Georgiana-Denisa Banulescu
email: georgiana.banulescu@univ-orleans.fr

Bertrand Candelon
email: bertrand.candelon@ipag.fr

Christophe Hurlin
email: christophe.hurlin@univ-orleans.fr

Sébastien Laurent
email: sebastien.laurent@iae-aix.com

Do We Need Ultra-High Frequency Data to Forecast Variances? *

Keywords: JEL classification: C22, C53, G12 Variance Forecasting, MIDAS, High-Frequency Data

In this paper we study various MIDAS models in which the future daily variance is directly related to past observations of intraday predictors. Our goal is to determine if there exists an optimal sampling frequency in terms of volatility prediction. Via Monte Carlo simulations we show that in a world without microstructure noise, the best model is the one using the highest available frequency for the predictors. However, in the presence of microstructure noise, the use of ultra high-frequency predictors may be problematic, leading to poor volatility forecasts. In the application, we consider two highly liquid assets (i.e., Microsoft and S&P 500). We show that, when using raw intraday squared log-returns for the explanatory variable, there is a "high-frequency wall" or frequency limit above which MIDAS-RV forecasts deteriorate. We also show that an improvement can be obtained when using intraday squared log-returns sampled at a higher frequency, provided they are pre-filtered to account for the presence of jumps, intraday periodicity and/or microstructure noise. Finally, we compare the MIDAS model to other competing variance models including GARCH, GAS, HAR-RV and HAR-RV-J models. We find that the MIDAS model provides equivalent or even better variance forecasts than these models, when it is applied on filtered data.

Introduction

The mixed data sampling (henceforth MIDAS) regression model, introduced in [START_REF] Ghysels | The MIDAS touch: Mixed data sampling regression models[END_REF], allows to forecast a measure of the daily variance (e.g., realized variance) by considering past intraday log-returns.

In their seminal paper, [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] consider various MIDAS regressions with different daily (squared returns, absolute returns, realized variance, realized power and return range) and intradaily regressors (squared returns, absolute returns), to examine whether one specification dominates the others. The goal of our study is different and consists in determining, for a given intradaily predictor, whether a sampling frequency (or a range of frequencies) dominates the others. 1 The objective is then to identify the best sampling frequency, using out-of-sample forecast evaluation criteria. This issue is not straightforward. On the one hand, not using the readily available high-frequency observations to perform variance forecasts implies a loss of information through the temporal aggregation. On the other hand, if the sampling frequency of the predictors is increased too much, the market microstructure noise (bid-ask bounce, screen fighting, jumps, and irregular or missing data) may lead to less accurate variance forecasts.

This question has to be distinguished from the well-documented discussion about the optimal sampling frequency of the returns used to compute realized estimators of daily variance (see [START_REF] Hansen | Technical appendix: An unbiased measure of realized variance[END_REF][START_REF] Aït-Sahalia | Out of sample forecasts of quadratic variation[END_REF][START_REF] Garcia | Comment on realized variance and market microstructure noise[END_REF]Ghysels et al., 2006, among others). Our goal consists in focusing on the optimal sampling frequency for the purpose of variance prediction, and not for variance measurement.

Consider a MIDAS variance model whose aim is to predict a measure of variance over some future horizon. This variance measure is typically a realized measure (realized variance, realized kernel etc.), based on intradaily returns sampled at a frequency m 2 . In order to forecast variance, we adopt exactly the same approach as [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] and consider intradaily predictors (absolute returns, squared returns etc.) sampled at a frequency m 1 , where m 1 may be different from m 2 . The discussion concerns only the sampling frequency of the predictors, m 1 .

In a related paper, [START_REF] Ghysels | Volatility forecasting and microstructure noise[END_REF] study a regression prediction problem with variance measures that are contaminated by market microstructure noise and examine optimal sampling for the purpose of variance prediction. They observe that, in general, discussions about the impact of microstructure have mostly focused on measurement. [START_REF] Ghysels | Volatility forecasting and microstructure noise[END_REF] focus instead on prediction in a regression framework, and therefore they can consider estimators that are suboptimal in the mean squared error (MSE) sense, since their covariation with the predictor is the object of interest. The authors consider univariate MIDAS regressions for the evaluation of the prediction performance and derive the optimal frequency in terms of prediction MSE. Their dependent variable is defined as the two scales estimator of the weekly variance [START_REF] Aït-Sahalia | How often to sample a continuous-time process in the presence of market microstructure noise[END_REF], and computed from the 5-minute, 1-minute or 2-second returns. One of the main differences with our study is that the authors consider various MIDAS specifications for which the predictors also correspond to realized estimators (plain vanilla, two scales estimator, Zhou, 1996, etc.), constructed using different sampling frequencies (from two seconds to ten minutes). Thus, high-frequency data are aggregated into daily realized measures, which are then used as predictors of future variance. In contrast, our goal is to analyze the direct impact of the intradaily predictors on the variance forecasts, and ultimately to evaluate the usefulness of the mixing of frequencies in this context. To this aim, we consider MIDAS models in which we directly project future realized variance onto high-frequency regressors, as in [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF]. 2To address these issues, we propose a Monte Carlo simulation study. Considering a noise-free diffusion process, we generate returns series at different sampling frequencies m 1 and daily realized variance measures, using the same set of continuous-time structural parameters. Then, we apply simple MIDAS specifications in which daily realized variance is predicted by past intradaily squared log-returns sampled at a frequency m 1 , ranging from one minute to three hours. The variance forecasts are compared based on the robust loss function proposed by [START_REF] Patton | Volatility forecast comparison using imperfect volatility proxies[END_REF] and the model confidence set (MCS) test introduced by [START_REF] Hansen | The model confidence set[END_REF]. This test aims at identifying among the set of competing models (i.e., sampling frequencies), the subset of models that are equivalent in terms of forecasting ability and their outperformance of all the other models for a given confidence level. Several results stand out. First, we observe that a higher sampling frequency for the regressors implies giving more weight to the most recent observations of the regressors. Second, we show that increasing the frequency of the regressors always improves the forecasting abilities of the MIDAS model. The average loss increases when the regressors are sampled less frequently, regardless of the choice of the loss function. These differences are statistically significant and the MCS test always concludes that the MIDAS model with the highest available sampling frequency significantly stands out in terms of forecasting performances. Nevertheless, opting for ultra-high frequency regressors is not optimal in the presence of microstructure noise. In this specific case, we need to use pre-averaged data to improve the MIDAS performances.

The sensitivity of MIDAS variance models to the choice of the sampling frequency m 1 is also investigated on real data, i.e., log-returns of the S&P 500 index and Microsoft over the period of October 29, 2004to December 31, 2008. The empirical results obtained for these two assets allow us to draw some interesting conclusions.

First, when using raw intraday returns, variance forecasts are not statistically different for sampling frequencies of the predictors (m 1 ) ranging from five minutes to one hour. Besides, it turns out that ultra-high-frequency regressors (i.e., m 1 < 5min) do not necessarily provide useful information to improve the variance forecasts because the loss function increases. The shape of the loss function indicates the presence of a "high-frequency wall", i.e., a limit frequency beyond which the quality of the forecasts deteriorates. This result is due to the presence of microstructure noise, jumps and intraday periodicity in the regressors. When the MIDAS regression model is applied to filtered data [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF][START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF][START_REF] Lahaye | Jumps, cojumps and macro announcements[END_REF], the conclusion in favor of the use of the highest available frequency remains valid. This point is crucial and indicates that the mixing frequency may require the use of filtered series. Indeed, the weighting scheme in MIDAS models does not allow, by itself, to underweight the observations affected by jumps or other market microstructure noise. These results are robust to the choice of variance measure (realized variance, realized kernel), forecasting horizon and sample period (calm/crisis).

Finally, we compare the performance of the MIDAS model (applied to filtered or unfiltered regressors) to other competing variance models, namely the GARCH(1,1), the Student Generalized Autoregressive Score (GAS) model [START_REF] Creal | Generalized autoregressive score models with applications[END_REF], the Heterogeneous Autoregressive Realized Volatility-based (HAR-RV) model [START_REF] Corsi | A simple approximate long-memory model of realized volatility[END_REF] and the HAR-RV adjusted for jumps [START_REF] Andersen | Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility[END_REF]. We show that MIDAS models are providing comparable or even better variance forecasts when filtered high-frequency data are used.

The chapter is structured as follows. Section 2 introduces the notations, the MIDAS model and the sampling frequency puzzle. Section 3 proposes a Monte Carlo simulation. In Section 4, we perform an empirical analysis and study the influence of the jumps and the intraday periodicity on the MIDAS performances. We also compare MIDAS to other competing variance models. Section 5 concludes.

Modeling Strategies

Notation

To set the notation, let p t denote the price for a financial asset sampled at daily frequency, and the corresponding daily return be defined by r t,t-1 ≡ log(p t ) -log(p t-1 ). The equally spaced series of continuously compounded returns is assumed to be observed m times per day (or to have an horizon of 1/m), and be computed as

r (m) t,t-1/m ≡ log(p t ) -log(p t-1/m
), where t = 1/m, 2/m, ... Throughout the analysis, we consider that the trading day spans the time period from 9:30 am to 16:00 pm, covering for instance m = 390 1-minute equally spaced intervals and m = 78 5-minute equally spaced intervals. r t,t-1/78 corresponds to the last 5-minute return of the day t -1, r t-1/78,t-2/78 corresponds to the return of the penultimate 5-minute period of day t -1, and so on.

MIDAS variance Models and Sampling Frequency

MIDAS models for variance predictions have been introduced in a number of recent studies, including [START_REF] Ghysels | There is a risk-return trade-off after all[END_REF][START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF], [START_REF] Ghysels | Volatility forecasting and microstructure noise[END_REF], [START_REF] Ghysels | Forecasting volatility with MIDAS[END_REF], [START_REF] Chen | News-good or bad-and its impact on volatility predictions over multiple horizons[END_REF], among others.

The general specification of the MIDAS variance model is given by:

σ 2 t+H,t = µ H,m1 + φ H,m1 Ω H,m1 (L 1/m1 )X (m1) t,t-1/m1 + ε t , (1)
where σ 2 t+H,t is a measure of variance evaluated over some future horizon H, and X (m1) t,t-1/m1 denotes an intradaily regressor sampled at frequency m 1 . The distributed lag polynomial is defined as:

Ω H,m1 (L 1/m1 ) = kmax k=0 L k/m1 ω H,m1 (k, θ H,m1 ) , (2)
where ω H,m1 (k, θ H,m1 ) corresponds to the lag coefficient associated with X (m1)

t,t-1/m1 , θ H,m1 is a finite set of parameters, L is the lag operator such that L 1/m1 X (m1) t,t-1/m1 = X (m1)
t-1/m1,t-2/m1 , and k max denotes the maximum number of lagged coefficients. In this specification, the low-frequency variance (for instance, daily variance if H = 1, weekly variance if H = 5, etc.) is predicted by the right-side intradaily forecasting factors which are sampled at a high-frequency m 1 (for instance, five minutes if m 1 = 78). Several intradaily regressors can be considered with this aim (e.g., intradaily squared returns, intradaily absolute returns, intradaily bipower variation). Following [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] we consider the intradaily squared returns r (m1)2 t,t-1/m1 , while the two other alternatives will be used to appraise the robustness of our results.

Since σ 2 t+H,t is unobservable, we rely on a proxy. For simplicity, we adopt the realized variance (Andersen and Bollerslev, 1998a), defined for the period t to t + H as following:

3 RV (m2) t+H,t = I H,m2 (L 1/m2 )r (m2)2 t+H,t+H-1/m2 , (3)
where the distributed lag polynomial in L 1/m2 is defined such that I H,m2 (L 1/m2 ) = Hm2 j=0 L j/m2 , and m 2 accounts for the sampling frequency of the squared returns used to compute the realized variance. Notice that the frequencies m 1 and m 2 may be different. In fact, the choice of m 2 is related to the variance measurement issue (see [START_REF] Hansen | Technical appendix: An unbiased measure of realized variance[END_REF][START_REF] Aït-Sahalia | Out of sample forecasts of quadratic variation[END_REF][START_REF] Garcia | Comment on realized variance and market microstructure noise[END_REF]Ghysels et al., 2006, among others), i.e., the consistency of the estimator defined by the realized measure. 4 On the contrary, the 3 A large number of alternative estimators (e.g., realized bipower variation, realized kernel, etc.), that deal with issues such as jumps and other market microstructure noise, have been proposed, especially by Barndorff-Nielsen and Shephard (2004a), [START_REF] Barndorff-Nielsen | Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise[END_REF], [START_REF] Zhang | Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach[END_REF], [START_REF] Hansen | Quadratic variation by markov chains[END_REF], inter alios. Some of them will be considered in the section devoted to the robustness analysis of our findings.

4 Since this study is not meant to determine the optimal sampling frequency m 2 , in the rest of the paper, the daily RV will always be computed by summing up 5-minute squared returns (i.e., m 2 = 78), as recommended by Andersen and Bollerslev (1998a).

choice of the sampling frequency of the predictors, m 1 , is related to the variance prediction issue. Indeed, this choice determines the regressors in the MIDAS variance model, and as a consequence, its forecasting abilities.

Under these assumptions, the MIDAS-RV regression becomes:

RV (m2) t+H,t = µ H,m1 + φ H,m1 Ω H,m1 (L 1/m1 )r (m1)2 t,t-1/m1 + ε t . (4)
One advantage of this specification is that it preserves the information contained in high-frequency data [START_REF] Ghysels | Forecasting volatility with MIDAS[END_REF] without computing daily aggregates such as realized variance for the regressors. In this context, we aim to determine the influence of the sampling frequency m 1 on the forecasting performances of the MIDAS model. In a related study, [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] compare several MIDAS specifications based on different intradaily or daily variance regressors (e.g., squared returns, absolute returns, realized volatility, realized power, and range). The logic is similar here, except that we consider the same intradaily regressor, i.e., X

t,t-1/m1 , for various sampling frequencies. For instance, we compare various MIDAS models where the same predictor is sampled at one minute (m 1 = 390), two minutes (m 1 = 195), five minutes (m 1 = 78), and so on. The question is whether increasing the sampling frequency m 1 systematically improves the quality of the variance forecasts, and ultimately if we need ultra-high frequency data in order to forecast daily variances.

Monte Carlo Simulation Study

We first propose a Monte Carlo simulation study in order to determine the influence of the sampling frequency of the regressors on the predictive abilities of MIDAS-RV models. We begin by describing the simulation setup and then we follow this by discussing the results.

Monte Carlo Design

Let us assume that instantaneous log-returns, dp t , are generated by the continuous-time martingale

dp t = σ t dW p,t , (5)
where W p,t denotes a standard Wiener process, and σ t is given by a separate continuous-time diffusion process.

For σ t , we use the diffusion limit of the GARCH(1,1) process introduced by [START_REF] Nelson | ARCH models as diffusion approximations[END_REF], i.e.,

dσ 2 t = θ(ω -σ 2 t )dt + (2λθ) 1/2 σ 2 t dW σ,t , (6)
where ω > 0, θ > 0, 0 < λ < 1, and the Wiener processes, W p,t and W σ,t , are independent. [START_REF] Drost | Closing the garch gap: Continuous time GARCH modeling[END_REF] and [START_REF] Drost | Temporal aggregation of GARCH processes[END_REF] prove that the exact discretization for stochastic variance processes is in line with the weak GARCH(1,1) representation meaning that a weak GARCH process can be identified at any discrete-time sampling frequency from the parameters of a continuous GARCH and vice versa.5 (ω, θ, λ) is set to (0.636, 0.035, 0.296) as in Andersen and Bollerslev (1998a) (these parameters have been calibrated by the authors to fit the daily GARCH estimates for the Deutschemark-U.S. Dollar (DM-$) spot exchange rates).

Given this data generating process (DGP), we draw large series of continuous-time log-returns and compute log-returns series sampled at different frequencies by applying the temporal aggregation proprieties of flow variables, i.e., r

(m) t,t-1/m = t t-1/m dp (τ ) dτ .
We consider various frequencies m 1 ranging from one minute to three hours, i.e., m 1 = {2, 3, 6, 13, 25, 39, 78, 130, 195, 390}. Next, we compute discrete realized variance series by summing up simulated 5-minute (m 2 = 78) squared return series (see Eq. ( 3)). In so doing, we obtain all the necessary elements to run MIDAS-RV regressions as defined in Eq. ( 4).

One notable advantage of this procedure is that it allows to generate intradaily log-returns series at different sampling frequencies, m 1 , and daily RVs, using the same data generating process and the same set of continuous time structural parameters. In addition, this process is calibrated to reproduce the main features of typical real financial series. The Monte Carlo simulation exercise is based on 10,000 replications and the daily/intradaily series are simulated for a period of 1,000 days. The parameters of all the competing MIDAS-RV models are estimated by Nonlinear Least Squares (NLS). 6 In order to allow for a fair comparison between models, the maximum lag order k max is fixed, such that the past information used to predict volatility covers a period of 30 days, whatever the sampling frequency of the regressor. For example, if m 1 = 78, the corresponding lag order k max is equal to 2, 340 (30 × 78), if m 1 = 3, k max is equal to 90 (30 × 3), and so on. A daily forecasting horizon (H = 1) is used in all simulations.

Weight Function and Sampling Frequency

One of the key features of MIDAS models is that it provides a parsimonious specification. This property is particularly important in our context, as the inclusion of high-frequency data might imply a significant increase in the number of lagged forecasting variables and hence the number of unrestricted parameters to be estimated [START_REF] Ghysels | Forecasting volatility with MIDAS[END_REF]. For instance, running unrestricted regressions based on the intraday information over the last 30 days implies estimating 30 × 390 parameters for a 1-minute regressor, 30 × 78 parameters for a 5-minute regressor, and so on. Nevertheless, the MIDAS model projects directly future variance onto an important number of high-frequency lagged regressors while considering a small number of parameters. The trick consists in using a suitable parametrization for the weights ω H,m1 (k, θ H,m1 ) to circumvent the problem of parameter proliferation. Therefore, as noted by [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF], the parametrization ω H,m1 (k, θ H,m1 ) becomes one of the most important ingredients in a MIDAS regression.

Two specifications of the weight function are generally considered, namely the exponential Almon lag and the Beta lag [START_REF] Ghysels | MIDAS regressions: Further results and new directions[END_REF]. These specifications have several interesting features: i) the distributed lag polynomial is tightly parameterized and prevents the proliferation of parameters as well as additional pretesting or lag-selection procedures; 7 ii) the coefficients are positive, which guarantees non-negative weights and consequently non-negative variance forecasts; iii) the data-driven weights are normalized to add up to one in order to identify the scale parameter φ H,m1 . There is no clear theoretical a priori for assuming that one specification is better than the other. However, [START_REF] Chen | Forecasting commodity prices with mixed-frequency data: An OLS-based generalized ADL approach[END_REF] and [START_REF] Frale | FaMIDAS: A mixed frequency factor model with MIDAS structure[END_REF] find that the Beta function is more suitable for an important number of time lags, as Almon could be very computationally demanding in such a context. For this reason, we adopt the Beta lag polynomial

ω H,m1 (k, θ H,m1 ) = f (k/k max , θ 1 ; θ 2 ) k max j=0 f (j/k max , θ 1 ; θ 2 ) , (7)
where 1h05,2h10,3h15 (Eq. 4). Column "Day 1" reports the sum of the weights associated with the first lagged day of the predictors, column "Days 2-5" present how much weight is given to the information of the second to the fifth lagged day of the predictors, and so on. The next two columns, namely %Q(12) and %Q 2 (12), correspond to the frequencies of rejection of the null hypothesis of no serial correlation at the 5% significance level for the Ljung-Box test applied on respectively residuals and squared residuals. The last column reports the average Mean Squared Error (MSE).

θ H,m1 = (θ 1 , θ 2 ) ′ is a vector of positive parameters, f (z, a, b) = z a-1 (1 -z) b-1 /B(a,
Table 1 presents the outline of the regression diagnostics for the 10,000 replications considered in the Monte

7
The selection of k max can be done by considering a large value and letting the weights vanish.

Carlo experiment. The first four columns represent the average values of the parameter estimates for various MIDAS-RV models, each associated with a particular frequency m 1 . Results first suggest that the estimates for the constant term of the model, µ H,m1 , and the first parameter of the weight function, θ 1 , decrease with m 1 . On the contrary, the estimates for the scale parameter, φ H,m1 , and the second parameter of the weight function, θ 2 , increase with the sampling frequency. These changes imply a deformation of the weight function that gives more weight to the most recent observations. This is confirmed by the next four columns of Table 1. Column "Day 1" reports K k=0 ω H,m1 (k, θH,m1 ) for a value of K corresponding to one day and θH,m1 = θ1 , θ2

′ is the vector containing the average estimates of the parameters θ 1 and θ 2 over the 10,000 replications. Similarly, Column "Days 2-5" presents how much weight is given to the information on the second to the fifth lagged day, and so on. Results confirm that more weight is given to the most recent observations of the variance predictor when the sampling frequency m 1 increases. The proportion of the weights allocated to the observations of the first lagged day represents 67.36% when the regressors are sampled at one minute, and this proportion decreases progressively to 9.59% when the regressors are sampled at 3h15.

To illustrate the deformation of the Beta polynomial shape, Figure 1 displays φ m1 ω m1 (k, θ m1 ), i.e., the product of the weight (determined by the Beta function) and the scale parameter estimate (see Eq. 1), as a function of the sampling frequency m 1 . For ease of comparison, the weights are displayed for a only 20 day window. First, in all of the cases, the weight function is gradually declining and there is no hump-shaped pattern.

Second, the slope of the weight function becomes smaller when the predictors are sampled at a lower frequency.

If the weights vanish after approximately three days in the case of a regressor sampled at one minute, the weights of the observations associated with the 20 th lagged day are still positive for a regressor sampled at three hours.

The sampling frequency of the predictors also has an impact on the quality of the fit of daily realized variances (i.e., H = 1). Columns %Q and %Q 2 in Table 1 report rejection frequencies of the null hypotheses of no serial correlation in the residuals and squared residuals (respectively) at the 5% significance level using a Ljung-Box test with 12 lags. Serial correlation and heteroskedasticity are detected in most cases. This is in line with [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF], who also find significant autocorrelation and heteroskedasticity in the residuals of daily MIDAS-RV regressions.8 However, results suggest that the problem of serial correlation in the residuals is less pronounced when using ultra-high frequency returns. For instance, when the regressors are sampled at five minutes, the residuals do not feature autocorrelation in 54.9% of the simulated samples, and this percentage decreases to 46.6% when the regressors are sampled at one minute. Finally, increasing the sampling frequency m 1 always tends to improve the in-sample goodness of fit, as indicated in the last column of Table 1 where the average MSE is reported. Note: This figure displays the scaled weights pattern based on average parameter estimates obtained in a Monte Carlo simulation study, for the nine MIDAS models with regressors sampled at a frequency ranging from one minute to 3h15. The scaled weights are obtained by multiplying the Beta weight function ωm 1 (k, θm 1 ) by the scale parameter, φm 1 (see Eq. 1). For ease of comparison, the weights are represented over the first 20 lagged days whatever the sampling frequency of the variance regressor.

Notice that all the MIDAS-RV models are directly comparable in terms of MSE since they all have the same number of estimated parameters whatever the frequency m 1 . When the sampling frequency increases from 78 (five minutes) to 390 (one minute), the gain in terms of average MSE reaches 27.70%. These gains are statistically significant. Since we have many competing models (i.e., frequencies m 1 ), we focus on multiple comparison-based tests and use the Model Confidence Set (MCS) approach introduced by [START_REF] Hansen | The model confidence set[END_REF]. This test allows identifying, among an universe of competing forecasting models, the subset of models that are equivalent in terms of forecasting ability, and which outperform all the other models at a confidence level α.

Interestingly, we find that the MCS test systematically selects the 1-minute MIDAS-RV specification, regardless of the loss-function used (KLIC, AIC or BIC).9

Out-of-Sample Analysis

To check whether the previous results remain valid out-of-sample, we subsequently focus on the influence of the sampling frequency of the variance predictors, m 1 , on the predictive abilities of the MIDAS-RV model.

At each replication, we compute a sequence of T = 500 daily realized variance forecasts, { RV (m2) t+1,t } T t=1 , for each MIDAS-RV specification. The forecasts sequences are obtained with a rolling window approach and the parameter estimates are updated every 50 days.

In order to compare these forecasts, we must use a loss function, defined as a general function of the variance forecasts and the true variance. In our simulation framework, the variance can be measured by the daily integrated variance, IV t,t-1 = t t-1 σ 2 (τ ) dτ . However, in practice, the integrated variance is not observable and we have to use a proxy. To reproduce the real conditions of application of the MIDAS-RV models, we also use a variance proxy to define the loss function, i.e., the realized variance RV (m2) t+H,t . 10 However, it is well known that the use of a proxy may distort the ranking of models based on loss functions. Andersen and Bollerslev (1998a) and [START_REF] Andersen | Correcting the errors: Volatility forecast evaluation using high-frequency data and realized volatilities[END_REF] show that the comparison of losses, even based on a conditionally unbiased proxy, may lead to a different outcome than the one obtained if the true latent variable had been used. More recently, Hansen and Lunde (2006a), [START_REF] Patton | Evaluating volatility and correlation forecasts[END_REF], [START_REF] Patton | Volatility forecast comparison using imperfect volatility proxies[END_REF], [START_REF] Laurent | Consistent ranking of multivariate volatility models[END_REF] have also insisted on the possible distortions observed in the ranking of volatility forecasts induced by the use of a noisy proxy. 11

For these reasons, we adopt the family of robust and homogeneous loss functions proposed by [START_REF] Patton | Volatility forecast comparison using imperfect volatility proxies[END_REF],

i.e., L( σ 2 , σ 2 ; b) =          1 (b+1)(b+2) ( σ 2(b+2) -σ 2(b+2) -1 b+1 σ 2(b+1) ( σ 2 -σ 2 ), for b / ∈ {-1, -2} σ 2 -σ 2 + σ 2 log σ 2 σ 2 , for b = -1 σ 2 σ 2 -log σ 2 σ 2 -1, for b = -2 (8)
with b a scalar parameter, σ 2 a measure of the true variance (i.e., the realized variance in our case) and σ 2 the predicted variance measure. This loss function encompasses in particular the MSE and the QLIKE loss functions when b = 0 and b = -2, respectively.

Evaluating the influence of the sampling frequency of the predictors on the predictive abilities of the MIDAS-RV model reduces to determining the sign of the derivative of the average loss function given by:

L m1 = T -1 ∂ T t=1 L( RV (m2) t+1,t , RV (m2) t+1,t ; b) ∂m 1 . ( 9
)
Since the sign of this derivative cannot be determined analytically we proceed by numerical analysis. Figure 2 displays the average (over the 10,000 replications) of the loss function

T -1 T t=1 L( RV (m2) t+1,t , RV (m2)
t+1,t ; b), as a function of the frequency m 1 .

In order to assess the robustness of our results, we consider three values for the parameter b, namely 0 (MSE),

-1 and -2 (QLIKE). The main conclusion is that the average loss decreases with the sampling frequency of the predictors, regardless of the loss function specification. For instance, the MSE increases progressively from 0.0085 (for a 1-minute regressor) to 0.0256 (for a regressor sampled twice a day). The use of the highest available frequency for the predictors is hence favored not only in-sample but also out-of-sample.

Besides, these gains are found to be statistically significant using the MCS test of [START_REF] Hansen | The model confidence set[END_REF] (as discussed in the previous section). Table 2 reports the MCS results for one replication of the Monte Carlo experiment. For each sampling frequency m 1 , we display the average loss function along with the corresponding MCS p-value. The entries in bold correspond to the cluster of the best MIDAS-RV models as identified by the MCS test. For each of the three loss functions, the MCS test confirms that the use of the 1-minute regressor leads to a significant improvement in the forecasting performances. This result is not specific for the particular replication reported in Table 2. The average value of the MCS p-values obtained over all the replications is not informative. Alternatively, it is possible to count the number of replications for which the MIDAS specification with the highest sampling frequency outperforms the other models. We find that in 65% of replications, the MCS approach selects the 1-minute MIDAS model to be the best. This proportion reaches 98% when we consider the clusters of outperforming models including also 2-minute and 5-minute MIDAS regressors. 8) for b = {0, -1, -2}.

For each MIDAS specification the average value of the loss function is reported (first column) along with the corresponding p-value (second column) resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples (with block length of five daily observations) are used to obtain the distribution under the null of equal predictive accuracy. The entries in bold refer to the best MIDAS-RV forecasts according to the MCS test.

DGP Sampling Frequency

In the previous experiment we considered a continuous data generating process and concluded in favor of the use of the highest frequency available for the predictors. Following [START_REF] Visser | GARCH parameter estimation using high-frequency data[END_REF] and [START_REF] Hecq | Common intraday periodicity[END_REF], we now consider a DGP for 1-second log-returns where the conditional variance varies every consecutive five minutes according to a discrete-time GARCH(1,1) with parameters (α 0 , α 1 , β) = (2.4693e -07, 0.0057, 0.9941) but is constant during every 5-minute intervals.

As in the previous simulation, out-of-sample forecasts as previously and analyze the forecasting accuracy of MIDAS variance models. Figure 3 displays the average loss functions (over 10,000 replications) as a function of the sampling frequency m 1 . Note: See Figure 2. Notice that the conditional variance of the simulated 1-second log-returns (before aggregation) varies every five minutes according to a discrete-time GARCH(1,1) but is constant during every consecutive 5-minute intervals.

Results suggest that using a sampling frequency m 1 greater than five minutes (i.e., one, two or three minutes)

does not improve much the quality of the fit. However, using data sampled at a much lower frequency than five minutes leads to a huge loss of information and therefore important increases of the average losses, irrespective of the choice of the loss function.

Microstructure Noise

The main conclusion of the previous Monte Carlo simulation is that ultra-high-frequency log-returns are not always useful in the context of MIDAS-RV. Another disadvantage of using ultra-high-frequency data is that at these frequencies, the true price process is likely to be contaminated by microstructure effects arising from market frictions, such as the bid-ask bounce or the discreteness of prices. This phenomenon produces spurious variations in asset prices and induces autocorrelation in high-frequency log-returns (see, Hansen and Lunde, 2006b;[START_REF] Zhou | High-frequency data and volatility in foreign-exchange rates[END_REF][START_REF] Aït-Sahalia | How often to sample a continuous-time process in the presence of market microstructure noise[END_REF].

The consequence of this noise on the realized variance is known (i.e., it is upward biased) but its impact on MIDAS-RV has not been investigated so far. We argue that this noise has an impact on the optimal frequency of the variance predictors when relying on raw data and therefore leads to a loss of information.

To study the impact of microstructure noise on MIDAS-RV models, we first simulate 1-second log-returns using the same approach that the one described in the previous simulation, except that the dynamics of the discrete-time GARCH(1,1) model is at the 1-minute frequency and not 5-minute. To contaminate the log-returns by noise, a normal random variable with mean 0 and variance 10 -3 × IntegratedQuarticity is added to every 1-second log-return.

This Monte Carlo experiment is based on 10,000 replications with the regressors and the realized variance simulated for 1,000 days (500 days are used for the purpose of the in-sample estimation and 500 days for the out-of-sample analysis). Figure 4 displays the average MSE for 500 out-of-sample one-step-ahead forecasts (over 10,000 replications) as a function of the sampling frequency m 1 . Three models are considered. The (black) dotted line corresponds to the case where the MIDAS-RV is estimated on non-contaminated log-returns. The (red) dashed line corresponds to the case where the MIDAS-RV is estimated on contaminated log-returns.

Recall that realized variance (the endogenous variable) is calculated on 5-minute returns, a frequency at which the simulated noise is negligible on realized variance.

Results clearly suggest that microstructure noise deteriorates the fit of MIDAS-RV models when using ultrahigh-frequency returns. The optimal frequency is between three and five minutes but the average MSE is about 40% greater than in the case without noise.

To account for the presence of microstructure noise in the context of non-parametric volatility estimators, it is standard practice to pre-filter ultra-high-frequency log-returns using the pre-averaging technique introduced by [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF] and [START_REF] Jacod | Microstructure noise in the continuous case: the pre-averaging approach[END_REF]. To the best of our knowledge, pre-averaging has never been used in the context of MIDAS models. The (blue) solid line corresponds to the case where the MIDAS-RV is estimated on contaminated but pre-averaged log-returns. Pre-averaging proves to be useful in the context of MIDAS models especially when relying on data sampled at frequencies higher than 15 minutes. Interestingly, for frequencies between 30 seconds and three minutes, the MSE of this model is stable and does not blow up, as in the case of the MIDAS-RV model estimated on contaminated log-returns (red dashed line).

Application

The main conclusion of Section 3 is that the choice of the optimal sampling frequency m 1 for the predictors in MIDAS-RV models is not obvious. We have seen two cases where the use of the highest available frequency does not necessarily improve the quality of the fit or the predictions. Therefore, a "high-frequency wall" might exist or frequency limit above which MIDAS-RV forecasts deteriorate or do not improve. In the Monte-Carlo simulation, only two features of the DGP have been considered to justify the presence of this "high-frequency wall", i.e.,

• that the process is not a pure continuous-time model but rather a model where the conditional variance is constant by pieces of for instance one or five minutes;

• and/or the presence of microstructure noise.

It has also been largely documented in the literature that high-frequency log-returns are characterized by the presence of strong intraday periodicity in volatility and jumps. Intraday periodicity [START_REF] Wood | An investigation of transactions data for NYSE stocks[END_REF][START_REF] Harris | A transaction data study of weekly and intradaily patterns in stock returns[END_REF]Andersen andBollerslev, 1997, 1998b;[START_REF] Hecq | Common intraday periodicity[END_REF] can be defined as the cyclical pattern of variance within the trading day, i.e., the fact that variance is typically more important at the opening and closing of the trading day and lower in the middle of the day while jumps correspond to large discontinuities in prices. Unlike intraday periodicity, jumps are not regular (most of the time they appear as the result of an unexpected news arrival) and are known to affect largely variance estimates and forecasts. For more details about the properties and the detection of jumps, see [START_REF] Bates | Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options[END_REF], Barndorff-Nielsen andShephard (2004b, 2006), [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF], [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF], [START_REF] Lahaye | Jumps, cojumps and macro announcements[END_REF], among many others.

In the application, we propose to investigate the impact of these two additional features of the data on MIDAS-RV models in an application on two highly liquid assets, one exchange-traded fund (ETF) and one quoted share.

The use of an ETF is justified by the increasing importance of these assets in the fund management industry.12

Data

The dataset consists of tick-by-tick prices and quotations from NYSE Trade and Quote (TAQ) database for Microsoft (MSFT) and one ETF (provided by SPDR ETFs) that tracks the S&P 500 index, spanning the period from September 2, 2004 to December 31, 2008. The price and quote series are reported every trading day from 9:30 am to 4:00 pm and rigorously cleaned using a set of baseline rules proposed by [START_REF] Barndorff-Nielsen | Realized kernels in practice: Trades and quotes[END_REF]. In order to avoid the effect of variance that comes from the overnight or holiday closures, all the variables are computed by using open-to-close data and focusing hence only on the effective trading day variance. The equally spaced intraday returns are subsequently derived from the high-frequency price series.

The dataset contains hence 1,101 trading days with 390/78/39/26/13/6/2 observations per day of respectively 1-minute/5-minute/10-minute/15-minute/30-minute/1h05/3h15 log-returns.

To compute the variance forecasts, we consider a rolling sample estimation scheme. The parameter estimates are updated every 50 days. For a fair comparison of the MIDAS models, the lag order k max is fixed such that the information used to estimate the parameters covers a period of 70 days, regardless of the sampling frequency of the regressors. For instance, for a 5-minute regressor we use a k max equal to 78 × 70 lags, where 78 represents the number of 5-minute intervals within a trading day.

Optimal sampling frequency for MIDAS-RV on raw data

We first consider one-step-ahead forecasts of MIDAS-RV models estimated on raw data, sampled at different frequencies m 1 ranging between one minute and three hours. Three horizons (H) are considered for the endogenous variable RV

t+H,t , i.e., one day (H = 1), one week (H = 5) and two weeks (H = 10).

Table 3 reports the MCS test for both the calm and crisis periods. For each horizon H, the average QLIKE is reported along with the p-value of the MCS test. The entries in bold correspond to the best models selected by the MCS procedure. The striking result is that the loss function does not smoothly decrease with the sampling frequency and seems to indicate the presence of a "high-frequency wall". In particular, the use of ultra-high-frequency regressors leads to a deterioration in the quality of variance forecasts.

Consider the example of S&P 500 during the calm period (Panel A). The loss function has a convex shape and its minimum is reached for a predictor sampled at five minutes, whatever the forecasting horizon considered.

Using 1-minute log-returns leads to a deterioration in the quality of the variance forecasts. This deterioration is statistically significant because MIDAS-RV estimated on 1-minute log-returns does not belong to the MCS set of optimal models. For the crisis period, the MCS test selects the 5-minute frequency as optimal for H = 1 and H = 5, and 10-and 15-minute frequencies for the two-week horizon. For Microsoft all the models but the one estimated on 1-minute log-returns are found to be statistically equivalent and superior during the calm period

(panel A) for H = 1.
All in all, these results question the usefulness of ultra-high-frequency data in the context of MIDAS-RV models. The results are reported for three forecasting horizons, i.e., one day (H = 1), one week (H = 5) and two weeks (H = 10). The QLIKE is reported along with the p-value of the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing models includes seven MIDAS specifications with regressors sampled at a frequency ranging from one minute to about three hours.

Breaking the Wall

We have suggested several explanations for the existence of this "high-frequency wall", i.e., an underlying DGP whose conditional variance is constant by pieces of e.g., one or five minutes or the presence of intraday periodicity, jumps and microstructure noise.

While no solution for breaking this wall might exist for the first one, filtering the raw lag-returns might help to improve the performance of MIDAS-RV in presence of intraday periodicity, jumps and microstructure noise. This is precisely the purpose of the this section.

Intraday periodicity

Figure 6 illustrates the intraday periodicity in the variance for the S&P 500 and Microsoft series, by plotting the average squared log-returns for each 1-minute, 5-minute, 30-minute and 1h05 interval, respectively. A clear Ushaped pattern is identifiable, as first noted by [START_REF] Wood | An investigation of transactions data for NYSE stocks[END_REF], suggesting that the variance is systematically high at the opening, declines to a low point at midday and then increases at the end of the trading day.

To estimate the intraday periodicity in volatility we rely on the non-parametric weighted standard deviation (WSD) of [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF], a non-parametric estimator that is robust to additive jumps. If r i denotes a raw return (sampled at a certain frequency), the corresponding periodicity adjusted return is obtained by dividing

r i by f WSD i , i.e., r i / f WSD i
, where f WSD i is the estimated WSD of [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] for the i th return.

Jumps

To filter out the jumps in the regressors of the MIDAS-RV model, we first apply a modified version of the jump test of [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] proposed by [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF]. More specifically, we assume that the log-price process log p(s) follows a Brownian SemiMartingale with Finite Activity Jumps (BSMFAJ) diffusion d log p(s) = µ(s)ds + σ(s)dw(s) + κ(s)dq(s), where µ(s) is the drift, σ(s) is the spot volatility, w(s) is a standard Brownian motion, the occurrence of jumps is governed by a finite activity counting process q(s) and the size of the jumps is given by κ(s).

The idea behind the jump test of [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] is that in the absence of jumps, instantaneous returns are increments of Brownian motion and, therefore, standardized returns that are too large to plausibly come from a standard Brownian motion must reflect jumps. In their original paper, [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] standardize every intraday returns r i by a robust estimate of the spot volatility, denoted ŝi , that assumes that the volatility is constant on a local window spanning between several hours to one or two days before or around the tested return. Their original statistic for jumps is J i = |ri| ŝi , where ŝi is the averaged bi-power variation belonging to the local window. To control for the size of the multiple jump tests [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] use the extreme value theory result that the maximum of n i.i.d. realizations of the absolute value of a standard normal random variable is asymptotically (for n → ∞) Gumbel distributed. More specifically, in the absence of jumps, the probability that the maximum of any set of n J-statistics exceeds g n,α = -log(-log(1 -α))b n + c n ,

with b n = 1/ √ 2 log n and c n = (2 log n) 1/2 -[log π + log(log n)]/[2(2 log n) 1/2
], is about α. [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF]'s proposal is that all returns for which the J test statistic exceeds this threshold g n,α should be declared to be affected by jumps. In the application, we set α = 1% and n to the total number of observations in the sample.

However, for such long windows, the assumption of constant volatility is at odds with the overwhelming empirical evidence that the intraday variation in market activity causes intraday volatility to be strongly timevarying and even displays discontinuities (see Figure 6). For this reason, we implemented the modified version proposed by [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] that accounts for the presence of intraday periodicity, i.e.,

FJ WSD i = |ri| f WSD i ŝWSD i ,
where f WSD i is the estimated WSD of [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] for the i th return (which is standardized such that its square has mean one in the local window).

Periodicity and jumps adjusted returns are computed as (r

i / f WSD i ) × I(FJ WSD i < g n,1% ) + I(FJ WSD i > g n,1%
), where I(.) is an indicator function.

Microstructure noise

As explained above, pre-averaging [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF][START_REF] Jacod | Microstructure noise in the continuous case: the pre-averaging approach[END_REF] is a powerful technique to robustify volatility estimators to the presence of microstructure noise.

Instead of noisy intraday returns (r t ), the authors suggest using pre-averaged returns (r t ) which, by the law of large numbers, asymptotically lose the noise component. More precisely, rt is approximated by an average of staggered returns r t in a neighborhood of t, the noise being hence averaged away. The pre-averaging approach depends on a bandwidth parameter, or window length, that increases with the sample and indicates the weighting scheme to be put into effect. The order of the window size is chosen to lead to optimal convergence rates (n -1/4 ).

To the best of our knowledge, pre-averaging has never been used in the context of MIDAS models. The balanced pre-averaging has been applied on 1-minute and 5-minute returns previously filtered for intraday periodicity and jumps, since it delivers according to [START_REF] Christensen | Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data[END_REF] the best rate of convergence.

Results

Figure 7 displays the filtered 1-minute return series for S&P 500 and Microsoft, adjusted for intraday periodicity and jumps. The correction procedure purges the intraday periodicity, identifies and smoothes the jumps, but preserves the variance dynamics. The procedure is applied to the 5-minute return series as well.

Jump-adjusted returns

Note: This figure displays the 1-minute intraday return series (in red) as well as the 1-minute return series filtered for intraday periodicity and jumps (in blue). [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF][START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF].

The MCS test is subsequently applied on the MIDAS out-of-sample obtained with both filtered and unfiltered data. The results are reported in Table 4. We notice a significant improvement in the MIDAS variance forecasts when using high-frequency predictors filtered for intraday periodicity and jumps. For the calm period (panel A), the S&P 500 forecasts obtained with filtered (both for jumps and periodicity) 1-minute predictors always belong to the set of superior forecasting models as identified by MCS. During the crisis period, similar results are obtained with filtered 5-minute regressors for short horizons (H = 1 or H = 5). These results prove the importance of using the filtered data, especially for short forecasting horizons. The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10). For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven MIDAS specifications with regressors sampled at a frequency ranging from one minute to three hours, as well as four MIDAS models with 1-and 5-minute regressors adjusted for intraday periodicity and jumps. Figures 8 and 9 display the average QLIKE for the calm and crisis periods, and the three forecasting horizons, for both S&P 500 and Microsoft. First, we remark that the gains related to the use of filtered data for intraday periodicity are generally lower than the gains related to data filtered both for periodicity and jumps. Second, considering filtered data during a relatively calm period, we get a loss function which smoothly decreases with the sampling frequency m 1 as in the Monte Carlo experiment.

To complete our analysis, we also perform MIDAS variance forecasts based on regressors filtered for periodicity, jumps and microstructure noise (through the pre-averaging technique). The results are available in Appendix 6.1 and Appendix 6.2. For instance, we observe that 1-minute pre-averaged regressors improve Microsoft variance forecasts during both the calm and crisis periods. These results apply also for the S&P 500 variance forecasts at short forecasting horizons. They become more puzzled for long forecasting horizons (e.g., two weeks), as well as for the crisis period.

MIDAS and Other Competing Variance Models

In this section, we compare the predictive accuracy of the MIDAS-RV forecasts with those obtained for four widely used variance models based on daily and/or intradaily data, i.e., the GARCH model, the Generalized Autoregressive Score (GAS), the Heterogeneous Autoregressive Realized Volatility-based model (HAR-RV) and the HAR-RV adjusted for jumps (HAR-RV-J).

i) The first competing model is the popular GARCH(1,1) model, pioneered by [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation[END_REF] and [START_REF] Bollerslev | Generalized autoregressive heteroskedasticity[END_REF], i.e.:

r t+1,t = c + z t+1,t h t+1,t , (10)
h t+1,t = α 0 + α 1 (r t,t-1 -c) 2 + β 1 h t,t-1 . ( 11
)
ii) The second model is the GAS model, recently introduced by Harvey (2013) and [START_REF] Creal | Generalized autoregressive score models with applications[END_REF]. This model is designed to better treat large outliers. We consider the Student GAS specification where the one step-ahead conditional variance is defined as follows:

h t+1,t = w 0 + a 1 u t,t-1 h t,t-1 + φ 1 h t,t-1 , (12)
with u t,t-1 = ((v + 1)z 2 t,t-1 )/(v -2 + z 2 t,t-1 ) -1, and z t ∼ t(0, 1, v).

Notice that for the GARCH and GAS models, the variance forecasts for H > 1 are obtained as

H i=1 h t+i,t
and not directly from r t+H,t as opposed to the MIDAS-RV model.

iii) The third competing model is the HAR-RV model, proposed by [START_REF] Corsi | A simple approximate long-memory model of realized volatility[END_REF]:

RV t+1,t = α 0 + α 1 RV t,t-1 + α 2 RV w t,t-1 + α 3 RV m t,t-1 + ε t+1 , (13)
where RV t+1,t is the daily realized variance (Eq. 3) and by convention, RV w t,t-1 = 1 5 4 i=0 RV t-i,t-i-1 and

RV m t,t-1 = 1 22 i=0 RV t-i,t-i-1
. This model is conceived as an additive cascade of different variance components defined over different time horizons of one day, one week (w), and one month (m), respectively. The HAR-

RV is therefore a constrained version of the MIDAS-RV model with intradaily squared return regressors and a particular weight structure. Indeed, given the definition of the realized volatility, Eq. ( 13) can be rewritten as a weighted sum of past observations of the intraday squared returns. For more details, see Appendix 6.3. iv) [START_REF] Andersen | Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility[END_REF] extended the classical HAR-RV framework by taking into account the lagged effect of jumps. The HAR-RV-J model (where J stands for jumps) is formally defined as following:

RV t+1,t = α 0 + α 1 RV t,t-1 + α 2 RV w t,t-1 + α 3 RV m t,t-1 + γ 1 J t,t-1 + γ 2 J w t,t-1 + γ 3 J m t,t-1 + ε t+1 , ( 14
)
where J t,t-1 = I t × (RV t,t-1 -BV t,t-1 ) is a random variable that is nonzero for the intervals in which jumps do occur and zero otherwise, BV is the daily realized bipower variation (Barndorff-Nielsen and Shephard, 2004b) which is defined as:

BV t = µ -2 1 m l=2 |r t,l ||r t,l-1 |, (15)
with µ 1 = (2/π) ≈ 0.79788, and I t ≡ I (Z t > Φ 0.999 ), where Z t is defined as:

Z t = m 2 (RV t,t-1 -BV t,t-1 )RV -1 t,t-1 [(µ -4 1 + 2µ -2 1 -5)max{1, T Q t,t-1 (m)BV -2 t,t-1 }] 1/2 , ( 16
)
with T Q t,t-1 the tri-power quarticity 13 , a robust estimator of the integrated variance, and Φ 0.999 the 99.9% quantile of the standard normal distribution.

The MCS procedure is now applied on 17 models, namely the seven MIDAS models with regressors sampled between 1-min and 3h15, the six MIDAS specifications with 1-and 5-minute regressors adjusted for intraday periodicity, jumps and/or microstructure noise, and the four competing variance models (HAR-RV, HAR-RV-J, Student GAS, GARCH). The results are summarized in Table 5. The global conclusion is that the MIDAS models provide (at least for these two assets) comparable, or even better, variance forecasts than the other competing models. In terms of the loss function, the models are dominated during the calm period by the MIDAS models, except for the daily Microsoft forecasts. For the calm period, we find that the forecasts provided by different MIDAS specifications are statistically comparable to those issued from the HAR-RV and HAR-RV-J models.

The GAS model provides comparable forecasts only in the case of S&P 500 for a forecasting horizon of two weeks. During the crisis, the best forecasts are generally provided by the MIDAS models with 1-or 5-minute filtered predictors, and the cluster of superior forecasting models no longer includes the HAR-RV and HAR-RV-J models. For longer horizons, the GAS and the GARCH provide similar results to those obtained with

13 T Qt ≡ mµ -3 4/3 m l=3 |r t,l | 4/3 |r t,l-1 | 4/3 |r t,l-2 | 4/3 , where µ 4/3 ≡ 2 2/3 Γ(7/6)Γ(1/2) -1 .
the MIDAS model. These findings confirm the intuition that high-frequency data can be used to successfully forecast volatility, provided that these data are filtered for periodicity and jumps. For these two assets, MIDAS models outperform in many cases standard variance models such as the GARCH model, or even the HAR-RV, HAR-RV-J or GAS models. Note: This table presents the MCS test results obtained for two assets (S&P 500 and Microsoft) during both calm and crisis periods. The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10). For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven MIDAS specifications with regressors sampled at a frequency ranging from one minute to about three hours, six MIDAS models with 1-and 5-minute regressors adjusted for intraday periodicity, jumps and/or microstructure noise, the HAR-RV, HAR-RV-J, GARCH and GAS models. For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test.

The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven log-MIDAS specifications with regressors sampled at a frequency ranging from one minute to about three hours, six log-MIDAS models with 1-and 5-minute regressors adjusted for intraday periodicity, jumps and/or microstructure noise, the log-HAR-RV, log-HAR-RV-J, GARCH and GAS models.

Another robustness check exercise consists in changing the measure of variance to be predicted. It is welldocumented that the realized variance estimator may become biased and inconsistent in the presence of market microstructure noise. A large number of alternative proxies of variance (e.g., realized bipower variation, realized kernel, etc.) that deal with issues such as jumps and other market microstructure noise, have consequently been introduced by Barndorff-Nielsen and Shephard (2004a), [START_REF] Zhang | Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach[END_REF], [START_REF] Barndorff-Nielsen | Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise[END_REF], [START_REF] Hansen | Quadratic variation by markov chains[END_REF], inter alios. To assess the robustness of our results, we also consider the realized kernel [START_REF] Barndorff-Nielsen | Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise[END_REF] as dependent variable in our specifications and obtain similar results (Appendix 6.4). Our results are also robust to the choice of the predictors of variance (absolute intradaily returns or intradaily bipower variation instead of squared intradaily returns). For a synthesis of all these extra findings see Appendix 6.5 and Appendix 6.6.

Conclusion

This paper analyses the forecasting performance of MIDAS-RV models in which future variances are directly related to past intraday log-returns. These predictors are usually constructed from tick-by-tick data and, consequently, the econometrician needs to choose a sampling frequency. The question we raise is whether ultra-high frequency data is needed to forecast variances.

The main findings of our study are the following. First, we show in a Monte Carlo simulation study that, in a world without jumps, periodicity in volatility and microstructure noise, there is an advantage in using the highest available frequency for the predictors. The information content of very high-frequency data improves significantly the quality of the MIDAS forecasts. Second, when considering two highly liquid assets (namely Microsoft and S&P 500) contaminated with typical market microstructure noise and intraday periodicity, we find that the use of very high-frequency predictors may become problematic. In particular, we show that there may exist a "high-frequency wall", i.e., a limit frequency above which the MIDAS forecasts may be less accurate. This result clearly illustrates the influence of the jumps and the intraday periodicity on the prediction of volatility, and not only on its measurement. Third, we discuss the potential solutions to combine the gains issued from high-frequency predictors and the negative impact of microstructure noise. A first solution consists in augmenting the MIDAS model by modifying the weighting scheme in order to limit the influence of the contaminated observations. A second solution consists in applying the MIDAS regression model on filtered data. Here, we adopt the latter solution and show that estimating MIDAS-RV models on filtered log-returns leads to significantly better out-of-sample forecasts. Finally, we compare the MIDAS model to other competing variance models including GARCH, GAS, HAR-RV and HAR-RV-J models. Results suggest that, for both assets, MIDAS models yield better forecasts in most cases and importantly never yield inferior forecasts, provided they are applied on filtered data.

A future research direction will be to compare the approach taken in this paper, where realized variance is directly related to past intraday data, as in [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF], with that of [START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] or [START_REF] Ghysels | Volatility forecasting and microstructure noise[END_REF], where daily realized measures (that are potentially robust to microstructure noise and jumps)

are introduced in a MIDAS-RV model.

Appendix C: HAR-RV versus MIDAS

In this appendix, we show that the HAR-RV model proposed by [START_REF] Corsi | A simple approximate long-memory model of realized volatility[END_REF] can be written as a weightconstrained form of the MIDAS model with regressors sampled at a frequency m 2 . The HAR-RV model is defined as:

RV (m2) t+1,t = α 0 + α 1 RV (m2) t,t-1 + α 2 RV (m2)w t,t-1 + α 3 RV (m2)m t,t-1 + ε t+1 , (21)
where RV t+1,t is the daily realized variance given by:

RV (m2) t+1,t = I m2 (L 1/m2 )r (m2)2 t+1,t+1-1/m2 , (22)
with

I m2 (L 1/m2 ) = m2-1 j=0 L j/m2
, and m 2 the sampling frequency of the squared returns used to compute the realized variance. By convention

RV (m2)w t,t-1 = 1 5 4 i=0 RV (m2) t-i,t-i-1 , (23)
and

RV (m2)m t,t-1 = 1 22 21 i=0 RV (m2) t-i,t-i-1 . (24)
To complete the explanation, we include Eq. ( 22), Eq. ( 23) and Eq. ( 24) into the definition of the model and obtain:

RV (m2) t+1,t = α 0 + (α 1 + 1 5 α 2 + 1 22 α 3 )RV (m2) t,t-1 + ( 1 5 α 2 + 1 22 α 3 ) 4 i=1 RV (m2) t-i,t-i-1 + 1 22 α 3 21 i=5 RV (m2) t-i,t-i-1 + ε t+1 = α 0 + (α 1 + 1 5 α 2 + 1 22 α 3 )I m2 (L 1/m2 )r (m2)2 t,t-1/m2 + ( 1 5 α 2 + 1 22 α 3 ) 4 i=1 I m2 (L 1/m2 )r (m2)2 t-i,t-i-1/m2 + 1 22 α 3 21 i=5 I m2 (L 1/m2 )r (m2)2 t-i,t-i-1/m2 + ε t+1 . (25)
Finally, the HAR-RV model takes the form of a daily MIDAS-RV model with squared return regressors sampled at a frequency m 2 :

RV (m2) t+1,t = α 0 + (α 1 + 1 5 α 2 + 1 22 α 3 ) m2-1 j=0 L j/m2 r (m2)2 t,t-1/m2 + ( 1 5 α 2 + 1 22 α 3 ) 4 i=1 m2-1 j=0 L j/m2 r (m2)2 t-i,t-i-1/m2 + 1 22 α 3 21 i=5 m2-1 j=0 L j/m2 r (m2)2 t-i,t-i-1/m2 + ε t+1 . (26)
6.4 Appendix D: MIDAS-RK Specification The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10). For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven MIDAS-RK specifications with regressors (squared return) sampled at a frequency ranging from one minute to about three hours, six MIDAS-RK models with 1and 5-minute regressors adjusted for intraday periodicity, jumps an/or microstructure noise, the HAR-RV, HAR-RV-J, GARCH and GAS models.

6.5 Appendix E: MIDAS with Absolute Return Regressors Note: This table presents the MCS test results obtained for S&P 500 and Microsoft during both calm and crisis periods. The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10). For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven MIDAS specifications with regressors (absolute return) sampled at a frequency ranging from one minute to about three hours, six MIDAS models with 1-and 5-minute regressors adjusted for intraday periodicity, jumps and/or microstructure noise, the HAR-RV, HAR-RV-J, GARCH and GAS models.

6.6 Appendix F: MIDAS with Bipower Variation Return Regressors Note: This table presents the MCS test results obtained for S&P 500 and Microsoft during both calm and crisis periods. The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10). For each of them, we present the average value of the QLIKE loss function along with the corresponding p-value resulting from the MCS test. The confidence level for the MCS test is set to α = 25% and 10,000 bootstrap resamples are used, with block length of five observations, to obtain the distribution under the null of equal predictive accuracy. The set of the competing variance models includes seven MIDAS specifications with regressors (bipower variation) sampled at a frequency ranging from one minute to about three hours, six MIDAS models with 1-and 5-minute regressors adjusted for intraday periodicity, jumps and/or microstructure noise, the HAR-RV, HAR-RV-J, GARCH and GAS models.

  b), with B(.) the Beta function defined as B(a, b) = Γ(a)Γ(b)/Γ(a + b), and Γ (.) representing the Gamma function. Depending on the value of the parameter θ 1 , this weight function can take many shapes, including flat weights, gradually declining weights, as well as hump-shaped patterns. The second parameter, θ 2 , determines the decreasing speed of the weighting shape. The smaller the parameter θ 2 , the smoother the weighting scheme. In other words, θ 2 determines the proportion of the total weight associated with the more recent past observations.

Figure

  Figure 1: Scaled weight function

Figure

  Figure 2: MIDAS average loss function

Figure

  Figure 3: 5-minute DGP: MIDAS-RV average loss function

Figure 4 :

 4 Figure 4: MIDAS-RV average MSE on 1-minute log-returns

Finally, the

  out-of-sample sample covers two years, i.e., 2007 and 2008. To test the robustness of the results upon the state of financial markets, the sample is split into two periods. The first one corresponds to the relatively calm variance period of 2007, and the second one to the financial crisis of 2008 (the end of this period corresponding to the peak of the crisis).

Figure

  Figure 5: Daily returns and realized variance

Figure

  Figure 6: Intraday periodicity

Figure 7 :

 7 Figure 7: Intraday returns and intraday jump-adjusted returns

Figure

  Figure 8: S&P 500 average QLIKE

Figure

  Figure 9: Microsoft average QLIKE

  S&P 500 -This figure displays the average QLIKE for the MIDAS-RV forecasts for various sampling frequencies (m1) of the predictors for three forecasting horizons.The left panel corresponds to the calm period (2007) and the right panel to the crisis period(2008). The solid blue line corresponds to the MIDAS-RV model with raw data sampled at frequencies 1-min to 3h15. The black, red and blue dotted lines correspond respectively to the MIDAS-RV models on periodicity, jumps and periodicity and noise, jumps and periodicity filtered log-returns sampled at frequencies of 1-min and 5-min.

Table 1 :

 1 Regression diagnostics and estimated weights of MIDAS models with intradaily regressors

	Frequency	µ	φ	θ 1	θ 2	Day 1 Days 2-5 Days 6-15 > 15 Days %Q(12) %Q 2 (12)	MSE
	1min	0.0280 372.3182 1.0702 66.6755 0.8830	0.1170	0	0	53.40	100	0.0231
	2min	0.0340 184.2469 1.0500 46.0871 0.7753	0.2244	0.0003	0	52.10	100	0.0252
	5min	0.0465 72.1241 1.0132 27.5914 0.6019	0.3914	0.0067	0	45.10	100	0.0295
	10min	0.0593 35.2540 1.0003 19.0019 0.4746	0.4940	0.0309	0.0005	100	100	0.0340
	15min	0.0688 23.1140 0.9904 15.3557 0.4100	0.5301	0.0580	0.0019	100	100	0.0372
	30min	0.0904 11.1019 0.9714 10.5584 0.3126	0.5471	0.1271	0.0132	100	100	0.0443
	1h05	0.1239	4.7946	0.9536 7.1060 0.2300	0.5099	0.2076	0.0524	100	100	0.0539
	2h10	0.1654	2.1969	0.9298 5.0132 0.1756	0.4477	0.2562	0.1205	100	100	0.0655
	3h15	0.1943	1.3731	0.9162 4.0938 0.1491	0.4060	0.2703	0.1745	100	100	0.0732
	Note: This table reports average values (over 10,000 replications) of the parameter estimates for the daily MIDAS-RV model (H = 1) with
	regressors sampled at one minute, two minutes, five minutes, 10 minutes, 15 minutes, 30 minutes,

Table 2 :

 2 Model Confidence Set test

	1min	0.0065	1	0.0108	1	0.0206	1
	2 min	0.0071	0.0012	0.0118	< 0.0001	0.0225	< 0.0001
	5min	0.0083	0.0001	0.0136	< 0.0001	0.0257	< 0.0001
	10min	0.0093	< 0.0001	0.0154	< 0.0001	0.0293	< 0.0001
	15min	0.0100	< 0.0001	0.0163	< 0.0001	0.0306	< 0.0001
	30min	0.0110	< 0.0001	0.0183	< 0.0001	0.0348	< 0.0001
	1h05	0.0144	< 0.0001	0.0236	< 0.0001	0.0441	< 0.0001
	2h10	0.0181	< 0.0001	0.0295	< 0.0001	0.0539	< 0.0001
	3h15	0.0199	< 0.0001	0.0328	< 0.0001	0.0601	< 0.0001

MSE (b = 0) b = -1 QLIKE (b = -2)

Frequency Av. loss MCS p-value Av. loss MCS p-value Av. loss MCS p-value Note: This table presents the Model Confidence Set (MCS) results obtained for three different loss functions, i.e., Eq. (

Table 3 :

 3 MIDAS sampling frequency puzzle

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2423	0.0871	0.2161	0.0177	0.2203	0.0621	0.0971	0.0829	0.2357	0.2016	0.1083	0.0062
	5min	0.2152	1.000	0.1369	1.0000 0.1904	1.0000 0.0823	0.9642 0.2061	0.7631 0.0861	0.1511
	10min 0.2203	0.3659 0.1412	0.6095 0.1968	0.3761 0.0878	0.4254 0.2013	1.0000 0.0950	0.1413
	15min 0.2265	0.0871	0.1447	0.5055 0.2060	0.0621	0.0881	0.4254 0.2245	0.2016	0.0948	0.0863
	30min 0.2152	0.9986 0.1407	0.6095 0.1920	0.8218 0.0833	0.9642 0.2140	0.6053 0.0874	0.1413
	1h05	0.2254	0.3659 0.1447	0.6095 0.2017	0.3761 0.0933	0.4254 0.2187	0.6053 0.0930	0.1413
	3h15	0.2713	0.0223	0.1449	0.6095 0.2471	0.0621	0.0815	1.0000 0.2518	0.2016	0.0732	1.0000
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.2185	0.1719	0.8640	0.1101	0.2337	0.1576	0.7399	0.1036	0.5308	0.0536	0.2231	0.1102
	5min	0.2033	1.0000 0.2891	0.1699	0.2182	1.0000 0.7152	0.1036	0.4144	0.1468	0.1914	1.0000
	10min 0.2166	0.1719	0.1848	1.0000 0.2340	0.1576	0.6791	0.1036	0.3245	0.4708 0.2839	0.0916
	15min 0.2172	0.1719	0.1960	0.6048 0.2306	0.1576	0.1820	1.0000 0.3146	1.0000 0.9670	0.0207
	30min 0.2317	0.0121	0.7694	0.1699	0.2662	0.0459	0.2397	0.1717	0.3408	0.1468	0.2670	0.0916
	1h05	0.2428	0.0121	0.2566	0.1699	0.3081	0.0459	0.2958	0.1036	0.3850	0.1468	0.5070	0.0916
	3h15	0.2468	0.1719	0.2612	0.1699	0.3437	0.0085	0.2485	0.1717	0.4352	0.0536	0.2213	0.3612

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value Note: This table presents the MCS test results for the S&P 500 and Microsoft.

Table 4 :

 4 MIDAS sampling frequency puzzle: intraday periodicity and jumps adjustments This table presents the MCS test results obtained for two assets (S&P 500 and Microsoft) during both calm and crisis periods.

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min RAW	0.2423	0.1328	0.2161	0.0076	0.2203	0.0905	0.0971	0.1130	0.2357	0.1235	0.1083	0.0104
	1min Per.Adj	0.2320	0.1328	0.1521	0.0563	0.2078	0.4787 0.0993	0.0825	0.2228	0.6122 0.1104	0.0096
	1min Jumps.Adj 0.2141	0.9970 0.1631	0.0076	0.1823	1.0000 0.1105	0.0066	0.2053	0.9283 0.1196	0.0014
	5min RAW	0.2152	0.9473 0.1369	1.0000 0.1904	0.7622 0.0823	0.9899 0.2061	0.9283 0.0861	0.2238
	5min Per.Adj	0.2138	1.0000 0.1399	0.7417 0.1884	0.8159 0.0831	0.9899 0.2063	0.9283 0.0873	0.2238
	5min Jumps.Adj 0.2152	0.9970 0.1380	0.7799 0.1850	0.8159 0.0830	0.9899 0.2029	0.9314 0.0888	0.2127
	10min	0.2203	0.3405 0.1412	0.7417 0.1968	0.4787 0.0878	0.5294 0.2013	1.0000 0.0950	0.2055
	15min	0.2265	0.1328	0.1447	0.6101 0.2060	0.0905	0.0881	0.5294 0.2245	0.2513 0.0948	0.1162
	30min	0.2152	0.9970 0.1407	0.7453 0.1920	0.8159 0.0833	0.9899 0.2140	0.8533 0.0874	0.2127
	1h05	0.2254	0.3405 0.1447	0.7417 0.2017	0.5530 0.0933	0.5294 0.2187	0.6568 0.0930	0.2127
	3h15	0.2713	0.0354	0.1449	0.7417 0.2471	0.0905	0.0815	1.0000 0.2518	0.1235	0.0732	1.0000
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.2185	0.0193	0.8640	0.0157	0.2337	0.0350	0.7399	0.0094	0.5308	0.0764	0.2231	0.0526
	1min Per.Adj	0.2112	0.0347	0.1635	0.1214	0.2227	0.3494 0.1465	0.1444	0.2975	0.0878	0.1679	1.0000
	1min Jumps.Adj 0.2042	0.0347	0.1587	1.0000 0.2026	0.4552 0.1362	1.0000 0.3172	0.0878	0.1758	0.6448
	5min	0.2033	0.0347	0.2891	0.0157	0.2182	0.3494 0.7152	0.0094	0.4144	0.0878	0.1914	0.2207
	5min Per.Adj	0.2008	0.0347	0.1673	0.0986	0.2142	0.4552 0.1534	0.1444	0.2737	1.0000 0.4426	0.0116
	5min Jumps.Adj 0.1842	1.0000 0.1598	0.8784 0.1956	1.0000 0.1528	0.1444	0.3073	0.0878	0.1735	0.6448
	10min	0.2166	0.0193	0.1848	0.0157	0.2340	0.0350	0.6791	0.0094	0.3245	0.0878	0.2839	0.0212
	15min	0.2172	0.0347	0.1960	0.0986	0.2306	0.0350	0.1820	0.0276	0.3146	0.0878	0.9670	0.0116
	30min	0.2317	0.0160	0.7694	0.0157	0.2662	0.0217	0.2397	0.0094	0.3408	0.0878	0.2670	0.0116
	1h05	0.2428	0.0160	0.2566	0.0157	0.3081	0.0217	0.2958	0.0094	0.3850	0.0878	0.5070	0.0116
	3h15	0.2468	0.0193	0.2612	0.0157	0.3437	0.0040	0.2485	0.0184	0.4352	0.0323	0.2213	0.2207
	Note:

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

Table 5 :

 5 Comparing competing variance models

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2423	0.1433	0.2161	0.0144	0.2203	0.1316	0.0971	0.1537	0.2357	0.1831	0.1083	0.0124
	1min Per.Adj	0.2320	0.1433	0.1521	0.0717	0.2078	0.6353 0.0993	0.1175	0.2228	0.6942 0.1104	0.0026
	1min Jumps.Adj	0.2141	0.9965 0.1631	0.0144	0.1823	1.0000 0.1105	0.0075	0.2053	0.9209 0.1196	0.0021
	1min Jumps.Adj-Preav 0.2155	0.1433	0.1560	0.0144	0.1900	0.7911 0.1000	0.0075	0.4243	0.1831	0.1054	0.0026
	5min	0.2152	0.9765 0.1369	0.7653 0.1904	0.7911 0.0823	0.9894 0.2061	0.9209 0.0861	0.2322
	5min Per.Adj	0.2138	1.0000 0.1399	0.7653 0.1884	0.8272 0.0831	0.9894 0.2063	0.9209 0.0873	0.2322
	5min Jumps.Adj	0.2152	0.9765 0.1380	0.7653 0.1850	0.8272 0.0830	0.9894 0.2029	0.9332 0.0888	0.2322
	5min Jumps.Adj-Preav 0.2149	0.9965 0.1560	0.0240	0.1899	0.7911 0.1001	0.0075	0.4286	0.1831	0.1052	0.0026
	10min	0.2203	0.1433	0.1412	0.7653 0.1968	0.6353 0.0878	0.5385 0.2013	1.0000 0.0950	0.2131
	15min	0.2265	0.1433	0.1447	0.2921 0.2060	0.1316	0.0881	0.4830 0.2245	0.3187 0.0948	0.1386
	30min	0.2152	0.9965 0.1407	0.7653 0.1920	0.8272 0.0833	0.9894 0.2140	0.8513 0.0874	0.2322
	1h05	0.2254	0.1433	0.1447	0.6464 0.2017	0.7046 0.0933	0.4830 0.2187	0.7082 0.0930	0.2322
	3h15	0.2713	0.0510	0.1449	0.7653 0.2471	0.1316	0.0815	1.0000 0.2518	0.1831	0.0732	1.0000
	HAR-RV	0.2176	0.1433	0.1345	1.0000 0.1941	0.7601 0.0868	0.4830 0.2172	0.7082 0.0943	0.1852
	HAR-RV-J	0.2187	0.1433	0.1359	0.7653 0.1973	0.7046 0.0883	0.4830 0.2226	0.4483 0.0960	0.1212
	GARCH	0.3240	0.0510	0.2208	0.0144	0.2849	0.1316	0.1677	0.0075	0.2935	0.1831	0.1747	0.0021
	GAS	0.3161	0.0510	0.1884	0.0144	0.2669	0.1316	0.1292	0.1175	0.2688	0.3187 0.1296	0.0564
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.2185	0.0296	0.8640	0.0246	0.2337	0.0501	0.7399	0.0141	0.5308	0.0250	0.2231	0.0899
	1min Per.Adj	0.2112	0.0713	0.1635	0.1816	0.2227	0.4943 0.1465	0.1426	0.2975	0.1171	0.1679	1.0000
	1min Jumps.Adj	0.2042	0.0713	0.1587	1.0000 0.2026	0.6264 0.1362	1.0000 0.3172	0.1171	0.1758	0.6378
	1min Jumps.Adj-Preav 0.2430	0.0296	0.1698	0.1816	0.2596	0.0269	0.1725	0.0309	0.3517	0.1171	0.1918	0.4482
	5min	0.2033	0.0713	0.2891	0.0246	0.2182	0.4943 0.7152	0.0141	0.4144	0.1171	0.1914	0.2987
	5min Per.Adj	0.2008	0.0713	0.1673	0.1429	0.2142	0.6264 0.1534	0.1426	0.2737	0.3803 0.4426	0.0214
	5min Jumps.Adj	0.1842	1.0000 0.1598	0.8742 0.1956	1.0000 0.1528	0.1426	0.3073	0.1171	0.1735	0.6378
	5min Jumps.Adj-Preav 0.2418	0.0296	0.1780	0.0330	0.2627	0.0269	0.1757	0.0237	0.3502	0.1171	0.1943	0.2987
	10min	0.2166	0.0296	0.1848	0.0246	0.2340	0.0501	0.6791	0.0141	0.3245	0.1171	0.2839	0.0214
	15min	0.2172	0.0713	0.1960	0.0330	0.2306	0.0501	0.1820	0.0309	0.3146	0.1171	0.9670	0.0214
	30min	0.2317	0.0296	0.7694	0.0246	0.2662	0.0269	0.2397	0.0141	0.3408	0.1171	0.2670	0.0214
	1h05	0.2428	0.0296	0.2566	0.0246	0.3081	0.0269	0.2958	0.0141	0.3850	0.0305	0.5070	0.0214
	3h15	0.2468	0.0296	0.2612	0.0246	0.3437	0.0063	0.2485	0.0211	0.4352	0.0250	0.2213	0.2987
	HAR-RV	0.2102	0.0713	0.1781	0.0330	0.2449	0.0501	0.1754	0.0211	0.3563	0.0305	0.2346	0.0214
	HAR-RV-J	0.2148	0.0713	0.1719	0.1429	0.2472	0.0501	0.1669	0.0675	0.3569	0.0250	0.2205	0.0330
	GARCH	0.2291	0.0713	0.2205	0.0246	0.2282	0.4943 0.2041	0.0211	0.2723	0.2751 0.2208	0.2987
	GAS	0.2363	0.0296	0.2447	0.0246	0.2173	0.6264 0.2142	0.0211	0.2348	1.0000 0.2226	0.2987

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

Table 6 :

 6 Log version -MCS TestThis table presents the MCS test results obtained for the two assets under analysis (S&P 500 and Microsoft) during both calm and crisis periods. The results are reported for three forecasting horizons, namely one day (H = 1), one week (H = 5) and two weeks (H = 10).

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2406	0.4797 0.1553	0.0850	0.2299	0.4750 0.0998	0.0826	0.2454	0.0388	0.1136	0.0146
	1min Per.Adj	0.2350	0.8721 0.1546	0.0850	0.2294	0.4750 0.0997	0.3562 0.2057	1.0000 0.1253	0.0065
	1min Jumps.Adj	0.2395	0.3885 0.1660	0.0850	0.2118	0.8000 0.1262	0.0019	0.2152	0.8574 0.1263	0.0005
	1min Jumps.Adj-Preav 0.2295	0.6477 0.1656	0.0850	0.2174	0.4805 0.1126	0.0633	1.0253	0.0011	0.1145	0.0146
	5min	0.2251	0.8721 0.1403	0.9503 0.1987	0.8000 0.0862	0.8052 0.2149	0.9264 0.0915	0.5738
	5min Per.Adj	0.2252	0.8721 0.1419	0.9151 0.2051	0.8000 0.0913	0.5983 0.2178	0.8574 0.0924	0.5738
	5min Jumps.Adj	0.2246	0.8721 0.1409	0.9503 0.2097	0.6152 0.0902	0.5983 0.2079	0.9822 0.0948	0.4988
	5min Jumps.Adj-Preav 0.2105	1.0000 0.1517	0.0850	0.1931	1.0000 0.1022	0.0826 0.2305	0.8574 0.1056	0.0230
	10min	0.2252	0.8721 0.1432	0.9151 0.2116	0.4805 0.0902	0.5983 0.2294	0.5842 0.0905	0.5738
	15min	0.2324	0.4797 0.1451	0.6570 0.2180	0.4750 0.0893	0.5579 0.2289	0.5842 0.0906	0.4988
	30min	0.2198	0.8721 0.1369	1.0000 0.2158	0.6152 0.0811	0.8052 0.2118	0.9822 0.0834	0.7378
	1h05	0.2348	0.5216 0.1420	0.9503 0.2193	0.4805 0.0837	0.8052 0.2104	0.9822 0.0800	0.8746
	3h15	0.2644	0.0522	0.1508	0.6570 0.2498	0.2066	0.0773	1.0000 0.2523	0.0388	0.0785	1.0000
	HAR-RV	0.2444	0.0959	0.1528	0.0850	0.2622	0.0112	0.1179	0.0007	0.3308	0.0011	0.1425	0.0004
	HAR-RV-J	0.2205	0.8721 0.1386	0.9503 0.2161	0.4805 0.0922	0.5579 0.2615	0.0338	0.1056	0.0146
	GARCH	0.3240	0.0522	0.2208	0.0631	0.2849	0.2066	0.1677	0.0007	0.2935	0.0388	0.1747	0.0005
	GAS	0.3161	0.0522	0.1884	0.0850	0.2669	0.4750 0.1292	0.0826	0.2688	0.5842 0.1296	0.0230
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.3628	0.0033	0.1881	0.0894	0.5074	0.0019	0.2244	0.0154	0.6579	0.0016	0.3851	0.0049
	1min Per.Adj	0.3553	0.0033	0.2575	0.0282	0.5223	0.0019	0.2093	0.0154	0.4240	0.1012	0.4045	0.0040
	1min Jumps.Adj	0.3369	0.0039	0.2407	0.0282	0.4799	0.0133	0.3244	0.0154	0.6406	0.0024	0.4123	0.0049
	1min Jumps.Adj-Preav 0.2926	0.0048	0.2369	0.0277	0.4875	0.0133	0.3434	0.0154	0.6936	0.0024	0.4136	0.0049
	5min	0.2530	0.0048	0.2085	0.0768	0.3928	0.0133	0.2029	0.0154	0.5136	0.0024	0.1682	1.0000
	5min Per.Adj	0.2544	0.0048	0.1712	0.0894	0.4154	0.0133	0.2683	0.0154	0.5072	0.0024	0.1737	0.3601
	5min Jumps.Adj	0.2083	0.0332	0.2011	0.0894	0.3872	0.0133	0.1986	0.0154	0.5550	0.0024	0.3325	0.0049
	5min Jumps.Adj-Preav 0.1514	1.0000 0.1471	1.0000 0.2404	0.5878 0.2175	0.0154	0.3625	0.1012	0.2851	0.1817
	10min	0.2323	0.0332	0.1699	0.0894	0.2952	0.0133	0.1624	0.0643	0.4903	0.0024	0.3197	0.0049
	15min	0.1930	0.0332	0.1591	0.3847 0.2065	0.9080	0.1946	0.0154	0.4679	0.0498	0.1820	0.2623
	30min	0.1926	0.0332	0.1725	0.0894	0.2992	0.0133	0.2407	0.0154	0.2814	0.3350 0.2797	0.1283
	1h05	0.1857	0.0332	0.2122	0.0282	0.2854	0.1674	0.2594	0.0154	0.3575	0.1012	0.2628	0.0950
	3h15	0.1993	0.0332	0.2173	0.0405	0.2552	0.0133	0.2300	0.0154	0.3809	0.0778	0.2717	0.1649
	HAR-RV	0.1748	0.1448	0.1662	0.0894	0.2649	0.0133	0.1821	0.0154	0.4382	0.0024	0.2644	0.0049
	HAR-RV-J	0.1667	0.2256	0.1497	0.8191 0.2039	1.0000 0.1426	1.0000 0.3294	0.1012	0.1965	0.1817
	GARCH	0.2291	0.0048	0.2205	0.0894	0.2282	0.5878 0.2041	0.0154	0.2723	0.2243	0.2208	0.1817
	GAS	0.2363	0.0332	0.2447	0.0282	0.2173	0.9080 0.2142	0.0154	0.2348	1.0000 0.2226	0.1817
	Note:

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

Table 7 :

 7 MIDAS-RK specification

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2488	0.1245	0.2008	0.1365	0.2165	0.1919	0.1096	0.0501	0.2306	0.1782	0.1132	0.0050
	1min Per.Adj	0.2403	0.1245	0.2029	0.0953	0.2039	0.5378 0.1115	0.0471	0.2199	0.2961 0.1142	0.0050
	1min Jumps.Adj	0.2211	0.8981 0.2164	0.0159	0.1738	1.0000 0.1244	0.0027	0.1960	0.7260 0.1302	0.0005
	1min Jumps.Adj-Preav 0.2209	0.1245	0.1993	0.1365	0.1792	0.6902 0.1087	0.0467	0.4065	0.1782	0.1086	0.0050
	5min	0.2204	0.8981 0.1841	0.5713 0.1836	0.6902 0.0914	0.9376 0.2044	0.4319 0.0903	0.2666
	5min Per.Adj	0.2192	1.0000 0.1886	0.3179 0.1810	0.6902 0.0929	0.9365 0.1998	0.6095 0.0900	0.2666
	5min Jumps.Adj	0.2231	0.1245	0.1870	0.3179 0.1771	0.7363 0.0929	0.9376 0.1944	1.0000 0.0921	0.2666
	5min Jumps.Adj-Preav 0.2200	0.9341 0.1993	0.1365	0.1787	0.7363 0.1090	0.0396	0.4063	0.1782	0.1084	0.0050
	10min	0.2254	0.1245	0.1888	0.3179 0.1905	0.5378 0.0993	0.4091 0.2127	0.1782	0.0980	0.1356
	15min	0.2313	0.1245	0.1899	0.3179 0.1988	0.1973	0.0967	0.6333 0.2194	0.2961 0.0953	0.1356
	30min	0.2228	0.1245	0.1852	0.3179 0.1866	0.6902 0.0928	0.9376 0.2079	0.6095 0.0912	0.2666
	1h05	0.2339	0.1245	0.1900	0.3179 0.1946	0.5940 0.1027	0.4091 0.2115	0.4319 0.0963	0.2666
	3h15	0.2766	0.0795	0.1857	0.3179 0.2366	0.1919	0.0883	1.0000 0.2417	0.1782	0.0762	1.0000
	HAR-RV	0.2242	0.1245	0.1809	1.0000 0.1916	0.5501 0.0979	0.4903 0.2127	0.2961 0.0982	0.1356
	HAR-RV-J	0.2292	0.1245	0.1837	0.3179 0.1953	0.5378 0.1007	0.1164	0.2178	0.1782	0.1030	0.0249
	GARCH	0.3195	0.0795	0.2203	0.0953	0.2663	0.1919	0.1265	0.0501	0.2708	0.1782	0.1236	0.0249
	GAS	0.3102	0.0795	0.2026	0.3179 0.2466	0.1919	0.1043	0.6333 0.2451	0.1782	0.0954	0.2666
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2342	0.0145	0.3377	0.0520	0.2458	0.0834	1.1943	0.0039	0.3280	0.0649	0.3076	0.0044
	1min Per.Adj	0.2255	0.0318	0.4581	0.0520	0.2353	0.4954 0.5386	0.0215	0.3120	0.0844	0.4533	0.0044
	1min Jumps.Adj	0.2119	0.0318	0.1886	0.8930 0.2155	0.5462 0.1502	1.0000 0.3344	0.0844	0.1797	1.0000
	1min Jumps.Adj-Preav 0.3830	0.0087	0.1976	0.2830 0.2973	0.0021	0.1858	0.0633	0.4030	0.0649	0.1991	0.4525
	5min	0.2193	0.0318	0.1976	0.1087	0.2301	0.4954 0.1963	0.0633	0.2995	0.0844	0.9667	0.0037
	5min Per.Adj	0.2150	0.0318	0.1932	0.2830 0.2280	0.5462 0.1646	0.1684	0.2873	0.3356 0.4717	0.0044
	5min Jumps.Adj	0.1805	1.0000 0.1878	1.0000 0.2065	1.0000 0.1649	0.1378	0.3243	0.0844	0.1805	0.9304
	5min Jumps.Adj-Preav 0.2587	0.0087	0.2083	0.0520	0.3011	0.0021	0.1880	0.0524	0.4033	0.0649	0.2012	0.3479
	10min	0.2306	0.0145	0.2083	0.0520	0.2423	0.0834	0.3447	0.0039	0.3428	0.0649	0.3525	0.0037
	15min	0.2319	0.0145	0.3678	0.0520	0.2448	0.0834	0.7383	0.0039	0.3320	0.0649	0.5074	0.0037
	30min	0.2491	0.0087	0.2585	0.0520	0.2887	0.0175	0.2169	0.0215	0.3600	0.0649	0.2417	0.0213
	1h05	0.2606	0.0087	0.2300	0.0520	0.3137	0.0364	0.6801	0.0039	0.4043	0.0649	0.3155	0.0037
	3h15	0.2635	0.0145	0.2847	0.0520	0.3670	0.0021	0.2551	0.0215	0.4520	0.0371	0.2275	0.3479
	HAR-RV	0.2355	0.0318	0.2022	0.1087	0.2735	0.0834	0.1890	0.0524	0.3950	0.0403	0.2519	0.0044
	HAR-RV-J	0.2418	0.0145	0.1968	0.2830 0.2742	0.0829	0.1818	0.0633	0.3943	0.0371	0.2397	0.0044
	GARCH	0.2445	0.0318	0.2285	0.0520	0.2385	0.4954 0.1907	0.1378	0.2772	0.3356	0.1976	0.7575
	GAS	0.2533	0.0145	0.2589	0.0520	0.2293	0.5462 0.2112	0.0524	0.2426	1.0000 0.2126	0.3479

Note: This table presents the MCS test results obtained for S&P 500 and Microsoft during both calm and crisis periods.

Table 8 :

 8 MIDAS with absolute return regressors

	Panel A: Calm period (2007)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
		QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value
	1min	0.2307	0.8883 0.1551	0.1123	0.1998	0.7409 0.0969	0.0549	0.2202	0.8202 0.1062	0.0059
	1min Per.Adj	0.2329	0.8883 0.1559	0.1002	0.1998	0.7409 0.0975	0.0490	0.2197	0.8965 0.1074	0.0022
	1min Jumps.Adj	0.2386	0.5373 0.1621	0.0597	0.2020	0.7409 0.1035	0.0084	0.2284	0.5357 0.1126	0.0019
	1min Jumps.Adj-Preav 0.2153	1.0000 0.1576	0.0186	0.2447	0.5811 0.0988	0.0084	0.3901	0.5357 0.1037	0.0022
	5min	0.2226	0.8883 0.1387	0.5411 0.2043	0.7409 0.0825	0.9210 0.2258	0.5357 0.0870	0.3491
	5min Per.Adj	0.2232	0.8883 0.1408	0.5411 0.2045	0.7409 0.0829	0.9210 0.2259	0.5357 0.0875	0.3491
	5min Jumps.Adj	0.2289	0.5373 0.1408	0.5411 0.2056	0.7409 0.0837	0.8663 0.2279	0.5357 0.0890	0.3491
	5min Jumps.Adj-Preav 0.2160	0.8883 0.1574	0.0186	0.2368	0.7007 0.0988	0.0084	0.3909	0.1500 0.1034	0.0026
	10min	0.2244	0.8883 0.1422	0.5411 0.2093	0.7409 0.0854	0.8663 0.2307	0.5357 0.0888	0.3491
	15min	0.2311	0.4120 0.1457	0.5411 0.2168	0.7007 0.0865	0.8663 0.2271	0.5357 0.0890	0.3491
	30min	0.2235	0.8883 0.1450	0.5411 0.2108	0.7409 0.0839	0.9210 0.2296	0.5357 0.0881	0.3491
	1h05	0.2383	0.4120 0.1440	0.5411 0.2204	0.7409 0.0882	0.8663 0.2314	0.5357 0.0911	0.3491
	3h15	0.2957	0.1030	0.1455	0.5411 0.2715	0.4483 0.0813	1.0000 0.2745	0.5357 0.0755	1.0000
	HAR-RV	0.2176	0.8883 0.1345	1.0000 0.1941	1.0000 0.0868	0.8663 0.2172	1.0000 0.0943	0.3491
	HAR-RV-J	0.2187	0.8883 0.1359	0.5411 0.1973	0.7409 0.0883	0.8663 0.2226	0.5357 0.0960	0.3491
	GARCH	0.3240	0.1937	0.2208	0.0186	0.2849	0.5811 0.1677	0.0084	0.2935	0.5357 0.1747	0.0019
	GAS	0.3161	0.1682	0.1884	0.0186	0.2669	0.6078 0.1292	0.0084	0.2688	0.5357 0.1296	0.0530
	Panel B: Crisis period (2008)
				H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.2047	0.5243 0.1606	0.5608 0.2195	0.4701 0.1391	0.4253 0.2789	0.5124 0.4582	0.0198
	1min Per.Adj	0.2048	0.5243 0.1612	0.5608 0.2176	0.4701 0.1386	0.4253 0.2769	0.5124 0.4573	0.0198
	1min Jumps.Adj	0.2096	0.5243 0.1595	1.0000 0.2228	0.4701 0.1373	1.0000 0.2866	0.5124 0.4128	0.0198
	1min Jumps.Adj-Preav 0.2069	0.5243 0.1757	0.4526 0.2317	0.4701 0.1796	0.0161	0.3016	0.5124 0.1920	0.0995
	5min	0.1994	1.0000 0.1643	0.5608 0.2165	0.4701 0.1491	0.3714 0.2714	0.6234 0.4339	0.0198
	5min Per.Adj	0.2001	0.5243 0.1660	0.5608 0.2150	1.0000 0.1473	0.4107 0.2706	0.6234 0.1638	1.0000
	5min Jumps.Adj	0.2033	0.5243 0.1656	0.5608 0.2176	0.4701 0.1472	0.4107 0.2789	0.5124 0.1646	0.7402
	5min Jumps.Adj-Preav 0.2160	0.5243 0.1788	0.1468 0.2323	0.4701 0.1814	0.0161	0.3012	0.5124 0.1941	0.0995
	10min	0.2087	0.5243 0.1782	0.1468 0.2351	0.4701 0.1724	0.0161	0.2948	0.5124 0.2268	0.0198
	15min	0.2145	0.5243 0.1701	0.5608 0.2235	0.4701 0.1578	0.1304	0.2904	0.5124 0.7828	0.0198
	30min	0.2165	0.5243 0.1962	0.1383	0.2352	0.4442 0.2109	0.0161	0.3041	0.2871 0.9052	0.0198
	1h05	0.2259	0.5243 0.2175	0.1383	0.2637	0.3085 0.2418	0.0161	0.3227	0.2871 0.2734	0.0198
	3h15	0.2604	0.0181	0.2554	0.0664	0.2836	0.0107	0.2449	0.0161	0.3364	0.0275	0.2334	0.0198
	HAR-RV	0.2102	0.5243 0.1781	0.4526 0.2449	0.4701 0.1754	0.0161	0.3563	0.2871 0.2346	0.0198
	HAR-RV-J	0.2148	0.5243 0.1719	0.5608 0.2472	0.4442 0.1669	0.3714 0.3569	0.2871 0.2205	0.0995
	GARCH	0.2291	0.5243 0.2205	0.1468	0.2282	0.4701 0.2041	0.0161	0.2723	0.5124 0.2208	0.0995
	GAS	0.2363	0.5243 0.2447	0.1383	0.2173	0.4701 0.2142	0.0161	0.2348	1.0000 0.2226	0.0995

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

Table 9 :

 9 MIDAS with bipower variation return regressors QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

	Panel A: Calm period (2007)
			H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	QLIKE p-value 1min 0.2506 0.1758	0.2468	0.0136	0.2298	0.1230	0.0972	0.0648	0.2417	0.2713 0.1089	0.0080
	1min Per.Adj	0.2391	0.2355	0.1534	0.0480	0.2158	0.6271 0.0994	0.0648	0.2307	0.4880 0.1112	0.0067
	1min Jumps.Adj	0.2127	0.9420 0.1589	0.0136	0.1816	1.0000 0.1047	0.0193	0.2062	1.0000 0.1149	0.0019
	1min Jumps.Adj-Preav 0.2155	0.2355	0.1560	0.0136	0.1900	0.8235 0.1000	0.0193	0.4248	0.2713 0.1053	0.0191
	5min	0.2089	1.0000 0.1363	0.7438 0.1890	0.8235 0.0803	0.7465 0.2095	0.9527 0.0860	0.5852
	5min Per.Adj	0.2097	0.9420 0.1379	0.7438 0.1887	0.8235 0.0796	1.0000 0.2086	0.9527 0.0849	0.5852
	5min Jumps.Adj	0.2145	0.8431 0.1386	0.7438 0.1885	0.8235 0.0815	0.7465 0.2085	0.9527 0.0881	0.2409
	5min Jumps.Adj-Preav 0.2137	0.9420 0.1566	0.0136	0.1891	0.8235 0.1003	0.0438	0.4424	0.2683 0.1046	0.0191
	10min	0.2171	0.2355	0.1397	0.7438 0.1987	0.7649 0.0888	0.1841	0.2161	0.6856 0.0934	0.1692
	15min	0.2212	0.2355	0.1507	0.0756	0.2016	0.7649 0.0928	0.0648	0.2230	0.6495 0.0969	0.1692
	30min	0.2200	0.2355	0.1499	0.2492	0.1970	0.8235 0.0916	0.0648	0.2302	0.4880 0.0957	0.1692
	1h05	0.2625	0.2355	0.1456	0.5857 0.2101	0.6271 0.0964	0.0648	0.2229	0.6856 0.1037	0.0191
	3h15	0.3410	0.0163	0.1502	0.2922 0.3065	0.1230	0.0833	0.7465 0.2993	0.2713 0.0766	1.0000
	HAR-RV	0.2176	0.2355	0.1345	1.0000 0.1941	0.8235 0.0868	0.0687	0.2172	0.7592 0.0943	0.1230
	HAR-RV-J	0.2187	0.2355	0.1359	0.7438 0.1973	0.7649 0.0883	0.0648	0.2226	0.4880 0.0960	0.0191
	GARCH	0.3240	0.1516	0.2208	0.0136	0.2849	0.1230	0.1677	0.0193	0.2935	0.2713 0.1747	0.0019
	GAS	0.3161	0.1419	0.1884	0.0136	0.2669	0.1230	0.1292	0.0193	0.2688	0.2713 0.1296	0.0191
	Panel B: Crisis period (2008)
			H=1				H=5			H=10
		S&P 500	MSFT	S&P 500	MSFT	S&P 500	MSFT
	1min	0.2261	0.0304	0.8523	0.0042	0.2428	0.0102	0.7263	0.0029	0.3122	0.1654	1.2599	0.0095
	1min Per.Adj	0.2200	0.0304	0.1990	0.1150	0.2330	0.0477	0.5763	0.0178	0.3000	0.5006 0.1664	1.0000
	1min Jumps.Adj	0.2095	0.0317	0.1605	1.0000 0.2031	0.5510 0.1343	1.0000 0.3207	0.5006 0.1748	0.5822
	1min Jumps.Adj-Preav 0.2431	0.0304	0.1717	0.3030 0.2609	0.0102	0.1741	0.0178	0.3524	0.1654	0.1927	0.3582
	5min	0.4821	0.0304	0.1722	0.1150	0.2314	0.4275 0.1586	0.0537	0.8549	0.0355	0.2625	0.0216
	5min Per.Adj	0.1858	0.0317	0.1663	0.3030 0.2025	0.5510 0.1508	0.1685	0.2866	0.5006 0.1702	0.6634
	5min Jumps.Adj	0.1747	1.0000 0.1620	0.8312 0.1937	1.0000 0.1519	0.1228	0.3097	0.5006 0.1721	0.6260
	5min Jumps.Adj-Preav 0.2465	0.0304	0.1927	0.1150	0.2712	0.0102	0.1848	0.0178	0.3625	0.1654	0.2002	0.1221
	10min	0.2233	0.0304	0.2059	0.0042	0.2293	0.1029	0.3507	0.0029	0.3277	0.1654	0.6099	0.0143
	15min	0.2139	0.0304	0.1743	0.3030 0.2249	0.1029	0.2972	0.0178	0.5412	0.0355	0.3097	0.0143
	30min	0.1937	0.0317	0.2547	0.0042	0.2454	0.0102	0.2603	0.0131	0.8991	0.0355	0.2655	0.0143
	1h05	0.2339	0.0304	0.2638	0.0042	0.2985	0.0102	0.3464	0.0029	0.3637	0.1654	1.0577	0.0143
	3h15	0.2937	0.0304	0.2880	0.0042	0.2843	0.0102	0.2497	0.0058	0.9281	0.0355	0.2327	0.0897
	HAR-RV	0.2102	0.0317	0.1781	0.1150	0.2449	0.0477	0.1754	0.0178	0.3563	0.0364	0.2346	0.0143
	HAR-RV-J	0.2148	0.0317	0.1719	0.3030 0.2472	0.0477	0.1669	0.0537	0.3569	0.0355	0.2205	0.0216
	GARCH	0.2291	0.0317	0.2205	0.1150	0.2282	0.4275 0.2041	0.0178	0.2723	0.5006 0.2208	0.1221
	GAS	0.2363	0.0304	0.2447	0.1150	0.2173	0.5510 0.2142	0.0178	0.2348	1.0000 0.2226	0.1221

QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value QLIKE p-value

In this study, we limit our analysis to the MIDAS specifications in which the future variance is directly related to past observations of intraday predictors, as in[START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF]. An alternative consists in using high-frequency data to compute daily realized measures (realized variance, two-scale estimator, realized kernel, etc.) which are introduced, in a second step, into a MIDAS regression model, as in[START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] and[START_REF] Ghysels | Volatility forecasting and microstructure noise[END_REF]. This choice will be discussed in Section 5.

Surprisingly,[START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] found that the forecasts directly using high-frequency data do not outperform those based on daily regressors (although the daily regressors are themselves obtained through the aggregation of high-frequency data). One related question is to understand whether this result depends on the sampling frequency of the high-frequency data.

[START_REF] Meddahi | Aggregations and marginalization of GARCH and stochastic volatility models[END_REF] show that the strong GARCH setting does not have a closed form with respect to temporal and contemporaneous aggregations.

For more details about the estimation procedure, see[START_REF] Ghysels | The MIDAS touch: Mixed data sampling regression models[END_REF].

However,[START_REF] Ghysels | Predicting volatility: getting the most out of return data sampled at different frequencies[END_REF] find no significant autocorrelation for longer forecasting horizons (from one week to four weeks). We obtain similar results in our simulations (not reported).

We set the significance level to α = 25% and use

10,000 bootsrap resamples (with block length of five observations) to obtain the distribution under the null of equal empirical fit. These results are available under request.10 Notice that the results obtained with the integrated variance (not reported) are qualitatively the same than those obtained with the realized variance.11 The robustness of the forecasts ranking has also an impact on the statistical inference used to asses the predictive accuracy. If the loss function ensures consistency of the ranking, the variability of the variance proxy is only likely to reduce the power of the test, but not its asymptotic size, which means that for a robust loss function it is always possible to recover asymptotically the true ranking. For more details, see[START_REF] Laurent | Consistent ranking of multivariate volatility models[END_REF].

At the end of August 2011, 2,982 ETFs worldwide were managing USD 1,348 bn, which represents 5.6% of the assets in the fund management industry. Additionally, the total ETF turnover on-exchange via the electronic order book was 8.5% of the equity turnover[START_REF] Fuhr | ETF landscape. Industry highlights[END_REF].

Robustness Check

In this section we examine the robustness of our results. The most frequent criticism on both MIDAS and HAR-RV models concerns the absence of some constraints ensuring the positivity of the variance process. A straightforward solution consists in predicting the logarithm of the variance proxy, i.e., Log-MIDAS:

Log-HAR-RV:

Log-HAR-RV-J:

To forecast the logarithm of the realized measure of volatility, we follow exactly the same procedure as for the level of variance. Next, in order to compare the log-variance forecasts with the level of variance proxy, the following transformation is required [START_REF] Andersen | Modeling and forecasting realized volatility[END_REF]:

where log( RV t+H,t ) is the forecast of the log of realized volatility and V ar(e t+H,t ) is the variance of the forecasting errors.

The results of the MCS-based comparison procedure are reported in Table 6. Once again, during the calm period, the standard models are generally dominated by the log-MIDAS models. For shorter forecasting horizons (one day and one week), the cluster also includes the log-HAR-RV-J model. Only in the case of S&P 500, the GAS model provides statistically comparable forecasts for an horizon of one and two weeks. During the crisis, the daily log-MIDAS-RV with 5-minute regressors pre-filtered for intraday periodicity, jumps and microstructure noise has the smallest QLIKE. For the one-week forecasting horizon the better forecast fit is given by the log-HAR-RV-J model for both Microsoft and S&P 500. The subset of superior forecasting models (as identified by the MCS) encompasses a smaller number of log-MIDAS specifications than in the calm period, the log-HAR-RV-J model (for one day and one week forecasting horizons) and the Student GAS model (for one and two week-ahead S&P 500 forecasts).