Abstract
We introduce noncausal counting processes, defined by time-reversing an INAR(1) process, a non-INAR(1) Markov affine counting process, or a random coefficient INAR(1) [RCINAR(1)] process. The noncausal processes are shown to be generically time irreversible and their calendar time dynamic properties are unreplicable by existing causal models. In particular, they allow for locally bubble-like explosion, while at the same time preserving stationarity. Many of these processes have also closed form calendar time conditional predictive distribution, and allow for a simple queuing interpretation, similar as their causal counterparts.
Highlights
Real-valued linear noncausal processes have raised much interest thanks to their ability to capture irreversible1, bubble-like phenomena, that are widely observed in financial applications [see e.g. [26, 20]]
We briefly review and illustrate this method using two further examples of noncausal INAR(1) processes with respectively negative binomial and Poisson-Inverse Gaussian distributed shocks
In this paper we have introduced the concept of noncausality for counting processes by time reversing a INAR(1), a Markov affine process, or a RCINAR(1)
Summary
Real-valued linear noncausal processes have raised much interest thanks to their ability to capture irreversible, bubble-like phenomena, that are widely observed in financial applications [see e.g. [26, 20]]. C) Alternatively, one can extend the INAR(1) model a) by relaxing the i.i.d. assumption of the Zi,t+1’s and allow them to be only conditionally i.i.d. given a stochastic probability parameter pt+1, which itself is an i.i.d. sequence that is independent of t+1, as well as of past observations Xt. In the counting process context, [28] shows that they have tractable marginal and predictive distributions at any horizons. We show that for noncausal INAR(1), an alternative, equivalent specification is based on the aforementioned queuing interpretation, but with a different set of distributional assumptions. This new queuing model is specified through the departure cohorts, rather than the arrival cohorts. As well as some results for noncausal RCINAR(1) processes are gathered in Appendices
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