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Asymptotic behavior of unstable INAR($p$) processes. (English) Zbl 1241.62122

The authors investigate the asymptotic properties of integer-valued autoregressive model $INAR\left(p\right)$ determined by the equation

${X}_{k}={\alpha }_{1}\circ {X}_{k-1}+\cdots +{\alpha }_{p}\circ {X}_{k-p}+{\epsilon }_{k},k\in ℕ,$

where $\alpha \circ X={\sum }_{j=1}^{X}{\xi }_{j}$ if $X>0$, $\alpha \circ X=0$ if $X>0$, ${\xi }_{j},\phantom{\rule{4pt}{0ex}}j\in ℕ$, are i.i.d. Bernoulli random variables with mean $\alpha \in \left[0,1\right]$, ${\epsilon }_{k}$ are i.i.d. non-negative integer-valued random variables with the mean ${\mu }_{\epsilon }$, and ${\alpha }_{1},\cdots ,{\alpha }_{p}\in \left[0,1\right]$. Under the assumption that the second moment of the innovation ${\epsilon }_{k}$ is finite the authors proved that the sequence of appropriately scaled random step functions ${X}_{t}^{n}={X}_{\left[nt\right]}/n$, $t\in {ℝ}_{+}$, $n\in ℕ$, formed from an unstable $INAR\left(p\right)$ process (${\alpha }_{1}+\cdots +{\alpha }_{p}=1$) converges weakly towards a squared Bessel process determined by the stochastic differential equation $d{X}_{t}=\left({\mu }_{\epsilon }dt+\sqrt{{\sigma }_{\alpha }^{2}{X}_{t}^{+}}d{W}_{t}\right)/{\varphi }^{\text{'}}\left(1\right)$, ${X}_{0}=0$, $t\in {ℝ}_{+}$, where ${\varphi }^{\text{'}}\left(1\right)={\alpha }_{1}+2{\alpha }_{2}+\cdots +p{\alpha }_{p}>0$, ${\sigma }_{\alpha }^{2}={\alpha }_{1}\left(1-{\alpha }_{1}\right)+2{\alpha }_{2}+\cdots +p{\alpha }_{p}\left(1-{\alpha }_{p}\right)$, ${W}_{t},\phantom{\rule{4pt}{0ex}}t\in {ℝ}_{+}$, is a standard Wiener process. This limit process is a continuous branching process also known as square-root process or Cox-Ingersoll-Ross process. This asymptotic behavior of unstable $INAR\left(p\right)$ models is quite different from that of familiar (real-valued) unstable autoregressive processes of order $p$ ($AR\left(p\right)$ models). An application to Boston armed robberies data is presented.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 60J80 Branching processes 60F99 Limit theorems (probability) 62P25 Applications of statistics to social sciences
##### References:
 [1] Al-Osh, M. A.; Alzaid, A. A.: First order integer-valued autoregressive $INAR\left(1\right)$ process, J. time ser. Anal. 8, No. 3, 261-275 (1987) · Zbl 0617.62096 · doi:10.1111/j.1467-9892.1987.tb00438.x [2] Al-Osh, M. A.; Alzaid, A. A.: An integer-valued pth-order autoregressive structure $\left(INAR\left(p\right)\right)$ process, J. appl. Probab. 27, No. 2, 314-324 (1990) · Zbl 0704.62081 · doi:10.2307/3214650 [3] M. Barczy, M. Ispány, G. Pap, Asymptotic behavior of unstable INAR(p) processes, 2009. Available on the arXiv: http://arxiv.org/abs/0908.4560. [4] Billingsley, P.: Convergence of probability measures, (1999) [5] Böckenholt, U.: Mixed $INAR\left(1\right)$ Poisson regression models: analyzing heterogeneity and serial dependencies in longitudinal count data, J. econometrics 89, 317-338 (1999) · Zbl 0958.62110 · doi:10.1016/S0304-4076(98)00069-4 [6] Box, G. E. P.; Tiao, G. C.: Intervention analysis with applications to economic and environmental problems, J. amer. Statist. assoc. 70, No. 349, 70-79 (1975) · Zbl 0316.62045 · doi:10.2307/2285379 [7] Brännäs, K.; Hellström, J.: Generalized integer-valued autoregression, Econometric rev. 20, 425-443 (2001) · Zbl 1077.62530 · doi:10.1081/ETC-100106998 [8] K. Brännäs, Q. Shahiduzzaman, Integer-valued moving average modelling of the number of transactions in stocks, Umea Economic Studies 637, University of Umeå, 2004. [9] Brualdi, R. A.; Cvetković, D.: A combinatorial approach to matrix theory and its applications, (2009) [10] Chan, N. H.; Wei, C. Z.: Limiting distributions of least squares estimates of unstable autoregressive processes, Ann. statist. 16, 367-401 (1988) · Zbl 0666.62019 · doi:10.1214/aos/1176350711 [11] Deutsch, S. J.; Alt, F. B.: The effect of massachusetts’ gun control law on gun-related crimes in the city of Boston, Eval. Q. 1, No. 4, 543-568 (1977) [12] Drost, F. C.; Den Akker, R. V.; Werker, B. J. M.: The asymptotic structure of nearly unstable non-negative integer-valued $AR\left(1\right)$ models, Bernoulli 15, No. 2, 297-324 (2009) · Zbl 1200.62105 · doi:10.3150/08-BEJ153 [13] Du, J. G.; Li, Y.: The integer valued autoregressive $\left(INAR\left(p\right)\right)$ model, J. time ser. Anal. 12, No. 2, 129-142 (1991) · Zbl 0727.62084 · doi:10.1111/j.1467-9892.1991.tb00073.x [14] Enciso-Mora, V.; Neal, P.; Rao, T. Subba: Efficient order selection algorithms for integer-valued ARMA processes, J. time ser. Anal. 30, No. 1, 1-18 (2009) · Zbl 1224.62053 · doi:10.1111/j.1467-9892.2008.00592.x [15] Ethier, S. N.; Kurtz, T. G.: Markov processes, (1986) [16] Franke, J.; Seligmann, T.: Conditional maximum-likelihood estimates for $INAR\left(1\right)$ processes and their applications to modelling epileptic seizure counts, Developments in time series, 310-330 (1993) · Zbl 0878.62080 [17] J. Franke, T. Subba Rao, Multivariate first order integer valued autoregressions, Technical Report, Math. Dep. UMIST, England, 1995. [18] Gauthier, G.; Latour, A.: Convergence FORTE des estimateurs des paramétres d’un processus $GENAR\left(p\right)$, Ann. sci. Math. Québec 18, No. 1, 49-71 (1994) · Zbl 0852.62082 · doi:http://www.lacim.uqam.ca/~annales/volumes/18-1/49.html [19] Gourieroux, C.; Jasiak, J.: Heterogeneous $INAR\left(1\right)$ model with application to car insurance, Insurance math. Econom. 34, 177-192 (2004) · Zbl 1107.62110 · doi:10.1016/j.insmatheco.2003.11.005 [20] Hay, R.; Mccleary, R.: Box–tiao times series models for impact assessment, Eval. Q. 3, No. 2, 277-314 (1979) [21] Hellström, J.: Unit root testing in integer-valued $AR\left(1\right)$ models, Econom. lett. 70, 9-14 (2001) · Zbl 0968.91029 · doi:10.1016/S0165-1765(00)00344-X [22] Horn, R. A.; Johnson, Ch.R.: Matrix analysis, (1985) [23] Ispány, M.; Pap, G.; Van Zuijlen, M. C. A.: Asymptotic inference for nearly unstable $INAR\left(1\right)$ models, J. appl. Probab. 40, No. 3, 750-765 (2003) · Zbl 1042.62080 · doi:10.1239/jap/1059060900 [24] Ispány, M.; Pap, G.: A note on weak convergence of step processes, Acta math. Hungar. 126, No. 4, 381-395 (2010) [25] Jacod, J.; Shiryaev, A. N.: Limit theorems for stochastic processes, (2003) [26] Jeganathan, P.: On the asymptotic behavior of least squares estimators in AR time series with roots near the unit circle, Econom. theory 7, 269-306 (1991) [27] Kallenberg, O.: Foundations of modern probability, (1997) [28] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) [29] Klimko, L. A.; Nelson, P. I.: On conditional least squares estimation for stochastic processes, Ann. statist. 6, No. 3, 629-642 (1978) · Zbl 0383.62055 · doi:10.1214/aos/1176344207 [30] Latour, A.: The multivariate $GINAR\left(p\right)$ process, Adv. in appl. Probab. 29, 228-248 (1997) · Zbl 0871.62073 · doi:10.2307/1427868 [31] Latour, A.: Existence and stochastic structure of a non-negative integer-valued autoregressive processes, J. time ser. Anal. 19, No. 4, 439-455 (1998) · Zbl 1127.62402 · doi:10.1111/1467-9892.00102 [32] B.P.M. McCabe, G.M. Martin, D. Harris, Optimal probabilistic forecasts for counts, Monash University, Working Paper 7/09, 2009. [33] Mckenzie, E.: Some simple models for discrete variate time series, Water resour. Bull. 21, 645-650 (1985) [34] O’donovan, T. M.: Short term forecasting: an introduction to the box–Jenkins approach, (1983) · Zbl 0565.62078 [35] Pavlopoulos, H.; Karlis, D.: $INAR\left(1\right)$ modeling of overdispersed count series with an environmental application, Environmetrics 19, 369-393 (2008) [36] Phillips, P. C. B.; Xiao, Z.: A primer on unit root testing, J. econom. Surv. 12, 423-470 (1998) [37] Revuz, D.; Yor, M.: Continuous martingales and Brownian motion, (2001) [38] Rudholm, N.: Entry and the number of firms in the swedish pharmaceuticals market, Rev. ind. Organ. 19, 351-364 (2001) [39] Steutel, F.; Van Harn, K.: Discrete analogues of self-decomposability and stability, Ann. probab. 7, 893-899 (1979) · Zbl 0418.60020 · doi:10.1214/aop/1176994950 [40] Steutel, F. W.; Van Harn, K.: Infinite divisibility of probability distributions on the real line, (2004) [41] Thyregod, P.; Carstensen, J.; Madsen, H.; Arnbjerg-Nielsen, K.: Integer valued autoregressive models for tipping bucket rainfall measurements, Environmetrics 10, 395-411 (1999) [42] Van Der Meer, T.; Pap, G.; Van Zuijlen, M. C. A.: Asymptotic inference for nearly unstable $AR\left(p\right)$ processes, Econom. theory 15, 184-217 (1999) · Zbl 0967.62072 · doi:10.1017/S0266466699152034 [43] Wei, C. Z.; Winnicki, J.: Some asymptotic results for the branching process with immigration, Stochastic process. Appl. 31, No. 2, 261-282 (1989) · Zbl 0673.60092 · doi:10.1016/0304-4149(89)90092-6 [44] Wei, C. Z.; Winnicki, J.: Estimation of the means in the branching process with immigration, Ann. statist. 18, No. 4, 1757-1773 (1990) · Zbl 0736.62071 · doi:10.1214/aos/1176347876 [45] Weiß, C. H.: Thinning operations for modelling time series of counts–a survey, Asta adv. Stat. anal. 92, No. 3, 319-341 (2008)