Exploring the Bivariate Processes—The Bivariate INAR (1) Model with Paired Poisson—Weighted Exponential Distributions

The growing number of applications that involve time series of counts has urged for more suitable integer-valued time series models that can handle the commonest phenomenon of over-dispersion, while taking into consideration the instances where two related series are collected Pedeli and Karlis [2011 ], Ristic et al. [ 2012 ], Pedeli and Karlis [2013aa ], Mamode Khan et al. [2016 ], Nastic et al. [2016 ], Sunechar et al. [2017 ], Jowaheer et al. [2017 ], Sunechar et al. [2018a ], Mamode Khan et al. [2019 ]. Admittedly, in the literature, since the pioneered work of Steutel and Van Harn [1979 ], McKenzie [1986 ], Al Osh and Alzaid [1987 ], McKenzie [1988 ], Alzaid and Al Osh [1990 ], Al Osh and Aly [1992 ], there has been significant progress in the development of simple integer-valued auto-regressive time series models of order 1 (INAR (1)) with different innovation distributions Ristic et al. [2009 ], Bakouch and Ristic [2010 ], Schweer and Weiß [2014 ], Bourguignon and Vasconcellos [2015 ], Livio et al. [2018 ], Mohammadpour et al. [2018 ], Bakouch et al. [2021 ], Sharafi et al. [2021 ] and the references therein. These various INAR (1)s compete to provide the most superior fitting criteria. Until recently, Altun [2019 ] proposed an alternative INAR (1) with PWE, that is deemed suitable to model over-dispersed series while providing lower Akaike Information Criterion. In his work, Altun [2019 ] applied the developed INAR (1) with PWE innovations to the Shots and Drugs data from the Police Car beat in Pittsburgh, and noted that the INAR (1) PWE yielded better AICs than the INAR (1) with Poisson Lindley and Generalized Poisson distributions and this conclusion is rather motivating to further explore the PWE in other important settings such as in the BINAR case, and thereon compare with other bivariate INAR (1) processes as discussed in Pedeli and Karlis [2011 ] and assess its performance versus the other popular BINAR (1) models. In this process, we first review the properties of the PWE and its adaption to the INAR (1) process. From thereon to mount the bivariate INAR (1) process with PWE innovations, we propose to induce a cross correlation between the counting series by assuming the paired PWE innovations are jointly distributed. Up to this extent, there is yet no bivariate PWE distribution, and in this book, we therefore propose to construct flexible bivariate PWE model using the approaches by Gomez Deniz et al. [2012 ] and Bermudez and Karlis [2021 ] that can account for both positive and negative cross correlation. The properties of the bivariate PWE models will be derived, followed by the BINAR (1) process where the paired innovations are assumed to follow the proposed bivariate PWE distributions. Since the bivariate PWE model is novel, we shall also present some Monte Carlo simulation results to assess the consistency of the bivariate PWE parameters. Estimation of the parameters will be done using the CML and CLS approaches and finally the application will be on real-life intra-day transaction series of some popular stocks and after comparing the new BINAR (1)s with PWE innovations with other popular BINAR (1) models, proper conclusion will be drawn.