Mixed causal-noncausal count process

Recently, Gouriéroux and Lu (Electron J Stat 15(2):3852–3891, 2021) introduced a class of (Markov) noncausal count processes. These processes are obtained by time-reverting a standard count process (such as INAR(1)), but have quite different dynamic properties. In particular, they can feature bubble-type phenomena, which are epochs of steady increase, followed by sharp decreases. This is in contrast to usual INAR and INGARCH type models, which only feature “reverse bubbles”, that are epochs of sharp increase followed by steady decreases. In practice, however, in many datasets, sudden jumps and crashes are rare, while it is more frequent to observe epochs of steady increase or decrease. This paper introduces the mixed causal-noncausal integer-valued autoregressive (m-INAR(1,1)) process, obtained by superposing a causal and a noncausal INAR(1) process sharing the same sequence of error terms. We show that this process inherits some key properties from the noncausal INAR(1), such as the bi-modality of the predictive distribution and the irreversibility of the dynamics, while at the same time allowing different accumulation and burst speeds for the bubble. We propose a GMM estimation method, investigate its finite sample performance, develop testing procedures, and apply the methodology to stock transaction data.