Abstract
A class of bivariate integer-valued time series models was constructed via copula theory. Each series follows a Markov chain with the serial dependence captured using copula-based transition probabilities from the Poisson and the zero-inflated Poisson (ZIP) margins. The copula theory was also used again to capture the dependence between the two series using either the bivariate Gaussian or “t-copula” functions. Such a method provides a flexible dependence structure that allows for positive and negative correlation, as well. In addition, the use of a copula permits applying different margins with a complicated structure such as the ZIP distribution. Likelihood-based inference was used to estimate the models’ parameters with the bivariate integrals of the Gaussian or t-copula functions being evaluated using standard randomized Monte Carlo methods. To evaluate the proposed class of models, a comprehensive simulated study was conducted. Then, two sets of real-life examples were analyzed assuming the Poisson and the ZIP marginals, respectively. The results showed the superiority of the proposed class of models.
Highlights
Following a similar framework of building bivariate models for ordinal panel data via [1], we constructed a class of bivariate integer-valued time series models using copula theory
To accurately study such data, one needs to account for the two types of dependence that emerge from the observed data by applying multivariate time series models
For each univariate time series, we considered a copula-based Markov model, where a copula family was used for the joint distribution of subsequent observations, and coupled these two time series using another copula at each time point
Summary
Following a similar framework of building bivariate models for ordinal panel data via [1], we constructed a class of bivariate integer-valued time series models using copula theory. Following the concepts of the integer-valued autoregressive moving average (INARMA) model, Reference [5] introduced the bivariate integer-valued moving average (BINMA) model, which allows for both positive and negative correlation between counts They presented an extension to the multivariate version starting from the BINMA model. The work of [8] presented a bivariate integer-valued autoregressive process (BINAR(1)) in which the cross-correlation was modeled using a copula to accommodate both positive and negative correlation They presented the use of a Frank and Gaussian copula to model dependence, and marginal time series were modeled using Poisson and negative binomial INAR(1) models.
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