Forecasting realized volatility: A review

Journal of the Korean Statistical Society (

)

–

Contents lists available at ScienceDirect

Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss

Review

Forecasting realized volatility: A review Dong Wan Shin Department of Statistics, Ewha University, Seoul, Republic of Korea

article

info

Article history: Received 7 June 2018 Accepted 8 August 2018 Available online xxxx AMS 2000 subject classifications: primary 91B84 secondary 62M20 Keywords: Asymmetry HAR model Long-memory Market microstructure noise Realized covariance Realized variance

a b s t r a c t Forecast methods for realized volatilities are reviewed. Basic theoretical and empirical features of realized volatilities as well as versions of estimators of realized volatility are briefly investigated. Major forecast models featuring the empirical aspects of persistency and asymmetry are discussed in terms of forecasting models for which the heterogeneous autoregressive (HAR) model is one of the most basic one in the recent literature. Forecast methods addressing the issues of jump, break, implied volatility, and market microstructure noise are reviewed. Forecasting realized covariance matrix is also considered. © 2018 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4. 5. 6. 7.

Introduction............................................................................................................................................................................................... Realized volatility ..................................................................................................................................................................................... Forecast evaluation ................................................................................................................................................................................... Realized volatility forecast models .......................................................................................................................................................... Issues in realized volatility forecasts ....................................................................................................................................................... Realized covariance forecasts................................................................................................................................................................... Conclusion ................................................................................................................................................................................................. Acknowledgments .................................................................................................................................................................................... References .................................................................................................................................................................................................

1 2 4 4 6 7 7 8 8

1. Introduction Forecasting volatility for financial assets is an essential part in risk management, derivative pricing, and portfolio allocation for which various methods have been employed. A class of widely adopted models is the GARCH-type models such as GARCH, EGARCH, CJR-GARCH models and others, which handle daily unobservable hidden conditional variance of asset returns. In the recent days, availability of high frequency intra-day asset price data sets endows us realization of the hidden volatility, called realized volatility. The daily realized variance is the sum of squares of intra-day high frequency returns for the given day and may be regarded as a realization of the underlying variance, daily integrated variance. In the recent literature, a large number of papers have appeared for the theoretical subjects of consistency, limiting distribution, consistent estimation, optimal sampling frequency, and others and for applications in various areas of finance, E-mail address: [email protected]. https://doi.org/10.1016/j.jkss.2018.08.002 1226-3192/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

2

D.W. Shin / Journal of the Korean Statistical Society (

)

–

business, and economics. Among the diverse research topics of realized volatility, forecast of it is one of the major issues because volatility forecast is related with future uncertainty of asset prices. During the last decade, a lots of forecast methods have appeared for realized volatility forecast, whose review is the aim of this paper. Some reviews on volatility forecasts are available in the literature. Poon and Granger (2003) made a comparative survey on methods for financial volatility forecasts: historical volatilities, GARCH-type forecasts, option-implied volatility based on Black–Scholes equation, and stochastic volatility model forecasts. McAleer and Medeiros (2008a, Section 5) made a brief review on realized volatility forecast stressing on long-memory features of realized volatilities. Andersen and Terasvirta (2009, Section 7.2) reviewed realized volatility forecast in terms of MIDAS and multiplicative error model, jump, and market microstructure noise. Recently, Bucci (2017) wrote a review paper of the same title of this paper. Bucci (2017) discussed diverse volatility forecasting models such as GARCH-type conditional heteroscedastic models, the stochastic volatility models, the time series models (the HAR-class) of the realized volatility, the MIDAS and GARCH–MIDAS models, the realized GARCH and HEAVY model as well as distributional properties of realized volatility and the forecast evaluation methods. The main difference between this paper and Bucci (2017)’s is that the former is focused on heterogeneous autoregressive (HAR) model and the latter is focused on the above-mentioned other models, especially GARCH-type models. In realized volatility forecasting, GARCH-type models, the stochastic volatility models, the MIDAS models, and HEAVY models have been much less frequently considered than HAR models. This paper therefore provides a comprehensive discussion on the most intensively considered HAR-type models and some discussions on GARCH-type models, stochastic volatility model and MIDAS models. Realized volatility forecasts have been considered as alternatives to GARCH-type daily volatility forecasts in risk management and portfolio allocation. For example, Giot and Laurent (2004), Louzis, Xanthopoulos-Sisinis, and Refenes (2014), and Wong, Chin, and Tan (2016) compared VaR (value-at-risk) forecasts based on GARCH-type models and realized volatility models. Čech and Barunik (2017) and Chiriac and Voev (2011) considered balancing of portfolio based on realized covariance matrix forecasts. The remaining of the paper discusses realized volatilities in terms of definition, estimation, basic theoretical properties, and empirical aspects in Section 2, forecast evaluation methods in Section 3, forecast models in Section 4, related forecast issues in Section 5, realized covariance forecasts in Section 6, and conclusion in Section 7. 2. Realized volatility Let Pt be a price process of an asset. Assume the log-price pt = ln Pt follows a generalized Ito process dpt = µt dt + σt dwt , t ∈ R,

(1)

where µt is a continuous and locally bounded function, σt is a càdlàg function, and wt is the standard Brownian motion. We follow the usual convention that an interval [t − 1, t ] of unit length corresponds to a day and t = 1, 2, . . . denote the beginning of day 1, 2, . . . . The integrated variance and the integrated volatility of the asset for day t are

∫

√∫

t

σ

2 s ds

,

t −1

t

σs2 ds, t = 1, 2, . . . t −1

Versions of realized volatility A simple version of realized variance, RV, is the sum of squares of intra-day log-returns. For a day t, assume a set of equally spaced intra-day price observations Pt −1+hj , j = 0, 1, . . . , N , h = 1/N is available. Then the h-spaced realized variance for day t is given by (h)

RVt

=

N ∑

(h)2

rtj

, t = 1, 2, . . . , h = 1/N

j=1

where (h)

rtj = pt −1+hj − pt −1+h(j−1) is the log-return for the h-spaced interval [t − 1 + h(j − 1), t − 1 + hj]. Realized volatility for day t is of

(h) RVt

√ such as

(h) RVt

and

(h) log RVt

√∑ N

(h)2 j=1 rtj

. Transforms

are frequently considered in practice. People use the notation RVt for both realized

variance and realized volatility and IVt for both integrated variance and integrated volatility, which will be also adopted in this paper. (h) (h) Since rtj , j = 1, 2, . . . , N in model (1) are uncorrelated, RVt converges in probability to IVt as h → 0, see Andersen and Bollerslev (1998) and Jacod and Shiryaev (1987, p. 55). A critical condition for consistency is that the log-price pt is observed free from noise, i.e., the market microstructure noise arising from bid–ask–spread, discreteness of price change, latency, and asymmetric information of traders. If the sampling interval h is small, then the market microstructure noise is Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

D.W. Shin / Journal of the Korean Statistical Society (

)

–

3

non-ignorable and RVt is subject to bias and inconsistency. In order to get consistent estimator of IVt from high frequency intraday log-returns, various methods were considered in the literature. If N is large, then the summing effect of noise ut in the observed price p˜ t = pt + ut is non-negligible in RVt even in (h) case of small ut . Then, the observed returns rtj = p˜ t −1+hj − p˜ t −1+h(j−1) = pt −1+hj − pt −1+h(j−1) + ut −1+hj − ut −1+h(j−1) , j = (h)

1, 2, . . . , N are serially correlated because of common noise terms for adjacent returns, rendering RVt to be biased and to be inconsistent. We discuss several widely-accepted bias-adjusted realized variances. Assume N is large. The realized kernel variance of Barndorff-Nielsen, Hansen, Lunde, and Shephard (2008, 2011) is RVtkernel = CRVt (0) + 2

Lker ∑

Kker (

ℓ−1 Lker

ℓ=1

)CRVt (ℓ),

where CRVt (ℓ) =

N ∑

(h) (h)

rtj rt ,j−ℓ , ℓ = 0, 1, . . . , Lker ,

j=1+ℓ

for some kernel function Kker (·) and a bandwidth Lker . The realized kernel variance is a long-run variance type estimator for ∑N (h) (h) j=1 rtj = pt − pt −1 , the daily log-return, based on high frequency intra-day returns rtj , j = 1, . . . , N which are serially correlated owning to market microstructure noise. Pre-averaged realized variance of Christensen, Kinnebrock, and Podolskij (2010) and Jacod, Li, Mykl, Podolskij, and Vetter (2009) is pre−av erage RVt

=

N −Lpre +1

1

∑

ψ2 Lpre

j=1

(h)2

r¯tj

−

ψ1 2N θ 2 ψ2

RVtall , RVtall = CRVt (0)

(2)

based on pre-averaged returns (h) rtj

¯

=

Lpre ∑

(h)

Kpre (ℓ/Lpre )rt ,j+ℓ ,

ℓ=1 L

where Kpre (·) is a kernel function and Lpre is a bandwidth, ψ1 , ψ2 are constants defined from Kpre , and θ = limN →∞ √pre . N In order to describe the subsample realized variance and the multi-scale realized variance of Zhang, Mykl, and Ait-Sahalia k (2005), assume we have a high-frequency intra-day observations of log returns at times t − 1 + MN , k = 1, . . . , MN for some positive integers M , N. The subsampling realized variance of Zhang et al. (2005) is the average av erage

RVt

M −1

= M −1

∑

(h)

RVt +ih/M , h = 1/N

(3)

i=0 (h)

(h)

(h)

of standard sparsely sampled, i.e., h-spaced, realized variances RVt , RVt +h/M , . . . , RVt +(M −1)h/M , with h of sparse sampling, (h) for example 5 or 10 min. If h is not small, each of RVt +ih/N

, i = 0, . . . , M − 1 is free from bias and so is RVtaverage . The multi-scale

realized variance of Zhang et al. (2005), in two-time-scale version, is av erage

RVtTS = (RVt

− RVtall /N). av erage

The term RVtall /N is consistent for the bias of RVt arising from market microstructure noise. Other noise-consistent realized volatilities are the multi-scale realized volatility of Zhang (2006, 2011), the wavelet estimator of Fan and Wang (2007), the quasi-maximum likelihood estimator of Aït-Sahalia, Fan, and Xiu (2010) and Xiu (2010), the local method of moments estimator of Bibinger, Hautsch, Malec, Reiss, et al. (2014), and the robust pre-averaging realized volatility estimator of Fan and Kim (2018). The Oxford-Man Institute’s Realized Library (https://realized.oxford-man.ox.ac.uk/) provides daily realized variances of h = 5 min, 10 min sampling intervals, realized subsampling variances of h = 5 min, 10 min sampling intervals, and realized kernel variances with several kernels for the world major 31 financial assets such as the S&P index, the KOSPI, and others Features of realized volatility Realized volatilities share the features of persistency, asymmetry in return leverage, heavy and asymmetric tails of other financial volatilities discussed in Engle and Patton (2007, Section 2) for example. Recent realized volatility forecast methods try to address these features for good forecast. The most dominant feature is persistency, or long-memory, as manifested by clustering of large moves and small moves. Essentially, almost all the recent realized volatility forecast methods address the long-memory feature. It is generally reported that the long-memoried realized volatility, even though persistent, is stationary and is mean reverting in that the order of fractional integration is typically less than 0.5 for realized volatilities on equities analyzed by many researchers, see Andersen and Benzoni (2001, Section 7.2). Also, many authors demonstrated that the augmented Dickey–Fuller test is generally rejected against stationarity for usual financial asset realized volatilities of stock prices, foreign exchange Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

4

D.W. Shin / Journal of the Korean Statistical Society (

)

–

rates, and others, see for example Andersen, Bollerslev, Diebold, and Ebens (2001). The real long-memory dynamics of persistent realized volatilities lie near the border of stationarity–nonstationarity in that some less frequent studies argue for nonstationarity rather than stationarity, see Cho and Shin (2016) and Shin and Shin (2018). Another important feature is asymmetry related with leverage of negative returns. It is well known that realized volatilities for financial asset prices are leveraged up by negative returns, see Bollerslev, Litvinova, and Tauchen (2006), Campbell and Hentschel (1992) and Dennis, Mayhew, and Stivers (2006). Many authors argued that the same is true for the realized volatilities, see for example Corsi and Reno (2009) and McAleer and Medeiros (2008a). The realized volatility has also asymmetric volatility clustering as discussed by Corsi, Kretschemer, Mittnik, and Pigorsch (2008) and Ning, Xu, and Wirjanto (2015) and has fat-tailed asymmetric distributions as discussed by Andersen, Bollerslev, Diebold, and Labys (1999) and Corsi et al. (2008). 3. Forecast evaluation Assume that we are interested in forecasting RVt , one of the realized variances discussed in Section 2, its square root transform, or logarithm transform. Forecast accuracies are usually compared by forecast measures computed from out-ofˆ T +H |T of RVT +H at forecast origin T is constructed by a forecast sample forecast errors. Assume an h-step ahead forecast RV method, H = 1, 2, . . .. Frequently-used forecast measures are based on losses such as ˆ T +H |T , RVT +H ) = (RV ˆ T +H |T − RVT +H )2 , Square error (SE) loss: ℓ(RV ˆ T +H |T , RVT +H ) = |RV ˆ T +H |T − RVT +H |, Absolute error (AE) loss: ℓ(RV ˆ T +H |T , RVT +H ) = 100 × (|RV ˆ T +H |T − RVT +H |/RVT +H ), Absolute percent error (APE) loss: ℓ(RV ˆ T +H |T , RVT +H ) = (log RVT +H + RV ˆ T +H |T /RVT +H ). Quasi likelihood (QLIKE) loss: ℓ(RV Assume we have a sample RVt , t = 1, . . . , n of size n. Starting from T0 , typically 0.8n or 0.85n, for T = T0 , ..., T1 = ˆ T +H |T is constructed from an expanding window sample or a moving window sample n − H, out-of-sample forecast RV up to time T . Then the out-of-sample MSE, MAE, MAPE, and QLIKE are computed by averaging the corresponding loss as ∑T ˆ T +H |T , RVT +H ). (T1 − T0 + 1)−1 T1=T ℓ(RV 0 Out-of-sample forecast measures can be statistically compared. The test of Diebold and Mariano (2002) can make a (k) ˆ T +H |T , RVT +H ), T = T0 , ..., T1 }, k = 1,2 pairwise statistical comparison. Given two sets of out-of-sample losses {ℓT = ℓ(k) (RV ¯ where obtained by two forecast methods k = 1,2, the Diebold and Mariano test is a paired two-sample t-test DM = d¯ /se(d), ∑T (2) (1) ¯ is an autocorrelation consistent standard error of d. ¯ The model d¯ = (T1 − T0 + 1)−1 T1=T dT , dT = ℓT − ℓT and se(d) 0 confidence set (MCS) analysis of Hansen, Lunde, and Nason (2011) can compare more than two forecast methods. The MCS is a set of superior models in the given forecast measure such as MSE, MAE, MAPE, and QLIKE, and others. For each forecast model, the MCS analysis produces the model forecast performance rank and the inclusion probability of the model in the MCS. When realized volatility forecasts are used for VaR (value-at-risk) forecasts, their forecast performances are usually compared by 1-step out-of-sample violations, i.e., the proportions of 1-step ahead actual returns smaller than the 1-step ahead VaR forecasts in the out-of-sample comparison: a method having out-of-sample violation closer to the given VaR level is a better one. See Giot and Laurent (2004), Louzis et al. (2014), and others. Performances of realized covariance matrix forecasts is evaluated by the forecast measures MSE, MAE, MAPE, QLIKE and more importantly by forecasted minimum variance of the minimum variance portfolio in which the minimum variance portfolio is constructed from covariance matrix forecasts, see Čech and Barunik (2017). 4. Realized volatility forecast models In the recent literature for realized volatility forecast, the heterogeneous autoregressive (HAR) model of Corsi (2009) is a key model from which many diverse modifications have appeared. The HAR model and its modifications will be reviewed in the rest of this section. Other methods than HAR methods are first reviewed. Long memory fractional integration models were widely applied in the past literature. For example, Deo, Hurvich, and Lu (2006) combined ARFIMA model with stochastic volatility model to develop a forecast method of realized volatility. The Mixed Data Sampling (MIDAS) regression method of Ghysels, Santa-Clara, and Valkanov (2004, 2006) combines volatility estimates at different frequencies and horizons. Engle and Gallo (2006) developed volatility forecast by an multivariate extension of the multiplicative error model by Engle (2002) to take advantage of different volatility measures. Various other methods were considered for realized volatility forecast: a Bayesian method by Liu and Maheu (2009); bagging by Hillebrand and Medeiros (2010); neural network by Kristjanpoller, Fadic, and Minutolo (2014) and Roh (2007). For forecasting return volatility, the GARCH-type models are modified to incorporate high frequency measures such as the realized volatility into the conditional return volatility model: the HEAVY model of Shephard and Sheppard (2010) and the realized GARCH model of Hansen, Huang, and Shek (2012). Recently, Kim and Wang (2016) modified the Ito-process model to a GARCH–Ito model by adding a GARCH-type volatility model in which the return square is replaced by integrated volatility square. Forecasts of large-sized volatility matrix were demonstrated by a high-dimensional extension of the GARCH–Ito model made by Kim and Fan (2018). Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

D.W. Shin / Journal of the Korean Statistical Society (

)

–

5

HAR-RV forecast model In the recent literature for realized volatility forecast, the most widely accepted model is the heterogeneous autoregressive realized volatility (HAR-RV) model proposed by Corsi (2009), which will be termed as HAR model. Let yt be one of RVt , RVt , log RVt , where RVt is a version of realized variances discussed in Section 2. We wish to construct forecasts of RVt . The HAR model is M yt +1 = β0 + β1 yDt + β2 yW t + β 3 y t + at + 1 ,

(4)

where yDt = yt ; yW t =

1 5

(yt + · · · + yt −4 ); yM t =

1 22

(yt + · · · + yt −21 )

(5)

and at is a sequence of regression error. The moving average definition in (5) is generic in that, for any sequence xt such 1 M D W M as realized volatility RVt , jump Jt , daily return rt , and others, xDt , xW t , xt denote xt = xt ; xt = 5 (xt + · · · + xt −4 ); xt = 1 D (x + · · · + xt −21 ), respectively. Forecast model (4) addresses volatility behaviors of short-term investors by yt , mid-term 22 t M investors by the weekly moving average yM t , and long-term investors by the monthly moving average yt . Therefore, the corresponding coefficients β1 , β2 , β3 have a clear economic interpretation in terms of a forecast term structure. At forecast horizon n, H-step ahead forecasts yˆ n+H |n of yn+H , H = 1, 2, ... are successively constructed from auto-regression (4), for t = n + 1, . . . , n + H in the same way that forecasts are constructed in autoregressive models. Note that model (4) is an autoregressive (AR) model of order 22. The dominant persistent features of realized volatilities are well captured by this long-lag structure of lag 22 by a parsimonious model having only 3 autoregressive parameters β1 , β2 , β3 . In the real world applications, the persistent long memory feature of realized volatilities are captured by this long-lag and by the fact of estimated value of β1 + β2 + β3 being close to 1, see Corsi (2009) and many others. Various modifications of HAR model have been proposed to take advantage of the dominating features discussed in the later part of Section 2 for improved forecasts. LHAR model Corsi and Reno (2009) try to address the well-known asymmetric leverage effect of negative return by a model called leverage HAR (LHAR) model M D− yt +1 = β0 + β1 yDt + β2 yW + γ2 rtW − + γ3 rtM − + at +1 , t + β3 yt + γ1 rt

(6)

where rt = pt − pt −1 is the log-return for day t, , , are constructed from rt as in (5), and xt = xt I(xt < 0) is the negative part of xt , and I(A) is the indicator function of an event A. In model (6), asymmetric leverage effects of return on realized volatility are reflected by the negative parts of daily, weekly, and monthly return rtD− , rtW − , rtM − . Similar leverage consideration was made by McAleer and Medeiros (2008b) who considered a logistic-type transition (1 + e−γ rt )−1 rather than the step-function type transition rt− = rt I(rt < 0). Choi and Shin (2018a) developed optimal forecast for an extended model of (6) with more general lag structure than the daily, weekly, and monthly moving average lags. Many studies reported that the LHAR models improves the HAR model in real world realized volatility forecasting, see Asai, McAleer, and Medeiros (2011), Audrino and Knaus (2016), Byun and Kim (2013), Corsi and Reno (2009), Liu and Maheu (2009), Patton and Sheppard (2015), Scharth and Medeiros (2009), and others. rtD

rtW

−

rtM

HAR(∞) and IHAR models The HAR model (4) has order 3 and may be termed as a HAR(3) model. The HAR(3) model, even having long lags yt , yt −1 , . . . , yt −21 , is theoretically short-memoried having exponentially decaying autocorrelation function (ACF) if the characteristic roots lie outside the unit circle. Hwang and Shin (2014) extended the HAR(3) model to an infinite order model, HAR(∞), which has regressors h1 (yt + · · · + yt −hj +1 ), j = 1, 2, . . . with exponentially increasing hj . They demonstrated j

real long-memoriness of algebraically decaying ACF of HAR(∞) model. They developed an asymptotic estimation theory and method. Functional central limit theorem for the HAR(∞) model was established by Lee (2014) by showing that the HAR(∞) process is near epoch dependent. Cho and Shin (2016), noting closeness of estimated value of β1 + β2 + β3 to 1 in most real data applications, considered the strategy of pushing β1 + β2 + β3 to 1 and termed the resulting model an integrated HAR(3), IHAR(3), model. It is a unit root model and hence is nonstationary. They justified nonstationarity by arguing that the KPSS unit root test of Kwiatkowski, Phillips, Schmidt, and Shin (1992) is usually rejected against nonstationary alternative hypothesis. They demonstrated that the IHAR(3) model produces better out-of-sample forecast, especially for multi-step ahead forecasts, than HAR(3) model in a diverse real data set applications. Song and Shin (2018) illustrated better out-of-sample forecast by the HAR(3) model having no intercept term than by the HAR(3) model. HARQ model Bollerslev, Pattona, and Quaedvlieg (2016) proposed an extension of the HAR model for the h-spaced realized variance (h) M yt = RVt in which the forecast regressors yDt , yW t , yt have varying coefficients according to the precision of the regressors.

√

(h)

d

By Barndorff-Nielsen and Shephard (2002), conditional on the realization of IQt , 1/h(RVt − IVt ) −→ N(0, 2IQt ), where ∫t (h) IQt = t −1 σs4 ds is the integrated quarticity. Precision of yDt = RVt is measured by its asymptotic variance 2IQt for which Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

6

D.W. Shin / Journal of the Korean Statistical Society ( (h)

∑N

(h)4

a consistent estimator RQt = N3 j=1 rtj model to have a varying coefficient as (h) 1/2

yt +1 = β0 + (β1 + β1Q RQt

)

–

, the realized quarticity, is available. Bollerslev et al. (2016) modified the HAR

M )yDt + β2 yW t + β3 yt + at +1 ,

(7)

which is termed a HAR quarticity (HARQ) model. In the real data analysis, estimated values of β1Q are usually negative, indicating that less weight is given to RVtD with larger variance 2IQt . Bollerslev et al. (2016) demonstrated improved performances over HAR model in forecasting realized volatilities of S&P500 equity index and individual constituents of the Dow Jones Industrial Vaerage. Vector models Souček and Todorova (2013) considered a vector version of the HAR(3) model (4) in which yt is a q-dimensional realized volatility vector. Cubadda, Guardabascio, and Hecq (2017) modified the vector HAR model to a parameter-parsimonious reduced rank model ′ M yt +1 = β0 + β1 ω′ yDt + β2 ω′ yW t + β3 ω yt + at +1 ,

(8)

where ω is a q × r full rank matrix. They call ft = ω yt the index. Shin and Shin (2018) extended the unit root strategy of IHAR model to a cointegration model, called a vector error correction HAR, VecHAR model, for a pair of realized volatility (y1t ) and volatility index (y2t ). Let yt = (y1t , y2t )′ . Following the spirit of the IHAR model of Cho and Shin (2016) and Shin and Shin (2018) regarded yt to be nonstationary and assume that yt has a stationary cointegration equilibrium zt = y1t − γ y2t . This assumption leads to a cointegration model, the VecHAR model, ′

D M D ∆yt = α zt + β0 + β2 (yW t − yt ) + β3 (yt − yt ) + at +1 .

Shin and Shin (2018) demonstrated substantially improved realized volatility forecasts for the VecHAR model over the HAR, IHAR, HARQ models for the S&P 500, RUSSELL 2000 indexes. 5. Issues in realized volatility forecasts Jump In spite of the nonstationary-persistency claimed by Cho and Shin (2016) and Shin and Shin (2018), persistency of realized volatility is generally accepted to be stationary. One reason for RVt to be empirically stationary is the sudden change of the underlying log-price processes which render RVt to be less persistent. Several authors try to address the sudden changes to improve realized volatility forecasts by decomposing realized variance into jump and continuous components. Andersen, Bollerslev, and Diebold (2007) pointed out improved forecasting performance if the jump and diffusive components are separated. Forsberg and Ghysels (2007) and Liu and Maheu (2005) demonstrated robustness of realized power variation in the presence of jumps as an estimator of integrated variance and it can improve volatility forecasts. The separated jump and continuous components are incorporated into the HAR model: the HAR-CJ models of Andersen et al. (2007) and Busch, Christensen, and Nielsen (2011), and Duong and Swanson (2011). Sudden changes in the log-price processes is usually considered by a jump diffusion process dpt = µt dt + σt dwt + κt dqt , which extends the Ito process (1) to have jumps of size κt at times t for τ1 < τ2 < · · · arriving according to an arrival process qt with dqt = 1 in times t for τ1 < τ2 < · · · of jumps and dqt =∫ 0 in other times. ∑q In the presence of jumps, the t (h) realized variance yt = RVt is consistent for the quadratic variation QVt = t −1 σs2 ds + j=t q κτ2j but not for the integrated t −1 ∑ (h) = π2 NN−k Nj=k+1 |rt(h) ,j−k rtj | is consistent for IVt , where k is any (h) (h) given positive integer. Now, a realized jump is J˜t = (RVt − BVt ). Statistical significance of J˜t are usually checked by the ˜ ˜ Z-test Zt = Jt /se(Jt ) of Barndorff-Nielsen and Shephard (2006) giving the ‘‘test-passed’’ jump, Jt = J˜t I(Zt > Φ −1 (1 − α )), where α ∈ (0, 0.5) is the size of the test and Φ is the standard normal distribution function. The HAR-CJ model is M D W M yt +1 = β0 +β1 yDt +β2 yW t +β3 yt +δ1 Jt +δ2 Jt +δ3 Jt + at +1 . The jump component is usually accompanied by leverage terms as D− W− M in Corsi and Reno (2009) and many others yt +1 = β0 +β1 yDt +β2 yW +γ3 rtM − +δ1 JtD +δ2 JtW +δ3 JtM +at +1 . t +β3 yt +γ1 rt +γ2 rt

(h)

variance IVt . Instead, the realized bi-power variation BVt

Break It is well known that long-memory is hardly distinguishable from occasional breaks, see for example, Granger and Hyung (2004) and the summary paper by Banerjee and Urga (2005) regarding structural breaks, long-memory and stock market volatility. Structural breaks in the coefficients of HAR models were studied by Liu and Maheu (2008) in terms of a Bayesian approach especially for the variance parameter and by Choi, Yu, and Zivot (2010) for mean in realized volatilities of some foreign exchange rates. The latter demonstrated better out-of-sample forecasts of ‘‘short-memory + break’’ models than ‘‘long-memory only’’ models in some known break cases. Song and Shin (2015) addressed both breaks and long memory by break dummies and HAR model. They demonstrated better out-of-sample performance of this ‘‘long-memory+break’’ method than the ‘‘short-memory+break’’ method, that is, the VAR+break method, of Choi et al. (2010). On the other hand, Yang and Chen (2014) argued that structural breaks explain only a part of long-memory of realized volatility in realized volatility forecasts. Gallo and Otranto (2015) considered slowly moving average type breaks by regime switching Markovian Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

D.W. Shin / Journal of the Korean Statistical Society (

)

–

7

dynamics and by multiplicative error models, which reveal satisfactory realized volatility forecasts. Structural breaks for the HAR(∞) model were studied by Hwang and Shin (2013, 2015, 2017, 2018b). Implied volatility Implied volatility has additional information in forecasting realized volatility. Christensen and Prabhala (1998) demonstrated usefulness of volatility implied by the S&P100 index option prices in forecasting realized volatility of the S&P100 index. Pong, Shackleton, Taylor, and Xu (2004) considered long term forecasts of monthly and quarterly forecasting realized volatilities of pound, mark, yen exchange rates against dollar and showed that implied volatility is at least as accurate as realized volatility. Implied volatility is demonstrated to have incremental information on realized volatility forecasts by Busch et al. (2011) for foreign exchange, stock, and bond markets and by Park and Shin (2014) for the S&P500 and the Korean stock price index. Shin and Shin (2018) took advantage of the additional information of implied volatility by a vector error correction model for the vector of implied volatility and realized volatility. Market microstructure noise Recalling that realized variance RVt is consistent for integrated variance IVt , we may regard the forecasts of realized variances as those for integrated variances. Since RVt is subject to bias owning to market microstructure noise, so are forecasts of IVt . The effects of market microstructure noise on realized volatility forecasts were discussed by, e.g., Aït-Sahalia and Mancini (2006), Andersen, Bollerslev, and Meddahi (2005, 2011), and Ghysels and Sinko (2006). Andersen et al. (2011) warned us the detrimental impact of market microstructure noise on forecast accuracy and recommended a linear forecast based on sub-sampled realized variance estimator (3) to be a robust one. Asai, McAleer, and Medeiros (2012) pointed out that even bias-corrected consistent realized volatility estimators such as the subsample estimator, kernel estimator, and twotime scale estimator can contain residual microstructure noise, which if neglected cause serious bias in integrated variance estimation but not its one-step ahead forecast in case of large intraday observations. Bandi, Russell, and Yang (2013) studied realized volatility forecasting in the situation of market microstructure noise with time-varying second moment. Others Tseng, Lai, and Lin (2012) incorporated overnight returns indicator into the HAR model to achieve forecast improvement. Ahoniemi and Lanne (2013) reported superior accuracy of the 1-day-ahead VaR forecasts for Russell 2000 index if overnight information is properly utilized. Audrino and Knaus (2016) discussed relation between lasso and the HAR model: lass recovers HAR model asymptotically if it is the true model; but not in a Monte Carlo finite sample, concluding that ‘‘HAR model may not be the true model but it captures a linear footprint of the true volatility dynamics’’. Choi and Shin (2018b) considered quantile forecasts for realized volatility in a LHAR+EGACH+skew-t model and demonstrated merits of specifying leverage, asymmetry for conditional variance of the LHAR regression (6) error at +1 , and skewness in the distribution of at +1 . 6. Realized covariance forecasts Forecasts of realized covariance is important especially in portfolio allocation. Now, consider a q-dimensional log-price vector pt = (p1t , . . . , pqt )′ derived by q dimensional Ito process dpt = µt dt + θt dwt , where µt = (µ1t , . . . , µqt )′ , θt = (θkℓt )q×q are unknown drift process and diffusion process, and wt = (w1t , . . . , wqt )′ is ∫ t a q-dimensional standard Brownian motion. The integrated covariance matrix over time [t − 1, t ] is IV t = (IV kℓt )q×q = t −1 θs′ θs ds. Assume we have (h)

equally spaced and synchronous intra-day returns rkjt = pk,t −1+jh − pk,t −1+(j−1)h , k = 1, . . . , q, h = 1/N . The realized (h)

∑N (h) (h) (h) (h) = (RVk(h) ℓt )q×q , where RVkℓt = j=1 rkjt rℓjt , k, ℓ = 1, ..., q. Note that RVkℓt is (h) the realized covariance of log returns of assets k, ℓ. A consistent estimator of the integrated covariance matrix IVt is St . covariance matrix is the q × q matrix St

In practice, if sampling interval h is small, then equally spaced and synchronous intra-day observations are unavailable. To handle the asynchronous observations, several strategies were considered for consistent realized covariance matrix. One strategy is joint consideration of the refresh time of Barndorff-Nielsen et al. (2011) and Bibinger (2012) and the previous-tick time of Wang and Zou (2010) and Zhang (2011), see for example Hwang and Shin (2018a). Let St be a version of realized covariance matrix. In constructing positive semi-definite forecasts of realized covariance matrix, Chiriac and Voev (2011) considered the Cholesky decomposition St = Ut′ Ut . Forecasts of U-component Ut is first constructed from which those of St is constructed. For the forecast models of the vector v ech(Ut ) of elements of Ut , they discussed joint prediction by VARFIMA, HAR, Wishart Autoregressive (WAR), (Fractionally Integrated) Dynamic Conditional Correlation Models. Čech and Barunik (2017) considered the HAR model in a seemingly unrelated regression setup which is termed as generalized HAR (GHAR) model. They demonstrated that the GHAR model produces good forecasts in terms of out-of-sample variance of minimum variance portfolio. One may consider any of the univariate forecast methods discussed in Sections 4, 5 for componentwise forecasts of Ut . Kim and Shin (2018) modified the GHAR model to GHARQ model by considering the HARQ model (7) instead of the HAR model to achieve better out-of-sample variance of minimum variance portfolio. 7. Conclusion We have reviewed recent developments in realized volatility forecasts which are rapidly growing up. Versions of realized volatilities were discussed in terms of market microstructure noise. Our focus lies on forecast methods especially those based Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

8

D.W. Shin / Journal of the Korean Statistical Society (

)

–

on the heterogeneous autoregressive model (HAR) whose variations feature the major aspects of realized volatilities of longmemory, asymmetry, jump, break, and market microstructure noise: LHAR, HAR(∞), IHAR, VecHAR, HARQ, HAR-CJ models. The roles of implied volatility and market microstructure in realized volatility forecasts are discussed. issues in forecasts of realized covariance matrix were addressed. Acknowledgments The author is very grateful to the Editor Professor Hee-Seok Oh for inviting him to this review and of the valuable comments of two unknown reviewers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, South Korea (2009-0093827). Editorial helps from Ms. Ji-Eun Choi and Ms. Bokyung Kim are also appreciated. References Ahoniemi, K., & Lanne, M. (2013). Overnight stock returns and realized volatility. International Journal of Forecasting, 29, 592–604. Aït-Sahalia, Y., Fan, J., & Xiu, D. (2010). High-frequency covariance estimates with noisy and asynchronous financial data. Journal of the American Statistical Association, 105, 1504–1517. Aït-Sahalia, Y., & Mancini, L. (2006). Out of sample forecasts of quadratic variation. Working Paper, Princeton University and University of Zürich. Andersen, T. G., & Benzoni, L. (2001). Stochastic volatility. In R. A. Meyers (Ed.), Complex systems in finance and economics (pp. 694–726). Springer-Verlag. Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Econometric Review, 39, 885–905. Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2007). Roughing it up including jump components in the measurement, modeling and forecasting. The Review of Economics and Statistics, 89, 701–720. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Ebens, H. (2001). The distribution of realized stock return volatility. Journal of Financial Economics, 61, 43–76. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (1999). The distribution of exchange rate volatility. Journal of the American Statistical Association, 96, 42–55. Andersen, T. G., Bollerslev, T., & Meddahi, N. (2005). Correcting the errors: volatility forecast evaluation using high-frequency data and realized volatilities. Econometrica, 73, 279–296. Andersen, T. G., Bollerslev, T., & Meddahi, N. (2011). Realized volatility forecasting and market microstructure noise. Journal of Econometrics, 160, 220–234. Andersen, T. G., & Terasvirta, T. (2009). Realized volatility. In T. Mikosch, J. P. Krei, R. Davis, & T. Andersen (Eds.), Handbook of financial time series. Berlin, Heidelberg: Springer. Asai, M., McAleer, M., & Medeiros, M. C. (2011). Asymmetry and long memory in volatility modelling. Journal of Financial Econometrics, 0, 1–18. Asai, M., McAleer, M., & Medeiros, M. C. (2012). Modelling and forecasting noisy realized volatility. Computational Statistics & Data Analysis, 56, 217–230. Audrino, F., & Knaus, S. (2016). Lassoing the HAR model: A model selection perspective on realized volatility dynamics. Econometric Review, 35, 1485–1521. Bandi, F. M., Russell, J. R., & Yang, C. (2013). Realized volatility forecasting in the presence of time-varying noise. Journal of Business & Economic Statistics, 31, 331–345. Banerjee, A., & Urga, G. (2005). Modelling structural breaks, long memory and stock market volatility: An overview. Journal of Econometrics, 129, 1–34. Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., & Shephard, N. (2008). Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76, 1481–1536. Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., & Shephard, N. (2011). Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Journal of Econometrics, 162, 149–169. Barndorff-Nielsen, O. E., & Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B., 64, 253–280. Barndorff-Nielsen, O. E., & Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics, 4, 1–30. Bibinger, M. (2012). An estimator for the quadratic covariation of asynchronously observed Ito processes with noise: asymptotic distribution theory. Stochastic Processes and their Applications, 122, 2411–2453. Bibinger, M., Hautsch, N., Malec, P., Reiss, M., et al. (2014). Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency. The Annals of Statistics, 42, 1312–1346. Bollerslev, T., Litvinova, J., & Tauchen, G. (2006). Leverage and volatility feedback effects in high-frequency data. Journal of Financial Econometrics, 4, 353– 384. Bollerslev, T., Pattona, A., & Quaedvlieg, R. (2016). Exploiting the errors: A simple approach for improved volatility forecasting. Journal of Econometrics, 192, 1–18. Bucci, A. (2017). Forecasting realized volatility: a review. Department of Economics and Social Sciences, Universita Politecnica delle Marche, Working paper. Busch, T., Christensen, B. J., & Nielsen, M. (2011). The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. Journal of Econometrics, 160, 48–57. Byun, S. J., & Kim, J. S. (2013). The information content of risk-neutral skewness for volatility forecasting. Journal of Empirical Finance, 23, 142–161. Campbell, J. Y., & Hentschel, L. (1992). No news is good news: a asymmetric model of changing volatility in stock returns. Journal of Financial Economics, 31, 281–318. Čech, F., & Barunik, J. (2017). On the modelling and forecasting of multivariate realized volatility: generalized heterogeneous autoregressive (GHAR) model. Journal of Forecasting, 36, 181–206. Chiriac, R., & Voev, V. (2011). Modelling and forecasting multivariate realized volatility. Journal of Applied Econometrics, 26, 922–947. Cho, S. J., & Shin, D. W. (2016). An integrated heteroscedastic autoregressive model for forecasting long-memory volatilities. Journal of the Korean Statistical Society, 45, 371–380. Choi, J., & Shin, D. W. (2018a). Forecasts for leverage heterogeneous autoregressive models with jumps and other covariates. Journal of Forecasting. http://dx.doi.org/10.1002/for.2530. Published online. Choi, J., & Shin, D. W. (2018b). Quantile forecasts for financial volatilities based on parametrically asymmetric models. Journal of the Korean Statistical Society. Choi, K., Yu, W. C., & Zivot, E. (2010). Long memory versus structural breaks in modelling and forecasting realized volatility. Journal of International Money and Finance, 29, 857–875. Christensen, B. J., & Prabhala, N. R. (1998). The relation between implied and realized volatility. Journal of Financial Economics, 50, 125–150.

Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

D.W. Shin / Journal of the Korean Statistical Society (

)

–

9

Christensen, K., Kinnebrock, S., & Podolskij, M. (2010). Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with nonsynchronous data. Journal of Econometrics, 159, 116–133. Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7, 174–196. Corsi, F., Kretschemer, U., Mittnik, S., & Pigorsch, C. (2008). The volatility of realized volatility. Econometric Reviews, 27, 46–78. Corsi, F., & Reno, R. (2009). HAR volatility modeling with heterogeneous leverage and jumps. Working Paper, University of Siena. Cubadda, G., Guardabascio, B., & Hecq, A. (2017). A vector heterogeneous autoregressive index model for realized volatility measures. International Journal of Forecasting, 33, 337–344. Dennis, P., Mayhew, S., & Stivers, C. (2006). Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. The Journal of Financial and Quantitative Analysis, 47, 381–406. Deo, R., Hurvich, C., & Lu, Y. (2006). Forecasting realized volatility using a long-memory stochastic volatility model: estimation, prediction and seasonal adjustment. Journal of Econometrics, 131, 29–58. Diebold, F. X., & Mariano, R. S. (2002). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, 134–144. Duong, D., & Swanson, N. (2011). Empirical evidence on jumps and large fluctuations in individual stocks. Department of Economics, Rutgers, The State University of New Jersey, Working Papers. Engle, R. F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics, 17, 425–446. Engle, R. F., & Gallo, G. M. (2006). A multiple indicators model for volatility using intra-daily data. Journal of Econometrics, 131, 3–27. Engle, R. F., & Patton, A. J. (2007). What good is a volatility model? In Forecasting volatility in the financial markets (3rd ed.). (pp. 47–63). Fan, J., & Kim, D. (2018). Robust volatility matrix estimation with factor-based diffusion model for high-frequency financial data. Journal of the American Statistical Association. Fan, J., & Wang, Y. (2007). Multi-scale jump and volatility analysis for high-frequency financial data. Journal of the American Statistical Association, 102, 1349–1362. Forsberg, L., & Ghysels, E. (2007). Why do absolute returns predict volatility so well? Journal of Financial Econometrics, 5, 31–67. Gallo, G. M., & Otranto, E. (2015). Forecasting realized volatility with changing average levels. International Journal of Forecasting, 31, 620–634. Ghysels, E., Santa-Clara, P., & Valkanov, R. (2004). The MIDAS touch: mixed data sampling regressions. Manuscript, University of North Carolina and UCLA. Ghysels, E., Santa-Clara, P., & Valkanov, R. (2006). Predicting volatility: how to get the most out of returns data sampled at different frequencies. Journal of Econometrics, 131, 59–95. Ghysels, E., & Sinko, A. (2006). Volatility forecasting and microstructure noise. Working Paper, University of North Carolina. Giot, P., & Laurent, S. (2004). Modelling daily value-at-risk using realized volatility and ARCH type models. Journal of Empirical Finance, 11, 379–398. Granger, C. W. J., & Hyung, N. (2004). Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance, 11, 399–421. Hansen, P. R., Huang, Z., & Shek, H. H. (2012). Realized GARCH: a joint model for returns and realized measures of volatility. Journal of Applied Econometrics, 27, 87–906. Hansen, P. R., Lunde, A., & Nason, J. M. (2011). The model confidence set. Econometrica, 79, 453–497. Hillebrand, E. P., & Medeiros, M. C. (2010). The benefits of bagging for forecast models of realized volatility. Econometric Reviews, 29, 571–593. Hwang, E., & Shin, D. W. (2013). A CUSUM test for a long memory heterogeneous autoregressive model. Economics Letters, 121, 379–383. Hwang, E., & Shin, D. W. (2014). Infinite-order long memory heterogeneous autoregressive models. Computational Statistics & Data Analysis, 76, 339–358. Hwang, E., & Shin, D. W. (2015). A CUSUMSQ test for structural breaks in error variance for a long 265 memory heterogeneous autoregressive model. Statistics & Probability Letters, 99, 167–176. Hwang, E., & Shin, D. W. (2017). Estimation of structural mean breaks for long-memory data sets. Statistics, 51, 904–920. Hwang, E., & Shin, D. W. (2018a). Two-stage stationary bootstrapping for bivariate average realized volatility matrix under market microstructure noise and asynchronicity. Journal of Econometrics, 202, 178–195. Hwang, E., & Shin, D. W. (2018b). Tests for structural breaks in memory parameters of long-memory heterogeneous autoregressive models. Communications in Statistics. Theory and Methods. http://dx.doi.org/10.1080/03610926.2017.1408827. Published online. Jacod, J., Li, Y., Mykl, P. A., Podolskij, M., & Vetter, M. (2009). Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Processes and their Applications, 119, 2249–2276. Jacod, J., & Shiryaev, A. N. (1987). Limit theorems for stochastic processes. Berlin: Springer-Verlag. Kim, D., & Fan, J. (2018). Factor GARCH-Ito models for high-frequency data with application to large volatility matrix prediction. Working paper, College of Business, Korea Advanced Institute of Science and Technology, Seoul, Korea. Kim, H., & Shin, D. W. (2018). Forecast of realized covariance matrix based on asymptotic distribution of the LU decomposition with an application for balancing minimum variance portfolio. Applied Economics Letters. http://dx.doi.org/10.1080/13504851.2018.1489108. Published online. Kim, D., & Wang, Y. (2016). Unified discrete-time and continuous-time models and statistical inferences for merged low-frequency and high-frequency financial data. Journal of Econometrics, 194, 220–230. Kristjanpoller, W., Fadic, A., & Minutolo, M. C. (2014). Volatility forecast using hybrid Neural Network models. Expert Systems with Applications, 41, 2437–2442. Kwiatkowski, D., Phillips, P., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159–178. Lee, O. (2014). The functional central limit theorem and structural change test for the HAR(∞) model. Economics Letters, 124, 370–373. Liu, C., & Maheu, J. M. (2005). Modeling and forecasting realized volatility: the role of power variation. Working Paper, University of Toronto. Liu, C., & Maheu, J. M. (2008). Are there structural breaks in realized volatility? Journal of Financial Econometrics, 6, 326–360. Liu, C., & Maheu, J. M. (2009). Forecasting realized volatility: A Bayesian model-averaging approach. Journal of Applied Econometrics, 24, 709–733. Louzis, D. P., Xanthopoulos-Sisinis, S., & Refenes, R. P. (2014). Realized volatility models and alternative Value-at-Risk prediction strategies. Economic Modelling, 40, 101–116. McAleer, M., & Medeiros, M. C. (2008a). Realized volatility: A review. Econometric Reviews, 27, 10–45. McAleer, M., & Medeiros, M. C. (2008b). A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries. Journal of Econometrics, 147, 104–119. Ning, C., Xu, D., & Wirjanto, S. T. (2015). Is volatility clustering of asset returns asymmetric? Journal of Banking & Finance, 52, 62–76. Park, S., & Shin, D. W. (2014). Modelling and forecasting realized volatilities of Korean financial assets featuring long memory and asymmetry. Asia-Pacific Journal of Financial Studies, 43, 31–58. Patton, A. J., & Sheppard, K. (2015). Good volatility, Bad volatility: signed jumps and the persistence of volatility. The Review of Economics and Statistics, 97, 683–697. Pong, S., Shackleton, M. B., Taylor, S. J., & Xu, X. (2004). Forecasting currency volatility: A comparison of implied volatilities and AR(FI)MA models. Journal of Banking & Finance, 28, 2541–2563. Poon, S., & Granger, W. C. J. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature, 41, 478–539. Roh, T. H. (2007). Forecasting the volatility of stock price index. Expert Systems with Applications, 33, 916–922.

Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.

10

D.W. Shin / Journal of the Korean Statistical Society (

)

–

Scharth, M., & Medeiros, M. C. (2009). Asymmetric effects and long memory in the volatility of Dow Jones stocks. International Journal of Forecasting, 25, 304–327. Shephard, N., & Sheppard, K. (2010). Realising the future: forecasting with high frequency- based volatility (HEAVY) models. Journal of Applied Econometrics, 25, 197–231. Shin, J., & Shin, D. W. (2018). Vector error correction heterogeneous autoregressive forecast model of realized volatility and implied volatility. Communications in Statistics. Simulation and Computation. http://dx.doi.org/10.1080/03610918.2017.1414250. Published online. Song, H., & Shin, D. W. (2015). Long-memories and mean breaks in realized volatilities. Applied Economics Letters, 22, 1273–1280. Song, H., & Shin, D. W. (2018). Do we need the constant term in the heterogeneous autoregressive model for realized volatilities? Communications in Statistics. Simulation and Computation, 47, 63–73. Souček, M., & Todorova, N. (2013). Realized volatilities transmission between crude oil and equity futures markets: A multivariate HAR approach. Energy Economics, 40, 586–597. Tseng, T. C., Lai, H. C., & Lin, C. F. (2012). The impact of overnight returns on realized volatility. Applied Financial Economics, 22, 357–364. Wang, Y., & Zou, J. (2010). Vast volatility matrix estimation for high-frequency financial data. The Annals of Statistics, 38, 943–978. Wong, Z. Y., Chin, W. C., & Tan, S. H. (2016). Daily value-at-risk modeling and forecast evaluation: The realized volatility approach. The Journal of Finance and Data Science, 2, 171–187. Xiu, D. (2010). Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics, 159, 235–250. Yang, K., & Chen, L. (2014). Realized volatility forecast: structural breaks, long memory, asymmetry, and day-of-the-week effect. International Review of Finance, 14, 345–392. Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli, 12, 1019–1043. Zhang, L. (2011). Estimating covariation: epps effect, microstructure noise. Journal of Econometrics, 160, 33–47. Zhang, L., Mykl, P. A., & Ait-Sahalia, Y. (2005). A tale of two time scales: determining integrated volatility with noisy high-frequency data. Journal of American Statical Association, 100, 1394–1411.

Please cite this article in press as: Shin, D.W., Forecasting realized volatility: A review. Journal of the Korean Statistical Society (2018), https://doi.org/10.1016/j.jkss.2018.08.002.