Return predictability of variance differences: A fractionally cointegrated approach

• Date: 01 July 2020
• Author: Xingzhi Yao,Marwan Izzeldin,Zhenxiong Li
• Published date: 01 July 2020
• DOI: http://doi.org/10.1002/fut.22110

J Futures Markets. 2020;40:1072–1089.wileyonlinelibrary.com/journal/fut1072

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© 2020 Wiley Periodicals, Inc.

Received: 24 April 2019

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Accepted: 17 February 2020

DOI: 10.1002/fut.22110

RESEARCH ARTICLE

Return predictability of variance differences:

A fractionally cointegrated approach

Zhenxiong Li

1

|Marwan Izzeldin

2

|Xingzhi Yao

3

1

Department of Economics, Dongwu

Business School, Soochow University,

Suzhou, China

2

Department of Economics, Lancaster

University, Lancaster, UK

3

Finance Department, International Business

School Suzhou (IBSS), Xi'an Jiaotong‐

Liverpool University, Suzhou, China

Correspondence

Xingzhi Yao, International Business School

Suzhou (IBSS), Xi'an Jiaotong‐Liverpool

University, 215123 Suzhou, China.

Email: Xingzhi.Yao@xjtlu.edu.cn

Funding information

Xi'an Jiaotong‐Liverpool University,

China, Grant/Award Number:

RDF‐17‐02‐07

Abstract

This paper examines the fractional cointegration between downside (upside)

components of realized and implied variances. A positive association is found

between the strength of their cofractional relation and the return predictability

of their differences. That association is established via the common long‐

memory component of the variances that are fractionally cointegrated, which

represents the volatility‐of‐volatility factor that determines the variance

premium. Our results indicate that market fears play a critical role not only in

driving the long‐run equilibrium relationship between implied‐realized

variances but also in understanding the return predictability. A simulation

study further verifies these claims.

KEYWORDS

fractional cointegration, return predictability, variance risk premium

JEL CLASSIFICATION

C14; C32; C51

1|INTRODUCTION

Numerous empirical studies suggest that, unlike the variance, the variance risk premium (

V

R

P

) carries nontrivial

predictive power for aggregate stock market returns over quarterly to annual horizons, where the degree of return

predictability is greater than that afforded by more conventional predictors, see, Bollerslev, Tauchen, and Zhou (2009),

Drechsler and Yaron (2010), Bollerslev, Marrone, Xu, and Zhou (2014), and Bali and Zhou (2016), among others. Those

studies also provide strong evidence that much of the predictability implicit in the

V

R

P

may be attributed to its close

relation with macroeconomic uncertainty and aggregate risk aversion.

The

V

R

P

is formally defined as the wedge between option‐implied and realized variances. It is measured as the

difference between (the square of) the CBOE VIX index and the statistical expectation of the future return variation. In

Bollerslev et al. (2009), Drechsler and Yaron (2010) and Bollerslev, Sizova, and Tauchen (2012), the return predictability

of the

V

R

P

is investigated using a self‐contained general equilibrium model. This is in the spirit of the long‐run risks

(LRR) model pioneered by Bansal and Yaron (2004). Specifically, Bollerslev et al. (2009) and Bollerslev et al. (2012)

extend the LRR model by allowing the time‐varying volatility‐of‐volatility (vol‐of‐vol) within the economy to be gen-

erated by its own stochastic process. They further show that the difference between the risk‐neutral and the objective

expectations of return variation isolates the vol‐of‐vol factor, which then serves as the sole source of the true

V

R

P

. The

V

R

P

, therefore, displays good predictive power for future returns in cases where the vol‐of‐vol plays a more dominant

role in determining variation in the equity premium. A direct link between the

V

R

P

and the vol‐of‐vol is also de-

monstrated in a purely probabilistic model introduced by Barndorff‐Nielsen and Veraart (2013).

By incorporating jumps in the LRR model, Drechsler and Yaron (2010) disentangle the difference in the drifts of

conditional variance from the

V

R

P

. That drift difference is related to the vol‐of‐vol associated with the level of

uncertainty, leading to the covariation between the

V

R

P

and the conditional equity premium that is partially dependent

on the vol‐of‐vol factor. As indicated by Bollerslev et al. (2009), the vol‐of‐vol is inherently latent and may be variously

defined across different volatility models. Recent literature has achieved limited progress in seeking an appropriate

proxy of the vol‐of‐vol factor which is necessary to explain the return predictability suggested by the

V

R

P

. A notable

exception is the work of Conrad and Loch (2015), where the vol‐of‐vol is represented by the long‐term volatility

component of the GARCH‐MIDAS model.

Consistent with the generalized LRR model discussed above, results suggested by Bollerslev, Osterrieder, Sizova,

and Tauchen (2013) provide new evidence for return predictability of the

V

R

P

based on the idea of fractional coin-

tegration. Given the mounting empirical studies that favor fractional integration in implied and realized variances (see

Bandi and Perron (2006) and Nielsen (2007) for instance), fast mean reversion in the

V

R

P

, that is the difference between

the two variances, points to the existence of fractional cointegration. Compared to the use of the univariate variances as

a proxy for risk, Bollerslev et al. (2013) show that the less persistent

V

R

P

rebalances the risk‐return regression in terms

of the order of fractional integration, and that this appears to be more informative for future returns. In addition, the

V

R

P

is estimated as the cointegrating relation between the two variances, which is highly linked to the vol‐of‐vol and

aggregate economic uncertainty.

Another strand of the literature on the properties of the

V

R

P

focuses on the decomposition of the

V

R

P

into its

downside and upside components, so differentiating between investors' attitudes toward uncertainty risks associated

with the left and right tail of the return distribution. Feunou, Jahan‐Parvar, and Okou (2017) find that the downside

V

R

P

often dominates total

V

R

P

in predicting future excess returns. They rationalize this result by showing (a) that the

two components exhibit opposite relationships to the equity premium and (b) that the link between the upside

V

R

P

and

the equity premium is insignificant. In similar vein, Kilic and Shaliastovich (2018) adopt an alternative decomposition

of the

V

R

P

: they argue that both good and bad components contain important information about future excess returns,

which results in higher return predictability than that inherent in the total

V

R

P

.

The difference between the components of the realized (implied) variance is also considered in several studies

where the main focus is on equity risk premium predictability. For instance, by decomposing the realized jump

volatility into negative jumps and positive jumps, Guo, Wang, and Zhou (2019) construct the signed jump (SJ) risk as

the difference between the two jump risk components. In addition, the SJ risk is found to contain information about

future market returns that is independent of that captured by the

V

R

P

. With an alternative decomposition approach,

Bollerslev, Li, and Zhao (2019) derive the SJ variation as the difference between the good and bad realized volatility and

show that this difference significantly predicts the variation in future returns. Under the risk‐neutral measure, Huang

and Li (2019) identify a significant relationship between future stock returns and implied variance asymmetry (IVA)

defined as the difference between the upside and downside semi‐variances derived from out‐of‐the‐money (OTM)

options.

Against this backdrop, the main contributions of this paper are threefold. First, by decomposing the implied and

realized variances into upside and downside components, we investigate the presence of fractional cointegration in

each pair of semi‐variances and establish new evidence for the long‐run relationship between different variance

components. Second, we show that the common long‐memory component in the fractionally cointegrated system, as

part of the conditional variance of market returns, is intimately associated with the vol‐of‐vol driving the variance

premium. Moreover, that common component plays a key role in connecting the strength of the fractional cointegration

between variances and the stock return predictability afforded by the variance differences. The dominant predictive

power of the downside

V

R

P

documented in the literature can therefore, be reasonably explained by the long‐run

equilibrium relation between the downside implied and realized semivariances. Third, to alleviate the impact on our

empirical findings of the limited availability of observed strikes in the construction of implied variances, we employ a

simulation study to verify the claims outlined above.

Following the procedures of Barndorff‐Nielsen, Kinnebrock, and Shephard (2010) and Andersen, Bondarenko, and

Gonzalez‐Perez (2015), we obtain upside and downside semivariances of the realized (RV ) and implied (IV ) variances,

which we refer to as RV RV I V,,

UD

U

, and

IVD

. The difference between IV IV()

UD

and

RV RV(

)

UD

is defined as the

upside (downside) variance risk premium, henceforth

V

RP VRP(

)

UD

. The SJ and IVA are, respectively, measured as the

difference between

RV

U

and

RV

D

and the difference between

IV

U

and

IVD

. Using a semiparametric approach, we

demonstrate the existence of a fractionally cointegrating relation in each pair of the semivariances listed above. In

addition, our results reveal the role of

IVD

as a long‐run upward biased forecast of future

RV

D

. This evidence is also

LI ET AL.

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