Count network autoregression

Abstract

We consider network autoregressive models for count data with a non‐random neighborhood structure. The main methodological contribution is the development of conditions that guarantee stability and valid statistical inference for such models. We consider both cases of fixed and increasing network dimension and we show that quasi‐likelihood inference provides consistent and asymptotically normally distributed estimators. The article is complemented by simulation results and a data example.

Inference for Network Count Time Series with the R Package PNAR

“…where n i = ∑ j̸ =i a ij is the total number of connections starting from the node i, such that i → j; called out-degree. We call (1) linear Poisson Network Autoregression of order 1, abbreviated by PNAR(1); (Armillotta and Fokianos, 2023a). From the left hand side equation of (1), we observe that the process Y i,t is assumed to be marginally Poisson but the joint process depends upon a copula function described in simulations at the end of the paper.…”

“…. , N. If p = 1 and q = 0 set Armillotta and Fokianos, 2023a). The linear form of (2) offers a great advantage interpreting the parameters but it accommodates positive covariates.…”

“…include exp(Z) instead of directly Z). This restriction is bypassed by the log-linear model (Armillotta and Fokianos, 2023a):…”

“…The unknown parameters of the model are estimated by Least Squares estimation (LS). This work was advanced by Armillotta and Fokianos (2023a) who developed linear and log-linear Poisson Network Autoregression model (PNAR) for multivariate count distributed data. The joint dependence among different variables is specified by a copula construction (Fokianos et al, 2020, Sec.…”

“…2). In addition, Armillotta and Fokianos (2023a) have further established parametric estimation under the framework of quasi maximum likelihood inference (see Wedderburn (1974) Gourieroux et al (1984) and associated asymptotic theory when the network dimension increases. In the context of epidemiology, related applied work has been developed by Held et al (2005) for the linear model only, and it was extended by Paul et al (2008); Paul and Held (2011); Held and Paul (2012); Meyer and Held (2014) and Bracher and Held (2020).…”