Modelling a non-stationary BINAR(1) Poisson process

A non‐stationary bivariate INAR(1) process with a simple cross‐dependence: Estimation with some properties

“…This section describes the new estimation approach to obtain the parameter estimators under the proposed bivariate model in (1)–(2). It is worth mentioning that under the non‐stationarity assumptions, the full likelihood is unfeasible due to the complex joint pgf whilst the existing popular approaches such as the CML, GMM and GLS are plausible to implement (see Mamode Khan, Sunecher & Jowaheer 2016b). We propose here a new formulated GQL to estimate the unknown regression and dependence effects.…”

“…This cycle continues until the convergence criterion . The R codes to implement the GQL iteration follows from Mamode Khan, Sunecher & Jowaheer (2016b). Under the GQL (20), the general form of the estimating equation is expressed as where ψ is the set of unknown parameters.…”

“…On the other side, for non‐stationary bivariate processes, Mamode Khan, Sunecher & Jowaheer (2016b) proposed the simple BINAR(1) with Poisson innovations under the assumption of cross‐correlated innovations and where the count series were influenced by time‐dependent covariates (NBINAR(1)P). In the same direction, Sunecher, Mamodekhan & Jowaheer () and Jowaheer, Mamode Khan & Sunecher () developed the non‐stationary BINAR(1)s with NB (NBINAR(1)N), COM‐Poison (NBINAR(1)C) and geometric (NBINAR(1)G) innovations to model the overdispersed series.…”

“…For the inferential procedures, it is important to note that under these non‐stationary BINAR(1)s the formation of the full likelihood equation becomes hard (see Nastic, Laketa & Ristic 2016a). However, the specification of the conditional maximum likelihood (CML) is possible for these non‐stationary cases as shown in Mamode Khan, Sunecher & Jowaheer (2016b). In addition, these researchers applied the generalised least squares (GLS) and generalised method of moment (GMM) approaches and proposed a novel generalised quasi‐likelihood (GQL) estimating equation.…”

“…The GQL is viewed as an attractive approach because it depends only on the correct specification of the mean scores and auto‐covariance structures or the first two moment specifications of the innovations which are derived from the model. This approach was initially proposed by Mamode Khan, Sunecher & Jowaheer (2016b) in the context of the BINAR(1)P where the authors clearly demonstrate that GQL yields estimators with lower bias than GLS and GMM and yield asymptotically equally efficient estimators as CML. In a recent paper by Mamode Khan et al .…”