Financial volatility modeling: The feedback asymmetric conditional autoregressive range model

• Published date: 01 January 2019
• Author: Haibin Xie
• DOI: http://doi.org/10.1002/for.2548
• Date: 01 January 2019

Received: 15 May 2017 Revised: 4 April 2018 Accepted: 3 August 2018

DOI: 10.1002/for.2548

RESEARCH ARTICLE

Financial volatility modeling: The feedback asymmetric

conditional autoregressive range model

Haibin Xie

School of Banking and Finance,

University of International Business and

Economics, Huixin East Road 10, 100029,

Beijing, China

Correspondence

Haibin Xie, University of International

Business and Economics, Huixin East

Road 10, Chaoyang District, Beijing

100029, China.

Email: hbxie@amss.ac.cn

Funding information

MOE (Ministry of Education in China)

Project of Humanities and Social Science,

Grant/AwardNumber: 14YJCZH167 ;

National Natural Science Foundation of

China, Grant/AwardNumber: 71401033

Abstract

An implied assumption in the asymmetric conditional autoregressive range

(ACARR) model is that upward range is independent of downward range.

This paper scrutinizes this assumption on a broad variety of stock indices.

Instead of independence, we find significant cross-interdependence between

the upward range and the downward range. Regression test shows that the

cross-interdependence cannot be explained by leverage effect. To include the

cross-interdependence, a feedback asymmetric conditional autoregressive range

(FACARR) model is proposed. Empirical studies are performed on a variety of

stock indices, and the results show that the FACARR model outperforms the

ACARR model with high significance for both in-sample and out-of-sample

forecasting.

KEYWORDS

ACARR, CARR, FACARR,price range, volatility forecasting

1INTRODUCTION

It has been known for a long time that high–low price

range is a far more efficient volatility estimator than

the commonly used return-based one (see, e.g., Alizadeh,

Brandt, & Diebold, 2002; Beckers, 1983; Garman & Klass,

1980; Kunitomo, 1992; Parkinson, 1980; Rogers, 1998;

Rogers & Satchell, 1991; Wiggins, 1991; Yang & Zhang,

2000). Degiannakis and Livada (2013) found that the

range-based volatility estimator was more accurate than

the realized volatility estimator.

Different models have been proposed to describe the

dynamics of the price range. Hsieh (1991, 1993) proposed

the autoregressive volatility model to capture the dynam-

ics of the range-based volatility. Chou (2005) proposed

the conditional autoregressive range (CARR) model and

found that it was a worthy candidate in volatility modeling

in comparison with the well-known generalized autore-

gressive conditional heteroskedasticity (GARCH) model.

Brandt and Jones (2006) provided a simple and effec-

tive framework for forecasting return volatility by com-

bining exponential GARCH (EGARCH) with data on the

range. They found the value of using range data in esti-

mation and out-of-sample volatility forecasting. Cheung,

Cheung, and Wan (2009) adopted the vector error cor-

rection model (VECM) to describe the range dynamics.

Li and Hong (2011) found that the range-based autore-

gressive volatility model gained good performance rela-

tive to the GARCH model. Xie and Wu (2017) proposed

a CARR model with gamma disturbance (GCARR) and

found that the GCARR model fitted the dynamics of price

range better than the commonly used CARR model with

Weibull disturbance. Liu, Wei, Ma, and Wahab (2017)

suggested forecasting the realized range-based volatility

using the dynamic model average approach. Applica-

tions of range-based volatility in the empirical literature

can be found in Chou and Liu (2010), Miao, Wu, and

Su (2013), Xie and Wang (2013), and Dimitrios, Spyros,

and Apostolos (2013). For a comprehensive review on

range-based volatility, see Chou, Chou, and Liu (2015).

For a comprehensive review on range-based volatility, see

Chou et al. (2015).

Journal of Forecasting. 2019;38:11–28. wileyonlinelibrary.com/journal/for © 2018 John Wiley & Sons, Ltd. 11

12 XIE

All the above-mentioned range models are symmetric

because they treat the maximum price and the minimum

price in a symmetric way. By allowing the dynamics of

the upward range to be different from that of the down-

ward range, Chou (2006) further proposed the asymmetric

conditional autoregressive range (ACARR) model which

treated the upward range movement and the downward

range movement in separate forms. Empirical studies per-

formed on the S&P 500 stock index reported consistent

evidence supporting the asymmetry in the upward range

and the downward range. Further, they found that the

ACARR model outperformed the CARR model with sig-

nificance. Chou and Wang (2014) combined the ACARR

model with the extreme value theory to estimate the

tail-related value-at-risk (VaR) measures and found their

approach gave better VaR estimates.

The ACARR model describes the dynamics of price

range by trying to treat upward range movement and

downward range movement separately. Treatment in this

way implies a default assumption that the movement of

the upward range is independent of that of the downward

range. This implied assumption should be scrutinized with

great care because invalidation of this assumption will

make the ACARRmodel report biased range-based volatil-

ity forecast.

We believe this implied assumption to be not right for at

least two reasons. First, investors trade on all the available

history price information. Thus it is counterintuitive to

assume that investors only use the history upward (down-

ward) range information to forecast the future upward

(downward) range.Second, instead of being instantaneous

and unbiased, the stock market is documented to over-

react to news (Debondt & Thaler, 1985, 1987). Overre-

action means that a current large decrease (increase) in

stock price predicts the next increase (decrease) in stock

price. Upward range measures the maximum increase

in stock price, while downward range gauges the maxi-

mum decrease. Market overreaction implies that upward

(downward) range helps to p redict the dow nward (upward)

range.

We test the implied assumption using the Granger

causality test. If the implied assumption holds, then

there would be no Granger causality between the upward

range and the downward range. Empirical studies are

performed on a variety of stock indices, and the evi-

dence consistently rejects the implied assumption. We

find significant cross-interdependence between the

upward range and the downward range. We also test

whether the cross-interdependence can be explained

by the well-documented leverage effect and find that

cross-interdependence remains significant even if the

leverage effect is controlled. This result indicates that the

forecasting power of the ACARR model can be further

improved if the cross-interdependence between upward

range and downward range is used.

Therefore, we propose the feedback asymmetric condi-

tional autoregressive range (henceforth FACARR) model

to describe the dynamics of the price range. The FACARR

model extends the ACARR model of Chou (2005) by

including the lagged upward range and downward range

as explanatory variables, and thus can explain the

cross-interdependence between the upward range and the

downward range. We do not use ACARRX (ACARRmodel

with exogenous variables) as in Chou (2006) for the rea-

son that, in the ACARRX model, the Xs usually refer to

the explanatory variables other than the explained variable

itself—stock returns (leverage effect), trading volume, and

so on. However, in this model we use the upward range

and the downward range as both the explanatoryvariables

and the explained variables.

A comprehensive empirical study is used to evaluate the

performance of the FACARR model. Empirical evidence

shows that cross-interdependence has a critical impact on

the persistence on both the upward range and the down-

ward range. Once cross-interdependence is included, per-

sistence on both the upward and the downward ranges

is reduced. Especially on the upward range, persistence

is reduced by 10–15%. Also, we find the FACARR model

proposed in this paper outperforms the ACARR model of

Chou (2006) with high significance for both in-sample and

out-of-sample forecasting.

The main contributions of this paper are summarized

as follows: (1) We are the first to test the indepen-

dence assumption in the ACARR model, and find signif-

icant cross-interdependence between the upward range

and the downward range, which invalidates the implied

assumption in the ACARR model; (2) we generalize the

ACARR model to the FACARR model by including such

cross-interdependence. The FACARR model includes the

ACARR model as a special case and thus is more flex-

ible for capturing the range dynamics; (3) comprehen-

sive empirical studies are performed on a variety of

stock indices, and the results show the dominance of the

FACARR model over the ACARR model. The FACARR

model provides a new benchmark for modeling the asym-

metry in range-based financial volatility. The implication

of this paper is clear: Range-based volatility modeling

should take into consideration the cross-interdependence

between the upward range and the downward range.

This paper is organized as follows. Section 2 presents a

brief review of the CARR and ACARR model. Section 3

introduces and discusses the econometric methodologies

used in this paper. Comprehensive empirical studies are

performed on a variety of stock indices in Section 4; in

that section we will present the detailed empirical results.

We conclude in Section 5 with a consideration of future

extensions.