Linear type conditional specifications for multivariate count variables
Yang Lu, Wei Sun
TL;DR
The paper addresses the compatibility of linear conditional specifications for multivariate count data, focusing on two families—the compound autoregressive model and the random coefficient integer autoregressive model—and their extensions to arbitrary dimensions. It shows that, under full conditional specification, only a handful of non-trivial solutions exist, with a canonical high-dimensional solution given by Poisson–gamma conjugacy; when the requirement is relaxed to linear conditional expectations alone, a much larger semi-parametric class of solutions emerges. The results indicate that for spatial count data, semi-parametric specifications based on linear conditional expectations are more flexible and potentially preferable to latent-factor or Gaussian-based approaches. These findings have practical implications for modeling, estimation, and computation, suggesting that linear-type conditional specifications can be both tractable and sufficiently expressive, and they hint at future exploration of non-linear dependency networks in count data contexts.
Abstract
This paper investigates conditional specifications for multivariate count variables. Recently, the spatial count data literature has proposed several conditional models such that the conditional expectations are linear in the conditioning variables. These models are much easier to estimate than existing spatial count models based on Gaussian random field. However, whether or not such conditional specifications are compatible have not been addressed. We investigate two large families of conditional models, that are the compound autoregressive model and the random coefficient integer autoregressive model. We characterize all the solutions to these two families of models at arbitrary dimensions, and find that only a handful of them admit non-trivial solutions. We then show that if we focus on the linearity condition of the conditional expectations only, a considerable larger family of solutions can be obtained. This suggests that for spatial count data modeling, semi-parametric type specifications that impose the conditional expectation structure is preferable.