Predictive likelihood for coherent forecasting of count time series

• Date: 01 April 2019
• DOI: http://doi.org/10.1002/for.2566
• Published date: 01 April 2019
• Author: Vurukonda Sathish,Siuli Mukhopadhyay

Received: 7 May 2018 Revised: 15 November 2018 Accepted: 19 November 2018

DOI: 10.1002/for.2566

RESEARCH ARTICLE

Predictive likelihood for coherent forecasting of count time

series

Siuli Mukhopadhyay1Vurukonda Sathish2

1Department of Mathematics, Indian

Institute of Technology Bombay, Mumbai,

India

2Department of Electrical Engineering,

Indian Institute of Technology Bombay,

Mumbai, India

Correspondence

Siuli Mukhopadhyay,Department of

Mathematics, Indian Institute of

Technology Bombay, Mumbai 400 076,

India.

Email: siuli@math.iitb.ac.in

Funding information

Council for Scientific and Industrial

Research

Abstract

A new forecasting method based on the concept of the profile predictive like-

lihood function is proposed for discrete-valued processes. In particular, gener-

alized autoregressive moving average (GARMA) models for Poisson distributed

data are explored in detail. Highest density regions are used to construct fore-

casting regions. The proposed forecast estimates and regions are coherent.

Large-sample results are derived for the forecasting distribution. Numerical

studies using simulations and two real data sets are used to establish the perfor-

mance of the proposed forecasting method. Robustness of the proposed method

to possible misspecifications in the model is also studied.

KEYWORDS

GARMA models, highest density regions, observation driven models, partial likelihood function,

profile likelihood

1INTRODUCTION

In contrast to time series for Gaussian responses, where

numerous forecasting methods are available, literature

on forecasting for count type time series is still very

sparse. However, time series data in the form of counts

is frequently measured in various fields such as finance,

insurance, biomedical, and public health. As an example,

consider a disease surveillance study, where health offi-

cials record the number of disease cases over a certain

time period to understand the disease trajectory. The

main interest in such surveillance studies is to forecast

disease counts in the future, so that public healthcare

providers are able to respond to disease outbreaks on time,

thereby reducing the disease impact and saving economic

resources (Myers, Rogers, Cox, Flahault, & Hay, 2000).

However, forecasting disease counts in these situations is

complex, due to the fact that the required forecasts have

to be consistent with the nonnegative and integer-valued

sample space of such count time series. Usage of the esti-

mated mean at a future time point which is a noninteger,

as a suitable point estimate for the future count (Andrade,

Andrade, & Ehlers, 2015; Davis, Dunsmuir,& Streett, 2003;

Jalalpour, Gel, & Levin, 2015) as practicedin usual autore-

gressive integrated moving average (ARIMA) forecasting

techniques gives rise to an incoherent forecast (Freeland &

McCabe, 2004).

In this article, we present a forecasting method based

on the profile predictive likelihood function for count

time series data. The proposed forecast estimates and

regions are coherent. The class of models we consider

is the observation-driven models (Benjamin, Rigby, &

Stasinopoulos, 2003; Fokianos & Kedem, 2004; Kedem &

Fokianos, 2005; Li, 1994; Zeger & Qaqish, 1988), where the

conditional distribution of counts given past information

belongs to the exponential family. As done in the avail-

able statistical literature (Agresti, 1990; Bishop, Fienberg,

& Holland, 1975; Cameron & Trivedi, 1998; Haberman,

1974; Winkelmann, 2000), we model the counts with a

Poisson-type distribution. These Poisson time series mod-

els allow inclusion of both autoregressive and moving aver-

age terms along with trend, seasonality, and dependence

222 © 2018 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journalof Forecasting. 2019;38:222–235.