Noncausal counting processes: A queuing perspective

Abstract

We introduce noncausal counting processes, defined by timereversing 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.

Noncausal affine processes with applications to derivative pricing

“…4 of Online Appendix 3 for the plot of FED target rate with bubble feature. This process, recently introduced by Gouriéroux and Lu (2021), has the stochastic representation: $$$$\begin{equation*} y_t=\sum _{i=1}^{y_{t+1}} Z_{i,t}+\epsilon _t, \end{equation*}$$$$where $$(Z_{i,t})$$ is an i.i.d. sequence of Bernoulli variable with parameter p , and $$\epsilon _t$$ is count‐valued, independent of $$(Z_{i,t})_i$$, with the discrete stable distribution characterized by its LT: $$\mathbb {E}[e^{v\epsilon _t}]=\exp [-\beta (1-e^v)^\nu ]$$.…”

Forecast performance and bubble analysis in noncausal MAR(1, 1) processes

“…(i) From a theoretical point of view, it can be shown that the predictive density l h tends to a discrete distribution with weights p h (ψ) = 1−ψ h at 0 and 1−p h (ψ) at 1/ψ h (see Fries, (2018), for a proof for α-stable AR(1) shocks 8 and Gourieroux and Lu, (2019), proposition 8, for a noncausal count process).…”