Realized Volatility Forecast: Structural Breaks, Long Memory, Asymmetry, and Day‐of‐the‐Week Effect

• DOI: http://doi.org/10.1111/irfi.12030
• Date: 01 September 2014
• Published date: 01 September 2014
• Author: Ke Yang,Langnan Chen

Realized Volatility Forecast:

Structural Breaks, Long

Memory, Asymmetry, and

Day-of-the-Week Effect*

KEYANG†AND LANGNAN CHEN‡

†College of Economics & Management, South China Agricultural University,

Guangzhou, China and

‡Lingnan College (University) and Institute for Economics, Sun Yat-Sen University,

Guangzhou, China

ABSTRACT

We investigate the properties of the realized volatility in Chinese stock

markets by employing the high-frequency data of Shanghai Stock Exchange

Composite Index and four individual stocks from Shanghai Stock Exchange

and Shenzhen Stock Exchange, and f‌ind that the volatility exhibits the

properties of long-term memory, structural breaks, asymmetry, and day-of-

the-week effect. In addition, the structural breaks only partially explain the

long memory. To capture these properties simultaneously, we derive an

adaptive asymmetry heterogeneous autoregressive model with day-of-the-

week effect and fractionally integrated generalized autoregressive conditional

heteroskedasticity errors (HAR-D-FIGARCH) and use it to conduct a forecast

of realized volatility. Compared with other heterogeneous autoregressive

realized volatility models, the proposed model improves the in-sample f‌it

signif‌icantly. The proposed model is the best model for the day-ahead real-

ized volatility forecasts among the six models based on various loss functions

by utilizing the superior predictive ability test.

JEL Classif‌ication: C22; C53; G10

I. INTRODUCTION

Accurately measuring and forecasting the volatility is central for asset pricing,

portfolio selection, and risk management. A number of volatility forecast

models have been developed for this purpose for last decades. Poon and Granger

(2003) provide a systematic review of these studies. As transaction data is widely

available, Andersen and Bollerslev (1998a) f‌ind that the daily unobserved vola-

tility can be represented by the sum of squared intraday returns, the so-called

* This research is supported by China Natural Science Foundation and Grant No. 71203067 and

71241019, Guangdong Soft Science under Grant No. 0204, Guangdong Social Science Foundation

under Grant No. GD10CYJ01, and Guangdong Social Science Project under Grant No.

08JDXM79001.

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International Review of Finance, 14:3, 2014: pp. 345–392

DOI: 10.1111/irf‌i.12030

© 2014 International Review of Finance Ltd. 2014

realized volatility,1which is a much more accurate measure of daily volatility

than the popular daily squared return.

A number of researchers have utilized the autoregressive fractionally inte-

grated moving average (ARFIMA) model to capture the properties of long

memory possessed by realized volatility. Based on Granger (1980) and Granger

and Joyeux (1980), Andersen et al. (2003) propose an ARFIMA model and use

it to forecast the realized volatility. They f‌ind that the ARFIMA model outper-

forms other standard methods based on squared returns. Beltratti and Morana

(2005) develop an ARFIMA-fractionally integrated GARCH (FIGARCH) model

for the logarithm of realized variance by incorporating the conditional

heteroskedasticity in the ARFIMA errors. In order to capture the leverage

effects, Giot and Laurent (2004) utilize the indicator functions and lagged

returns as explanatory variables of the mean to construct an ARFIMAX model.

Degiannakis (2008) integrates an ARFIMAX model into a Threshold ARCH

(TARCH) for the residuals variance. The combined mode proves to be a superior

forecast compared with an ARFIMAX model.

As an alternative model, Corsi (2009) develops a heterogeneous autoregres-

sive realized volatility model (henceforth HAR model) based on the heteroge-

neous market hypothesis and the Heterogeneous Autoregressive Conditional

Heteroskedasticity (HARCH) model of Muller et al. (1997), which employs vola-

tility components at different time resolutions so as to reproduce the long

memory property of realized volatility directly. The HAR model’s tractable

estimation and superior forecast performance lead to several similar studies

such as Andersen et al. (2007), Forsberg and Ghysels (2007), and Louzis et al.

(2012). More recently, Corsi et al. (2008) consider the time varying conditional

heteroskedasticity of the normally distributed HAR errors and improve the

accuracy of the estimated parameters and the forecast performance of the HAR

model. Corsi and Reno (2009) develop a new HAR model by incorporating past

daily, weekly, and monthly negative returns as regressors in the volatility

process in order to capture the leverage effects.

However, f‌inancial time series are subject to occasional structural breaks

because of various factors such as f‌inancial crisis, major changes in market

sentiments, generation of speculative bubbles, and regime switches in

monetary and debt management policies (Stock and Watson 1996; Pesaran and

Timmermann 2002). Volatility forecast without taking into account the breaks

will result in overestimations on long memory parameter and poor performance

(Rapach and Strauss 2008; Choi et al. 2010). Some studies have been doubtful

about the adequacy of long memory models for volatility series as the long

memory can be easily affected by the structural breaks. In particular, it is

suggested that various types of structural changes can induce a strong persis-

tence in the autocorrelation function, and hence generate “spurious” long

memory. Granger and Hyung (2004) present the theoretical and simulation

1 The sum of squared intraday returns is actually the realized variance. Realized volatility is

def‌ined as the square root of realized variance. Thus, many authors use the term realized

volatility interchangeably with the term realized variance.

International Review of Finance

346 © 2014 International Review of Finance Ltd. 2014

evidence that spurious long memory can be detected from a short memory

process with breaks in the mean. Starica and Granger (2005) f‌ind that a

nonstationary model allowing for breaks outperforms a long memory model in

forecasting only for long run. Diebold and Inoue (2001) reveal how Markov

switching processes generate long memory while Granger and Terasvirta (1999)

f‌ind that a process that switches in sign can display the property of long memory.

Moreover, Choi et al. (2010) f‌ind that an increase of the number of mean breaks

makes the memory of the process more persistent. Choi et al. (2010) utilize a

break-adjusted forecast method to reduce the forecast error for several realized

volatility series, which are modeled as long memory processes. However, their

methods depend on the accurate estimate of the break date. More recently,

several studies suggest that an appropriate model for the volatility of f‌inancial

returns should include the joint occurrence of long memory and structural

changes. Hillebrand and Medeiros (2008) and Martens et al. (2009) build from

an ARFIMA model by allowing for smooth level shifts, day-of-the-week effects

and leverage. McAleer and Medeiros (2008) propose a different strategy by

presenting a multiple regime smooth transition extension of HAR, whereas Lux

and Morales-Arias (2010) develops a regime-switching multifractal model.

The proposed model is an extended HAR model where the structural changes

in log realized volatility are modeled by allowing the intercept to follow a

smooth deterministic process as represented by Gallant’s (1984) f‌lexible func-

tional form, which does not require pretesting for numbers of break points, nor

does it require any smooth transition between volatility regimes, and has the

advantage of being computationally straightforward. In addition, we utilize the

daily dummy variables to capture the day-of-the-week effects. Based on Louzis

et al. (2012), the asymmetries are modeled as lagged standardized returns and

absolute standardized returns, like a more f‌lexible Exponential Generalized

Autoregressive Conditional Heteroskedasticity (EGARCH)-type model, occur-

ring at different time intervals (daily, weekly, and monthly). In addition, based

on a slow decay in the autocorrelation of the squared residuals of an ARFIMA

realized volatility model proposed by Beltratti and Morana (2005), we derive a

FIGARCH for the conditional heteroskedasticity of the residuals, which is called

adaptive asymmetric HAR-D-FIGARCH model. The biggest difference between

our model and Martens et al. (2009) is the approach to modeling the structural

break and that Martens et al. (2009) do not take into consideration the long-

range dependence on the heteroskedastic variance of the residuals. By taking

into consideration the structural breaks, day-of-the-week effects, the asymmet-

ric responses to negative and positive shocks, and the long memory in the

heteroskedastic variance of the residuals, we intend to improve the volatility

forecast performance of the HAR models.

By employing the 9 years of 5-min intraday Shanghai Stock Exchange Com-

posite Index (SSEC) and four individual stocks returns, we conduct the volatility

forecast and f‌ind that the proposed model is superior in-sample f‌itting to the

other HAR models. And then we employ Hansen’s (2005) superior predictive

ability (SPA) to assess the day-ahead volatility forecast accuracy. The results

Realized Volatility Forecast

347© 2014 International Review of Finance Ltd. 2014