand the third-order moments of the data:

3 = [ (0, 1) (0,p) ]T




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3,X = H


3 = [ (0, 1) (0,p) ]T

Least Squares estimator of using HOS (LS_HOS)

= min{L()} = min

{(3H)

T(3 H)}




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3,X = H


3 = [ (0, 1) (0,p) ]T

Least Squares estimator of using HOS (LS_HOS)

= min{L()} = min

{(3H)

T(3 H)}

In practice: = min{ L()} = min

{(3 H)

T(3 H)}




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Monte Carlo results

To examine the small sample properties of the LS_HOS

To compare its performance with other estimation methods: YW, CLS and WHT

Monte Carlo results and application to real data August, 2008 8 / 1



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Monte Carlo results



1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,

for several orders, sample sizes and parameter values




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Monte Carlo results





Sample properties of the LS_HOS estimator

The sample bias, variance and mean square error decrease as the sample size

increases

Distribution of the estimators is consistent and symmetric




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Monte Carlo results





Sample properties of the LS_HOS estimator

The sample bias, variance and mean square error decrease as the sample size

increases

Distribution of the estimators is consistent and symmetric

For small sample size: evidence of departure from symmetry in the marginal

distributions, specially for values of the parameter near the non-stationary region




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Monte Carlo results

YW CLS WHT LS_HOS YW CLS WHT LS_HOS

0.4

0.2

0

0.2

0.4

0.6

Bias()

YW CLS WHT LS_HOS YW CLS WHT LS_HOS4

2

0

2

4

Bias()

N=50 N=200

Figure: Boxplots of the sample bias for the estimates obtained in 1000 realizations of 50 and 200 observations of the

INAR(1) model: Xt = 0.9Xt1 + et, where etPo(1)




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Application to real data

1920 1930 1940 1950 1960 1970 19815

10

15

20

25

30

35

40

45

50

Numbero

fplants

Figure: The number of Swedish mechanical paper and pulp mills, from 1921 to 1981 [Brnns (1995) and Brnns and

Hellstrm (2001)]




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Simple INAR(1)

It is not assumed the Poisson distribution for the innovation process:

X= 20.40 and S2 = 155.16



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Simple INAR(1)


X= 20.40 and S2 = 155.16

Method e 2e x 2

x MSE

CLS 0.9591 0.2017 15.2268 4.9315 192.2764 8.5494

LS_HOS 0.9269 1.3635 19.2253 18.6525 145.4513 7.4465

Table: The parameter estimates of the number of Swedish mechanical paper and pulp mills

Mean and variance of the estimated models: x =e

1 and 2x =

(1 )(e+ 2e )(1 )2(1 + )




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Simple INAR(1)


X= 20.40 and S2 = 155.16

Method e 2e x 2

x MSE

CLS 0.9591 0.2017 15.2268 4.9315 192.2764 8.5494

LS_HOS 0.9269 1.3635 19.2253 18.6525 145.4513 7.4465

Table: The parameter estimates of the number of Swedish mechanical paper and pulp mills

Mean and variance of the estimated models: x =e

1 and 2x =

(1 )(e+ 2e )(1 )2(1 + )

MSE between the observations and the fitted models based on LS_HOS and CLS

estimates




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1920 1930 1940 1950 1960 1970 19815

10

15

20

25

30

35

40

45

50

Numberofplants

Real dataCLS

LS_HOS

Figure: The number of plants and the fitted values considering the LS_HOS and CLS estimates




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Final remarks

Advantage of HOS: capability to detect and characterize the deviations from

Gaussianity and non-linearity of the processes

Final remarks August, 2008 13 / 1



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Final remarks


Gaussianity and non-linearity of the processesINAR processes are non-Gaussian

Parameter estimation method: Least squares using HOS

Minimize the errors between the third-order moment of the observations and of

the fitted model




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Final remarks





the fitted model

Monte Carlo results: LS_HOS provides good results, in terms of sample bias,

variance and mean square error




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Final remarks





the fitted model

Monte Carlo results: LS_HOS provides good results, in terms of sample bias,

variance and mean square error

When used in the context of a non-Poisson real dataset the LS_HOS estimates

provide a model with mean, variance and autocorrelations closer to the sample

values



f


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ReferencesBRNNS, K. (1995).

Explanatory Variables in the AR(1) Count Data Model.

Ume Economic Studies 381.

BRNNS, K. and HELLSTRM, J. (2001).

Generalized Integer-Valued Autoregression.

Econometric Reviews 20 (4), 425-443.

GAUTHIER, G. and LATOUR, A. (1994).

Convergence forte des estimateurs des paramtres dtun processus GENAR(p).

Annales des Sciences Mathmatiques du Qubec 18, 49-71.

LATOUR, A. (1998).

Existence and stochastic structure of a non-negative integer-valued autoregressive process.

Journal of Time Series Analysis 19, 439-455.

SILVA, I. (2005).

Contributions to the analysis of discrete-valued time series.

PhD Thesis. Universidade do Porto, Portugal.

SILVA, M. E. and OLIVEIRA, V. L. (2004).

Difference equations for the higher-order moments and cumulants of the INAR(1) model.


SILVA, M. E. and OLIVEIRA, V. L. (2005).

Difference equations for the higher-order moments and cumulants of the INAR(p) model.


STEUTEL, F. W. and VAN HARN, K. (1979).

Discrete analogues of self-decomposability and stability.

The Annals of Probability 7, 893-899.

References August, 2008 14 / 1

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