Modelling non‐stationary multivariate time series of counts via common factors

We develop a new parameter‐driven model for multivariate time series of counts. The time series is not necessarily stationary. We model the mean process as the product of modulating factors and unobserved stationary processes. The former characterizes the long‐run movement in the data, whereas the latter is responsible for rapid fluctuations and other unknown or unavailable covariates. The unobserved stationary processes evolve independently of the past observed counts and might interact with each other. We express the multivariate unobserved stationary processes as a linear combination of possibly low dimensional factors that govern the contemporaneous and serial correlation within and across the observed counts. Regression coefficients in the modulating factors are estimated via pseudo‐maximum‐likelihood estimation, and identification of common factor(s) is carried out through eigenanalysis on a positive definite matrix that pertains to the autocovariance of the observed counts at non‐zero lags. Theoretical validity of the two‐step estimation procedure is documented. In particular, we establish consistency and asymptotic normality of the pseudo‐maximum‐likelihood estimator in the first step and the convergence rate of the second‐step estimator. We also present an exhaustive simulation study to examine the finite sample performance of the estimators, and numerical results corroborate our theoretical findings. Finally, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings awarded to seven research universities from January 2001 to December 2012.