Realized Volatility Forecast of Stock Index Under Structural Breaks

• DOI: http://doi.org/10.1002/for.2318
• Author: Fengping Tian,Langnan Chen,Ke Yang
• Published date: 01 January 2015
• Date: 01 January 2015

Realized Volatility Forecast of Stock Index Under Structural Breaks

KE YANG,

1

LANGNAN CHEN

2

*AND FENGPING TIAN

3

1

College of Economics and Management, South China Agricultural University, Guangzhou

510642, China

2

Lingnan College, Sun Yat-sen University, Guangzhou 510275, China

3

International Business School, Sun Yat-sen University, Guangzhou 510275, China

ABSTRACT

We investigate the realized volatility forecast of stock indices under the structural breaks. We utilize a pure multiple

mean break model to identify the possibility of structural breaks in the daily realized volatility series by employing

the intraday high-frequency data of the Shanghai Stock Exchange Composite Index and the five sectoral stock indices

in Chinese stock markets for the period 4 January 2000 to 30 December 2011. We then conduct both in-sample tests

and out-of-sample forecasts to examine the effects of structural breaks on the performance of ARFIMAX-FIGARCH

models for the realized volatility forecast by utilizing a variety of estimation window sizes designed to accommodate

potential structural breaks. The results of the in-sample tests show that there are multiple breaks in all realized volatility

series. The results of the out-of-sample point forecasts indicate that the combination forecasts with time-varying

weights across individual forecast models estimated with different estimation windows perform well. In particular,

nonlinear combination forecasts with the weights chosen based on a non-parametric kernel regression and linear com-

bination forecasts with the weights chosen based on the non-negative restricted least squares and Schwarz information

criterion appear to be the most accurate methods in point forecasting for realized volatility under structural breaks. We

also conduct an interval forecast of the realized volatility for the combination approaches, and find that the interval

forecast for nonlinear combination approaches with the weights chosen according to a non-parametric kernel regres-

sion performs best among the competing models. Copyright © 2014 John Wiley & Sons, Ltd.

key words realized volatility; ARFIMAX-FIGARCH model; structural breaks; combination forecasts;

estimation window

INTRODUCTION

Accurate measurement and forecast of volatility are important for asset pricing, portfolio selection and risk manage-

ment. Many volatility forecast models based on low-frequency data, such as generalized autoregressive conditional

heteroskedasticity (GARCH) models and stochastic volatility (SV) models, have been proposed over the last 30 years.

As high-frequency data have recently become widely available, a new observable measure of volatility, the so-called

realized volatility, derived from the sum of intraday squared returns, has been developed in the literature. The realized

volatility has proved to be a much more accurate measure of the daily unobserved volatility than the popular daily

squared return.

As realized volatility is an observable variable, standard time series models can be used for forecast purposes. A

number of researchers have utilized long-memory models such as the autoregressive fractionally integrated moving

average (ARFIMA) model to forecast realized volatility, since a popular property of realized volatility is its strong

serial dependence. Andersen et al. (2003) employ the ARFIMA model to forecast several exchange rate realized vol-

atility series, and find that the ARFIMA model improves the performance of out-of-sample volatility forecast signif-

icantly compared to other standard methods based on squared returns such as the GARCH model and RiskMetrics.

Beltratti and Morana (2005) combine the ARFIMA model with a FIGARCH specification for the residuals to develop

an ARFIMA-FIGARCH model for the logarithm of realized variance. Giot and Laurent (2004) develop an

ARFIMAX model to capture leverage effects and long memory simultaneously by incorporating indicator functions

and lagged returns into the ARFIMA model. Degiannakis (2008) mixes the ARFIMAX model with a threshold

ARCH (TARCH) specification for the residuals, which is found to be a superior forecast model to a pure ARFIMAX

model. As an alternative approach, Corsi (2009) proposes a heterogeneous autoregressive realized volatility model

(henceforth HAR model) derived from the heterogeneous market hypothesis and the HARCH model of Muller

et al. (1997), which utilizes volatility components at different time resolutions in order to regenerate the long-memory

property of realized volatility directly. The tractable estimation and superior forecast performance of the HAR model

make it the subject of similar studies such as Andersen et al. (2007), Forsberg and Ghysels (2007) and Louzis et al.

(2012). Corsi et al. (2008) develop a HAR-GARCH model to improve the accuracy of the estimated parameters and

*Correspondence to: Langnan Chen, Lingnan College, Sun Yat-sen University, Guangzhou, Guangdong 510275, China. E-mail: lnscln@mail.

sysu.edu.cn

Journal of Forecasting,J. Forecast. 34,57–82 (2015)

Published online 28 November 2014 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2318

Copyright © 2014 John Wiley & Sons, Ltd.

the forecast performance by taking into account time-varying conditional heteroskedasticity in the HAR model’s re-

siduals. Corsi and Reno (2009) construct an asymmetry HAR model to capture the asymmetries and long-range de-

pendence simultaneously by incorporating past daily, weekly and monthly negative returns into the HAR model, and

Louzis et al. (2012) develop an asymmetry HAR model with a FIGARCH specification for the residuals to forecast

the realized volatility. Both models improve the forecast performance.

Although the ARFIMA-type and HAR-type models above have been widely used to forecast the realized volatility

in recent literature, little attention has been paid to the role of structural breaks. In fact, researchers normally assume

explicitly or implicitly the existence of a stable ARFIMA process or HAR process when forecasting the realized vol-

atility and hence use a recursive or fixed window size when estimating the realized volatility model used to generate

out-of-sample realized volatility forecasts. However, the financial markets are periodically buffeted by suddenly im-

portant events, such as the stock market crash in 1987, the Russian and East Asian financial crises in 1997, the Inter-

net bubble in 1995–2000 and the American subprime mortgage crisis in 2008. These important events can lead to

sharp breaks in the financial markets and thus breaks in parameters governing the volatility models. Liu and Maheu

(2008) provide a Bayesian analysis of structural breaks in the daily S&P 500 realized volatility in 1993–2004 by

using the HAR models and find strong evidence of a structural break in early 1997. Thus ignoring this break will lead

to biased persistence estimates in the variance of log realized volatility (too high) and poor density forecasts. Choi

et al. (2010) also find strong evidence of structural breaks in the daily realized volatility of the

Deutschemark/dollar, yen/dollar and yen/Deutschemark spot exchange rates, and these breaks can partly explain

the persistence of realized volatility. Yang and Chen (2014) obtain similar empirical results for realized volatility se-

ries in the Chinese stock market. Therefore, structural breaks have important implications for forecasting the realized

volatility.

We investigate the realized volatility forecast of the stock index under structural breaks. We start our forecast by

estimating ARFIMAX-FIGARCH models using a variety of estimation window sizes, including models based on an

expanding estimation window and those based on short, medium and long rolling estimation windows, as well as an

estimation window whose size is identified by using the pure multiple mean break model. We use these last four win-

dows to reflect structural breaks in the data-generating process during the estimate of a realized volatility forecast

model. We also compare forecasts using various window sizes with the forecasts using an expanding window, and

intend to confirm whether it is worth accommodating potential structural breaks.

Furthermore, we consider that the combination forecast may be an effective method for forecasting the realized

volatility of stock index under structural breaks. In light of Clark and McCracken (2009) and Pesaran and

Timmermann (2007), we consider a number of linear combination forecasts and nonlinear combination forecasts

based on the individual ARFIMAX-FIGARCH model forecasts generated using different estimation window sizes.

Finally, we use the MSE loss function, QLIKE loss function and a homogeneous robust loss function (Patton,

2011) and the model confidence set (MCS) test to evaluate and compare the point forecasting performance of these

realized volatility forecast methods. At the same time, we also conduct an interval forecast of the realized volatility

for the combination approaches.

The rest of this paper is organized as follows. The next section presents the methodologies. The third section dis-

cusses the in-sample test and out-of-sample forecasts. The fourth section concludes.

METHODOLOGIES

Multiple structural break model

Andersen and Bollerslev (1998) reveal that realized volatility, constructed from the sum of squared intraday returns,

is a more accurate measure of daily unobserved volatility than squared daily returns. As it is an observable variable,

standard time series models can be used for forecast purposes. But this estimator for daily volatility does not contain

the volatilityinformation during non-trading hours.Thus we use Martens’(2002) method to correctthe bias and estimate

realized volatility for the stock index. Similar to Pesaran and Timmermann (2005) and Choi et al.(2010), we employ the

pure multiple meanbreak method of Bai and Perron (1998, 2003) to test for the possibility of multiplebreaks and deter-

mine the break dates for all realized volatility series. The pure multiple mean break model is defined as

zt¼mjþεt;

t¼Tj1þ1;Tj1þ2;…Tj

j¼1;2;…;Mþ1

(1)

where z

t

is the log realized variance, m

j

is the mean of z

t

and the error term ε

t

may be serially correlated and

heteroskedastic.The break dates (T

1

,T

2

,…,T

M

) are unknown. Model (1) can be estimated by utilizing the least squares

58 L. Chen, K. Yang and F. Tian

Copyright © 2014 John Wiley & Sons, Ltd. J. Forecast. 34,57–82 (2015)