Coherent forecasting in integer time series models

Abstract

Our principal focus is on forecasting methods suitable for a certain class of observation-driven time series models for counts. Integer-valued autoregressive (INAR) models may be attractive when the data exhibit a significant serial dependence structure. Having briefly reviewed the familiar first order Markov model, we give an account of the extension of the method of moments estimation procedures to higher order INAR models, concentrating on the second order case. We provide means of obtaining estimated standard errors which are not easily found by analytical methods. Throughout the paper the methods are illustrated using a well known test data set. These models seem particularly useful in the context of forecasting, especially if the integer nature of the data is to be acknowledged in the modelling exercise. A computer intensive method for generating coherent, integer out-of-sample predictions is proposed and used in the context of the data. Distributions are generated for multi-step and also for sequences of one-step rolling/recursive forecasts. Block-of-blocks bootstrap techniques are used for estimating asymptotic standard errors and the results of the exercise are central in allowing for parameter uncertainty in the forecast distributions.