Research Article Integer-Valued Moving Average Models with Structural Changes

Transcription

1 Matheatical Probles in Engineering, Article ID 21592, 5 pages Research Article Integer-Valued Moving Average Models with Structural Changes Kaizhi Yu, 1 Hong Zou, 2 and Daiin Shi 1 1 Statistics School, Southwestern University of Finance and Econoics, Chengdu 61110, China 2 School of Econoics, Southwestern University of Finance and Econoics, Chengdu 61110, China Correspondence should be addressed to Kaizhi Yu; kaizhiyu.swufe@gail.co Received 2 March 2014; Revised 22 June 2014; Accepted 7 July 2014; Published 21 July 2014 Acadeic Editor: Wuquan Li Copyright 2014 Kaizhi Yu et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. It is frequent to encounter integer-valued tie series which are sall in value and show a trend having relatively large fluctuation. To handle such a atter, we present a new first order integer-valued oving average odel process with structural changes. The odels provide a flexible fraework for odelling a wide range of dependence structures. Soe statistical properties of the process are discussed and oent estiation is also given. Siulations are provided to give additional insight into the finite saple behaviour of the estiators. 1. Introduction Integer-valued tie series occur in any situations, often as counts of events in consecutive points of tie, for exaple, the nuber of births at a hospital in successive onths, the nuber of road accidents in a city in successive onths, and big nubers even for frequently traded stocks. Integer-valued tie series represent an iportant class of discrete-valued tie series odels. Because of the broad field of potential applications, a nuber of tie series odels for counts have been proposed in literature. McKenzie [1] introducedthe first order integer-valued autoregressive, INAR(1), odel. The statistical properties of the INAR(1) are discussed in McKenzie [2], Al-Osh and Alzaid []. The odel is further generalized to a pth-order autoregression, INAR(p), by Alzaid and Al-Osh [4] andduandli[5]. The qthorder integer-valued oving average odel, INMA(q), was introduced by Al-Osh and Alzaid [6] and in a slightly different for by McKenzie [7]. Ferland et al. [8] proposed an integer-valued GARCH odel to study overdispersed counts, and Fokianos and Fried [9], Weiß [10], and Zhu and Wang [11 1] ade further studies. Györfi et al. [14] proposed a nonstationary inhoogeneous INAR(1) process, where the autoregressive type coefficient slowly converges to one. Bakouch and Ristić[15] introduced a new stationary integervalued autoregressive process of the first order with zero truncated Poisson arginal distribution. Kachour and Yao [16] introduced a class of autoregressive odels for integer-valued tie series using the rounding operator. Ki and Park [17] proposed an extension of integer-valued autoregressive INAR odels by using a signed version of the thinning operator. Zheng et al. [18] proposed a first order rando coefficient integer-valued autoregressive odel and got its ergodicity, oents, and autocovariance functions of the process.goesandcantoecastro[19]presentedarando coefficient autoregressive process for count data based on a generalized thinning operator. Existence and weak stationarity conditions for these odels were established. A siple bivariateinteger-valuedtieseriesodelwithpositivelycorrelated geoetric arginals based on the negative binoial thinning echanis was presented by Ristić et al.[20], and soe properties of the odel are also considered. Pedeli and Karlis [21] considered a bivariate INAR(1) (BINAR(1)) process where cross correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations. Structural changes in econoic data frequently correspondtoinstabilitiesintherealworld.however,ostworkin this area has been concentrated on odels without structural changes. It sees that the integer-valued autoregressive oving average (INARMA) odel with break point has not attracted too uch attention. For instance, a new ethod for odelling the dynaics of rain sapled by a tipping bucket rain gauge was proposed by Thyregod et al. [22].

2 2 Matheatical Probles in Engineering Theodelstaketheautocorrelationanddiscretenatureof the data into account. First order, second order, and threshold odels are presented together with ethods to estiate the paraeters of each odel. Monteiro et al. [2] introduced a class of self-exciting threshold integer-valued autoregressive odels driven by independent Poisson-distributed rando variables. Basic probabilistic and statistical properties of this class of odels were discussed. Moreover, paraeter estiation was also addressed. Hudecová [24] suggested a procedure for testing a change in the autoregressive odels for binary tie series. The test statistic is a axiu of noralized sus of estiated residuals fro the odel, and thus it is sensitive to any change which leads to a change in the unconditional success probability. Structural change is a stateent about paraeters, which only have eaning in the context of a odel. In our discussion, we will focusonstructuralchangeinthesiplecountdataodel, the first order integer-valued oving average odel, whose coefficient varies with the value of innovation. One of the leading reasons is that piecewise linear functions can offer a relatively siple approxiation to the coplex nonlinear dynaics. The rest of this paper is divided into four sections. In Section 2, we give the definition and basic properties of the new INMA(1) odel with structural changes. Section discusses the estiation of the unknown paraeters. We test the accuracy of the estiation via siulations in Section 4. Section 5 includes soe concluding rearks. 2. Definition and Basic Properties Definition 1. Let {X t } be a process with state space N 0 ;let 0 < α i < 1, i 1,...,,andτ i, i 1,..., 1,be positive integers. The process {X t } is said to be first order integer-valued oving average odel with structural change (INMASC(1))ifX t satisfies the following equation: α 1 ε t 1 +ε t, for τ 0 ε t 1 τ 1 ε t 1 +ε t, for τ 1 <ε t 1 τ 2 { X t. { α 1 ε t 1 +ε t, for τ 2 <ε t 1 τ 1 { α ε t 1 +ε t, for τ 1 <ε t 1 <τ, where {ε t } is a sequence of independent and identically distributed Poisson rando variables with ean λ and τ 0 : 0, τ :. The ai of this section is to provide expressions for the oents and stationary of INMASC(1) odel. For this purpose, we introduce the following notations: p i : P (τ i 1 <ε t τ i ), u i : E (τ i 1 <ε t 1 τ i ), σ 2 i : Var (τ i 1 <ε t 1 τ i ), q i : 1 p i, I t 1,i : { 1, if τ i 1 <ε t 1 τ i 0, otherwise,,...,. (1) (2) Theore 2. The nuerical characteristics of {X t } are as follows: (i) μ X : E (X t ) (ii) σ 2 X : Var (X t) p i α i u i +λ, p i α i [α i (u 2 i +σ 2 i )+(1 α i)u i ] ( p i α i u i ) (iii) γ X (k) : cov (X t,x t k ) 2 +λ, { p i α i (u 2 i +σ 2 i λu i ), { k1 { 0, k 2. Proof. (i)itiseasytogettheeanandvarianceofx t by using the law of iterated expectations: E(X t )E[I t 1,1 (α 1 ε t 1 )+ +I t 1, (α ε t 1 )+ε t ] E{E[I t 1,1 (α 1 ε t 1 ) + +I t 1, (α ε t 1 ) ε t 1 ]} + E (ε t ) α 1 E(I t 1,1 ε t 1 )+ +α E(I t 1, ε t 1 )+E(ε t ) (ii) Moreover, Var (X t ) p i α i u i +λ. Var (I t 1,1 (α 1 ε t 1 )+ +I t 1, (α ε t 1 )+ε t ) Var (I t 1,i (α i ε t 1 )) + Var (ε t ) +2cov (I t 1,i (α i ε t 1 ),I t 1,j (α j ε t 1 )) {Var (E [I t 1,i (α i ε t 1 ) ε t 1 ]) +E(Var [I t 1,i (α i ε t 1 ) ε t 1 ])} + λ +2{E [I t 1,i (α i ε t 1 )I t 1,j (α j ε t 1 )] 2E[I t 1,i (α i ε t 1 )] E [I t 1,j (α j ε t 1 )]} [ i Var (I t 1,iε t 1 )+α i (1 α i )E(I t 1,i ε t 1 )]+λ 2E{E[I t 1,i (α i ε t 1 ) ε t 1 ]} E{E[I t 1,j (α j ε t 1 ) ε t 1 ]} () (4)

3 Matheatical Probles in Engineering {[ i E(I2 t 1 ε2 t 1 ) E2 (I t 1,i ε t 1 )] +p i α i (1 α i )u i }+λ 2α i α j E(I t 1,i ε t 1 )E(I t 1,j ε t 1 ) {[p i i (u2 i +σ 2 i ) p2 i α2 i u2 i ]+p iα i (1 α i )u i }+λ 2p i α i u i p j α j u j p i α i [α i (u 2 i +σ 2 i )+(1 α i)u i ] ( p i α i u i ) 2 +λ. (iii) Note the correlation between α i ε t 1 and ε t 1 ;we have cov (X t,x t 1 ) cov ( I t 1,i (α i ε t 1 )+ε t, j1 cov (I t 1,i (α i ε t 1 )+ε t,ε t 1 ) I t 2,j (α j ε t 2 )+ε t 1 ) (5) {E [I t 1,i (α i ε t 1 )ε t 1 ] E[I t 1,i (α i ε t 1 )] E (ε t 1 )} {E [E (I t 1,i (α i ε t 1 )ε t 1 ε t 1 )] E[E(I t 1,i (α i ε t 1 ) ε t 1 )] E (ε t 1 )} α i [E (I t 1,i ε 2 t 1 ) λα ie(i t 1,i ε t 1 )] p i α i (u 2 i +σ 2 i λu i ). Theore. Let X t be the process defined by the equation in (1); then the {X t } is a covariance stationary process. Proof. Both the unconditional ean and the unconditional variance of the {X t } are finite constant. And the autocovariance function does not change with tie. Thus {X t } is a stationary process. Theore 4. Suppose {X t } is INMASC(1) process. Then (i) T(X μ X ) N(0,σ L 2 X +2γ X(1)); (ii) E(X k t I t 1,i 1)<, k1,2,,,...,. (6) Proof. (i) Fro definition and Theore 2, we have that (X 1,...,X i ) and (X j,x j+1,...) are independent whenever j i > 1. According to Theore 9.1 of DasGupta [25], the process {X t } is a stationary 1-dependent sequence. Therefore we can coplete the proof. (ii) For k1,itfollowsthat E(X t ) ax {E [I t 1,i (α i ε t 1 )+ε t ],,...,} ax {E (α i ε t 1 )+E(ε t ),,...,} λ(α ax +1)<,α ax ax (α 1,...,α ). For k2, E(X 2 t ) ax {E[I t 1,i(α i ε t 1 )+ε t ] 2,,...,} For k, ax {E[I t i,1 (α i ε t 1 )] 2 +E(ε 2 t ) +2E [I t i,1 (α i ε t 1 )ε t ],,...,} ax {E[(α i ε t 1 )] 2 +E(ε 2 t ) +2E [(α i ε t 1 )ε t ],,...,} ax {[(λ + λ 2 ) i +λα i (1 α i )] +(λ+λ 2 )+λ 2 α i,,...,} 2(λ+λ 2 )α ax λ + λ 2 α ax <. E(X t ) ax {E[I t 1,i (α i ε t 1 )+ε t ],,...,} ax {E[I t 1,i (α i ε t 1 )] +E(ε t ) +E[I 2 t 1,i (α i ε t 1 ) 2 ε t ] +E[I t 1,i (α i ε t 1 )ε 2 t ],,...,} ax {E(α i ε t 1 ) +E(ε t ) + E [(α i ε t 1 ) 2 ε t ] +E [(α i ε t 1 )ε 2 t ],,...,} ax {[α i τ 1 + i (1 α i)τ 2 +(α i i (1 α i) α i )λ] +τ 1 +{[ i τ 2 +α i (1 α i )λ]λ} +λα i τ 2,,...,} λα ax [ ax (τ 1 1 λ) +τ 2 (α ax +1)+λ+1]+τ 1 <, where τ 1 : λ +λ 2 +λ, τ 2 : λ 2 +λ,andα ax ax(α 1,...,α ).ThennotethatE(X k t )< iplies E(Xk t I t 1,i 1)< for k1,2,,,2,...,. (7) (8) (9)

4 4 Matheatical Probles in Engineering Theore 5. Let {X t } be a INMASC(1) process according to Definition 1. LetX be the saple ean of {X t }; then the stochastic process {X t } is ergodic in the ean. Proof. Since γ X (k) 0, k. Fro Theore in Brockwell and Davis [26], we get Var (X T ) E(X T μ X ) 2 0. (10) Then X T converges in probability to μ X. Therefore, the process {X t } is ergodic in the ean. Theore 6. Suppose {X t } is a INMASC(1) process; then P( γ X (k) γ X (k) ε) P 0, (11) where γ X (k) : (1/T) T k t1 (X t+k X T )(X t X T ). The proof of Theore 6 issiilartotheore4givenin Yu et al. [27]. It is easy to verify; we skip the details.. Estiation of Paraeters In this paper, we consider one ethod, naely, oent estiation. An advantage of the ethod is that it is siple and often produces good results. The estiation proble of INMASC(1) paraeters is coplex. In fact, for the INMASC(1) processes, the conditional distribution of the X t given ε t 1 is the convolution of the distribution of the arrival process ε t and one thinning operation α i ε t 1. On the other hand, there are too any unknown paraeters of the odel, such as λ, α i, p i, u i,andσ 2 i,,...,,whereasthenuber of oent conditions is sall. Therefore we cannot estiate all the paraeters unless additional assuptions are ade. Then, we assue that the nuber of break point is two and assue that the value of break point τ i, i 1,...,,andtheeanofinnovationλ arealsoknown.thus,hereweestiateinmasc(1) odel with two break points. Under these assuptions, all the paraeters λ, p i, u i,andσ 2 i, i 1, 2,, areknown.weonly need to estiate the autoregressive coefficients α 1,,and α. Using the saple ean and saple covariance function, we can get the oent estiators via solving the following equations: γ (0) γ (1) X ( p i α i [α i (u 2 i +σ 2 i )+(1 α i)u i ] p i α i u i ) 2 +λ p i α i (u 2 i +σ 2 i λu i ) p i α i u i +λ. (12) Table 1: and ean square error for odels A, B, and C. Model A B C Paraeter α (0.948) (0.471) α (0.2908) α (0.462) (0.2806) α (0.408) α (1.045) (0.4127) (0.205) Saple size (0.075) (0.045) (0.114) (0.076) (0.0811) (0.042) (0.180) (0.0745) (0.449) (0.0847) (0.072) (0.0165) (0.562) (0.057) (0.04) (0.0212) (0.0547) (0.0274) If you want to estiate all paraeters, you can use GMM ethod based on probability generating functions introduced by BräKnnäK and Hall [28]. But they found covariance atrix of estiators depends on z and the orders besides the odel paraeters in a highly coplex way. Thus we do not use this ethod here. In next section, siulations are provided to give insight into the finite saple behaviour of these estiators. 4. Siulation Study Consider the following INMASC(1) odel: α { 1 ε t 1 +ε t, for ε t 1 τ 1 X t α { 2 ε t 1 +ε t, for τ 1 <ε t 1 τ 2 { α ε t 1 +ε t, for τ 2 <ε t 1, (1) where {ε t } is a sequence of i.i.d. For fixed t, ε t follows a Poisson distribution with ean λ. The paraeters values considered in this odel are listed as follows: (odel A) (α 1,,α ) (0.1, 0.1, 0.1), withτ 1, τ 2 10, λ1; (odel B) (α 1,,α ) (0.2, 0., 0.1), withτ 1 8, τ 2 17, λ10; (odel C) (α 1,,α ) (0.4, 0., 0.1), withτ 1 21, τ 2 4, λ50. We use the above odels to generate data and then use oent ethods to estiate the paraeters. We coputed the epirical bias and the ean square error () based on 00 replications for each paraeter cobination. These values are reported within parenthesis in Table 1. FrotheresultsinTable 1,wecanseeoentestiation is good estiation ethods producing estiators whose bias

5 Matheatical Probles in Engineering 5 and s are sall when the saple sizes are larger. In addition, this ethod is fast and easy to ipleent. It is perhaps not surprising that the s are larger when these saple sizes are saller. As to be expected, both the bias and thesconvergetozerowithincreasingsaplesizet. 5. Conclusion Based on soe liitations of the present count data odels, a new INMA odel is introduced to odel structural changes. Expressions for ean, variance, and autocorrelation functions are given. Stationary and other basic statistical properties are also obtained. We derived oent estiators of the unknown paraeters. Furtherore, we constructed several siulations to evaluate the perforance of the estiators of odel paraeters. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgents This work is supported by the National Natural Science Foundation of China (no ), the Science Foundation of Ministry of Education of China (no. 12XJC910001), and the Fundaental Research Funds for the Central Universities (nos. JBK and JBK120405). References [1] E. McKenzie, Soe siple odels for discrete variate tie series, Water Resources Bulletin,vol.21,no.4,pp ,1985. [2] E. McKenzie, Autoregressive oving-average processes with negative-binoial and geoetric arginal distributions, Advances in Applied Probability, vol.18,no.,pp , [] M. A. Al-Osh and A. A. Alzaid, First order integer-valued autoregressive INAR(1) process, Tie Series Analysis, vol.8,no.,pp ,1987. [4] A. A. Alzaid and M. Al-Osh, An integer-valued pth-order autoregressive structure INAR(p) process, Applied Probability,vol.27,no.2,pp.14 24,1990. [5] J. G. Du and Y. Li, The integer-valued autoregressive (INAR(p)) odel, Tie Series Analysis, vol. 12, no. 2, pp , [6] M. Al-Osh and A. A. Alzaid, Integer-valued oving average (INMA) process, Statistical Papers, vol.29,no.4,pp , [7] E. McKenzie, Soe ARMA odels for dependent sequences of Poisson counts, Advances in Applied Probability,vol.20,no. 4, pp , [8] R.Ferland,A.Latour,andD.Oraichi, Integer-valuedGARCH process, Tie Series Analysis, vol. 27, no. 6, pp , [9] K. Fokianos and R. Fried, Interventions in INGARCH processes, Tie Series Analysis, vol. 1, no., pp , [10] C. H. Weiß, The INARCH(1) odel for overdispersed tie series of counts, Counications in Statistics: Siulation and Coputation,vol.9,no.6,pp ,2010. [11] F. Zhu, A negative binoial integer-valued GARCH odel, Tie Series Analysis,vol.2,no.1,pp.54 67,2011. [12] F. Zhu, Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued GARCH odels, Matheatical Analysis and Applications,vol.89,no. 1,pp.58 71,2012. [1] F. Zhu and D. Wang, Diagnostic checking integer-valued ARCH(p) odels using conditional residual autocorrelations, Coputational Statistics and Data Analysis, vol.54,no.2,pp , [14] L. Györfi, M. Ispány, G. Pap, and K. Varga, Poisson liit of an inhoogeneous nearly critical INAR(1) odel, Acta Universitatis Szegediensis,vol.7,no.-4,pp ,2007. [15] H. S. Bakouch and M. M. Ristić, Zero truncated Poisson integer-valued AR(1) odel, Metrika, vol.72,no.2,pp , [16] M. Kachour and J. F. Yao, First-order rounded integer-valued autoregressive ((RINAR(1)) process, Tie Series Analysis, vol. 0, no. 4, pp , [17] H.-Y. Ki and Y. Park, A non-stationary integer-valued autoregressive odel, Statistical Papers, vol. 49, no., pp , [18] H. Zheng, I. V. Basawa, and S. Datta, First-order rando coefficient integer-valued autoregressive processes, Statistical Planning and Inference, vol.17,no.1,pp , [19] D. Goes and L. Canto e Castro, Generalized integer-valued rando coefficient for a first order structure autoregressive (RCINAR) process, Statistical Planning and Inference,vol.19,no.12,pp ,2009. [20] M. M. Ristić, A. S. Nastić,K.Jayakuar,andH.S.Bakouch, A bivariate INAR(1) tie series odel with geoetric arginals, Applied Matheatics Letters,vol.25,no.,pp ,2012. [21] X. Pedeli and D. Karlis, A bivariate INAR(1) process with application, Statistical Modelling, vol.11,no.4,pp.25 49, [22] P. Thyregod, J. Carstensen, H. Madsen, and K. Arnbjerg- Nielsen, Integer valued autoregressive odels for tipping bucket rainfall easureents, Environetrics,vol.10,no.4,pp , [2] M. Monteiro, M. G. Scotto, and I. Pereira, Integer-valued selfexciting threshold autoregressive processes, Counications in Statistics: Theory and Methods, vol.41,no.15,pp , [24] Š. Hudecová, Structural changes in autoregressive odels for binary tie series, Statistical Planning and Inference, vol. 14, no. 10, pp , 201. [25] A. DasGupta, Asyptotic Theory of Statistics and Probability, Springer,NewYork,NY,USA,2008. [26] P. J. Brockwell and R. A. Davis, TieSeries:TheoryandMethods, Springer,NewYork,NY,USA,1991. [27] K. Yu, D. Shi, and H. Zou, The rando coefficient discretevalued tie series odel, Statistical Research, vol. 28, no. 4, pp , [28] K. BräKnnäK and A. Hall, Estiation in integer-valued oving average odels, Applied Stochastic Models in Business and Industry,vol.17,no.,pp ,2001.

6 Advances in Operations Research Advances in Decision Sciences Applied Matheatics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Subit your anuscripts at International Advances in Cobinatorics Matheatical Physics Coplex Analysis International Matheatics and Matheatical Sciences Matheatical Probles in Engineering Matheatics Discrete Matheatics Discrete Dynaics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optiization