Does Volatility Proxy Matter in Evaluating Volatility Forecasting Models? An Empirical Study

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1 Does Volatility Proxy Matter in Evaluating Volatility Forecasting Models? An Empirical Study Zhixin Kang 1 Rami Cooper Maysami 1 First Draft: August 2008 Abstract In this paper, by using Microsoft stock s daily return data and intra-day high frequency data, we investigate the impacts of the two different daily volatility proxies on the evaluation of the forecasting performances of the well-known volatility models. The two volatility proxies, one of which is the conventional realized volatility estimator, and the other of which is the two-scale realized volatility estimator, are estimated from intra-day high frequency data set. Our results show that, different volatility proxies generate quite different RMSE and MAE values in evaluating the same volatility time series model. The volatility proxy with microstructure noise correction gains lowest RMSE and MAE across all the models evaluated. For the same volatility proxy, the F IGARCH outperforms other GARCH-class models and stochastic volatility (SV ) model in the volatility forecasting. Our study confirms that the microstructure noise imposes significant impact on the estimation of the realized volatility using intra-day high frequency data set. Further, our results suggest that the long memory pattern existed in the return volatility series may be better captured by a long-memory volatility model such as F IGARCH. We conclude that the validity of evaluation of volatility forecasting performances should take consideration of the volatility proxy used in the evaluation. Keywords: Intra-day high frequency data; Long-memory volatility models; Short-memory volatility models; Daily volatility forecasting 1 Introduction Financial assets return volatility, the gauge of the fluctuation of a return time series, is directly related to option s pricing and trading, volatility derivatives, and risk management. In recent years, some volatility-related products, such as options on V IX, have been intensively traded in the exchange platform. (later will add the statistics about the trading volumes increases). Further, the different volatility-related derivatives, such as volatility corridor, volatility swap, and variance swap are being traded in the Over-the-Counter (OT C) market. Therefore, studying and understanding the return volatility are very important to researchers, practitioners, and supervisory organizations. Since early 1 Assistant Professor, Depart of Economics, Finance, and Decision Sciences, School of Business, University of North Carolina at Pembroke, One University Drive, Pembroke, NC zhixin.kang@uncp.edu 1 Professor, Depart of Economics, Finance, and Decision Sciences, School of Business, University of North Carolina at Pembroke, One University Drive, Pembroke, NC maysami@uncp.edu 1

2 80s, modeling and forecasting the return volatility has drawn significant attention in both academia and industry. There are numerous papers in the literature about the new developments in modeling and forecasting return volatility. Among them, the GARCH-class models proposed by Engle [10], Bollerslev [5], and other researchers are widely used. Further, Another type of volatility model, the stochastic volatility (SV ) model, emerges as an Bayesian approach in estimating the return volatility, see Jacquier et al. [13], Kim et al. [14] for details. All these models use daily return series as the input and generate the daily instantaneous volatility measurements. Along with the development of these models, research on forecasting of the return volatility using these models are significantly pursued. See Figlewski [11], Anderson et al. [1], and Koopman et al. [15]. However, due to the unobservability of the return volatility, the volatility proxy must be generated in order for the forecasting performances from these standard models can be evaluated. In the literature, different volatility proxies are used in evaluating the forecasting performances from a certain model. And we see controversial results simply because of the different volatility proxies implemented in the forecasting evaluation. For example, Figlewski use daily squared return as the volatility proxy to evaluate the forecasting performance of the GARCH model, and obtained very small R 2 when regressing the squared returns against the forecasted volatility. This finding lead to the conclusion of poor forecasting capability from the famous GARCH model. On the other hand, Anderson et al. use the same GARCH model, while employing the realized volatility obtained from the intra-day high frequency data set as the volatility proxy, and obtained a very high R 2 using the same OLS regression framework. Obviously, further research will be necessary and helpful in understanding the roles of a volatility proxy in evaluating the volatility forecasting performance from different time series models. In recent years, using the intra-day high frequency data set to estimate a financial asset s daily realized volatility has drawn intensive research. In the literature, it is showed by Anderson et al. [2] that the daily realized volatility obtained from the intra-day high frequency data set is an consistent and robust estimator of the true volatility. As such, it is used as a volatility proxy to evaluate the daily return volatility forecasting performances using different models, see Anderson et al. [1]. However, one of the remarkable characteristics embedded in the high frequency data set is the microstructure noise, which becomes more prominent in estimating the daily realized volatility using intra-day high frequency data set. Zhang et al. [22], aiming at alleviating the negative impacts of the microstructure noise on the estimation precision, propose a two-scale realized estimator from using the intra-day high frequency data set. These two methods are based on a non-parametric, model-free fashion. As such, we have different candidates for the daily realized volatility estimation using the intra-day high frequency data set. Then there are more choices of the volatility proxies in evaluating the forecasting performances of different standard volatility models. Unfortunately, there are no published results to document the comparison of the evaluation of the forecasting performances using above-mentioned two volatility estimators with regard to the different volatility models. Our empirical study attempts to fill this void. In our study, we employ different GARCH-class models, such as basic GARCH, EGARCH, F IGARCH, F IEGARCH models, and a SV model, to model and forecast the daily return volatility of an individual stock: the Microsoft Corporation s stock (ticker: MSF T ) in the period of 1990 to In order to implementing the comparison of the forecasting evaluations, we generate the two volatility estimators using the intraday high frequency data, and use these volatility proxies to evaluate the forecasting performances of 2

3 the employed volatility models. The remaining of the paper is structured as follow: Section 2 briefly review the volatility time series models implemented in our study. Section 3 introduces the two realized volatility estimator obtained from using the intra-day high frequency data set. The following section describe the data set we used in this paper. Section 5 presents the methodology in forecasting the different steps ahead volatility. Section 6 documents our empirical results using the Microsoft stock s data set. Section 7 concludes. 2 Models for Forecasting Volatility 2.1 GARCH-class Models GARCH(p, q) Model: Generalized autoregressive conditional heteroskedasticity (GARCH) model was proposed by Bollerslev [5] in This model is the extension of Autoregressive conditional heteroskedasticity (ARCH) model, which was proposed by Robert Engle in his seminal paper [10] in A general GARCH model assumes a form of: y t = σ t ε t, t = 1,..., T σ 2 t = α 0 + α 1 y 2 t α p y 2 t p + β 1 σ 2 t β q σ 2 t q (1) where p is the maximum order of the ARCH term, q the maximum order of the GARCH term, y t the demeaned return time series of a financial asset, and σ t is the instantaneous volatility of the return at time t. The model is established to capture the time-varying variance, which is quite common in financial market. EGARCH Model: A general exponential GARCH (EGARCH) model is proposed by Nelson [18], and we adopt a form proposed by Bollerslev and Mikkelsen [6] in 1996: y t = σ t ε t, t = 1,..., T q p log(σt 2 ) = α 0 + [1 β j B j ] 1 [1 + α i B i ]g(z t 1 ) j=1 i=1 (2) where y t is the demeaned returns. g(z t ) = λ 1 z t + λ 2 ( z t E[z t ]). p and q are non-negative integers. z t is a standard normal distribution, α 0, α, β, and λ are parameters in the EGARCH model. Different from a basic GARCH model, an EGARCH model incorporates the leverage term to capture the different responses to good news and bad news in the market.in this model, the asymmetry of the volatility due to the different direction in returns are captured. F IGARCH Model: The fractional integrated GARCH (F IGARCH) model is proposed by Baillie et al. [19]. This model capture the long memory property in the return volatility. According to Baillie, Bollerslev, and Mikkelsen in 1996 ([19]), the general form of a F IGARCH model may be written as: 3

4 y t = σ t ε t, t = 1,..., T q σt 2 = α 0 [1 β j B j ] 1 + j=1 [ 1 [1 q β j B j ] 1 j=1 p α i B i (1 B) ]ε d 2 t i=1 (3) where y t is the demeaned returns. ε t follows an i.i.d. standard normal distribution. B j and B i are the backshift operators defined as: x t B j = x t j. d is called fractional differencing parameter, and d is a real number in ( 0.5, 0.5). F IEGARCH Model: The fractionally integrated exponential GARCH (F IEGARCH) model is proposed by Bollerslev et al. [6] in The variance equation in a F IEGARCH model assumes a form of: y t = σ t ε t, t = 1,..., T q p log(σt 2 ) = α 0 + [1 β j B j ] 1 [1 + α i B i ]g(z t 1 ) j=1 i=1 (4) where B j and B i are the backshift operators defined as: x t B j = x t j. d is called fractional differencing parameter, and d is a real number in ( 0.5, 0.5). y t is the demeaned returns. And g(z t ) = λ 1 z t + λ 2 ( z t E[z t ]). Similar to the EGARCH model, the F IEGARCH model allows an asymmetric information in the market. Further, the F IEGARCH model incorporates the long memory dynamics in the volatility series. All the above-mentioned GARCH-class models are estimated using the maximum likelihood function. We use ox software to conduct the estimation and volatility forecasting procedures. For details, see Laurent and Peters [16]. It is noteworthy that we use student distribution to calculate the likelihood function, given the fact that the return distribution exhibits a fat-tail property. 2.2 Stochastic Volatility (SV ) Model A basic stochastic volatility (SV thereafter) model is in the following form: y t = σ t ε t σt 2 = exp(h t ) (5) h t = µ + ϕh t 1 + σ η η t where y t is the demeaned returns of a financial asset, and σ 2 t is the instantaneous variance of the return process y t at time t. Both ε t and η t follow i.i.d. standard normal distributions, and they are assumed to be independent. h t, the logarithm of the instantaneous variance, is modeled as a latent variable, where ϕ is a persistence parameter, and it is assumed to be positive and less than 1. Several methods have been developed to estimate the SV model. These methods include quasimaximum likelihood (QML) proposed by Ruiz [20], simulated maximum likelihood (SML) proposed 4

5 by Danielsson [7], generalized method of moments (GMM) proposed by Anderson et al. [3]. We adopt a Makov Chain Monte Carlo (MCMC) technique proposed by Kim et al. [14] to estimate the SV model. Following Kim et al.. [14], the draws for h t are implemented using accept/reject method. After a h t draw is carried out, the parameters µ, ϕ will be jointly sampled from a bivariate normal distribution. We use a drawing method proposed by Jacquier et al. [13]. From the equation: h t = µ+ϕh t 1 +σ η η t, it can be seen that a linear relationship exists between h t and h t 1, which is connected by the parameters µ, ϕ and σ η. Therefore, if one treats σ η, h t and h t 1 as fixed values at time t, then µ, ϕ can be generated from f N (Ξ, Σ), the density function of a multivariate normal distribution with mean Ξ and variance Σ, where And Ξ = (W W ) 1 (W Z), Σ = σ 2 η(w W ) 1 (6) Z = h 2. (7) W = h T 1 h 1.. (8) 1 h T 1 Therefore, the draws for µ, ϕ can be implemented through following steps: Step 1: Construct W vector and Z matrix based on the current draws for h t and previous draws for µ and σ η ; Step 2: Calculate Ξ and Σ; Step 3: Jointly draw µ and ϕ from f N (Ξ, Σ) Regarding the draw for ση, 2 we follow a methodology suggested by Kim et al. [14]. Assume that a conjugate prior for ση ϕ, 2 µ follows IG(γ 1 /2, γ 2 /2), where IG stands for a inverse-gamma distribution. Given the IG prior, the ση 2 can be sampled from following distribution: σ 2 η y, h, ϕ, µ IG((T 2)/2, SSQ/2) where SSQ = (W Zβ) (W Zβ), β = (µ, ϕ), and Z and W are obtained from Equation 7 and Equation 8. T is the observation number in the return data set. In order to draw a random number from a inverse gamma distribution, one may at first draw a random number x from a gamma distribution with the parameters γ 1 /2 and γ 2 /2. Then the inverse of the random number, 1/x, is equivalent to a draw from a inverse gamma distribution with the same parameters γ 1 /2 and γ 2 /2. In accordance with the industry practice measuring σ t instead σt 2, in each sampling iteration, we transform the generated h t vector to σ t vector, which is linked by the second equation in Equation 5. The final estimated σ t vector is the average of the σ t s across the full sampling paths. 5

6 3 Two Volatility Proxies In order to investigate the volatility forecasting evaluations from using different true volatility proxies, we employ the daily realized volatility estimated from intra-day high frequency data set. In the literature, it has been documented that these non-parametric, model-free estimators of the daily volatility from intra-day high frequency data set is robust and consistent estimators of the true volatility. See Anderson et al. [2], Barndorff-Nielsen et al. [4], and Zhang et al. [22] for details. In this paper we employ two different realized volatility estimation methodologies, which are proposed by Anderson et al., and Zhang et al., respectively. The method proposed by Anderson is to aggregate the squared returns obtained in a tiny time intervals from intra-day high frequency data set during a certain trading day. This is the realized volatility estimator (We call it RV thereafter). And the method proposed by Zhang et al. at first divides the intra-day high frequency data set into subgrids, then combines a subsampling within each sub-grid, an averaging across whole data set to obtain the estimated daily realized volatility. This estimator is called two-scale realized volatility estimator (We call it T SRV thereafter). Both methods assume that the log prices of a financial asset follow a semi-martingale. RV is motivated by approximating the integrated variance in a continuous process using a quadratic variation obtained from a discrete process. Without loss of generality, suppose in a certain trading day t, the total trading period is T. We split [0, T ] into n equally spaced sub-intervals. And y j s is the log prices of a financial asset, where j [0, T ]. Then the RV is calculated as follow: n+1 RV 2 = (y j y j 1 ) 2 (9) j=2 This formula implies that, within a collection of the high frequency data set in a certain day, only n+1 observations are used for the realized volatility computation, and most of the data set is abandoned. A typical trading period for a stock/stock index is from 9:30 am to 16:00 pm in the U.S. This is 6.5 hours of trading period. In our study, we use a 5-minute sub-interval to split a trading day, thus the total number of the sub-intervals is 78. The T SRV is motivated by the prominent microstructure noise existed in an intra-day high frequency data set. This estimator combines sub-sampling, averaging, and bias correction mechanism together to reduce the impacts of the microstructure noise on the realized volatility estimation. Suppose a full set of log prices y R + is our interest for the volatility estimation. The corresponding time points in this data set is then: T = { t 0,..., t n }. In the sub-sampling procedure, T is partitioned into K non-overlapping sub collections of time points T k, where k = 1,..., K, and T k is constructed as: T k = { t k 1, t k 1+K,..., t t 1+nk K } where n k is the largest integer to make the t k 1+nk K th element to be included in the sub-grid T k. Then, the T SRV is calculated as follow: where T SRV 2 = RV 2 avg RV 2 adj (10) 6

7 RV 2 k = RV 2 adj = n n RV 2 avg = 1 K K RV k (11) k=1 t i,t i+1 T (k) (y ti+1 y ti ) 2 (12) t j,t j+1 T (y tj+1 y tj ) 2 (13) where n = n K+1 K. It can be seen that this term takes a portion of the realized variance obtained from the total observations to correct the bias existed in sub-sampling procedure of estimating the overall realized variance. The extent of the correction of the bias depends on the n, which is determined by an optimal procedure. In our calculation, we set K = 200. The realized volatility obtained from above-mentioned methodology only reflects the volatility during the trading time period. However, over-night price changes may contribute to an asset s realized volatility, either. In order to convert the trading-time volatility to the day-to-day volatility, an adjustment is needed. We employ a method proposed by Martens [17] to adjust the trading time realized variance to the daily realized variance. The coefficient is calculated as: σ 2 t = σ2 1 + σ 2 2 σ 2 1 D i=1 RV 2 t,i (14) where σ 2 1 is the intra-day open-to-close variances and σ 2 2 is the overnight close-to-open variances. They are calculated as follows: σ 2 1 = σ 2 2 = 10, 000 T 10, 000 T T (log P t,d log P t,0 ) 2 t=1 (15) T (log P t,0 log P t 1,D ) 2 t=1 where t = 1,..., T, and T is the total observations in the data set. D is the numbers of sub-intervals designated in a trading day. For a 5-minute sub-interval, D = Data Set As a successful high-tech company, Microsoft Corporation (M SF T thereafter) has been significantly traded since its stock was issued to the public. Its stock s performance represents the booming period of high-tech in the late 90 s, and also experienced debacle of the high-tech bubbles. We use the Microsoft s daily returns in the GARCH-class models, and the SV model. This data set is downloaded from Wharton Research Data Service (W RDS). The data set spans from January 02, 1990 to December 31, 2003 with 3532 observations. As mentioned earlier, we make the forecasts of daily return volatility from the different time series models, and evaluate the forecasting performance using the true volatility benchmark. The true volatility we use in our research is the realized volatility obtained from the intra-day high frequency 7

8 data set, which is the daily Trade and Quote (T AQ) data set downloaded from W RDS. We make a total of 252 out-of-sample forecasts for the daily return volatility using the different models in the time period of January 02, 2003 to December 31, As such, we estimate the RV and T SRV using the intra-day high frequency data set in this time period. Table 2 lists the basic statistical analysis of the daily returns of MSF T s stock. As can be seen that, the mean return during this period was positive, implying a positive gain in investing the MSF T. However, the median is almost 0, reflecting frequent trading days with zero return. The high kurtosis indicates a sharp peakedness in the distribution. The highest daily return reached to 19.57%, and highest daily loss is 15.60%. Figure 1 is the time plot of the daily MSF T returns. It is clear that, during 2001 and 2002, along with the debacle of the high-tech bubble in the market, the return series exhibits much more intensive volatility. Further, Figure 2 documents the density of the daily MSF T returns. As can be seen that, it approximately follows a normal distribution with slightly higher peakedness. In the literature, it has been documented a so-called long memory pattern existed in some financial series, such as returns. See Baillie [19], Ding and et al. [8]. A long memory process is characterized by very persistent and slow declining autocorrelations (ACF ) within the time series. Using the absolute return and squared returns of the MSF T, we calculate the sample ACF s of these two time series and plot them. As Figure 3 shows that, the ACF s for both absolute return and the squared return exhibit very strong and persistent auto-correlations. This finding adds the evidence to the literature about the long memory characteristic embedded in financial return series. 5 Forecasting Methodology In order to investigate the forecasting performances, we generate the 1-day, 5-day, 10-day, 15-day, 20-day, and 25-day ahead forecasts for the MSF T s daily volatility using the standard models introduced in previous section. A rolling window forecasting methodology is employed for the out-ofsample forecasting of M SF T daily volatility. In each rolling window, with a window size of 3280 M SF T daily return observations and rolling size of 1, we entertain GARCH(1, 1), EGARCH(1, 1), F IGARCH(1, d, 1), F IEGARCH(1, d, 1), and SV models to estimate the parameters. Then the outof-sample forecasts with different steps ahead are made based on the estimated models. For each of the different step ahead forecasting procedures, a total of 252 forecasts for M SF T s daily volatility, spanning from 01/02/2003 to 12/31/2003, are generated. It is worth noting that, in using the SV model to make the out-of-sample forecasts, the forecasts of σ 2 for l steps ahead, the ˆσ t+l 2, are obtained from generating the random values for the η t after the parameters are estimated in each iteration of the Gibbs sampler, as showed in the following equation. It is assumed that the η t follow a Gaussian distribution. ˆσ t+l 2 = exp(ĥt+l) (16) ĥ t+l = ˆµ + ˆϕĥt 1+l + ˆσ η η t whereˆrepresents the estimated values or forecasted values in each Gibbs sampler iteration. 8

9 6 Empirical Results Using the Microsoft s daily return data set, we estimate the different volatility models introduced in Section 2. Table 1 list the estimates from different GARCH-class models. As can be seen that, using different ARCH and GARCH orders in the GARCH class models indeed results in different likelihood values. However, the differences among these likelihood values are very tiny. Therefore, considering the parsimony of the GARCH class models, we choose both ARCH and GARCH orders as 1 in the forecasting procedures. Regarding the SV model, in implementing the Gibbs sampler technique, a total of 50,000 samplings are made, and the first 2,000 burn-ins are abandoned. The estimated values for the parameters are: ˆµ = , ˆϕ = , and ˆσ η = Following the literature, in evaluating the forecasting performances of the different models, we use RMSE and MAE to measure the forecasting precision with regard to MSF T s volatility forecasts. (1). Root Mean Square Error is defined as: RMSE = 1 M (ˆσ j σ j ) M 2 (17) where M is the total number of the forecasts, σ is either the realized volatility (RV ), or the historical volatility (HV ). ˆσ is the forecasted MSF T volatility. The RMSE measures the local fluctuations of the forecasted realized volatility from the benchmark volatility. (2). Mean Absolute Error is defined as: j=1 RMSE = 1 M M ˆσ j σ j (18) j=1 where M, σ, and ˆσ are the same as in the RMSE calculation. Similar to RMSE, the MAE measure the point-to-point forecasting precision. By comparing the different out-of-sample forecasts for M SF T with the corresponding RV and HV proxies, we calculate the RMSEs and MAEs for each of the time series models. Table 3 to Table 6 list the forecasting evaluation for the different time series models. The results show that, given the same model and same step ahead volatility forecasts, the RV and T SRV proxies yield different RMSEs and MAEs. As can be seen that, the T SRV, with microstructure noise correction, consistently generates lower RM SEs and M AEs comparing with those generated from RV. It is showed from the results that for same step ahead forecast, the F IGARCH model outperforms other models, since this model gains the lowest RMSEs and MAEs. And this pattern are clearly showed in both RV and T SRV proxies. As showed in the section 4, the MSF T s volatility series exhibit distinct long-memory pattern, thus the F IGARCH, as a long memory volatility model, seems to capture this property, and as a result, yield more precise forecasts than other short memory GARCH-class models. However, it is worth noting from the results that, the EGARCH and F IEGARCH model perform very poorly in forecasting the daily volatility, especially in the higher steps forecasts. The SV model, as a short memory volatility model, performs quite well in doing the volatility forecasting, since the RM SEs and M AEs are lower than the GARCH-class counterparts. 9

10 From above results, it implies that the microstructure noise may worsen the estimation precision of the realized volatility using the intra-day high frequency data set, thus the correspondingly obtained RM SEs and M AEs are higher. This confirms the findings obtained from the simulation documents in Zhang et al. [22]. 7 Conclusions In this paper, we empirically investigate the impacts of the two different volatility proxies on the evaluation of the volatility forecasting models using the Microsoft stock s inter-daily and intra-day return data set. Due to the unobservability of the volatility measurements, the evaluation of the forecasting precision is dependent on the estimation of a volatility proxy. Recently, new methods in estimating the true volatility are growing, and it is documented that the magnitudes of the estimation vary from the different estimation methods. This in turn, may impose challenge in the forecasting evaluation using these differently estimated volatility proxies. Our empirical results reveal that, using the conventional realized volatility estimator (RV ) without the microstructure noise correction as the volatility proxy in evaluating the volatility forecast, the corresponding RM SEs and M AEs are higher than those obtained from us the two-scale realized volatility estimator (T SRV ) with microstructure noise correction. Even though both of these two volatility proxies are obtained from the same intra-day high frequency data set, our results suggest that the estimated forecasting precision may be far different, not because of the volatility models, but because of the volatility proxies. Our results consistently show that, given the same model and same step ahead volatility forecasts, the RV and T SRV proxies yield different RMSEs and MAEs. As can be seen that, the T SRV, with microstructure noise correction, consistently generates lower RM SEs and MAEs comparing with those generated from RV. From the ACF s of the Microsoft s volatility series, it suggests that this volatility series follows a long memory, or so-called fractionally integrated pattern. The forecasting evaluation from using the fractionally integrated GARCH model, the F IGARCH model yields the lowest RM SEs and M AEs, indicating that a long memory time series model may be more appropriate in forecasting the return volatility. Based on these results, we conclude that the volatility proxy does matter in leading to the evaluation of volatility forecasting models. Our results may add our understanding of the volatility modeling and forecasting, and may have practical implication to the practitioners who use forecasted volatility to guide their trading activities. 10

11 References [1] T. G. Anderson and T. Bollerslev, Answering the skeptics: Yes, standard volatility models do provide accurate forecasts, International Economic Review 39 (1998), [2] T. G. Anderson, T. Bollerslev, F. X. Diebold, H. Ebens, and P. Labys, Modeling and forecasting realized volatility, Econometrica 71 (2003), [3] T. G. Anderson and B. Sorensen, Gmm estimation of a stochastic volatility model: A monte carlo study, Journal of Business and Economics Statistics 14 (1996), [4] Ole E. Barndorff-Nielsen and Neil Shephard, Econometric analysis of realized volatility and its use in estimating stochastic volatility methods, Journal of Royal Statistical Society, B 64 (2002), [5] Tim Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986), [6] Tim Bollerslev and H.O. Mikkelsen, Modeling and pricing long memory in stock market volatility, Journal of Econometrics 73 (1996), [7] J. Danielsson, Stochastic volatility in asset prices: Estimation with simulated maximum lieklihood, Journal of Econometrics 64 (1994), [8] Z. Ding, C. W. J. Granger, and R. F. Engle, A tale of two time scales: Determining integrated volatility with noisy high-frequency data, Journal of Empirical Finance 1 (1993), [9] J.A. Doornik and M. Ooms, Inference and forecasting for arfima models with an application to us and uk inflation, Studies in Nonlinear Dynamics & Econometrics 8 (2004), Article 14. [10] Robert F. Engle, Autoregressive conditional heteroskedasticity with estimates of variance of the u.k. inflation, Econometrica 50 (1982), [11] S. Figlewski, Forecasting volatility, Financial Markets, Institutions and Instruments 6 (1997), [12] N. Gospodinov, A. Gavala, and D. Jiang, Forecating volatility, Journal of Forecasting 25 (2006), [13] Eric Jacquier, Nicholas G Polson, and Peter E. Rossi, Bayesian analysis of stochastic volatility models, Journal of Business and Economic Statistics 12 (1994), [14] Sangjoon Kim, Neil Shephard, and Siddhartha Chib, Stochastic volatility: Likelihood inference and comparison with arch models, Review of Economic Studies 65 (1998), [15] S.J. Koopman, B. Jungbacker, and E. Hol, Forecasting daily variability of the s & p 100 stock index using historical, realized and implied volatility measurements, Working Paper (2004). [16] S. Laurent and J. P. Peters, G@rch 4.2, estimating and forecasting arch models, London:Timberlake Consultants Press,

12 [17] M. Martens, Measuring and forecasting s&p 500 index-futures volatility using high-frequency data, Journal of Futures Markets 22 (2002), [18] D.B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59 (1991), [19] T. Bollerslev R.T. Baillie and H.O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 74 (1996), [20] E. Ruiz, Quasi-maximum likelihood estimation of stochastic volatility models, Journal of Econometrics 63 (1994), [21] F.B. Sowell, Maximum likelihood estimation of stationary univariate fractionally integrated time series models, Journal of Econometrica 53 (1992a), [22] Lan Zhang, Per Mykland, and Y. Aït-Sahalia, A tale of two time scales: Determining integrated volatility with noisy high-frequency data, Journal of American Statistical Association 100 (2005),

13 MSFT Daily Return: % /02/ /02/ /03/ /02/ /02/ /03/ /02/ /31/2003 Time Figure 1: Time Plot of Daily MSFT Returns: 01/02/ /31/

14 Table 1: Estimation Results of the GARCH Class Models for MSF T Model C α0 α1 α2 β1 β2 λ1 λ2 ˆd Likelihood GARCH(1, 1) ( ) ( ) ( ) ( ) GARCH(2, 1) ( ) ( ) ( ) ( ) ( ) GARCH(1, 2) ( ) ( ) ( ) ( ) ( ) GARCH(2, 2) ( ) ( ) ( ) ( ) ( ) ( ) EGARCH(1, 1) ( ) ( ) ( ) ( ) ( ) ( ) EGARCH(2, 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) EGARCH(1, 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) EGARCH(2, 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F IGARCH(1, 1) ( ) ( ) ( ) ( ) ( ) F IGARCH(2, 1) ( ) ( ) ( ) ( ) ( ) ( ) F IGARCH(1, 2) ( ) ( ) ( ) ( ) ( ) ( ) F IGARCH(2, 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F IEGARCH(1, 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F IEGARCH(2, 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F IEGARCH(1, 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F IEGARCH(2, 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 14

15 Table 2: Statistical Description of the DAILY M SF T Returns: 01/02/ /31/2003 Mean Median Standard deviation Skewness Kurtosis Maximum Minimum Q Q Range Normal Distribution MSFT Kernel Density Density MSFT Return Figure 2: Density Plot of Daily MSF T Returns: 01/02/ /31/ Sample Autocorrelation 0.05 Sample Autocorrelation Lag Lag Figure 3: Sample ACF s of MSF T Volatility Proxies 15

16 Table 3: Evaluation of MSF T s Forecasts with Different Steps Ahead: RMSE from RV Proxy Models 1-day 5-day 10-day 15-day 20-day 25-day GARCH(1, 1) EGARCH(1, 1) F IGARCH(1, d, 1) F IEGARCH(1, d, 1) SV Table 4: Evaluation of MSF T s Forecasts with Different Leads Ahead: RMSE from T SRV Proxy Models 1-day 5-day 10-day 15-day 20-day 25-day GARCH(1, 1) EGARCH(1, 1) F IGARCH(1, d, 1) F IEGARCH(1, d, 1) SV Table 5: Evaluation of MSF T s Forecasts with Different Steps Ahead: MAE from RV Proxy Models 1-day 5-day 10-day 15-day 20-day 25-day GARCH(1, 1) EGARCH(1, 1) F IGARCH(1, d, 1) F IEGARCH(1, d, 1) SV Table 6: Evaluation of MSF T s Forecasts with Different Steps Ahead: MAE from T SRV Proxy Models 1-day 5-day 10-day 15-day 20-day 25-day GARCH(1, 1) EGARCH(1, 1) F IGARCH(1, d, 1) F IEGARCH(1, d, 1) SV