Bootstrapping INAR models

Integer-valued autoregressive (INAR) time series form a very useful class of processes suitable to model time series of counts. In the common formulation of Du and Li (1991,JTSA), INAR models of order p share the autocorrelation structure with classical autoregressive time series. This fact allows to estimate the INAR coeffcients, e.g., by Yule-Walker estimators. However, contrary to the AR case, consistent estimation of the model coeffcients turns out to be not suffcient to compute proper 'INAR residuals' to formulate a valid model-based bootstrap scheme. In this paper, we propose a general INAR-type bootstrap procedure. Based on mild regularity conditions and suitable meta assumptions, we prove bootstrap consistency of our pro- cedure for statistics belonging to the class of functions of generalized means. In particular, we discuss parametric and semi-parametric implementations of the INAR bootstrap scheme. The latter approach is based on a semi-parametric estimator suggested by Drost, van den Akker and Werker (2009, JRSSB) to estimate jointly the INAR coeffcients and the distribution of the innovations. In an extensive simulation study, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed INAR bootstrap over some obvious competitors. We illustrate our method by an application to a real data set about iceberg orders for the Lufthansa stock.