Nearly unstable integer-valued ARCH process and unit root testing. (English) Zbl 07819235
Summary: This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroscedastic (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process weakly converges to a Cox-Ingersoll-Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type-I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID-19 in the United Kingdom.
© 2023 The Authors. Scandinavian Journal of Statistics
© 2023 The Authors. Scandinavian Journal of Statistics
MSC:
| 62-XX | Statistics |
Keywords:
count time series; Cox-Ingersoll-Ross diffusion process; inference; limit theorems; stochastic integralSoftware:
RReferences:
| [1] | Al‐Osh, M. A., & Alzaid, A. A. (1987). First‐order integer valued autoregressive (INAR(1)) process. Journal of Time Series Analysis, 8, 261-275. · Zbl 0617.62096 |
| [2] | Barczy, M., Ispány, M., & Pap, G. (2011). Asymptotic behavior of unstable INAR(p) processes. Stochastic Processes and their Applications, 121, 583-608. · Zbl 1241.62122 |
| [3] | Barczy, M., Ispány, M., & Pap, G. (2014). Asymptotic behavior of conditional least squares estimators for unstable integer‐valued autoregressive models of order 2. Scandinavian Journal of Statistics, 41, 866-892. · Zbl 1305.62321 |
| [4] | Barczy, M., Körmendi, K., & Pap, G. (2016). Statistical inference for critical continuous state and continuous time branching processes with immigration. Metrika, 79, 789-816. · Zbl 1346.62046 |
| [5] | Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307-327. · Zbl 0616.62119 |
| [6] | Chan, N. H., Ing, C.‐K., & Zhang, R. (2019). Nearly unstable processes: A prediction perspective. Statistica Sinica, 29, 139-163. · Zbl 1412.62124 |
| [7] | Chan, N. H., & Wei, C.‐Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics, 15, 1050-1063. · Zbl 0638.62082 |
| [8] | Christou, V., & Fokianos, K. (2015). Estimation and testing linearity for non‐linear mixed Poisson autoregressions. Electronic Journal of Statistics., 9, 1357-1377. · Zbl 1327.62456 |
| [9] | Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385-407. · Zbl 1274.91447 |
| [10] | Davis, R. A., & Liu, H. (2016). Theory and inference for a class of nonlinear models with applications to time series of counts. Statistica Sinica, 26, 1673-1707. · Zbl 1356.62137 |
| [11] | Deelstra, G., & Delbaen, F. (1998). Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Applied Stochastic Models and Data Analysis, 14, 77-84. · Zbl 0915.60064 |
| [12] | Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-431. · Zbl 0413.62075 |
| [13] | Drost, F. C., Van Den Akker, R., & Werker, B. J. M. (2009). The asymptotic structure of nearly unstable non‐negative integer‐valued AR(1) models. Bernoulli, 15, 297-324. · Zbl 1200.62105 |
| [14] | Ethier, S. N., & Kurtz, T. G. (1986). Markov processes: Characterization and convergence. Wiley. · Zbl 0592.60049 |
| [15] | Ferland, R., Latour, A., & Oraichi, D. (2006). Integer‐valued GARCH process. Journal of Time Series Analysis, 27, 923-942. · Zbl 1150.62046 |
| [16] | Fokianos, K., & Fried, R. (2010). Interventions in INGARCH processes. Journal of Time Series Analysis, 31, 210-225. · Zbl 1242.62095 |
| [17] | Fokianos, K., Rahbek, A., & Tjøstheim, D. (2009). Poisson autoregression. Journal of the American Statistical Association, 104, 1430-1439. · Zbl 1205.62130 |
| [18] | Fokianos, K., & Tjøstheim, D. (2011). Log‐linear Poisson autoregression. Journal of Multivariate Analysis, 102, 563-578. · Zbl 1207.62165 |
| [19] | Gonçalves, E., Mendes‐Lopes, N., & Silva, F. (2015). Infinitely divisible distributions in integer‐valued GARCH models. Journal of Time Series Analysis, 36, 503-527. · Zbl 1325.62165 |
| [20] | Guo, H., & Zhang, M. (2014). A fluctuation limit theorem for a critical branching process with dependent immigration. Statistics & Probability Letters, 94, 29-38. · Zbl 1320.60143 |
| [21] | Hellström, J. (2001). Unit root testing in integer‐valued AR(1) models. Economics Letters, 70, 9-14. · Zbl 0968.91029 |
| [22] | Ispány, M., Körmendi, K., & Pap, G. (2014). Asymptotic behavior of CLS estimators for 2‐type doubly symmetric critical Galton‐Watson processes with immigration. Bernoulli, 20, 2247-2277. · Zbl 1321.60178 |
| [23] | Ispány, M., Pap, G., & Van Zuijlen, M. C. A. (2003). Asymptotic inference for nearly unstable INAR(1) models. Journal of Applied Probability, 40, 750-765. · Zbl 1042.62080 |
| [24] | Ispány, M., Pap, G., & Van Zuijlen, M. C. A. (2005). Fluctuation limit of branching processes with immigration and estimation of the means. Adv. in Appl. Probab., 37, 523-538. · Zbl 1069.62065 |
| [25] | McKenzie, E. (1985). Some simple models for discrete variate time series. Water Resources Bulletin, 21, 645-650. |
| [26] | Phillips, P. C. B. (1987). Towards a unified asymptotic theory for autoregression. Biometrika, 74, 535-547. · Zbl 0654.62073 |
| [27] | R Development Core Team. (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing. http://www.R‐project.org |
| [28] | Rahimov, I. (2007). Functional limit theorems for critical processes with immigration. Advances in Applied Probability, 39, 1054-1069. · Zbl 1205.60150 |
| [29] | Rahimov, I. (2008). Asymptotic distribution of the CLSE in a critical process with immigration. Stochastic Processes and their Applications, 118, 1892-1908. · Zbl 1213.60135 |
| [30] | Rahimov, I. (2009). Asymptotic distributions for weighted estimators of the offspring mean in a branching process. Test, 18, 568-583. · Zbl 1203.60131 |
| [31] | Rogers, L. C. G., & Williams, D. (2000). Diffusions, Markov processes and martingales. Cambridge University Press. · Zbl 0949.60003 |
| [32] | Silva, R. B., & Barreto‐Souza, W. (2019). Flexible and robust mixed Poisson INGARCH models. Journal of Time Series Analysis, 40, 788-814. · Zbl 1431.62350 |
| [33] | Sriram, T. N. (1994). Invalidity of bootstrap for critical branching process with immigration. Annals of Statistics, 22, 1013-1023. · Zbl 0807.62063 |
| [34] | Steutel, F. W., & vanHarn, K. (1990). Discrete analogues of self‐decomposability and stability. Annals of Probability, 7, 893-899. · Zbl 0418.60020 |
| [35] | Tjøstheim, D. (1986). Estimation in nonlinear time series models. Stochastic Processes and their Applications, 21, 251-273. · Zbl 0598.62109 |
| [36] | Wei, C. Z., & Winnicki, J. (1989). Some asymptotic results for the branching process with immigration. Stochastic Processes and their Applications, 31, 261-282. · Zbl 0673.60092 |
| [37] | Wei, C. Z., & Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Annals of Statistics, 18, 1757-1773. · Zbl 0736.62071 |
| [38] | Weiß, C. H. (2010). The INARCH(1) model for overdispersed time series of counts. Communications in Statistics: Simulation and Computation, 39, 1269-1291. |
| [39] | Weiß, C. H. (2015). A Poisson INAR(1) model with serially dependent innovations. Metrika, 78, 829-851. · Zbl 1333.62218 |
| [40] | Weiß, C. H., Zhu, F., & Hoshiyar, A. (2022). Softplus INGARCH models. Statistica Sinica, 32, 1099-1120. · Zbl 1524.62457 |
| [41] | Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Probability Theory and Related Fields, 88, 77-106. · Zbl 0736.62072 |
| [42] | Zhu, F. (2011). A negative binomial integer‐valued GARCH model. Journal of Time Series Analysis, 32, 34-67. · Zbl 1290.62092 |
| [43] | Zhu, F. (2012). Modeling overdispersed or underdispersed count data with generalized Poisson integer‐valued GARCH models. Journal of Mathematical Analysis and Applications, 389, 58-71. · Zbl 1232.62120 |
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