Dimitrios P Louzis

Spyros Xanthopoulos-Sisinis

Apostolos P Refenes

Time-Series Models < C2 -Econometric Methods: Single Equation Models < C -Mathematical and Quantitative Methods, C51 -Model Construction and Estimation < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods, C53 -Forecasting and Other Model Applications < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods

Keywords: Volatility forecasting, Leverage effects, Long memory, High frequency data

come

Stock index realized volatility forecasting in the presence of heterogeneous leverage effects and long range dependence in the volatility of realized volatility

shocks is also identified. Although the inclusion of leverage effects in the HAR regression reduces both the skewness and the heteroskedasticity of the error term, it does not eliminate the ARCH effects. Through Exact Local Whittle (ELW) and Maximum Likelihood Estimation (MLE) integration order estimations, the suspected long range dependence in the volatility of realized volatility is also verified.

The out-of-sample one day ahead, five and twenty-two days ahead forecasting performance is then evaluated for seven established loss functions, as well as with [START_REF] Hansen | A test for superior predictive ability[END_REF] Superior Predictive Ability (SPA) test. The proposed specification minimizes the majority of the loss functions, for both indices and for all the forecasting horizons. Its volatility forecasting performance is further improved when the RPV is included as a regressor, while its superiority is also confirmed by the SPA test p-values.

Finally, the TTS estimated realized volatility forecasting results underline its robustness against the microstructure noise in the returns process.

The remaining of this article is organized as follows: In Section II we introduce the realized volatility measures and the mathematical notations and definitions used throughout this article. In Section III we present the HAR and ARFIMA based models, while the data set, descriptive statistics and the in-sample maximum likelihood models estimation are shown in Section IV. In Section V, we present the out-of sample realized volatility forecasting evaluation methodology and results. Section VI summarizes and concludes this article. In [START_REF] Andersen | Answering the sceptics: yes, standard volatility models do provide accurate forecasts[END_REF], the authors defined realized variance as the sum of squared intraday returns and proved it is an unbiased and less noisy estimator for the daily unobserved volatility, than the squared daily returns proxy (for a good review on realized volatility see [START_REF] Mcaller | Realized volatility: A review[END_REF]. Let us define the m th intraday return for day t as ( ) ( ) ( ) ( ) (see [START_REF] Martens | Measuring and forecasting S&P 500 index-futures volatility using high-frequency data[END_REF][START_REF] Koopman | Forecasting daily variability of the S&P100 stock index using historical, realised and implied volatility measurements[END_REF][START_REF] Angelidis | Volatility forecasting: Intra-day versus Interday models[END_REF]. The realized volatility is simply the square root of the realized variance, i.e. The intraday sampling frequency used in this article is five minutes, which for liquid assets like the S&P 500 and the DJIA stock indices it has been found to be the highest sampling frequency with acceptable market microstructure bias (see Andersen et al., 2001a;[START_REF] Koopman | Forecasting daily variability of the S&P100 stock index using historical, realised and implied volatility measurements[END_REF]Corsi et al., 2008). Moreover, in order to verify the robustness of our findings against microstructure noise, we repeat our analysis volatility estimator, which has been shown to produce consistent realized volatility estimates in the presence of microstructure noise in the returns process. Here, the TTS volatility estimates are computed using a fifteen minute sampling interval as in [START_REF] Martens | Forecasting S&P 500 volatility: Long memory, level shifts, leverage effects, day of the week seasonality and macroeconomic announcements[END_REF].

II. Realized volatility measures

,
2 2 2 2 2 , , 1
M RV t oc co oc m t m r σ σ σ σ =   = +   ∑ (1)

III. The realized volatility models

Heterogeneous Autoregressive (HAR) models for realized volatility

Based on the Heterogeneous Market Hypothesis4 and the HARCH model5 , Corsi is that short-term investors interpret the level of long-term volatilities as predictions of future volatility and adjust their trading strategies accordingly, while short-term volatility is irrelevant to investors with longer holding periods. Corsi showed that by aggregating daily, weekly and monthly volatility components in an autoregressive structure, one could capture the heterogeneity of realized volatility, whilst approximating its long range dependence properties.

Here, in order to mitigate any positivity restrictions on the model's parameters and error term (e.g. see [START_REF] Andersen | Modelling and forecasting realized volatility[END_REF], we will use the logarithm of the realized volatility in the HAR implementation:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 1 d d w m t t t t t d w m lrv a a lrv a lrv a lrv u - - - = + + + + (2)
where ( )

( ) 2 , log d t RV t lrv σ =
is the daily logarithmic realized variance, t u are the errors distributed as ( )

2 0, u N σ since ( ) d t
lrv is approximate normal (e.g. see [START_REF] Andersen | Modelling and forecasting realized volatility[END_REF],

( ) ( ) ( ) 5 1 1 , 5 1 w d t t t i i lrv lrv - - = = ∑ and ( ) ( ) ( ) 22 1 1 , 22 1 m d t t t i i lrv lrv - - = = ∑
are the lagged weekly and monthly volatility components respectively, computed for each day t as a moving average of the previous five and twenty-two days daily logarithmic realized volatilities respectively. We anticipate that some informational content in the individual lags of the logarithm of realized volatility could as a result of the averaging be lost, however empirical evidence has shown that the embedded long lag structure of the HAR model, equivalent to a restricted AR(22) 6 , is capable of reproducing the long memory behavior of realized volatility (Corsi, 2009;[START_REF] Martens | Forecasting S&P 500 volatility: Long memory, level shifts, leverage effects, day of the week seasonality and macroeconomic announcements[END_REF]. Moreover, the simple 6 For the use of AR models in realized volatility forecasting see [START_REF] Hooper | Optimal modelling frequency for foreign exchange volatility forecasting[END_REF] F o r P e e r R e v i e w autoregressive functional form of the HAR model requires no more than OLS for the estimation of its parameters.

In Corsi et al. (2008), the authors proposed a GARCH(p,q) error process in order to account for the time varying conditional heteroscedasticity of the normally distributed HAR errors, i.e. the so called "volatility of realized volatility":

( ) ( ) ( ) ( ) 0 ( ) 1 ( ) 1 ( ) 1 d d w m t d t w t m t t lrv a a lrv a lrv a lrv u - - - = + + + + (3) , t u t t u σ ε = and 2 2 2 , , ( ) ( )
u t t u t L u L σ ω α β σ = + + (4)
where

1 | t t I ε -~(0,1) N , 1 t I -is the information available until 1 t -, L is the lag operator ( ) 1 t t
Ly y - =

, ( )

2 1 2 p p L L L L α α α α = + + + K and ( ) 2 1 2 q q L L L L β β β β = + + + K
are the lag polynomials of orders p and q respectively.

Corsi and Reno ( 2009) also included past negative daily, weekly and monthly returns as regressors in the HAR model, aiming to capture the leverage effects in the volatility process, plus a jump component. Finally, the authors in [START_REF] Andersen | Roughing it up: Including Jump components in the measurement, modelling and forecasting of return volatility[END_REF] proposed a HAR model with a jump component and found that the latter had restricted persistence compared with the continuous part of the quadratic variation, i.e. its contribution to forecasting volatility was limited.

The Asymmetric HAR-(FI)GARCH models

In this article, we propose extending the HAR specification towards three directions.

Firstly, we adopt a more flexible EGARCH-type structure for implementing the asymmetries in the volatility process. We expand the HAR model of Equation (2) in order to include standardized and absolute standardized returns aggregated over different time resolutions. Here, we consider the complete returns dataset in the analysis, thus allowing for asymmetric responses to both negative and positive shocks.

Secondly, through a FIGARCH specification, we account for the long memory of the residual's variance in Equation ( 4). Finally, we use the Realized Power Variation (RPV) as a regressor, which has been shown to be robust to jumps and a more persistent and accurate predictor of future volatility than realized volatility, see [START_REF] Ghysels | Predicting volatility: how to get most out of returns data sampled at different frequencies[END_REF] and Forsberg and Ghysels (2007).

Initially, the asymmetric dynamics of past daily positive and negative returns are introduced. The Asymmetric (daily) HAR (hereafter AdHAR) model with daily leverage effects is defined as follows: 5) can be extended in order to account for the heterogeneity in asymmetric effects, i.e. asymmetric volatility reactions not only to past daily but also to weekly and monthly standardized returns. The Asymmetric HAR (AHAR) is given by:

( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) 1 ( ) 1 ( ) 1 ( ) 1
( ) ( ) ( ) ( ) 0 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) 1
where ( )

2 1 1 1 , 1 h h h i i t t i RV t i z r σ = = -+ -+ = ∑ ∑ are the daily ( 1 h d = = ), weekly ( 5 h w = = )
and monthly ( 22 h m = = ) standardized returns. The response of the logarithmic realized variance to past positive and negative standardized returns is given by:

( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 1 , if >0 , if 0 d t t t t z lrv z z ϑ γ ϑ γ ⋅ ⋅ ⋅ - ⋅ ⋅ ⋅ ⋅ - -  + ∂  =  - < ∂   (7)
The leverage effects are captured by the coefficient ( ) ϑ ⋅ which is expected to be negative and statistically different from zero, should past negative shocks yield a greater impact on future volatility. Although accounting for leverage effects in Equation ( 6), may lead to some reduction in the skewness of the errors, the heteroscedasticity in the residuals is expected to remain due to the variance of the realized volatility estimator (Corsi et al., 2008). A straightforward approach, is to implement a GARCH(p,q) error process to account for the conditional heteroscedasticity of the HAR residuals, in an AHAR-GARCH model.

We suspect however that the residuals could still retain the long memory property of realized volatility. Motivated by the findings of [START_REF] Beltratti | Statistical Benefits of Value-at-risk with long memory[END_REF] (see ARFIMA models below), we propose to model the residuals with a FIGARCH( , , ) u m d q specification (see [START_REF] Baillie | Fractionally integrated generalized autoregressive conditional heteroskedasticity[END_REF], implemented as:

( ) ( ) ( )( ) { } 2 2 2 , , 1 1 u d u t u t t L L L L u σ ω β σ β ϕ = + + - - - (8)
The FIGARCH model captures the long memory behavior of the variance process through the long memory, or fractional differencing parameter, u d , and is essentially an ARFIMA implementation of the squared residuals in Equation ( 6). For values of the differencing parameter u d between 0 and 1, the autocorrelation of the volatility process exhibits a slow hyperbolic rate of decay. As the term ( )

1 u d L -
in Equation ( 8) is an infinite summation, the FIGARCH obtains an infinite order specification, which in practice is truncated at 1000 lags, as suggested in [START_REF] Baillie | Fractionally integrated generalized autoregressive conditional heteroskedasticity[END_REF].

The Realized Power Variation

Recently, the Realized Power Variation (RPV) proposed by [START_REF] Barndorff-Nielsen | Power and bipower variation with stochastic volatility and jumps[END_REF], has been found to produce superior realized volatility forecasts when implemented as a reggresor in a HAR model. The RPV of order p , is defined as: ( )

1 (1 / 2) , 1 M p p t z m t m RPV p M r µ --- = = ∑ (9) where 0 2 p < < , ( ) ( ) ( ) / 2 1 1 2 2 2 1 / p p z E z p µ = = Γ + Γ with z ~(0,1) N 7 . Forsberg
and Ghysels (2007), [START_REF] Ghysels | Predicting volatility: how to get most out of returns data sampled at different frequencies[END_REF] and Ghysels and Sinko (2006) demonstrated the ability of realized absolute variation, i.e. RPV(1), to produce superior volatility forecasts compared to the squared return volatility measures. They argued that the RPV is a better predictor of realized volatility because of its robustness to jumps, its smaller sampling error and its improved predictability. Here, following [START_REF] Liu | Forecasting realized volatility: A Bayesian modelaveraging approach[END_REF] and Fuertes et al. (2009), who showed that an RPV of order other than one can significantly improve the accuracy of volatility forecasts, we use a RPV of order 1.5 as a regressor in the HAR models presented above. Hence, the simple HAR-RPV model is defined as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 1 d d w m t t t t t d w m lrv a a lrpv a lrpv a lrpv u - - - = + + + + (10)
where ( )

( ) log d t t lrpv RPV =
is the RPV daily logarithm and

( )

1 w t lrpv -and ( ) 1 m t lrpv -are
the respective five and twenty days logarithm RPV moving average components for each day t. The other HAR models are analogously defined.

ARFIMA models for realized volatility

The application of ARFIMA models for realized volatility modelling and forecasting purposes was first proposed by Andersen et al. (2001a) and [START_REF] Andersen | Modelling and forecasting realized volatility[END_REF], based on the analysis of [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF] and [START_REF] Granger | An introduction to the long memory time series models and fractional differencing[END_REF]. Since 7 Note that when 2 p = , the RPV is by definition equal to the realized volatility (i.e. ). In this case, the RPV is not robust to jumps and converges to the integrated volatility plus the jump component. 12 then, a number of ARFIMA based models have been proposed for volatility forecasting applications, most with good results (e.g. see [START_REF] Andersen | Modelling and forecasting realized volatility[END_REF]Pong et al., 2004 and[START_REF] Koopman | Forecasting daily variability of the S&P100 stock index using historical, realised and implied volatility measurements[END_REF]. We will thus include in this study some of the more relevant ARFIMA models, in order to provide a straightforward comparison to the HAR based model proposed above.

ARFIMA models are genuine long memory time series models which describe the stochastic behavior of a fractionally integrated variable using autoregressive and moving average components. The basic ( )

ARFIMA , ,

RV r d l model for the logarithm of the realized volatility, in terms of deviations from the mean, µ , is defined as:

( ) ( )( ) ( ) ( ) ( ) ( ) 1 1 1 RV d d t t L L lrv L u ψ µ δ - - - = + (11)
where RV d is the fractionally differencing parameter of the logarithmic realized variance and t u ~( ) 0, u N σ as in (2). [START_REF] Beltratti | Statistical Benefits of Value-at-risk with long memory[END_REF] first considered the conditional heteroscedasticity in the ARFIMA errors by proposing an

( ) ( ) ARFIMA , , FIGARCH , , RV u r d l m d q - model: ( ) ( )( ) ( ) ( ) ( ) ( ) 1 1 1 RV d d t t L L lrv L u ψ µ δ - - - = + (12) , t u t t u σ ε = (13)
where ,

u t
σ is modeled as in Equation ( 8). In order to capture the asymmetric effects, lagged returns and indicator functions have been utilized as explanatory variables of the mean (ARFIMAX models), e.g. see Andersen et al. (2001a), Thomakos and Wang ( 2003), [START_REF] Giot | Modelling daily value-at-risk using realized volatility and ARCH type models[END_REF], [START_REF] Angelidis | Volatility forecasting: Intra-day versus Interday models[END_REF]. Here, we implement lagged (absolute) daily standardized returns, along with a FIGARCH structure for the conditional heteroscedasticity of the errors, thus defining the Andersen et al. (2001b).

( ) ( ) ARFIMAX , , FIGARCH , , RV u r d l m d q - as: ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 1 1 1 1 RV d d d d t d t d t t L L lrv z z L u ψ µ ϑ γ δ - - - - -- - = + (
Insert Table 1 The fractional integration order of the logarithmic realized variance series is calculated using the Exact Local Whittle (ELW) estimator proposed by Shimotsu and Philips (2005). The estimator relies on the frequency domain representation of the observed series, as expressed by its Discrete Fourier Transform (DFT) and evaluated at m Fourier frequencies from the spectrum's origin. A widely adopted functional form for m is m T µ = with 0 1 µ < < , while the empirical evidence suggests that values for µ in the interval [ ] 0.5, 0.6 limit the bias and variance of the integration order estimate. The logarithmic realized variance ELW integration order estimates for both indices are shown in Fig. 1 for

[ ] 0.4, 0.7 µ ∈
. The RV d estimates vary between 0.55 and 0.6, suggesting that the realized variance follows a covariance non-stationary fractionally integrated process.

Insert Fig. 1 Finally, the descriptive statistics and stylized facts for the two indices for the TTS estimated realized variance show no significant departure from the aforementioned squared intraday returns observations and are available from the authors upon request. 2 and3 for the S&P 500 and DJIA indices respectively.

Estimation of the HAR based models

Insert Table 2 Insert Table 3 For the HAR, HAR-GARCH and AdHAR models, the coefficient of the lagged weekly volatility component, ( ) w a , bears the greatest impact on current volatility, followed by the daily volatility component, while the monthly one influences the total volatility significantly less. Hence, the day-ahead volatility appears to be the aggregate effect of short and mostly medium term volatility components and much less of longer term volatility factors. However, when the heterogeneity of the leverage effects is taken into consideration in the AHAR based models (i.e. in the AHAR, AHAR-GARCH and AHAR-FIGARCH models), the balance shifts drastically. Now, past longer horizon volatility events (weekly and monthly volatility components) appear to primarily shape the indices future volatility, a conclusion which is in agreement with the economic rationale laid out in Section III. The discussion of the estimation results in terms of the leverage effects and the presence of long memory in the residuals is next presented.

9 All estimates were deduced by numerical optimization of the log likelihood function (Maximum Likelihood Estimation, MLE) and they were conducted with the Ox Metrics G@RCH 4.2 package developed by [START_REF] Laurent | G@RCH 2.2: an Ox package for estimating and forecasting various ARCH models[END_REF]. The coefficients of the lagged daily, weekly and monthly standardized returns, ( ) ϑ ⋅ , in the AHAR based models are all statistically significant at a 1% significance level, confirming that future market volatility will react asymmetrically not only to yesterday's negative returns, but also to past weekly and monthly returns. Their negative weighting also suggests that past negative shocks induce more market volatility than past positive ones. This is clearly depicted in Fig. 2 below, where the impact of past daily, weekly and monthly shocks on future realized volatility is shown.

Insert Fig. 2 It is clear that past negative return events, irrespective of the time horizon, subscribe to future volatility variations more than positive ones. The volatility contribution hierarchy is analogous to that of the volatility components, with the weekly standardized return being the prevailing contributor to the overall volatility, followed by the monthly one. However, it is interesting to note that past positive monthly shocks will also tend to increase volatility, while past positive daily and weekly shocks will have a negative impact on volatility. As far as we are aware, this is a novel finding, underlining the importance of including in the analysis the complete returns dataset and not just past negative returns.

Long range dependence in the residuals

The presence of long memory in the residuals' variance is depicted in Fig. 3 are available from the authors upon request).

Estimation of the ARFIMA based models

The estimation results for the ARFIMA based models are shown in Table 4. The lag structures minimizing the A(S)IC information criteria for the ARFIMA models were again determined through testing various lag combinations for

{ } 0,1, 2 r = , { } 0,1, 2 l = , { } 0,1 m = and { } 0,1 q =
. For both indices, we adopted the following specifications for the respective models: an

( ) ARFIMA 1, , 0 RV d , an ( ) ARFIMA 1, , 0 RV d - ( ) FIGARCH 0, , 0 u d and an ( ) ARFIMAX 0, ,1 RV d - ( ) FIGARCH 0, , 0 u d .
Insert Table 4 The Overall, amongst the nine HAR and ARFIMA realized volatility models estimated here, the proposed model exhibits for both indices the best overall goodness of fit, as measured by the AIC and SIC criteria. Moreover, for both the HAR and ARFIMA models, when the conditional heteroscedasticity of the residuals and more significantly so the leverage effects are accounted for, the in-sample fitting as determined by the AIC and SIC criteria is considerably improved. Finally, the excess kurtosis values suggest that a more fat tailed distribution than the normal, might have been more appropriate.

V. Realized volatility forecasting and evaluation

In order to evaluate the realized volatility forecasting performance, a rolling window of * T observations was used to re-estimate the models and produce n out-of-sample day-ahead realized volatility forecasts calculated as:

( ) ( ) ( ) 1/ 2 2 , 1 , / 1 / 1 , / 1 ˆ/ exp 0.5 j j j j RV t t RV t t t t u t t E I lrv σ σ σ - - - - = = + (15)
where / 1 ˆj t t lrvis the day-ahead logarithmic realized variance forecast and 2 , / 1 j u t t σ -is the model j residuals variance 10 , while n=1,000, 996 and 979 observations for the day ahead, five days and twenty-two days ahead realized volatility forecasts respectively, spanning from the 31 st December 2002 to the 29 th December 2006. For the five and twenty two days ahead realized volatility forecasts, the corresponding realized volatilities were computed as the square root of the sum of daily realized variances over each forecasting period. 10 The transformation in Equation ( 15) is derived from the realized variance lognormality assumption: A random variable t y is lognormally distributed if log t t

x y = is normally distributed. Then, the expectation

of t y is ( ) ( ) 2 exp 0.5 t E y µ σ = +
, with µ and 2 σ denoting the mean and the variance of t y respectively, e.g. see [START_REF] Beltratti | Statistical Benefits of Value-at-risk with long memory[END_REF] and [START_REF] Giot | Modelling daily value-at-risk using realized volatility and ARCH type models[END_REF].

Realized volatility forecasting evaluation

In order to evaluate the model's out-of sample realized volatility forecasting performance over the three forecasting horizons, we used the seven loss functions shown in Table 5 below.

Insert (1994) is the loss implied by a Gaussian likelihood. The R2LOG loss function of [START_REF] Pagan | * and ** indicate statistical significance at 1%, and 5% significance levels respectively. LogL is the optimized value of the log likelihood. A([END_REF] is equivalent to the MSE, but for the logarithm of realized volatility. This loss function applies a greater penalty when forecasting errors occur in low volatility periods, than when they occur in high volatility periods. Using the MSE (R2LOG) criteria is equivalent to testing whether the R-square of the Mincer-Zarnowitz regressions for the (logarithmic) realized volatility of the benchmark model, is always greater than that of its counterparts [START_REF] Marcucci | Forecasting stock market volatility with regime-switching GARCH models[END_REF]. Finally, the HMSE is the Heteroscedastic Mean Squared Error proposed by Bollerslev and Ghysels (1996).

The predictive ability of the realized volatility models was also assessed via [START_REF] Hansen | A test for superior predictive ability[END_REF] f , we expect that on average the forecasting loss function of the benchmark model will be smaller, or at least equal to that of model k . Thus, the null hypothesis can be stated as:

( ) 0 , 1... : max 0 k t k k l H E f µ = =
≤ and can be tested through the following test statistic:

( ) var 1... max k k n f SPA n n f k l T = = (16)
where The SPA test analysis focuses only on the MSE and QLIKE loss functions as these two measures have been shown to be robust against volatility proxy noise (see Patton, 2006). Since realized volatility is a proxy for the true unobservable volatility, the aforementioned two loss functions yield consistent model rankings, without however diminishing the informative power of the other loss functions in Table 5.

( ) 1 , 1/ n t k t k f n f = = ∑

The models' loss function performance

In Tables 6 and7, the 2 and3 and Fig. 3), the benefit of implementing an explicit long memory volatility specification for the residuals is moderated, especially for longer term forecasts. The AHAR-RPV-GARCH model according to most of the loss functions ranks second for the one and five days ahead forecasting horizons, followed by the AHAR-FIGARCH and the AHAR-GARCH models. For the S&P 500 index and for the twenty-two days ahead horizon, the AHAR-FIGARCH ranks second (first for the QLIKE and HMSE loss functions), followed by the AHAR-RPV-GARCH and AHAR-GARCH models.

As for the rest HAR and ARFIMA model variations, the HAR models with leverage effects (AdHAR, AHAR) typically outperform the more advanced ARFIMA models (ARFIMA-FIGARCH and ARFIMAX-FIGARCH). However, for the S&P500 index twenty-two days ahead forecasts, the aforementioned performance ranking is reversed.

In turn, the AHAR model consistently outperforms the AdHAR one, underlining again the importance of considering the heterogeneity in the leverage effects.

It is nonetheless clear that even though the inclusion of daily, weekly and monthly (absolute) standardized returns in the HAR regression reduces the heteroscedasticity in the residuals (see the ARCH-LM tests in Tables 2 and3), a GARCH, or significantly more so, a FIGARCH implementation for the residuals invariably enhances the forecasting performance. Finally, the AFRIMA model performs, par from a few exceptions, better than the HAR-GARCH model, while the basic HAR model is the overall worst performing specification.

Insert 7 From the results synopsis presented above, the following conclusions can be drawn:

Firstly, regardless if an RPV regressor is used or not, accounting for asymmetric effects is as important as accounting for their heterogeneity: the AHAR model always outperforms the AdHAR one and the latter consistently outperforms the HAR.

Secondly, as expected, we confirm that considering the conditional heteroscedasticity of the realized volatility residuals is essential in volatility modelling and forecasting applications, even with the presence of asymmetric effects in the volatility equation.

However, it is now evident that the heteroscedasticity is better accounted for with a FIGARCH implementation which captures the long memory of the variance residuals.

The models with a FIGARCH specification for their residuals outperform the respective models with GARCH ones, as almost all the loss functions results suggest. Finally, the RPV is a better predictor of realized volatility than the squared returns measure, significantly improving the volatility forecasting performance when added as a regressor.

The TTS volatility forecast rankings

The out-of-sample volatility forecasting analysis for the HAR based models is once again evaluated using the TTS realized volatility estimates (see Table 8). For both indices and across all forecasting horizons, the AHAR-FIGARCH model outperforms all the other models as it minimizes almost all the loss functions. These results confirm that the aforementioned findings are robust to microstructure noise bias. However, in Insert Table 8 The SPA test results

The SPA test p-values for the MSE and QLIKE loss functions are shown in Table 9 and align with the aforementioned findings. When the AHAR-RPV-FIGARCH or the AHAR-RPV-GARCH model is the benchmark model, the null hypothesis of superior performance is strongly accepted (at a 10% significance level), for both indices and both loss functions, for the short and mid term forecasting horizons (one and five days ahead). When an alternative realized volatility model is chosen as the benchmark model, then the null hypothesis is rejected, implying that another model, or models, produces statistically significant better forecasts.

For the twenty-two days ahead forecast horizon, the null hypothesis is also accepted for the AFRIMA-FIGARCH and ARFIMAX-FIGARCH models, indicating that for longer term forecasts the ARFIMA models are competitive to the HAR models, potentially benefiting by their genuine long memory structure. Overall, the asymmetric RPV models exhibit the best out-of-sample forecasting performance, yielding for both functions the highest p-values for almost all the forecast horizons.

Insert 2009), we captured the leverage effects in the volatility process using (absolute) standardized returns of daily, weekly and monthly frequencies. The HAR error's heteroscedasticity and long memory was also accounted for with a FIGARCH implementation. Moreover, we introduced the RPV as a regressor, which has been shown to be robust to jumps, has a smaller sampling error and is more predictable. In order to examine the robustness of our findings to microstructure noise, we also calculated the realized variance with a TTS estimator and then re-evaluated our models.

The proposed AHAR-FIGARCH model produced the best in-sample fitting against the alternative HAR and ARFIMA based realized volatility models. The estimation results confirmed the appropriateness of our modelling as heterogeneity in the asymmetric effects was established, along with a long range dependence in the volatility's residuals. Rankings of each model's forecasting performance for seven established loss functions were also produced. Overall, the proposed model with the RPV as a regressor (i.e. the AHAR-RPV-FIGARCH), minimized the majority of the forecast loss functions, across all forecasting horizons and indices. The SPA test pvalues also confirmed that the AHAR-RPV-GARCH model was not, for the most part, outperformed by any other model. The TTS estimated realized volatility forecasting results demonstrated that the proposed model specification is also robust against the microstructure noise in the returns process.

The published evidence so far concurs that despite the predominant economic significance of producing accurate volatility forecasts, there is no single "ideal" volatility model for all markets and for all financial applications. Here, we showed that the proposed HAR specification can significantly improve the stock index volatility forecasting performance. We have no reason to doubt that similar improvements can also be realizable for other liquid stock indices. However, further investigation is necessary into the performance of the proposed specification in key financial applications like risk management, but also for capital allocation, derivatives pricing and hedging and for other asset classes, such as bonds and currencies.

  than positive ones, suggesting a heterogeneous component structure in asymmetric effects. Moreover, an interesting contribution of past monthly positive

σ

  are the "open-to-close" and "close-to-open" sample variances respectively, which in turn are computed from

  variance with the Zhang et al. (2005) Two-Times Scale (TTS)

( 2009 )

 2009 proposed an approximate long memory model for realized volatility, the Heterogeneous Autoregressive (HAR) Realized Volatility model. The author suggested that a significant contributor to the market's heterogeneity was the presence of three types of market agents with different time investment horizons: short (daily), medium (weekly) and long term (monthly) investment horizons. Short-term traders (such as hedge funds, FX and statistical arbitrage traders) typically adjust their market positions intradaily, swiftly reacting to any relevant new information. Μedium and long-term investors (such as commercial banks and pension funds) have longer holding periods and restructure their trading portfolios according to lower frequency information flow.This asymmetry leads to a hierarchic structure of volatility components, where low (e.g. volatility components should yield a greater impact on the overall volatility than high (e.g. daily) frequency volatility components. The economic rationale

  FIGARCH(0,d u ,0) for the S&P 500 and DJIA indices respectively. The coefficient

  For both indices, accounting for daily leverage effects in the AdHAR model leads to a reduction in the residuals' skewness and a significant improvement in the goodness of fit indicators (i.e. A(S) Information Criteria). However, there is still evidence of ARCH effects in the residuals of both indices, as the ARCH-LM tests suggest. For the DJIA index, the inclusion of weekly and monthly standardized returns in the AHAR model reduces the skewness of the errors, while there is also evidence in favor of the rejection of the ARCH effects hypothesis. This is also reflected in the GARCH coefficients estimates of the AHAR-GARCH model for the DJIA index, where none of them are statistical significant at a 5% significance level.

  , where the ELW fractional differencing parameter estimates, u d , for the HAR model squared residuals are shown for For both indices, the ELW u d estimates are statistically significant, thus confirming that not only realized volatility, but also the "volatility of realized volatility" is autocorrelated for longer time periods. The presence long memory process in the HAR residuals confirms the suitability of the proposed FIGARCH implementation for their modelling. The AHAR-FIGARCH residuals u d MLE estimates shown in Tables 2 and 3 are very close to the respective HAR squared residuals u d ELW estimates, as expected.Finally for both indices, there is no noteworthy difference between the TTS realized volatility model coefficient estimates and the ones presented above, confirming the findings in the presence of microstructure noise (the estimation results

  FIGARCH and ARFIMAX-FIGARCH models range from 0.07 to 0.09 and are

  the test statistic p-values are consistently estimated via stationary bootstrapping as inPolitis and Romano (1994).

  the authors but not shown here, we noted that the inclusion of the RPV regressor did not significantly improve the model's forecasting ability, when the dependent variable is the TTS realized volatility.

  . (2006) Volatility forecast comparison using imperfect volatility proxies, Working paper 175, Quantitative Finance Research Centre, University of Technology, Sydney. Politis, D.N. and Romano, J.P. (1994) The stationary bootstrap, Journal of the American Statistical Association, 89, 1303-1313. Pong, S., Shackleton, M.B., Taylor, S.J. and Xu, X. (2004) Forecasting currency volatility: A comparison of implied volatilities and AR(FI)MA models, Journal of Banking and Finance, 28, 2541-2563. Poon, S.H. and Granger, C.W.J. (2003) Forecasting volatility in financial markets: A review, Journal of Economic Literature, 41, 478-539. Shimotsu, K. and Phillips, P.C.B. (2005) Exact local whittle estimation of fractional integration, The Annals of Statistics, 33, 1890-1933. Thomakos, D. and Wang, T. (2003) Realized volatility in the futures markets, Journal of Empirical Finance, 10, 321-353. Zhang, L., Mykland, P. and Ait-Sahalia, Y. (2005) A tale of two time scales: determining integrated volatility with noisy high frequency data, Journal of the American Statistical Association, 100, 1394-1411.

Fig. 1 .

 1 Fig. 1. S&P 500 and DJIA indices log realized variance ELW fractional differencing parameter estimates

  Both original return series have negative skewness and fat tails, a departure from normality which can be attributed to mainly negative price shocks near the end of 1997 and 1998, all through 2000 and towards the end of 2002. The skewness and kurtosis of the standardized returns and of the logarithmic variance series as well as

	where ( ) d t z the normality tests suggest that the respective distributions are approximately normal, a , t RV t r σ = are the daily standardized returns and , u t σ is a FIGARCH
	conclusion similar as in Andersen et al. (2001a) and
	F o r F o r
	there were interpreted as T = M 2,508 78 =	14) trading days per index, with six and a half trading hours per day, P e e r R e v i e w intraday returns. Each full data set, was divided into 1,508 ( ) ( ) , , 1 100 log log t M t M t r P P -= × -where ( , , 1 , M t M t P P -P e e r R e v i e T ( ) w	)
		13 14

specification (see Equation (

8

)). The asymmetric effect is captured by ( ) d ϑ , which is expected to be negative and statistically different from zero. IV. The data set, descriptive statistics and model estimation The data set was obtained from Tick Data and consists of five minutes previous tick interpolated prices, , m t P , for the S&P 500 and the DJIA cash indices over a ten year period, from 1.1.1997 to 12.31.2006. After adjustments for holidays and half-holidays, * = in-sample observations, from 31.01.1997 to 30.12.2002 and * 1, 000 n T T = -= out-ofsample observations.

Descriptive statistics and stylized facts

The descriptive statistics for the daily logarithmic returns 8 , daily standardized returns, realized variance and logarithmic realized variance for the two full data sets are shown in Table

1

.

8

The daily logarithmic returns are calculated as is the closing price of day t, (t-1).

Table 5

 5 MSE and MAE stand for the Mean Square Error and Mean Absolute Error standard loss functions respectively. MAPE is the Mean Absolute Percentage Error and MLAE is

the Mean Logarithm of Absolute Errors. The QLIKE criterion proposed by Bollerslev et al.

Table 5

 5

	loss functions results, as well as each model's relative
	performance rankings (in parenthesis) are shown for the S&P 500 and DJIA stock
	indices respectively. Across all forecasting horizons, the proposed AHAR-RPV-

FIGARCH model nearly always ranks first amongst the alternative models, minimizing the respective loss functions, with the exception of the DJIA index twenty-two days

Table 6

 6

	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 9

 9

	VI. Conclusions
	Building on the HAR realized volatility model proposed by Corsi (
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 1 :

 1 Descriptive statistics and stylized facts for the S&P 500 and DJIA indices

				S&P 500								DJIA
		t r	/ r σ t	, RV t	σ	2 RV t ,	log( σ	2 RV t ,	)	t r	/ r σ t	, RV t	σ	2 RV t ,	log( σ	2 RV t ,	)
	Mean	0.0261	0.0882	0.9718	-0.433		0.0263	0.0742	1.0537	-0.3416
	Median	0.0492	0.0835	0.6224	-0.4741		0.0336	0.0458	0.6888	-0.3728
	Maximum	5.308	3.5254	13.2346	2.5828		6.2002	3.0564	17.6652	2.8716
	Minimum -7.1127	-3.208	0.0543	-2.9133		-7.439	-3.0309	0.0555	-2.8915
	Std. Dev.	1.1459	1.1499	1.1658	0.8648		1.1075	1.0608	1.2942	0.8443
	Skewness -0.0777	0.0977	4.4241	0.277		-0.1945	0.0789	4.6922	0.3079
	Kurtosis	6.015	2.5655	32.235	3.0848		7.0752	2.608	35.5938	3.3817
							Normality tests
	Jarque-Bera	952	24	97,496	33		1751	19	120,219	55
	[p-value] [0.0000]	[0.0000]	[0.0000]	[0.0000]	[0.0000] [0.0001] [0.0000]	[0.0000]
	Lilliefors	0.0493	0.0176	0.2189	0.0213		0.0517	0.0153	0.2264	0.0299
	[p-value] [0.0000]	[0.0697]	[0.0000]	[0.0106]	[0.0000]	[>0.1]	[0.0000]	[0.0000]
	Anderson-Darling	14.5295	1.7548	256.8931	2.3783		15.0759	1.4858 269.2609	3.2387
	[p-value] [0.0000]	[0.0002]	[0.0000]	[0.0000]	[0.0000] [0.0008] [0.0000]	[0.0000]

Table 2 :

 2 The HAR based realized volatility models estimation results for the S&P 500 index

					HAR-			AHAR-	AHAR-
				HAR	GARCH	AdHAR	AHAR	GARCH	FIGARCH
	0 a	-0.006 (0.013)	-0.008 (0.013)	-0.074* (0.022)	-0.174* (0.032)	-0.177* (0.031)	-0.178* (0.031)
	( ) d a	0.301* (0.030)	0.293* (0.035)	0.213* (0.032)	0.140* (0.032)	0.139* (0.031)	0.144* (0.032)
	( ) w a	0.422* (0.050)	0.454* (0.052)	0.507* (0.050)	0.363* (0.053)	0.381* (0.053)	0.375* (0.052)
	( ) m a	0.170* (0.044)	0.146* (0.044)	0.171* (0.042)	0.352* (0.046)	0.344* (0.045)	0.345* (0.045)
	( ) d ϑ	---	---	-0.137* (0.011)	-0.089* (0.012)	-0.083* (0.012)	-0.083* (0.012)
	( ) w ϑ	---	---	---	-0.106* (0.015)	-0.104* (0.015)	-0.103* (0.015)
	( ) m ϑ	---	---	---	-0.056* (0.016)	-0.057* (0.016)	-0.058* (0.015)
	γ	( ) d	---	---	0.081* (0.020)	0.060* (0.019)	0.059* (0.019)	0.059* (0.018)
	γ	( ) w	---	---	---	0.055* (0.020)	0.055* (0.020)	0.055* (0.020)
	γ	( ) m	---	---	---	0.092* (0.022)	0.096* (0.022)	0.096* (0.021)
	ω	0.280* (0.010)	0.030 (0.023)	0.255* (0.010)	0.240* (0.010)	0.005 (0.004)	0.017 (0.010)
	d	u	---	---	---	---	---	0.222* (0.077)
	1 α	---	0.057** (0.025)	---	---	0.021** (0.009)	---
	1 β	---	0.832* (0.104)	---	---	0.955* (0.022)	0.736* (0.070)
	1 ϕ	---	---	---	---	---	0.570* (0.085)
	LogL			-1,163	-1,153	-1,095	-1,049	-1,042	-1,040
	AIC			1.573	1.563	1.483	1.427	1.420	1.419
	SIC			1.591	1.588	1.508	1.466	1.467	1.469
	Skewness	0.333	0.309	0.307	0.308	0.306	0.294
	Excess Kurt.	0.529	0.559	0.610	0.766	0.828	0.784
				45	43	51	59	65

Table 3 :

 3 The HAR based realized volatility models estimation results for the DJIA index LogL is the optimized value of the log likelihood. A(S)IC is the Akaike (Schwartz) Information Criterion. Q(h) and Q 2 (h) are the Ljung-Box statistics for h th order serial correlation for the standardized and squared standardized residuals respectively. The p-values for the Jarque Bera (JB), Q( . ), Q 2 ( . ) and the ALCH-LM( . ) test statistics are depicted in brackets.

					HAR-			AHAR-	AHAR-
				HAR	GARCH	AdHAR	AHAR	GARCH	FIGARCH
	0 a	0.002 (0.013)	-0.006 (0.013)	-0.070* (0.022)	-0.168* (0.031)	-0.168* (0.030)	-0.170* (0.030)
	( ) d a	0.291* (0.030)	0.291* (0.034)	0.224* (0.032)	0.167* (0.032)	0.170* (0.035)	0.169* (0.032)
	( ) w a	0.450* (0.050)	0.450* (0.053)	0.510* (0.051)	0.389* (0.054)	0.400* (0.056)	0.403* (0.053)
	( ) m a	0.144* (0.044)	0.146* (0.044)	0.154* (0.044)	0.301* (0.045)	0.291* (0.045)	0.291* (0.046)
	( ) d ϑ	---	---	-0.114* (0.012)	-0.073* (0.013)	-0.070* (0.013)	-0.070* (0.013)
	( ) w ϑ	---	---	---	-0.092* (0.015)	-0.088* (0.016)	-0.089* (0.016)
	( ) m ϑ	---	---	---	-0.050* (0.016)	-0.050* (0.016)	-0.050* (0.016)
	γ	( ) d	---	---	0.091* (0.021)	0.067* (0.020)	0.067* (0.020)	0.067* (0.020)
	γ	( ) w	---	---	---	0.068* (0.021)	0.065* (0.021)	0.066* (0.021)
	γ	( ) m	---	---	---	0.089* (0.022)	0.087* (0.022)	0.088* (0.022)
	ω	0.265* (0.009)	0.086** (0.042)	0.249* 0.010	0.237* (0.010)	0.078 (0.074)	0.162* (0.029)
	d	u	---	---	---	---	---	0.050** (0.024)
	1 α	---	0.087* (0.031)	---	---	0.046 (0.038)	---
	1 β	---	0.585* (0.178)	---	---	0.621 (0.331)	---
	1 ϕ	---	---	---	---	---	---
	LogL			-1,121	-1,113	-1,076	-1,040	-1,037	-1,037
	AIC			1.517	1.508	1.458	1.414	1.414	1.412
	SIC			1.535	1.533	1.483	1.454	1.460	1.455
	Skewness	0.300	0.265	0.254	0.236	0.237	0.244
	Excess Kurt.	0.767	0.653	0.796	1.019	0.921	0.911
	JB			58 [0.000]	43 [0.000]	55 [0.000]	78 [0.000]	[0.000]	66 [0.000]
	Q(5)			[0.255]	[0.422]	[0.136]	[0.651]	[0.715]	[0.721]
	Q(50)			[0.074]	[0.078]	[0.049]	[0.051]	[0.050]	[0.056]
	Q 2 (5)			[0.005]	[0.207]	[0.088]	[0.534]	[0.669]	[0.953]
	Q 2 (50)		[0.201]	[0.639]	[0.035]	[0.512]	[0.631]	[0.781]
	ARCH-LM(2)	[0.005]	[0.832]	[0.057]	[0.342]	[0.632]	[0.724]
	ARCH-LM(5)	[0.009]	[0.485]	[0.131]	[0.556]	[0.908]	[0.953]
	Notes: Standard errors are presented in parentheses. * and ** indicate statistical significance at 1%, and 5%
	significance levels respectively.

Table 4 :

 4 The ARFIMA realized volatility models estimation results for the S&P 500 and DJIA indices

						S&P 500			DJIA
						ARFIMA-	ARFIMAX-		ARFIMA-	ARFIMAX-
					ARFIMA	FIGARCH	FIGARCH	ARFIMA	FIGARCH	FIGARCH
	d	RV	0.497*	0.502*	0.584*	0.511*	0.509*	0.582*
	µ	(0.028) -0.565	(0.029) -0.551	(0.047) -0.669**	(0.028) -0.356	(0.029) -0.370	(0.046) -0.442
					(0.268)	(0.272)	(0.314)	(0.264)	(0.254)	(0.291)
	ψ	1	-0.107*	-0.109*	---	-0.129*	-0.121*	---
					(0.037)	(0.038)	---	(0.036)	(0.038)	---
	δ	1	---	---	-0.255*	---	-0.244*
					---	---	(0.060)	---	(0.059)
	( ) d ϑ	---	---	-0.091*	---	-0.076*
					---	---	(0.010)	---	(0.011)
	γ	( ) d	---	---	0.051*	---	0.063*
					---	---	(0.017)	---	(0.018)
	d	u	---	0.093*	0.077*	---	0.078*	0.069*
	ω	---0.281*	(0.025) 0.138*	(0.024) 0.148*	---0.265*	(0.025) 0.147*	(0.025) 0.152*
					(0.010)	(0.025)	(0.027)	(0.009)	(0.027)	(0.028)
	LogL				-1,166	-1,156	-1,118	-1,123	-1,117	-1,091
	AIC				1.575	1.562	1.516	1.518	1.510	1.479
	SIC				1.590	1.580	1.541	1.532	1.527	1.504
	Skewness	0.328	0.305	0.330	0.279	0.261	0.253
	Excess Kurt.	0.516	0.564	0.565	0.767

Table 6 :

 6 Forecast loss functions for the S&P 500 index

	1 day ahead

Table 7 :

 7 Forecast loss functions for the DJIA index

	1 day ahead

The Heterogeneous Market Hypothesis[START_REF] Muller | Fractals and intrinsic time -A challenge to econometricians[END_REF] states that market agents differ with respect to their investment horizon, risk aversion, degree of available information, institutional constraints, transaction costs, etc. This diversity is identified as the root cause of asset volatility, as market agents aim to settle at different asset valuations, according to their individual market view, preferences and expectations.

For practical applications of the HARCH model seeMcMillan and Speight (2006a) andMcMillan and Speight (2006b).

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Bollerslev, T., Engle, R.F. and Nelson, DB. (1994) ARCH models. Handbook of Econometrics, Vol IV, 2959 -3038. Bollerslev, T. and Ghysels, E. (1996) Periodic Autoregressive Conditional Heteroskedasticity, Journal of Business and Economic Statistics, 14, 139-151. Clements, M., Galvao, A. and Kim, J. (2008) Quantile forecasts of daily exchange rate returns from forecasts of realized volatility, Journal of Empirical Finance, 15, 729-750. Corsi, F. (2009) A simple approximate long memory model of realized volatility. Journal of Financial Econometrics, 7, 174-196. Corsi, F. and Reno, R. (2009) HAR volatility modelling with heterogeneous leverage and jumps. Working Paper. Corsi, F., Mittnik, S., Pigorsch, C. and Pigorsch, U. (2008) The volatility of realized volatility, Econometric Reviews, 27, 46-78. Degiannakis, S. (2008) ARFIMAX and ARFIMAX-TARCH realized volatility modelling, Journal of Applied Statistics, 35,1169-1180. Engle, R.F. and Patton, A.J. (2001) What good is a volatility model? Quantitative Finance, 1, 237-245. Forsberg, L. and Ghysels, E. (2007) Why do absolute returns predict volatility so well?, Journal of Financial Econometrics, 5, 31-67. Fuertes, A.M., Izzeldin, M. and Kalotychou, E. (2009) On forecasting daily stock volatility: The role of intraday information and market conditions, International Journal of Forecasting, 25, 259-281. Ghysels, E. and Sinko, A. 2006. Comment on Hansen and Lunde JBES paper, Journal

of Business and Economic Forecasting, 26,[192][193][194]. Page 29 of 43 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The benchmark model, model 0, in the first column is tested against its competitors, models k, with k = 1…l. for the specific loss functions denoted in the subsequent columns. The null hypothesis that the benchmark model is not outperformed by any of its competitors for the specific loss function is accepted at a 10% significance level when the p-value is greater than 0.10. The bold face fonts indicate acceptance of the null at a 10% significance level.