comparison of realized measure and implied volatility in forecasting volatility

*CorrespondenceRepublic of Kor

Journal of Forecasting J. Forecast. 32, 522–533 (2013)Published online 7 June 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2253

Copyright © 2

Comparison of Realized Measure and Implied Volatility inForecasting Volatility

HEEJOON HAN1* AND MYUNG D. PARK2

1 Department of Economics, Kyung Hee University, Seoul, Republic of Korea2 Department of Economics, National University of Singapore, Singapore

ABSTRACTThis paper compares the information content of realized measures constructed from high-frequency data and impliedvolatilities from options in the context of forecasting volatility. The comparison is based on within-sample and out-of-sample(over horizons of 1–22 days) forecasts of daily S&P 500 index return volatility. The paper adds to the findings of previousstudies, by considering recent developments in the related practice and the literature. It is shown that, for within-samplefitting, the realized measure is more informative than the implied volatility. In contrast, the implied volatility is more infor-mative than the realized measure for out-of-sample forecasting, in particular for multi-step-ahead forecasting. Moreover,we show that it is helpful to use all the information provided by the realized measure and the implied volatility forthe within-sample fitting. For multi-step-ahead forecasting, however, it is better to use only the implied volatility.Copyright © 2013 John Wiley & Sons, Ltd.

key words volatility forecast; realized kernel; VIX; GARCH-X; HEAVY models

INTRODUCTION

The literature on realized measures of volatility constructed from high-frequency data has been rapidly growing,particularly in the last decade. Recently there have been attempts to use a realized measure as an exogenous covariatein the framework of the GARCH-X model:

s2t ¼ oþ ay2t�1 þ bs2t�1 þ gRMt�1 (1)

where (yt) is a demeaned return series, s2t� �

is its variance conditional on the information available at time t-1 andRMt� 1 is a realized measure of volatility based on high-frequency data. The multiplicative error model (MEM) byEngle (2002) first used the realized variance as the covariate in (1) (see also Engle and Gallo, 2006; Barndorff-Nielsenand Shephard, 2007; Cipollini et al., 2007; Shephard and Sheppard, 2010; Hansen et al., 2012).

In particular, the HEAVY models (High-frEquency-bAsed VolatilitY models) by Shephard and Sheppard (2010,henceforth SS) and the realized GARCH model by Hansen et al. (2012) specify conditional variance as theGARCH-X model with the restriction of a = 0 in (1). This means that the volatility of the return series can be fullyexplained by only the information provided by its past realized measures. Moreover, these two models providesystematic frameworks that enable us to obtain multi-step-ahead volatility forecasts using realized measures. Forexample, the HEAVY models are defined as

var yt FHFt�1

�� ¼ s2t ¼ oþ bs2t�1 þ gRMt�1�

(2)

E RMt FHFt�1

�� ¼ mt ¼ oR þ bRmt�1 þ gRRMt�1�

(3)

where FHFt�1 denotes the information of the past of yt and RMt. SS refer to (2) as the HEAVY-r model and to (3) as the

HEAVY-RM model.On the other hand, researchers have also been using the implied volatilities from options to forecast volatility.

Related studies date back to the 1970s. See Latane and Rendleman (1976) and Chiras and Manaster (1978) for earlyexamples. For recent studies, see Christensen and Prabhala (1998), Fleming (1998), Blair et al. (2001), Giot (2003)and Koopman et al. (2005). It is shown in most cases that the models based on implied volatilities provide bettervolatility forecasts of returns on stock indices. Poon and Granger (2003) also reported that, in general, models usingimplied volatilities show better forecasting ability.

Among these studies, we are particularly interested in Blair et al. (2001, henceforth BPT) and Koopman et al.(2005, henceforth KJH) because both papers compare the information content of realized measures and impliedvolatilities in the context of volatility forecasting. BPT showed that the evidence for incremental forecasting information

to: Heejoon Han, Department of Economics, Kyung Hee University, 1 Hoegi-dong, Dongdaemoon-gu, Seoul, 130-701,ea. E-mail: [email protected]

013 John Wiley & Sons, Ltd.

Comparison of Realized Measure and Implied Volatility 523

in the realized measure (realized variance) is insignificant, while the implied volatility (VIX index) provides nearly all therelevant information. In contrast, KJH showed that models with the realized measure are superior.

However, there are shortcomings in the analyses of BPT and KJH. The study by BPT is based on the framework ofthe GARCH-X model using either the realized measure or implied volatility as an exogenous covariate. They alsoconsider multi-step-ahead volatility forecasts. However, when conducting the multi-step-ahead forecasts, they assumethat there is a constant proportional relationship between squared returns and realized measures, which is unrealisticbecause the ratio between squared returns and realized measures is in fact volatile. Meanwhile, KJH only comparedone-step-ahead forecasts without considering multi-step-ahead forecasts.

In addition to these analytical shortcomings, there are a number of recent developments in the related practice andthe literature that we need to consider. First, the new VIX index, introduced in 2003, is based on the S&P 500 index.The VIX index is the main source of implied volatility used in the literature, and the foregoing previous studies usedthe old VIX index, which is based on the S&P 100 index. Second, among the realized measures of volatilityconstructed from high-frequency data, Barndorff-Nielsen et al. (2008) introduced the realized kernel, which issomewhat robust to market microstructure effects. Third, there has been a recent development on the evaluationmethod of volatility forecasts that is robust to the noise in volatility proxies (see Hansen and Lunde, 2006; Patton,2009; Patton and Sheppard, 2009). Fourth, a recent development on volatility modeling uses realized measures inthe framework of the GARCH-X model, e.g. the HEAVY models and the Realized GARCH model, such thatmulti-step-ahead forecasting is available in a systematic manner.

These new developments motivate us to reinvestigate the issue studied by BPT and KJH. Considering these newdevelopments, we compare the information content of the realized measure and implied volatility, in within-sample andout-of-sample (over horizons of 1–22 days) forecasts of daily S&P 500 index return volatility. Our study is based onthe framework of the GARCH-X model in which either the realized measure or implied volatility is added as a covariate.

For the realized measure and implied volatility, we use the realized kernel and the VIX index based on the S&P 500index, respectively. For the multi-step-ahead forecasts, we adopt the procedure in the HEAVYmodels and consider bothiterated and direct forecasting. To evaluate volatility forecasts, we focus on the QLIKE loss function. Patton and Sheppard(2009) showed in their simulation study that the QLIKE loss function, which is defined in the next section, has the highestpower among the loss functions robust to the use of a noisy volatility proxy. The significance of any difference in theQLIKE loss is tested via a Diebold–Mariano and West test (see West, 2006; Patton and Sheppard, 2009).

The main results of our paper are as follows. First, we compare three models, which use the information provided bythe realized measure, implied volatility and squared returns, respectively. For the within-sample fitting, the model usingthe realized measure outperforms the implied volatility and squared returns models in terms of the log-likelihood valueand the QLIKE loss. This implies that the realized measure is more informative than implied volatility and squared returnsin the within-sample fitting. In contrast, for the out-of-sample forecasting, the model using implied volatility outperformsthe models using either the realized measure or squared returns in terms of the QLIKE and MSE losses. In particular,implied volatility is more informative for the multi-step-ahead forecasts. Considering the characteristics of the realizedmeasure and the implied volatility, these results are not surprising. By construction, the realized measure is backwardlooking and implied volatility is forward looking.

Second, we also investigate the performance of the model using all the information provided by the realizedmeasure, implied volatility and squared returns. For the within-sample fitting, the model using all the informationperforms better than the three models using information provided by only one source. Evidently it is helpful to useall the information provided by the realized measure, implied volatility and squared returns in this case. However,for multi-step-ahead forecasting, the general model performs worse than the model using only the implied volatility.The results evidently show that the model using only the information provided by the implied volatility is preferred formulti-step-ahead forecasting.

The rest of this paper is organized as follows. The next section describes the data and models, and also explainsthe forecasting procedure and evaluation criteria. The third section reports the main results of the paper. We evaluatethe within-sample and out-of-sample forecasting performance of the models, and investigate the performance of themodel using all the information provided by the realized measure, implied volatility and squared returns. The fourthsection concludes the paper.

MODELS AND FORECASTING METHODOLOGY

Data and within-sample modelsWe consider the daily S&P 500 index returns from 3 January 1996 to 27 February 2009 (3260 trading days). Previousstudies, such as BPT and KJH, investigated the S&P 100 index returns because the VIX index used in their studieswas based on the S&P 100 index. However, the new VIX index, which was introduced in 2003, is based on theS&P 500 index. Therefore, we consider the new VIX index and the return series of the S&P 500 index.

Copyright © 2013 John Wiley & Sons, Ltd. J. Forecast. 32, 522–533 (2013)

524 H. Han and M. D. Park

The VIX index represents a measure of the expected volatility of the S&P 500 index over the next 30 days. Hence it isforward looking, as opposed to the realized measures. Realized measures are constructed from observed high-frequencydata and are therefore, by definition, backward looking. The VIX methodology involves calculating an estimate of ‘fairvariance’ for near-term and next-term options, weighting these two values to construct a constant 30-day variance, andthen taking the square root to produce a value for the VIX index.1

For the realized measure, BPT adopted a realized volatility, denoted as INTRAt, using 5-minute squared returns andovernight squared returns. Instead, we use the realized kernel, introduced by Barndorff-Nielsen et al. (2008), because ithas some robustness to the noise of market microstructure effects. See Andersen et al. (2011) for volatility forecasting andmarket microstructure noise. The realized kernel has the familiar form of an HAC type estimator:

RMt ¼XHh¼�H

Kh

H þ 1

� �gh; gh ¼

Xnj¼ hj jþ1

xj;txj� hj j;t

whereK(�) is the Parzen kernel function and xj,t is the jth high-frequency return on the tth day. For the bandwidth choice ofH and other details, we refer to Barndorff-Nielsen et al. (2009) and Heber et al. (2009). The realized kernel is computed intick time using every available data point, after cleaning. See the appendix of Shephard and Sheppard (2010) for datacleaning. The realized kernel of the S&P 500 index return is available at the database, ‘Oxford-Man Institute’s realizedlibrary’, produced by Heber et al. (2009).2

Table 1 gives summary statistics for the squared returns, the realized kernel and the VIX index. The variance of dailyS&P 500 index returns from January 1996 to February 2009 is equivalent to an annualized volatility of 20.8%. The squareroot of the average value of the realized kernel is equivalent to an annualized volatility of 15.9% during the same period.The average value of the VIX index is equivalent to an annualized volatility of 21.7% during the same period. Asmentioned in SS, the approximate annualized volatility for the realized kernel is lower because the realized kernel missesout on the overnight return.

The SD figures are standard deviations. The VIX index shows much higher standard deviation than the squaredreturns and the realized kernel. The realized kernel exhibits the lowest standard deviation. The ACF figures are theserial correlation coefficients at one lag. The degree of serial correlation of squared returns is modest (0.209), whilethe realized kernel is more persistent (0.700). The ACF is close to unity (0.981) for the VIX index. The GPH d figuresare the log-periodogram estimates of the order of fractional integration. The estimated d for squared returns is 0.290,which implies the squared return series is stationary. Meanwhile, the estimated d for the realized kernel is 0.559,which suggests the realized kernel is non-stationary. The estimated d for the VIX index is 0.867.

These results show that the degrees of persistence in squared returns, the realized kernel and implied volatility are alldifferent, with implied volatility being the most persistent and the squared return the least persistent. When these variablesare used as exogenous covariates in the GARCH-X model, their effects will be different from each other because eachvariable exhibits a different degree of persistence. Han (2010) analytically shows how the time series properties of theGARCH-X process depend on the degrees of persistence in the exogenous covariates in the GARCH-X model.

We evaluate the within-sample performance using the framework of the GARCH-X model. The most generalspecification is as follows:

rt ¼ cþ etet ¼ stzt; zt � i:i:d: 0; 1ð Þs2t ¼ oþ a1e2t�1 þ a2st�1e2t�1 þ bs2t�1 þ gRMt�1 þ dVIX2

t�1:

(4)

Here, rt is the return series of the S&P 500 index (100 times differences of the log price); st is 1 when et< 0, and zerootherwise; RMt is the realized kernel; and VIXt is defined asVIX=

ffiffiffiffiffiffiffiffi252

p, as in BPT, to consider annualization. Following

BPT, we estimate model (4) with restrictions on its parameters and obtain the results for the following seven models:

1. The GARCH(1,1) model: a2 = g = d = 0.2. The GJR model of Glosten et al. (1993): g = d= 0.3. The HEAVY-r model, which uses the information provided by RM: a1 = a2 = d = 0.4. The VIX model, which uses the information provided by VIX: a1 = a2 = g= 0.5. A volatility model that uses the information provided by RM and VIX: a1 = a2 = 0.6. A volatility model that uses all the available information in daily returns, RM and VIX, but excludes the GJR

term: a2 = 0.7. The unrestricted model using all the available information in daily returns, RM and VIX.

1See http://www.cboe.com/VIX for more details of the VIX index. The VIX index is also available at the web site.2See http://realized.oxford-man.ox.ac.uk/.


http://realized.oxford-man.ox.ac.uk/

Table I. Summary statistics for the return series of the S&P 500 index

r2t Realized kernel VIX

Avol 20.8 15.9 21.7SD 5.46 2.14 8.83ACF1 0.209 0.700 0.981GPH d 0.290 0.559 0.867

Note: The calculations use 100 times differences of the log price (i.e. roughly percentage changes) from 3 January 1996 to 27 February 2009 (3260trading days). Avol is the approximate annualized volatility and is the square root of the mean of 252 times either the squared returns or the realizedkernel. Avol for the VIX index is the sample mean of the VIX index. The SD figures are standard deviations. The ACF figures are the serialcorrelation coefficients at one lag. GPH d is the log-periodogram estimate of the order of fractional integration.


We use the quasi-maximum likelihood estimation method to estimate the models. We compare the within-sampleperformance of the models and investigate the degree of incremental information from daily returns, therealized measure and the VIX index.

Out-of-sample models and forecasting methodologyFor the out-of-sample forecast evaluation, we focus on the following three models among the seven models considered inthe previous subsection:

s2t ¼ oþ a1e2t�1 þ a2st�1e2t�1 þ bs2t�1 GJR models2t ¼ oþ bs2t�1 þ gRMt�1 HEAVY� r models2t ¼ oþ bs2t�1 þ dVIX2

t�1 VIX model

The first model, GJR, uses only the information provided by daily returns.3 The second model, HEAVY-r, usesonly the information provided by the realized measure. The third model, VIX, uses only the information providedby the VIX index.

We adopt the rolling window forecast procedure with moving windows of four years (1008 trading days). This meansthat we obtain one-step-ahead (N=1) forecasts of the models for the period from 9 February 2000 to 27 February 2009(2252 trading days). Additionally, we also obtain multi-step-ahead (N= 10 and N=22) forecasts. We let var rTþh F Tj Þðdenote the h-step-ahead (pointwise) forecast, and the N-step ahead cumulative forecast is defined as

var rTþ1 þ rTþ2 þ . . .þ rTþN jF Tð Þ ¼XNh¼1

var rTþhjF Tð Þ; N ¼ 10; 22

where T is the last day of the moving window of 1008 days andF T contains all available information at time T, includingthe realized measure and the VIX index.

To conduct a multi-step-ahead forecast, BPT assumed that there is a proportional relationship between squaredreturns and realized measures, so that

E RMTþhjF Tð Þ ¼ cTE rTþh � cð Þ2jF T

� ¼ cTE s2TþhjF T

� �for some constant cT that is the same for positive h. Here, c is the sample mean of the return series. To calculate themulti-step-ahead forecasts at time T, the constant cT is estimated by the ratio

cT ¼XT

t¼T�999

RMt=XT

t¼T�999

rt � cð Þ2:

However, it is unrealistic to assume that a constant proportional relationship between squared returns and realizedmeasures holds for a certain time period. We can easily confirm this by drawing a graph of the ratio between squaredreturns and realized measures, which shows that the relationship is volatile.

Iterated forecastingFor the multi-step-ahead forecast of the HEAVY-r model, we follow the method in SS. The HEAVY models, (2) and(3), can be written as

3We also tried the GARCH(1,1) model. Its forecasting performance is equivalent or worse than the GJR model for our data.



s2tmt

� �¼ wþ B

s2t�1mt�1

� �þ g 0

0 gR

� �RMt�1 � mt�1

RMt�1 � mt�1

� �

where

w ¼ ooR

� �and B ¼ b g

0 bR þ gR

� �

See SS for the estimation of the HEAVY models. Let Atjt�1 ¼ s2t ; mt� �0

. Then, we can obtain the one-step-aheadforecast at time T from

ATþ1jT ¼ wþ BAT jT�1 þ g 00 gR

� �RMT � mTRMT � mT

� �

Note that RMt� mt is a martingale difference sequence with respect to F t�1 , which impliesE RMTþh � mTþh F Tj Þ ¼ 0�

for h≥ 2. Hence we can produce multi-step-ahead forecasts at time T following

ATþhþ1 T¼wþBATþh Tj

��for h≥ 2.

For the multi-step-ahead forecast of the VIX model, we use the following assumption:

VIXTþh ¼ VIXT for h ¼ 1; 2; . . . ; 21: (5)

As explained in the previous section, the VIX index is the forecast of a constant 30-day volatility implied byoptions. Therefore, (5) corresponds to the definition of the VIX index. Instead, we could follow the same methodas the HEAVY models using

E VIXt F t�1j Þ ¼ mt ¼ aR þ bRmt�1 þ gRVIXt�1:ð (6)

We tried both methods, and (5) produces slightly better multi-step-ahead forecasts than (6).

Direct forecastingThe above ‘iterated’ forecasts are made using a one-period-ahead model, iterated forward for the desired number ofperiods, whereas ‘direct’ forecasts are made using a horizon-specific estimated model, where the dependent variableis the multi-period-ahead value being forecasted. Direct forecasting generally produces different coefficients atdifferent horizons. As SS adopted this direct forecasting method4 for their out-of-sample forecast evaluation, we alsocompare the direct forecasts.

As explained in SS, an h-step-ahead direct forecast of RMT + h|T is made using the horizon-specific estimateθR,h = (oR,h,bR,h,gR,h), which is based on the following log-likelihood function of the entire sample (RM1, . . .,RMT):

XT�hþ1

t¼2

‘RMt;h θR;h� � ¼ � 1

2

XT�hþ1

t¼2

logmtþh�1jt�1 θR;h� �þ RMtþh�1

mtþh�1jt�1 θR;h� �

!

where

mtþh�1 t�1¼oRþbRmt�1þgRRMt�1:j

This direct forecast of RMT + h|T is used for the h-step-ahead direct forecast of s2TþhjT in the HEAVY-r model

optimized for the one-step horizon, as in SS.Insteadof using thedirect forecast ofRMT+h|T, it is possible toobtain thedirect forecast of theHEAVY-rmodelwithout using

the HEAVY-RMmodel. It is based on the horizon-specific estimate from (7) where s2tþh�1jt�1 ¼ oþ bs2t�1 þ gRMt�1.

4See Sections 2.4.4 and 4.2 in SS for the discussions on direct forecasting.



However, the forecasts provided by this method are worse than the forecasts of the HEAVY-r model using thedirect forecast of RMT + h|T. Therefore, we do not report it.

For the GJR and VIX models, we also produce direct forecasts. An h-step-ahead direct forecast of s2TþhjT for the

GJR model or the VIX model is made using the horizon-specific estimate θh(=(oh,a1,h,a2,h,bh) or (oh,bh,gh)) fromthe following log-likelihood function of the entire sample (r1, . . .,rT);

XT�hþ1

t¼2

‘t;h θhð Þ ¼ � 12

XT�hþ1

t¼2

logs2tþh�1jt�1 θhð Þ þ rtþh�1 � cð Þ2s2tþh�1jt�1 θhð Þ

!(7)

where

s2tþh�1jt�1 ¼ oþ a1e2t�1 þ a2st�1e2t�1 þ bs2t�1 GJR model

s2tþh�1jt�1 ¼ oþ bs2t�1 þ dVIX2t�1 VIX model

Forecast evaluation criteriaWe use the realized kernel as the proxy for actual volatility. It is known that realized measures are better proxies foractual volatility than squared return series. Moreover, the realized kernel is robust to the noise from marketmicrostructure effects. However, a better volatility proxy is still imperfect and a noisy proxy for actual volatility. Itis possible, due to noisy proxies, that the evaluation based on some loss functions may identify an inferior volatilitymodel as the ‘best’ and the inferior model may spuriously be found to be ‘significantly’ better than all other models.Hence there has been research on loss functions that are robust to the use of a noisy volatility proxy (see Hansen andLunde, 2006; Patton and Sheppard, 2009; Patton, 2010).

Patton (2010) provided necessary and sufficient conditions for the functional form of the loss function to ensure theranking of various forecasts is preserved when using a noisy volatility proxy, and showed that the MSE and QLIKEare robust. Patton and Sheppard (2009) showed that the use of loss functions that are ‘non-robust’ can yield perverseranking of forecasts, even when very accurate volatility proxies are employed. Moreover, their simulation studyshowed that the QLIKE loss function defined below has the highest power. The QLIKE loss function is defined as

L s2t ; s

2t

� � ¼ s2ts2t

� logs2ts2t

� 1 (8)

where s2t is a volatility proxy and s2t is a volatility forecast. As the simulation results by Patton and Sheppard (2009)

point to this QLIKE as the preferred choice amongst the loss functions, including the MSE, that are robust to noise inthe proxy, we use the QLIKE as the main loss function. In addition, we also report the MSE in the out-of-sampleforecast evaluation.

The significance of any difference in the QLIKE loss is tested via a Diebold–Mariano and West (henceforth DMW)test (see Diebold and Mariano, 1995; West, 2006; Patton and Sheppard, 2009). A DMW statistic is computed usingthe difference in the losses of two models:

dt ¼ L s2t;1; s

2t

� � L s2

t;2; s2t

�

DMWT ¼ffiffiffiffiT

p�dTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

^avarffiffiffiffiT

p�dT

� �q (9)

where �dT is the sample mean of dt and T is the number of forecasts. The asymptotic variance of the average iscomputed using a Newey–West variance estimator, with the number of lags set to [T1/3].

ESTIMATION AND FORECAST EVALUATION

Within-sample estimationTable 2 reports the within-sample estimation results of the models. The estimation results for the GARCH(1,1) model

and the GJR model are similar to what we commonly observe in the literature. In the GARCH(1,1) model, a1 þ b isvery close to unity (0.99), and a2 is significant in the GJR model, which implies an asymmetric relationship betweenstock return and its volatility.

The realized measure and the VIX index are estimated to be significant in model 3 (the HEAVY-r model) andmodel 4 (the VIX model), respectively. However, there is a difference in the estimation results between these two



models. b is insignificant in the VIX model, while it is significant in the HEAVY-r model. This indicates that the pastvalues of the realized measure (RMt� 2,RMt� 3,RMt� 4, . . .) are necessary to explain the next day’s volatility in theHEAVY-r model, but the past values of the VIX series VIX2

t�2;VIX2t�3;VIX

2t�4; . . .

� �are unnecessary in the VIX

model.5 This result implies that the options market on the S&P 500 index is informationally efficient because the latestoption prices contain all relevant information about the next day’s volatility. This is different from the result in BPT,

where b is significant in the VIX model. Considering that BPT uses the old VIX index based on the S&P 100 index,the difference between our result and theirs may imply that the options market on the S&P 500 index isinformationally more efficient than the options market on the S&P 100 index.

We also estimated the following GARCH-X models without the constraint of a1 = 0:

s2t ¼ oþ a1e2t�1 þ bs2t�1 þ gRMt�1;

s2t ¼ oþ a1e2t�1 þ bs2t�1 þ dVIX2t�1:

In both cases, the coefficient of the squared return a1 turns out to be insignificant. This result is well reported in SS,which is why their HEAVY-r model excludes the term e2t�1 by having the constraint of a1 = 0. Moreover, the log-likelihoodvalues of both GARCH-X models without the constraint of a1 = 0 are similar to those of models 3 and 4, respectively.

Now we compare the three models (GJR, HEAVY-r, VIX) based on the log-likelihood value and QLIKE loss, tosee which one is the more informative. Recall that the GJR model contains only the information given by the squaredreturn series, the HEAVY-r model contains only the information given by the realized measure and the VIX modelcontains only the information provided by the implied volatility.

The results show that the HEAVY-r model outperforms the GJR and VIX models in the within-sample fitting for bothcriteria. If we consider the log-likelihood values of the three models, they are all higher than the log-likelihood of theGARCH(1,1) model. In particular, the HEAVY-r model exhibits the highest increase of 103.1 compared to GARCH(1,1), in the log-likelihood value among the three models. Meanwhile, the VIX model shows the second highest increaseof 76.7 and the GJR model shows the least increase of 61.3. If we compare the QLIKE losses6 of the three models, theresults are similar. The HEAVY-r model has the smallest QLIKE of 0.278, while the GJR and VIX models both havea QLIKE of 0.303.The DMW test shows that the within-sample forecast of the HEAVY-r model is significantly betterthan that of the GJR and VIX models.

This implies that the realized measure is more informative than squared returns and implied volatility in the within-sample fitting. Our result is different from the result of BPT, who showed that the VIX index is more informative thanthe realized measure in the within-sample fitting. The difference could be a result of the use of different realizedmeasures.As their realized measure, BPT used a simple realized variance using 5-min squared returns and overnight squaredreturns. In contrast, we use the realized kernel, which is robust to the noise of market microstructure effects. Besides, thereare other differences between BPT and our study, which are explained in the Introduction, such as the data (S&P 500index instead of S&P 100 index) and the method used to measure the implied volatility VIX index. These differencesmay, together, have led to us having a different within-sample result from BPT.

Model 5 combines the realized measure and the VIX index. Compared to the log-likelihood value of the HEAVY-rmodel, that of model 5 is higher by 15.3. As the models become more general (models 6 and 7) by including theinformation provided by squared returns, the log-likelihood value increases. When we compare the QLIKE losses, theresult is similar. The QLIKE of model 7 is 0.263, which is smaller than the QLIKE of the HEAVY-r model. Consideringthese results on the log-likelihood and QLIKE loss, the most general model outperforms the rest of the models inthe within-sample fitting. This implies that it is better to use all the information provided by squared returns, the realizedmeasure and implied volatility. Therefore, it will be interesting to investigate whether this general model usingall information can provide better out-of-sample forecasts than models 3–5, which is done in the last part of this section.

Out-of-sample forecast evaluationTables 3 and 4 report the out-of-sample forecast evaluation results of the three models (GJR, HEAVY-r, VIX). Weconsider multi-step-ahead (10 and 22) forecasts as well as the one-step-ahead forecast. Table 3 considers the iteratedforecasts and Table 4 compares the direct forecasts. For the multi-step-ahead forecasts, we conduct both pointwisecomparison and cumulative comparison. Tables 3 and 4 report the QLIKE and MSE losses, and the DMW tests arecorrespondingly conducted.

5As b is insignificant, the estimation result of the VIX model implies that s2t can be explained only by a nonlinear function of the VIX index.Considering that the VIX index can be modeled as a near-integrated process, the VIX model without the s2t�1 term is an example of the nonlinearnon-stationary heteroskedasticity model by Park (2002). See Park (2002) and Han and Park (2008) for the time series properties of such models.6The proxy for actual volatility is the realized kernel and the fitted values of the models are their within-sample forecasts. While the QLIKE isrelated to the log-likelihood, it provides some information that is not covered by the log-likelihood. The QLIKE is based on the realized kernel,which is not used in the log-likelihood.


Table II. Within-sample estimation results of the models

Parameter

Model

(1) (2) (3) (4) (5) (6) (7)

a1 0.077 -0.011 -0.05 -0.07(6.60) (-0.97) (-1.80) (-5.80)

a2 0.150 0.10(7.75) (5.03)

b 0.916 0.923 0.77 0.004 0.44 0.35 0.77(87.47) (86.35) (17.12) (0.02) (4.09) (3.33) (15.54)

g 0.36 0.49 0.62 0.29(4.31) (4.17) (5.21) (3.80)

d 0.77 0.20 0.24 0.06(4.09) (3.24) (3.82) (2.75)

Log-L -4844.5 -4783.2 -4741.4 -4767.8 -4726.1 -4718.0 -4699.3Excess log-L 61.3 103.1 76.7 118.4 126.5 145.2QLIKE 0.329 0.303 0.278 0.303 0.269 0.271 0.263

Note: The table reports the quasi-maximum likelihood estimation results of the following model with restrictions on certain parameters:

rt ¼ cþ etet ¼ stzt ; zt � i:i:d: 0; 1ð Þs2t ¼ oþ a1e2t�1þa2st�1e2t�1þbs2t�1þgRMt�1þdVIX2

t�1

where (rt) is the S&P 500 index return series; RMt� 1 is its realized kernel based on high-frequency data; VIXt� 1 is the VIX index divided byffiffiffiffiffiffiffiffi252

pto consider annualization; and st is 1 when et< 0 and zero otherwise. The t-statistics based on the QMLE standard errors are reported in

parentheses. Log-L is the log-likelihood, and excess log-L is the difference of the model’s log-L compared to the log-L of the GARCH(1,1) model.The QLIKE loss is defined in (8).


Table 3 shows that, in general, the VIX model performs the best in almost all cases.7 The VIX model has the smallestlosses both in the QLIKE andMSE, except for the QLIKE of the one-step-ahead forecast. For the one-step-ahead forecast,even though the HEAVY-r model has the smallest QLIKE of 0.232, the VIXmodel has a similar QLIKE of 0.234 and theDMW test shows that their forecasts are not significantly different. In terms of the MSE loss, the VIX model performs thebest (significant at 10% for the DMW tests) for the one-step-ahead forecast. For the multi-step-ahead forecasts, the DMWtest for the QLIKE loss shows that the out-of-sample forecasts of the VIXmodels are significantly better than those of theGJR and VIX models, regardless of pointwise or cumulative comparison. The DMW test results for the MSE loss aremostly insignificant in the multi-step-ahead forecasts.

Even when we compare the direct forecasts instead of the iterated forecasts, the VIX model still performs the best.Table 4 shows that the VIX model has the smallest QLIKE and MSE losses for all cases, regardless of pointwise orcumulative comparison. In the pointwise comparison, the DMW test results are mostly significant for the QLIKE andMSE losses. In the cumulative comparison, the DMW test results are mostly significant, while the tests against theHEAVY-r model are insignificant in terms of the QLIKE loss.

In general, Tables 3 and 4 show that the VIX model outperforms the rest of the models in the out-of-sample forecastevaluation. Our result that the VIX model outperforms the GJR and HEAVY-r models is the same as the out-of-sampleforecast result in BPT.8 The VIX model has the smallest losses in the QLIKE andMSE, regardless of iterated forecastingor direct forecasting. The out-of-sample forecast evaluation shows a different result from the within-sample performance.While the HEAVY-r model outperforms the VIX model in the within-sample, the VIX model outperforms the HEAVY-rmodel in the out-of-sample forecast. This difference can be explained by the differences in the nature of the realized-measure and implied volatility. By construction, the realized measure is backward looking and the implied volatility isforward looking. As high-frequency data are used for constructing the realized measure, the realized measure containsalmost all the information available at the moment. This may be the reason why the realized measure provides betterwithin-sample performance. Meanwhile, as mentioned above (‘Data and within-sample models’), implied volatilitymeasures the expected volatility over the future 30 days. Hence, considering the forward-looking nature of impliedvolatility, it is no surprise that implied volatility provides better out-of-sample forecasts, in particular multi-step-aheadforecasts, than the realized measure.

It is interesting to compare iterated forecasts with direct forecasts. For the VIX model and GJR model, iteratedforecasts have smaller losses in QLIKE and MSE than direct forecasts in all cases. Therefore, it is clearly better to adopt

7Even when we use method (6) instead of (5) for the VIX model, the forecast evaluation results are similar.8BPT also compared the historic volatility model, which is the sample variance of returns over the last 100 samples. We also considered the historicvolatility model, but we do not report the result, to save space, because its performance is worse than the three models in all cases as in BPT.


Table III. Out-of sample forecast evaluation for iterated forecasting

Model

Pointwise comparison Cumulative comparison

QLIKE(DMW) MSE(DMW) QLIKE(DMW) MSE(DMW)

N= 1GJR model 0.245(1.88*) 4.35(1.68*)HEAVY-r model 0.232 5.23(1.95*)VIX model 0.234(0.32) 2.80s2Tþ10, N= 10

GJR model 0.344(3.07***) 5.93(1.17) 0.200(2.75***) 317(1.32)HEAVY-r model 0.375(6.18***) 6.83(1.82*) 0.206(3.76***) 405(1.81*)VIX model 0.309 4.67 0.175 182s2Tþ22, N= 22

GJR model 0.441(2.75***) 6.46(0.69) 0.222(3.01***) 1569(1.21)HEAVY-r model 0.518(6.58***) 6.62(1.11) 0.253(5.66***) 1790(1.58)VIX model 0.394 6.09 0.187 1024

Note: The QLIKE loss is defined in (8) and the DMW test statistic is defined in (9). Asterisks indicate rejection of the null hypothesis of equalpredictability for *10%, **5% and ***1% tests.

Table IV. Out-of sample forecast evaluation for direct forecasting

Model



s2Tþ10, N= 10

GJR model 0.465(5.34***) 12.58(2.39**) 0.256(7.87***) 534(1.97**)HEAVY-r model 0.345(1.71*) 7.82(1.61) 0.198(1.50) 450(1.75*)VIX model 0.326 5.09 0.185 200s2Tþ22, N= 22

GJR model 0.567(7.16***) 15.98(2.24**) 0.296(8.25***) 3369(2.06**)HEAVY-r model 0.440(1.82*) 10.10(1.75*) 0.221(1.48) 2597(1.74*)VIX model 0.418 8.04 0.206 1329

Note: The QLIKE loss is defined in (8) and the DMW test statistic is defined in (9). Asterisks indicate rejection of the null hypothesis of equalpredictability for *10%, **5% and ***1% tests.


the iterated forecasting procedure in multi-step-ahead forecasts using the VIX and GJR models. However, the HEAVY-rmodel shows mixed results. Iterated forecasts have smaller MSEs but larger QLIKEs than direct forecasts.

Effect of using all informationIt is shown above (‘Within-sample estimation’)that the general model using all the information provided by squaredreturns, the realized measure and implied volatility performs the best in the within-sample fitting. Now we investigateits performance in the out-of-sample forecast. Table 5 reports the results.9 We compare models 5 and 7 with the VIXmodel. Model 5 contains the information provided by the realized measure and implied volatility, and model 7contains all the information provided by squared returns, the realized measure and implied volatility.

The three models considered ‘Out-of-sample forecast evaluation’ are non-nested and the DMW test statistic is as-ymptotically normal for non-nested models. However, we consider nested models in Table 5, and the asymptotic dis-tribution of the test statistic is not standard normal for nested models (see Clark and McCracken, 2001; Clark andWest, 2007; and references therein). One can adopt a bootstrap procedure to inference (see Clark and McCracken,2012), but we simply provide the DMW test statistic as a reference in Table 5. Clark and McCracken (2001) and Clarkand West (2007) showed that critical values of the test statistic for nested models are actually smaller than standardnormal critical values. Therefore, when the DMW statistics are very large in Table 5 (e.g. multi-step forecasts in termsof the QLIKE loss), we expect the rejection of equal predictability would be still valid.

9The evaluations are based on iterated forecasts. The previous subsection shows that, for the VIX model, iterated forecasts are better than directforecasts in terms of both QLIKE and the MSE losses. Hence we compare only the iterated forecasts of the VIX model. Even if compared tothe direct forecasts of models 5 and 7, the iterated forecasts of the VIX model are still better.


Table V. Out-of sample forecast evaluation of general models (iterated forecasting)

Model



N= 1VIX model 0.234(4.08) 2.80Model 5 0.212(2.18) 3.33(1.56)Model 7 0.210 3.42(1.65)s2Tþ10, N= 10

VIX model 0.309 4.67 0.175 182Model 5 0.344(5.85) 5.15(1.45) 0.187(2.59) 233(1.38)Model 7 0.345(5.95) 5.17(1.49) 0.187(2.36) 235(1.41)s2Tþ22, N= 22

VIX model 0.394 6.09 0.187 1024Model 5 0.463(6.62) 6.22(0.80) 0.223(5.62) 1229(1.41)Model 7 0.466(6.80) 6.22(0.84) 0.224(5.69) 1238(1.45)

Note: Model 5 includes the RM and VIX, and model 7 includes all the RM, VIX and GJR. The QLIKE loss is defined in (8) and the DMW teststatistic is defined in (9).


For the one-step-ahead forecast, the evaluation result depends on a loss function. In terms of the QLIKE loss,models 5 and 7 outperform the VIX model. However, in terms of the MSE loss, the VIX model outperforms thetwo models. Comparing model 5 and model 7, the results still depend on a loss function. In the QLIKE loss,model 7 produces slightly better forecasts than model 5. On the other hand, model 5 has a smaller MSE loss thanmodel 7.

When we consider the multi-step-ahead forecasts, the evaluation results clearly support the VIX model. Forboth N = 10 and N = 22, the VIX model outperforms models 5 and 7 in terms of both the QLIKE and MSE losses,regardless of pointwise or cumulative comparison. This implies that, for multi-step-ahead forecasting, it is detrimentalto add the information provided by squared returns and the realized measure in the VIX model. When we comparemodel 5 with model 7, their forecasts are similar to each other.

A natural question is whether it is better or worse to use the VIX index in the HEAVY-r model for multi-step-aheadforecasting. Comparing model 5 (see Table 5) and the HEAVY-r model (see Table 3), model 5, with the VIX index inthe HEAVY-r model, significantly outperforms the HEAVY-r model in all cases (N = 10, 22). This means that addingthe information provided by implied volatility to the HEAVY-r model appears to be helpful in multi-step-aheadforecasting.

For the one-step-ahead forecast, the results are mixed depending on the loss functions. On the other hand, for themulti-step-ahead forecasts, our results clearly show that it is best to use only the information provided by the VIXindex and worst to use only the information provided by the realized measure. When we add the realized measureto the VIX model, it produces worse multi-step-ahead out-of-sample forecasts. On the other hand, when we includethe VIX index in the HEAVY-r model, it produces better multi-step-ahead out-of-sample forecasts. Our results showthat the VIX index contains useful information, in particular, for multi-step-ahead forecasting, which is not surprisingconsidering the VIX index represents a measure of the expected volatility over the next 30 days.

CONCLUSION

This paper compares the information content of a realized measure constructed from high-frequency data and the im-plied volatility from options. The comparison is based on within-sample and out-of-sample (one-step and multi-step)forecasts. This issue has been widely explored in the literature. The new feature of our paper is that we consider recentdevelopments in the related practice and the literature. First, as the implied volatility, we use the new VIX index,introduced in 2003, which is based on the S&P 500 index. Hence we consider the return series of the daily S&P500 index instead of the S&P 100 index used in previous studies. Second, as the realized measure, we use the realizedkernel instead of the realized variance because it is somewhat robust to market microstructure effects. Third, as theforecast evaluation criterion, we focus on the QLIKE loss because it is known to have the highest power among lossfunctions that are robust to the noise in volatility proxies. Moreover, the significance of any difference in the QLIKEloss is tested via the DMW test. Fourth, we adopt the HEAVY models because multi-step-ahead forecasting isavailable in a systematic manner. Finally, we consider the direct forecasting procedure, as well as the usual iteratedforecasting procedure.

Our results show that the realized measure is more informative than implied volatility for the within-sample fitting.This is contrary to the results in BPT. They showed that implied volatility is more informative than the realizedmeasure in both the within-sample and out-of-sample forecasts. The difference between our results and theirs could



be due to the use of the realized kernel, the new VIX index or the S&P 500 index with a different sample period. Onthe other hand, our results for the out-of-sample forecast are the same as those in BPT. The implied volatility is moreinformative than the realized measure, in particular for multi-step-ahead forecasting. Our results for the within-sampleand out-of-sample forecasts are compatible with the fact that the realized measure is backward looking and impliedvolatility is forward looking by construction.

We also investigate the effect of using all the information provided by the realized measure and the impliedvolatility on volatility forecasting. For the within-sample fitting, the general model using all the information performsthe best. However, for the multi-step-ahead out-of-sample forecasting, the general model using all the informationperforms worse than the model using only the implied volatility. Our results clearly show that it is preferable touse only the implied volatility for multi-step-ahead forecasting.

ACKNOWLEDGEMENTS

We thank Shen Zhang for her excellent research assistance. We are grateful to Ruey S. Tsay and anonymous refereesfor useful comments. Han gratefully acknowledges the research support from the NUS Risk Management Institute.

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Authors’ biographies:

Heejoon Han, is currently an associate professor in the department of Economics at Kyung Hee University, Republic of Korea.His research focuses on non-linear non-stationary time series econometrics, non-parametric econometrics and volatility modelingand forecasting. His work has been published in Journal of Econometrics and the Econometrics Journal.

Myung D. Park, is a Visiting Fellow in the department of Economics at National University of Singapore. His research interestsare in time series analysis, forecasting and applied econometrics.

Authors’ addresses:

Heejoon Han, Department of Economics, Kyung Hee University, 1 Hoegi-dong, Dongdaemoongu,Seoul, 130-701, Republic of Korea.

Myung D. Park, Department of Economics, National University of Singapore, Singapore.


comparison of realized measure and implied volatility in forecasting volatility

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