Bivariate rounded Z-valued autoregressive models with flexible auto- and cross-correlations

Abstract

Discrete time series data include count values and $Z$ values. Count time series models have been well studied in the literature, and univariate $Z$-valued models also receive their own advances and progress, but the bivariate $Z$-valued models are rare. Existing bivariate $Z$-valued models are constructed using the thinning operator, which limits their flexibility. The recently rediscovered univariate mean-preserving rounded operator can break these limits and we extend it to the bivariate case. This paper introduces a novel class of bivariate $Z$-valued autoregressive models based on a new kind of bivariate Skellam distribution and the bivariate mean-preserving rounded operator, a pth-order model without pairwise interaction and a first-order model with pairwise interaction, which provides very flexible range of auto- and cross-correlations. The stationarity conditions and some stochastic properties (including expressions for unconditional mean, variance, autocorrelation function, cross-correlation function, and conditional probability distribution) of the new models are given. An easy-to-implement two-step conditional least squares estimation procedure for the parameters and related large-sample asymptotic properties are provided. Simulations and three real data examples from different fields are illustrated to demonstrate the superiority of the new models compared with existing ones.