The Non-Stationary BINARMA(1,1) Process with Poisson Innovations: An Application on Accident Data

Authors: Y. Sunecher, N. Mamode Khan, V. Jowaheer

Abstract:

This paper considers the modelling of a non-stationary bivariate integer-valued autoregressive moving average of order one (BINARMA(1,1)) with correlated Poisson innovations. The BINARMA(1,1) model is specified using the binomial thinning operator and by assuming that the cross-correlation between the two series is induced by the innovation terms only. Based on these assumptions, the non-stationary marginal and joint moments of the BINARMA(1,1) are derived iteratively by using some initial stationary moments. As regards to the estimation of parameters of the proposed model, the conditional maximum likelihood (CML) estimation method is derived based on thinning and convolution properties. The forecasting equations of the BINARMA(1,1) model are also derived. A simulation study is also proposed where BINARMA(1,1) count data are generated using a multivariate Poisson R code for the innovation terms. The performance of the BINARMA(1,1) model is then assessed through a simulation experiment and the mean estimates of the model parameters obtained are all efficient, based on their standard errors. The proposed model is then used to analyse a real-life accident data on the motorway in Mauritius, based on some covariates: policemen, daily patrol, speed cameras, traffic lights and roundabouts. The BINARMA(1,1) model is applied on the accident data and the CML estimates clearly indicate a significant impact of the covariates on the number of accidents on the motorway in Mauritius. The forecasting equations also provide reliable one-step ahead forecasts.

Keywords: Non-stationary, BINARMA(1, 1) model, Poisson Innovations, CML

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3593124

References:

[1] E. McKenzie, “Some ARMA models for dependent sequences of Poisson counts,” Advances in Applied Probability, vol. 20, pp. 822–835, 1988.
[2] M. Al Osh and A. Alzaid, “First-order integer-valued autoregressive process,” Journal of Time Series Analysis, vol. 8, pp. 261–275, 1987.
[3] K. Brannas, “Explanatory variable in the AR(1) count data model,” Umea University, Department of Economics, vol. No.381, pp. 1–21, 1995.
[4] V. Jowaheer and B. Sutradhar, “Fitting lower order nonstationary autocorrelation models to the time series of Poisson counts,” Transactions on Mathematics, vol. 4, pp. 427–434, 2005.
[5] N. Mamode Khan and V. Jowaheer, “Comparing joint GQL estimation and GMM adaptive estimation in COM-Poisson longitudinal regression model,” Commun Stat-Simul C., vol. 42(4), pp. 755–770, 2013.
[6] K. Brannas and A. Quoreshi, “Integer-valued moving average modelling of the number of transactions in stocks,” Applied Financial Economics, vol. No.20(18), pp. 1429–1440, 2010.
[7] M. Al Osh and A. Alzaid, “Integer-valued moving average (INMA) process,” Statistical Papers, vol. 29, pp. 281–300, 1988a.
[8] A. Nastic, P. Laketa, and M. Ristic, “Random environment integer-valued autoregressive process,” Journal of Time Series Analysis, vol. 37(2), pp. 267–287, 2016.
[9] X. Pedeli and D. Karlis, “Bivariate INAR(1) models,” Athens University of Economics, Tech. Rep., 2009.
[10] X. Pedeli and D.Karlis, “Some properties of multivariate INAR(1) processes.” Computational Statistics and Data Analysis, vol. 67, pp. 213–225, 2013.
[11] P. Popovic, M. Ristic, and A. Nastic, “A geometric bivariate time series with different marginal parameters,” Statistical Papers, vol. 57, pp. 731–753, 2016.
[12] M. Ristic, A. Nastic, K. Jayakumar, and H. Bakouch, “A bivariate INAR(1) time series model with geometric marginals,” Applied Mathematical Letters, vol. 25(3), pp. 481–485, 2012.
[13] Y. Sunecher, N. Mamodekhan, and V. Jowaheer, “A gql estimation approach for analysing non-stationary over-dispersed BINAR(1) time series,” Journal of Statistical Computation and Simulation, 2017.
[14] A. Quoreshi, “Bivariate time series modeling of financial count data,” Communication in Statistics-Theory and Methods, vol. 35, pp. 1343–1358, 2006.
[15] A.M.M.S.Quoreshi, “A vector integer-valued moving average model for high frequency financial count data,” Economics Letters, vol. 101, pp. 258–261, 2008.
[16] Y. Sunecher, N. Mamodekhan, and V. Jowaheer, “Estimating the parameters of a BINMA Poisson model for a non-stationary bivariate time series,” Communication in Statistics: Simulation and Computation, vol.
[Accepted for Publication on 27 June 2016], 2016.
[17] C. Weib, M. Feld, N. Mamodekhan, and Y. Sunecher, “Inarma modelling of count series,” Stats, vol. 2, pp. 289–320, 2019.
[18] S. Kocherlakota and K. Kocherlakota, “Regression in the bivariate Poisson distribution,” Communications in Statistics-Theory and Methods, vol. 30(5), pp. 815–825, 2001.
[19] F. Steutel and K. Van Harn, “Discrete analogues of self-decomposability and statibility,” The Annals of Probability, vol. 7, pp. 3893–899, 1979.
[20] I. Yahav and G. Shmueli, “On generating multivariate Poisson data in management science applications,” Applied Stochastic Models in Business and Industry, vol. 28(1), pp. 91–102, 2011.