A simple parameter‐driven binary time series model

• Date: 01 March 2020
• Author: Yang Lu
• Published date: 01 March 2020
• DOI: http://doi.org/10.1002/for.2621

Received: 9 September 2016 Revised: 16 April 2018 Accepted: 5 July 2019

DOI: 10.1002/for.2621

RESEARCH ARTICLE

A simple parameter-driven binary time series model

Yan g Lu

Department of Economics (CEPN),

University of Paris 13, Villetaneuse, France

Correspondence

Yang Lu, Department of Economics,

University of Paris 13, 99 Avenue Jean

Baptiste Clement, 93430 Villetaneuse,

France.

Email: yang.lu@univ-paris13.fr

Abstract

We introduce a parameter-driven, state-space model for binary time series data.

The model is based on a state process with a binomial-beta dynamics, which has

a Markov, endogenous switching regime representation. The model allows for

recursive prediction and filtering formulas with extremely low computational

cost, and hence avoids the use of computational intensive simulation-based

filtering algorithms. Case studies illustrate the advantage of our model over

popular intensity-based observation-driven models, both in terms of fit and

out-of-sample forecast.

KEYWORDS

conjugate prior, state-space model, switching regime

1INTRODUCTION

Binary (or binomial, categorical) time series data

appear in many statistical and economic applications.

This paper proposes a parameter-driven, state-space

model for binary process. The model is based on an

underlying binomial-beta process, which is Markov of

finite-dimensional dependence with latent switching

regime. The model allows for simple recursive formulas

for forecasting and filtering, without the use of simulation

methods.

Following Cox, Gudmundsson, Lindgren, Bondes-

son, Harsaae, Laake, and Lauritzen (1981), time series

models can be divided into observation-driven and

parameter-driven models. Observation-driven models

assume that predictive probability is a simple function of

the past observations. Although these models have been

largely used in the literature (see, e.g., Chavez-Demoulin,

Davison, & McNeil, 2005; Estrella & Mishkin, 1998;

Fokianos, Rahbek, & Tjøstheim, 2009; Hausman, Lo, &

MacKinlay, 1992; Kauppi & Saikkonen, 2008; Nyberg,

2013) because of the simplicity of the likelihood func-

tion, parameter-driven models are usually more intuitive.

These latter assume that the process of probability of tak-

ing response 0, say,is driven by an unobserved, exogenous

process. Moreover, in the context of binary time series,

it has been shown by Keenan (1982) that any stationary

binary time series has such a parameter-driven represen-

tation. Moreover, parameter-driven models allow us to

tackle problems such as missing observations. Neverthe-

less, the estimation and forecasting of parameter-driven

models is usually computationally intensive (see, e.g.,

Abanto-Valle & Dey, 2014; Czado, Kastenmeier, Brech-

mann, & Min, 2012; Dueker, 2005). Our paper proposes

a model that allows for a explicit formula for forecasting,

which renders the estimation computationally cheap. The

model is based on a state variable with beta stationary

distribution, whose serial correlation is introduced via a

latent switching regime. We also propose a comparison

between parameter-driven and a popular intensity-based

observation-driven binary time series model. By doing

so we contribute to the recent literature on the compar-

ison of forecasting abilities of observation-driven and

parameter-driven models (see, e.g., Koopman, Lucas, &

Scharth, 2016), which has, up to now, omittedbinary data.

The paper is organized as follows. Section 2 presents

the state-space model. We first analyze the dynamics of

the state process, called binomial-beta process, before dis-

cussing its implications for the dynamics of the observ-

able variable. Section 3 provides the recursive formulas

for prediction, filtering as well as smoothing. Section 4

illustrates the model on two data sets. Section 5 dis-

cusses the generalization of the model by introducing

covariates or considering categorical time series. Section 6

Journal of Forecasting. 2020;39:187–199. wileyonlinelibrary.com/journal/for © 2019 John Wiley & Sons, Ltd. 187