Robust estimation of conditional variance of time series using density power divergences

• Author: T. N. Sriram,Jin‐Hong Park
• Published date: 01 September 2017
• Date: 01 September 2017
• DOI: http://doi.org/10.1002/for.2465

Received: 18 March 2015 Revised: 19 November 2016 Accepted: 31 January 2017

DOI: 10.1002/for.2465

RESEARCH ARTICLE

Robust estimation of conditional variance of time series using

density power divergences

Jin-Hong Park1T. N. Sr ira m2

1Department of Mathematics, College of

Charleston, Charleston, SC, USA

2Department of Statistics, University of

Georgia, Athens, GA, USA

Correspondence

T. N. Sriram, Department of Statistics,

University of Georgia, Athens, GA 30602,

USA.

Email: tn@stat.uga.edu

Funding information

National Science Foundation, Grant/Award

Number: 1309665

Abstract

Suppose Ztis the square of a time series Ytwhose conditional mean is zero. We do

not specify a model for Yt, but assume that there exists a p×1 parameter vectorΦsuch

that the conditional distribution of ZtZt−1is the same as that of ZtΦTZt−1,where

Zt−1=(Zt−1,…,Zt−p)Tfo r some lag p⩾1. Consequently, the conditional variance

of Ytis some function of ΦTZt−1. To estimate Φ, we propose a robust estimation

methodology based on density power divergences(DPD) indexed by a tuning param-

eter ∈[0,1], which yields a continuum of estimators, {

Φ;∈[0,1]},where

controls the trade-off between robustness and efficiency of the DPD estimators. For

each ,

Φis shown to be strongly consistent. We develop data-dependent criteria

for the selection of optimal and lag pin practice. Weillustrate the usefulness o f our

DPD methodology via simulation studies for ARCH-type models, where the errors

are drawn from a gross-error contamination model and the conditional variance is

a linear and/or nonlinear function of ΦTZt−1. Furthermore, we analyze the Chicago

Board Options Exchange Dow Jones volatility index data and show that our DPD

approach yields viable models for the conditional variance, which are as good as,

or superior to, ARCH/GARCH models and two other divergence-based models in

terms of in-sample and out-of-sample forecasts.

KEYWORDS

conditional variance, dimension reduction, density power divergence, tuning parameter, estimating

functions, robustness, optimal estimator

1INTRODUCTION

In this article, we present an alternative approach to mod-

eling the conditional variance of a time series Ytwhose

conditional mean is zero. We consider the squared series

Zt(= Y2

t)and adopt a sufficient dimension reduction

(SDR) approach (see Park, Sriram, & Yin, 2010), which

aims at reducing the dimension in the conditional vari-

ance of Ytwithout imposing a model assumption. More

specifically, our SDR approach assumes that there exists

ap×1 parameter vector Φsuch that the conditional

distribution of ZtZt−1is the same as that of ZtΦTZt−1,

where Zt−1=(Zt−1,…,Zt−p)Tfor som e lag p⩾1.

Consequently, the conditional variance of Yt, denoted by 2

t,

is some function of ΦTZt−1.SinceΦis unknown and there is

no model assumption, we estimate Φusing a nonparametric

method.

To estimate Φ, we propose a robust estimation methodol-

ogy based on density power divergence (DPD) indexed by a

tuning parameter ∈[0,1]. This yields a continuum of DPD

estimators, {

Φ;∈[0,1]},wherecontrols the trade-off

between robustness and efficiency of the estimators. In fact,

for ∈[0,1], the DPD provides a smooth bridge between

the Kullback–Leibler (KL) divergence (=0) and the L2

distance (=1). Rather than preselecting any one value

of and considering the corresponding DPD, we propose a

Journal of Forecasting.2017;36:703–717. wileyonlinelibrary.com/journal/for Copyright © 2017 John Wiley & Sons, Ltd. 703