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In particular, this paper addresses solutions to the computational challenges encountered in estimating parameters in non-stationary over-dispersed bivariate integer-valued autoregressive of order 1 (BINAR(1)) model with Negative Binomial (NB) innovations. In this BINAR(1) model, the cross-correlation is induced through the paired NB innovations which follows a recently introduced bivariate NB model under different over-dispersion indices. The estimation of the model parameters is conducted via a two-phased generalized quasi-likelihood (GQL) approach but the second GQLs auto-covariance structure constitutes of higher-order moment entries which are not readily available in closed form. In this context, two GQL approaches: GQL MVN based on approximating the higher-order covariances through the multivariate normality structure and GQL BT based on deriving the exact higher-order covariances by some novel high-dimensional thinning properties are proposed and compared. The asymptotic properties of the respective GQLs are derived. Monte-Carlo simulation experiments are implemented to investigate on the performance of the GQLs, the consistency and the asymptotic efficiency of the estimates. The proposed model and the estimation methodologies are applied to a real-life time series data in the Transport sector in Mauritius. The root mean square error based on some out-sample statistics are also computed to assess the reliability of the model. [ABSTRACT FROM AUTHOR]