Model averaging estimation for conditional volatility models with an application to stock market volatility forecast

• DOI: http://doi.org/10.1002/for.2659
• Author: Qingfeng Liu,Qingsong Yao,Guoqing Zhao
• Published date: 01 August 2020
• Date: 01 August 2020

Received: 3 October 2019 Accepted: 12 January 2020

DOI: 10.1002/for.2659

RESEARCH ARTICLE

Model averaging estimation for conditional volatility models

with an application to stock market volatility forecast

Qingfeng Liu1Qingsong Yao2Guoqing Zhao2

1Department of Economics, Otaru

University of Commerce, Otaru City,

Hokkaido, Japan

2School of Economics, Renmin University

of China, Beijing, China

Correspondence

Qingsong Yao,School of Economics,

Renmin University of China, Beijing

100872, China.

Email: greatpine@ruc.edu.cn

Funding information

JSPS KAKENHI Grant (C), Grant/Award

Number: JP16K03590 and JP19K01582;

Outstanding Innovative Talents

Cultivation Funded Programs 2018 of

Renmin University of China

Abstract

This paper is concerned with model averaging estimation for conditional volatil-

ity models. Given a set of candidate models with different functional forms, we

propose a model averaging estimator and forecast for conditional volatility, and

construct the corresponding weight-choosing criterion. Under some regulatory

conditions, we show that the weight selected by the criterion asymptotically

minimizes the true Kullback–Leibler divergence, which is the distributional

approximation error,as well as the Itakura–Saito distance, which is the distance

between the true and estimated or forecast conditional volatility. Monte Carlo

experiments support our newly proposed method. As for the empirical applica-

tions of our method, we investigate a total of nine major stock market indices

and make a 1-day-ahead volatility forecast for each data set. Empirical results

show that the model averaging forecast achieves the highest accuracy in terms

of all types of loss functions in most cases, which captures the movement of the

unknown true conditional volatility.

KEYWORDS

conditional volatility, forecast, model averaging, optimality

1INTRODUCTION

Conditional volatility models are important tools for the

analysis of financial markets as well as many other

economic phenomena. Such a model family depicts how

the information and random shocks from the past exert

influences on the current market volatility. Engle (1982)

first proposed the well-known autoregressive conditional

heteroskedasticity (ARCH) model, which was then gen-

eralized to GARCH by Bollerslev (1986). Inspired by

the pioneering works, various types of volatility models

with distinctive functional forms have been proposed—for

example, EGARCH (Nelson, 1991); APGARCH (Ding,

Granger, & Engle, 1993); GJR (Glosten, Jagannathan, &

Runkle, 1993)—greatly contributing to people's under-

standing of the financial markets. Interested readers are

referred to Engle (2002), Morimune (2007), and Bollerslev

(2008) a comprehensive literature review. Given a set

of alternative volatility models, the researcher needs to

select one model to describe the unknown data-generating

process. The estimation and forecast accuracy of the

conditional volatility can be very different when different

models are applied. Moreover, the optimal model usually

appears to be data specific (P. R. ; Hansen & Lunde, 2005).

As a result, although abundant volatility models enrich the

choices of the researchers, much confusion related to how

to choose an optimal model from the various alternatives

arises at the same time. Generally, prior knowledge of the

financial markets or high-dimensional data can provide

significant guidance to selection of the candidate models,

but such information is not always available. To make the

data speak more, the data-driven model selection proce-

dure based on information criteria such as the Akaike

information criterion (Akaike, 1973) or Bayesian infor-

mation criterion (Schwarz, 1978) is commonly applied in

empirical studies. Despite its great popularity, the model

Journal of Forecasting. 2020;39:841–863. wileyonlinelibrary.com/journal/for © 2020 John Wiley & Sons, Ltd. 841

LIU ET AL.

selection procedure suffers from some drawbacks. On the

one side, model selection results based on information

criteria lack stability, especially when the sample size is

small. A slight perturbation of the data set can lead to

a totally different selection result (Yang, 2001), making

subsequent economic interpretation based on the selection

results risky. Moreover, when the true data-generating

process is not included in the model space, model selection

may display poor performance (Granger & Jeon, 2004).

On the other side, different conditional volatility mod-

els depict different mechanisms through which previous

information influences current volatility. When the true

data-generating process is unknown, selecting a unique

model instead of averaging all the alternatives in a reason-

able way causes loss of information.

Motivated by the above considerations, this paper

applies the frequentist model averaging (FMA) technique

to the estimation of the conditional volatility model fam-

ily and out-of-sample forecast of conditional volatility. The

FMA was proposed by Hansen (2007) for linear regres-

sion models and it is widely generalized to various model

families including linear regression models (Dardanoni,

Luca, & Modica, 2015; Hansen & Racine, 2012; Liu &

Okui, 2013; Liu, Okui, & Yoshimura,2016; Zhang & Liang,

2011), nonlinear regression models (Liu, Yao, & Zhao,

2017), structural break models (Hansen, 2009), near unit

root models (Hansen, 2010), instrumental variable mod-

els (Kuersteiner & Okui, 2010), quantile regression models

(Lu & Su, 2015), and generalized linear models (Zhang,

Yu, Zou, & Liang, 2016). The key feature of the FMA lies

in that the researcher does not select a unique model from

all the alternatives. After the estimation of all the can-

didate models, the estimators from different models are

combined in a weighted manner,where the optimal weight

is obtained based on some specific weight-choosing cri-

terion. This effectively controls the risks resulting from

basing all the economic interpretations on a single model.

More recently,some researchers have obtained the asymp-

totic distribution of the model averaging estimator (C.-A. ;

Liu, 2015; Zhang & Liu, 2017) and proved the dominance

of model averaging estimation over least squares estima-

tion (Zhang, Ullah, & Zhao, 2016; Zhao, Ullah, & Zhang,

2018).

The main work of this paper is described as follows.

First, we propose the model averaging estimator and fore-

cast for conditional volatility under a framework where

the conditional mean of the data is constantly zero. We

construct a feasible weight-choosing criterion, and under

some regulatory conditions we show that the weight

minimizing the weight-choosing criterion asymptotically

minimizes the unknown Kullback–Leibler (KL) diver-

gence, which is a measurement of the approximation

error of the conditional distribution. Wefurther investigate

the estimation accuracy of the conditional volatility. We

show that the optimal model averaging weight also asymp-

totically minimizes the Itakura–Saito distance (Itakura

& Saito, 1968), which is a measurement of the distance

between the estimated and true conditional volatility. Sec-

ond, to make our method more general, we lose the restric-

tion of the zero conditional mean and allow both the

conditional mean and volatility to be influenced by a set

of variables. In this situation the specification of the con-

ditional mean has influences on the estimation accuracy

of the conditional volatility, and vice versa. Thus we pro-

pose to apply the model averaging technique to both the

conditional mean and volatility processes, where separate

weights are assigned to the conditional mean and vari-

ance estimators. We call the above weighting scheme the

double model averaging estimation procedure (DMAE).

Asymptotic optimality in terms of the KL divergence and

Itakura–Saito distance under the DMAE is proved under

some regulatory conditions. Monte Carlo experiments are

conducted to evaluate our newly proposed method. The

simulation results are supportive in most cases.

As an empirical application, we apply the model

averaging method to forecast the daily volatility of stock

markets. We consider a total of nine major stock market

indices and make a 1-day-ahead volatility forecast for

each data set. Realized volatility constructed by 5-minute

high-frequency data is used to evaluate the forecast accu-

racy of different methods. The empirical results indicate

that, compared with the forecast based on an individual

model, equal-weight forecast, and forecast based on model

selection and model smoothing, the model averaging fore-

cast generally leads to the highest accuracy even when

different types of loss functions are considered.

Pasaran, Schleicher,and Zaffaroni (2016) considered the

model averaging problem for multivariate volatility mod-

els and proposed to average the density functions to over-

come model uncertainties. Their weighting schemes are

based on “thick modeling” (where models are first ranked

according to AIC or BIC criteria, and then a simple average

is taken among the top-percentile models) and Bayesian

model averaging (where the smoothed BIC is applied to

obtain an approximation of the BMA weight), which are

totally different from ours. Moreover,though useful in the

evaluation of the value at risk, their method cannot be

applied to the point forecast of conditional volatility,which

isthecasewemainlyconsiderinthispaper.

The remainder of the paper is arranged as follows.

In Section 2, we introduce the model averaging method

under the framework of zero conditional mean and discuss

its statistical properties. In Section 3, we generalize our

method and deal with the case of time-varying conditional

mean. The DMAE is proposed and its statistical properties

are provided. In Section 4, we present some Monte Carlo

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