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

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.

[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,
[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.