Computing value at risk with high frequency data

Introduction

One of the most interesting developments in empirical finance is given by the availability of long data sets recording prices of assets prevailing at various points of time during a trading day. The raw data are often obtained tick by tick and present various opportunities and challenges from the point of view of the econometric models which can treat observations with unequal time spacing.

( )

Organizing the data on the basis of a fixed time span, for example every 5 minutes, allows the use of more traditional econometric models based on the Ž . assumption of a stable recording of the observations. Dacorogna et al. 1993 presents a detailed and clear explanation of how the raw data may be filtered to produce fixed-frequency returns in the case of foreign exchange rates.

The availability of high frequency data has offered new insights on the volatility structure of financial returns, particularly on its persistence. Analyses based on daily data have traditionally found a very high degree of persistence, see Ž . for example Baillie and Bollerslev 1989 who find integration in variance for daily exchange rates. More recently, persistence has been interpreted in the context of a Fractionally Integrated Generalized Autoregressive Conditional Heteroskedas-Ž . Ž . ticity FIGARCH model, see Baillie et al. 1996 . See also Dacorogna et al. Ž . 1993 for a similar finding of hyperbolic, rather than exponential, decay in the autocorrelogram of volatility of exchange rates. Ž . Andersen and Bollerslev 1997b estimate the degree of fractional differencing applying spectral analysis methods to high frequency absolute returns data and find a good characterization of the hyperbolic decay. They also note that standard time-domain estimation procedures based on the autocorrelation function can produce a similar estimate of the long run volatility persistence only after a preliminary filtering of the raw data. This observation must be cast more generally Ž . in the stream of research, see for example Bollerslev 1997a andŽ . Dacorogna et al. 1993 , which has uncovered the existence of various intra-day empirical regularities in asset returns, taking the form of cycles repeating within the day and the week. The econometric analyses have clarified the connection between the high degree of volatility persistence traditionally obtained from daily data and the lower degree of persistence inherent in the high frequency data. In particular, it has been shown that taking the intra-day seasonality into account is of key importance for estimating the relevant parameters from the high frequency data.

The recent developments appear to be very important for various financial applications, among which is portfolio risk measurement. Most models widely used among practitioners assume that the distribution of returns is conditionally normal. It follows that the conditional variance is the key element for measuring risk. Moreover, most models assume that shocks to volatility are permanent. As we have just pointed out, this characterization is compatible with the results obtained from daily data but not with those obtained with high frequency data. Models assuming a non-stationary volatility process may be grossly misleading when the true volatility dynamics is stationary, particularly when the decision maker is interested in measurement of multi-period volatility. This is the common case for banks, which use their internal risk models for forecasting 10-day volatility.

The main goal of this paper is to evaluate the use of high frequency data for risk measurement. We shall estimate volatility models on the basis of high-( ) Ž . frequency half-hour data for the Deutsche mark-US dollar exchange rate and compare the results to those obtained from volatility models estimated on the basis of daily data. In particular, we want to evaluate whether the results obtained by analyzing volatility with high frequency data have implications for measurement of volatility at lower frequencies, e.g., daily or weekly. Such a possibility is suggested by the new framework emerging from the various studies with high frequency data, emphasizing the importance of long memory components connected with the slow decay in the autocorrelation of volatility, see Ž . Ž . Bollerslev 1997b andBaillie et al. 1996 . In the following we shall innovate on the existing literature in terms of the econometric modeling of high frequency data. We shall consider two methodologies for taking into account the intra-daily seasonal, comparing the deterministic seasonal model of Andersen and Bollerslev with a model resulting from the Ž . application of the structural methodology developed by Harvey 1989 . We shall then compare the risk measurements obtained from the models at various multi-Ž . period horizons. Beltratti and Morana 1998 explores further the validity of the Ž . structural approach by comparing it with that of Andersen and Bollerslev 1997a Ž . and with that advocated by Dacorogna et al. 1993 .

The plan of the paper is as follows. After this introduction, Section 2 discusses models for volatility measurement and existing empirical results from the GARCH literature. Section 3 describes two models for taking into account intra-day seasonal, one with a deterministic cycle and one with a stochastic cycle. Section 4 presents the econometric results. Section 5 evaluates the ability of the stochastic volatility model to forecast multi-period volatility at various horizons. Section 6 concludes.

The evidence in favour of transitory shocks to volatility is reinforced by the use of intra-day data. Various papers using high frequency data have proposed a different picture of the persistence of the volatility process. Baillie and Bollerslev Ž . 1990 show that a GARCH model with several dummy variables for the hourly exchange rate does not show signs of integration in variance. Other papers find that the half-lives of most intra-daily volatility shocks are very low and disappear in few hours. The coexistence of such conflicting results is explained by Andersen Ž . Ž . and Bollerslev 1997b , see also Granger and Ding 1996 , on the basis of a model where volatility is the sum of various components, some having a short-lived and others having a long-lived effect. When allowance is made for these different Ž . processes the volatility dynamics measured in terms of absolute returns is well described by an ARFIMA process, characterized by a slow hyperbolic decay of the Ž . autocorrelation function. The empirical results of Andersen and Bollerslev 1997b suggest that the estimate of the coefficient of fractional integration for the time series of absolute returns, close to 0.36, is stable across estimation methodologies and frequencies of data collection.

On the basis of the existing results it seems therefore important to use an estimation method which at least allows for stability in the variance process. The assumption of permanent shocks to volatility is likely to overestimate the risk measurement implicit in any portfolio especially when the daily estimated models are used for long run estimates of volatility. However, in order to proceed to the empirical estimation of the volatility process one needs to take into account the intra-day seasonality existing in the foreign exchange market.

One disadvantage of the stochastic volatility model with respect to the model of the GARCH family is that maximum likelihood based estimation can only be Ž . computed via simulation methods see Shephard, 1993;Jacquier et al., 1993 . Ž . Ž . However, as suggested in Harvey et al. 1994 andRuiz 1994 a quasi-maximum Ž . likelihood method QML , computed using the Kalman filter, yields reasonably efficient estimates. In practice quasi-maximum likelihood estimation is carried out by treating the disturbances of the measurement and transition equations as they were normal and maximizing the prediction error decomposition form of the likelihood function obtained via the Kalman filter. In a Gaussian model the Ž . Kalman filter is the minimum mean square error estimator MMSE within the class of all of the estimator. In the framework of quasi-maximum likelihood estimation, however, the Kalman filter is still the MMSE within the class of the Ž . linear estimators MMSLE . QML estimators are asymptotically normal with Ž . covariance matrix computed as in Dunsmuir 1979 .

Volatility and risk measurement

There are various approaches to portfolio risk measurement. One that is particularly popular among practitioners is based on the assumption of conditionally normal returns, which opens the way for the use of various analytical results. In the context of conditionally normal returns, the main issue is the measurement of the time-varying variance of the rate of return of the portfolio, s 2 . A popular t Ž Ž . volatility model is the GARCH see Engle and Bollerslev 1986, Bera and Ž . Ž . Higgins 1993, Bollerslev et al. 1994 for a general discussion and a survey of . the literature .

Financial investors are often interested in evaluating the variance over a future horizon rather than simply over the current short period of time. In order to do that one can use the volatility model to project forward the expectation of the variance.

Ž .

2

For example, the multi-period conditional forecast of a GARCH 1,1 is E s s . as the integrated GARCH IGARCH process. In this case one expects the variance to be forever equal to the most recent conditional value. This case is of particular importance from the point of view of practitioners as it can be shown that the IGARCH is equivalent to the exponentially weighted moving average Ž . model popularized by J.P. Morgan 1995 in its RiskMetrics portfolio evaluation framework.

An incorrect assumption of a unit root in the variance process may be associated with an overestimate both of the long run cumulated variance and of the Ž . risk of the current portfolio. Ding and Granger 1996 observe that in an IGARCH model a shock may permanently affect the expectation of a future conditional variance without affecting the true conditional variance. For example, for a Ž .

it can be shown that

Ž .

t tqk t

Es 2 rE´2 s ab ky 1 .

tqk t

Ž .

Notice that for an IGARCH 1,1 model the expected future variance reduces to

Ž . It follows that in the case of an IGARCH 1,1 a shock to the rate of return affects the expected variance independently of the horizon while the actual future variance tends to revert to zero if b -1. It is therefore of great importance for risk measurement to assume whether the conditional variance process is integrated or not.

Is there any statistical basis for the assumption of non-stationary variance? Literature has found signs of very strong persistence in volatility using data Ž Ž . sampled at daily or lower frequencies see Baillie and Bollerslev 1989 for Ž . . exchange rates and Bollerslev et al. 1994 for stock prices . On the other hand Ž . Baillie and Bollerslev 1989 find that GARCH properties disappear for exchange rate series evaluated at monthly and quarterly frequencies. This result is coherent with the theoretical results on temporal aggregation of stable GARCH processes Ž . and is in contrast with the IGARCH representation. Baillie et al. 1996 apply the class of FIGARCH processes to the analysis of the daily Deutsche mark-U.S. dollar exchange rate for the period March 14, 1979-December 30, 1992. Their Ž . preferred model is a FIGARCH 1, d, 0 with an estimate of d equal to 0.823. The important practical difference between a GARCH process and a FIGARCH process is the rate of absorption of a shock. For a GARCH process shocks die out at a quick exponential rate while for a FIGARCH process shocks die out at a slower hyperbolic rate.

Models of cyclical volatility

In this section we compare the model for intra-day seasonality proposed by Ž . Andersen and Bollerslev 1997a with a stochastic volatility model. Andersen

We believe there is no reason to believe that the daily or weekly seasonal patterns in foreign exchange volatility repeat themselves exactly. The geographical ( ) Ž . model of Dacorogna et al. 1993 predicts that volatility depends on the level of activity in the market. Such a level of activity is however, certainly stochastic, depending on how many traders actually participate in the market, a factor which may well change from one day to the other. Moreover, using quotes rather than actual transaction prices, as done for example in the Olsen and Associates dataset that is used in this paper, may imply various measurement errors of market volatility. Finally, the intensity of some of the intra-day causes of volatility, for example the intensity of overlapping between different markets, may well depend on the overall level of volatility.

For these reasons we propose a model including some stochastic volatility terms. This idea is certainly not new to the analysis of financial markets. Harvey et Ž . al. 1994 have applied a stochastic volatility model based on the structural time Ž . series approach to analysis of daily exchange rate returns. Payne 1996 has extended this methodology to incorporate an intra-day fixed seasonal component. In this paper we further extend the model to account for stochastic intra-daily cyclical components. We consider the following model for exchange rate returns:

the non-stationary volatility component:

h is the stochastic stationary acyclic volatility component: taking logs, the model may be rewritten as

The c component is broken into a number of cycles corresponding to the t fundamental daily frequency and its intra-daily harmonics c s Ý 2 c . We

model c , the fundamental daily frequency, as stochastic while its harmonics 1, t, n Ž . are modelled as deterministic as in Andersen and Bollerslev 1998 . By writing the generic cyclical component as a linear combination of sine and cosine terms, that is

where t s 1, . . . ,T indicates the number of trading days, n s 1, . . . , N is the Ž . number of intraday periods, l s 2prN is the frequency measured in radians,

where 0 F r F 1 is a damping factor, c

Assuming that k and k are not correlated is sufficient to

achieve identification of the two components c and c U . In addition, the

restriction of equal variances of the noise components,

parsimony reasons. We model c as a deterministic flexible Fourier form:

. Therefore, the model encompasses the one of Andersen and Bollerslev 1997a , which can be considered as a special case of ours when r s 1 and

The full stochastic volatility model in state space form reads as follows:

and where I , i s 1, . . . ,k, is the scaling variable, n and n , and a number of

dummy variables that take into account day-of-the-week effect, outliers and Tokyo

. selection matrix and m is a 4 q P y 1 q k = 1 selection vector. The stochastic disturbances of the measurement and transition equations are assumed to be uncorrelated with each other in each time period. The state space formulation shows that the various components may be estimated simultaneously.

The empirical analysis

The daily data used in this section cover the period December 31, 1972-January 31, 1997, corresponding to 6545 observations; the source is Datastream. The high-frequency data are for 1996, from the Olsen dataset HFDF-96, containing 12 576 observations excluding week-ends. The estimation method for both the GARCH and the FIGARCH processes is maximum likelihood. Following Baillie Ž . Ž . et al. 1996 , the FIGARCH 1, d, 1 has been estimated in terms of its infinite

and where´s r y m q u´to take into account possible autocorrelation due to

microstructure effects. The model has been estimated by imposing the non-nega-Ž . tivity constraints described in Bollerslev and Mikkelsen 1996 ,

which are sufficient to ensure that the parameters of the ARCH representation are all nonnegative. Following the standard procedure in the literature, initial condi-( ) tions have been set to´s 0 for t s 0,y 1,y 2 . . . ,y 1000 and´2 s T y1 Ý T r 2 t t t s1 t for t s 0,y 1,y 2 . . . y1000. The unobserved components model has been esti-Ž . mated following the Kalman filter methodology described by Harvey 1989 . Estimation of the unobserved components model has been performed by means of the software STAMP 5.0. All the other estimation has been performed by means of GAUSS codes.

Ž . Ž . The daily returns are analyzed first with a MA 1 -GARCH 1,1 model and then Ž . Ž . with a MA 1 -GARCH 2, 1 model. As the moving average coefficient was not Ž . significantly different from zero, we report only the results for GARCH p, q Ž . models Table 1 .

The results are similar to those presented in the literature. In particular one finds a very large degree of persistence. The sum of a and b is virtually identical Ž . to the value reported by Andersen and Bollerslev 1997a . Taking into account the existence of a dummy variable for Monday does not change the main results. We Ž . have also estimated a GARCH 2, 1 model to allow for a slower decay of the Ž . autocorrelation coefficient, see e.g., Ding and Granger 1996 , but the Ž . GARCH 1, 1 previously reported is preferred by the Akaike criterion. The main Ž . advantage of the GARCH 2, 1 specification is the reduction of the degree of skewness and kurtosis of the standardized residuals. Estimation of a Ž . FIGARCH 1,d,1 model yields the estimates shown in Table 2. Ž . The estimates are very close to the model estimated by Baillie et al. 1996 . A coefficient of fractional differencing equal to 0.6, significantly different from both zero and one, means that the process is non-stationary but that the decay of the autocorrelation function is much slower than the one implied by the GARCH process. The standardised residuals produced by the FIGARCH are worse, in terms of both skewness and kurtosis, than those produced by the GARCH models.

It is of interest to compare the results for the daily data with those obtained from the analyses of the high frequency data. We follow the other researchers in the high frequency field and define the rate of return as the difference between two Ž . averages of log of the bid and ask prices measured at different times, see e.g., Ž . Ž . Ž . Dacorogna et al. 1993 . First we estimate an MA 1 -GARCH 1,1 and an Ž . Ž . MA 1 -GARCH 2,1 models for the half-hour returns. The results can be seen in Table 3.

The results reported in Table 3 confirm previous findings according to which the process for volatility of high frequency exchange rate returns shows signs of Ž . stability. The GARCH 2,1 is now preferred by the Akaike criterion even though the performance of the two models is similar as far as the other tests are concerned. Before investigating the importance of the seasonal intra-daily component we estimated a MA-FIGARCH model. The results are shown in Table 4.

The coefficient f, which can be interpreted as the sum of the coefficients a Ž . and b of the GARCH 1,1 , is now well below unity and the coefficient of fractional integration is lower than the values reported by Andersen and Bollerslev in estimating an ARFIMA model for the time series of absolute returns. This may ( ) Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž .

2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests. be consistent with their results because their estimate is obtained by the application of spectral methods to the raw returns. The FIGARCH now marginally improves upon the GARCH in terms of both skewness and kurtosis and is very similar as far as skewness and kurtosis are concerned.

Next we analyze the intra-daily seasonal component, both with the Fourier flexible form and with the stochastic structural time series model of Harvey . AIC is the Akaike Information Criterion computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the Ž .

2 Ž . standardized residuals and the squared standardized residuals are denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of the previous effects jointly. Definition of Ž . this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests.

Ž

. 1989 . In terms of diagnostics of the unobserved components model we simply note that the estimate of the coefficient of the autoregressive part is equal to 0.284. The R 2 is equal to 9.5%, slightly larger than the R 2 of the Fourier flexible form, equal to 7.5%. The Durbin-Watson for the Fourier flexible form is 1.77 while that for the unobserved components model is 1.90. Fig. 1 compares the deterministic cycles over the 48 half-hours with the Ž . average over the 262 days in our sample of the stochastic cycles. 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž . 2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests. Fig. 1 shows that on average the stochastic cycle and the deterministic cycle are fairly close. It is at the day to day level that the two specifications differ, with the stochastic cycle being more suited to model intradaily heterogeneity in the volatility process than its deterministic counterpart. Table 5 reports some summary statistics about the statistical properties of the raw and the filtered returns r rs . 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž . 2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 distribution. The correlation between the raw returns and the deterministically detrended returns is 0.92 and that between the deterministically detrended returns and the stochastically detrended returns is 0.95. Figs. 2-4 analyze the correlogram for absolute raw and filtered returns, < < < < respectively r and r rs . returns filtered with the stochastic cycle is lower than the autocorrelation function of the returns filtered with the deterministic cycle.

. that abstracts from intra-daily irregularities due to stochastically repetitive events. The volatility estimate obtained from our filtered returns is indeed an estimate of such a measure of risk and has therefore been used in what follows as a comparison for the estimates obtained from daily data.

In the following we will consider the volatility estimates obtained for horizons of 1 day, 5 days and 10 days from GARCH, IGARCH and FIGARCH models, estimated with both daily and high frequency data. The IGARCH models have ( ) 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž .

2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 been estimated with the two datasets imposing the condition that a q b s 1 in the GARCH representation, obtaining a s 0.12 and b s 0.88 for the daily data and a s 0.03 and b s 0.97 for the half-hour data.

In all cases we have used the appropriate forecasting equations to obtain the estimates of the total variance over the horizon. As a simple benchmark against which to consider the performances of the various models we have computed the historical standard deviation of returns over the previous one hundred observations ( ) and we have then counted the frequency of future T observations falling below y1.65 times the computed standard deviation. We have used the square root rule ' to forecast the multi-period standard deviation, i.e., T s is the time t estimate of t the standard deviation of returns for the period from time t to time t q T. The percentage should be 5% for a normal distribution. The empirical percentage is 5.4% for 1 day, 6.6% for 5 days and 7.2% for 10 days when daily data are used 1 and 4.3% for half-hour, 3.6% for 2 hours, 3.7% for 5 hours. There is therefore a 2 difference between the daily data and the high frequency data, pointing to possible difference in the distribution of the variable at different frequencies.

We then report the correlation coefficients between the historical standard deviation computed from the simple estimator and that computed from the various Ž . econometric models Table 10 .

The correlation of GARCH and IGARCH is rather similar, while the FI-GARCH shows a substantially higher correlation. The percentages of observations below y1.65 standard deviations for the various horizons are as indicated in Table 11.

The numbers are very close to each other, with a slight improvement for the FIGARCH. In all cases the estimated standard deviation from the GARCH models seems to be a better guide to the actual distribution of returns. For the high frequency data evaluated at the high frequency, see Table 12.

The performance is slightly worse than before. The percentile computed from the high frequency data is too conservative. The actual percentage of observations below the percentile is too low. From this point of view the GARCH model seems to be the best among the three under observation for the half-hour horizon, while the FIGARCH is better for the other horizons. Finally, when the high frequency data are evaluated at the daily horizon we obtain the following results indicated in Table 13.

There is now a large deterioration in the performance, especially for longer horizon. The percentiles obtained from high frequency data are much too conservative. The actual returns have never been lower than the values calculated on the basis of the recursive equations. Moreover, there is not much difference across the three econometric models in this respect. From this point of view, it seems that high frequency data provide overly conservative measures of risk, even though the finding needs further study and testing with different returns and time periods.

An intuitive stylized explanation of the results is the following. Consider a Ž . GARCH 1, 1 model for high frequency returns with a s 0.03, like the one we have actually estimated. The fact that the variance forecasts from the GARCH are very similar to the variance forecasts from the IGARCH can be explained by noting that the sum of the coefficients a and b in the GARCH is strictly less than one but very close to one. In terms of volatility forecasts then there is not much difference between the two as shown by the example of the second section of this paper. The similarity between the estimated GARCH and the IGARCH also explains the large overestimate of the total multi-period variance. The comparison between the expected variance and the actual variance performed in the second section shows how GARCH models are over-sensitive to the current information set. Such an over-sensitivity may be at the basis of the findings of this paper.

t, n tn

The filtered returns r rs are then analyzed on the basis of GARCH t, n tn processes. The results from applying GARCH to the deterministically detrended returns are shown in Table 6. The results are in line with the previous estimates. In particular the sum of the coefficients a and b is larger than the one obtained with the raw data and the Ljung-Box statistics does not reject its null hypothesis more strongly than before. Estimation of FIGARCH yields is shown in Table 7. The FIGARCH model points out that persistence has been dramatically increased by the filtering in that the estimate of the coefficient of fractional integration goes from 0.18 obtained with the raw data to the current 0.21. When GARCH is applied to the stochastically detrended returns one obtains the data reflected in Table 8.

One notices here a strong increase in the sum a q b. The stochastic detrending therefore seems more useful to take out the infra-day components and to allow the long run component of volatility to emerge from the analysis of the high frequency data. Estimation of FIGARCH yields the data in Table 9.

The FIGARCH model confirms that persistence has been increased by the stochastic filter even more than what was done by the deterministic filter. The coefficient of fractional differencing is now larger than previous estimates. This confirms the results obtained with different methods and a different data set from Andersen and Bollerslev, and suggests that taking care of the intra-day seasonal cycle is essential in estimating parameters connected with long run persistence. A comparison between Tables 7 and 9 also suggests that the stochastic model allows a more precise estimate of such parameters.

Value at risk estimation

Variance is an essential component of risk measurement models based on the assumption of conditional normality of returns. The typical applications of such ( ) , 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž . 2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests. models is to daily and 2-week horizons. The daily variance is estimated from past daily data, perhaps with some system of weights which magnify the role of the most recent observations, and then extrapolated with the square root rule based on the assumption of uncorrelated returns. In this section we use the various estimates obtained in the previous section to compare the risk measurements for various horizons obtained with both daily and high frequency data. The aim is to see whether high frequency data may be useful not only for more frequent but also for better measurement of risk at standard multi-day horizons. 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž .

2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests.

An issue of great relevance is which sort of high frequency data should one use, i.e., the raw data or the filtered data? On the basis of the evidence that we have reported in the previous section and in a related paper, see Beltratti and Morana Ž . 1998 , we believe that filtered returns not only have better statistical properties, as shown in the previous section, but also offer a better picture of the risk practitioners are interested in. In a multi-day context one is interested in a measure of risk 1996, to December 31, 1996 week-ends and holidays, for a total of 12,576 observations. Robust standard errors are reported in the table. AIC is the Akaike Information Criterion Ž . computed as y 2 ln Ly2 K r T where ln L is the maximized log likelihood function, K is the number of estimated parameters and T is the number of observations. The Ljung-Box portmanteau tests for up to K th-order serial correlation in the standardized residuals and the squared standardized residuals are Ž . 2 Ž . denoted by Q K and Q K , respectively. The sign bias is a test for asymmetric effects of lagged positive and negative innovations on the conditional variance that have not been taken into account in the specification of the model. The negative size bias is a test to evaluate whether large negative innovations have a stronger impact on the conditional variance process than small negative innovations. The positive size test evaluates whether large positive innovations have a stronger impact on the conditional variance process than small positive innovations. The joint test evaluates the presence of Ž . the previous effects jointly. Definition of this latter test can be found in Engle and Ng 1991 . The table reports the p-values of the various tests.

Conclusions

The paper has analyzed the value at risk measurements that can be obtained from different volatility models. In particular we have compared GARCH and FIGARCH models and computed the multi-horizon volatility forecasts for periods up to 10 days. The volatility models have been estimated both with daily and with half-hour data. The high frequency data have been filtered to take into account of intra-day seasonality. The filter is compatible with the existence of stochastic intra-day seasonality and seems to compare favourably with deterministic filters.

It has been shown that filtering the data is of great importance for estimation of volatility models. For example the coefficient of fractional integration is better estimated with the filtered rather than with the raw data. It has also been shown Ž . that even at the high half-hour frequency the coefficients of the GARCH volatility model are not very different from those estimated on the basis of an IGARCH model. While the econometric estimates of the FIGARCH model are different, they generate multi-period forecasts of volatility which are fairly similar to those generated by simpler GARCH models. The empirical implications for practical risk measurement is therefore that there is validity in the use of computationally simple models with forecasting properties mimicking GARCH.

This work has raised in our opinion a few suggestions for future research. First is the application of similar methods to time series of other assets for time periods different from the one used in this paper. A second possible topic for future research lies in a thorough comparison between deterministic and stochastic models for intra-day seasonal. For example, it seems interesting to evaluate the differences between the estimated stochastic cycles during announcement days or days in which some important news has been revealed to the market.

A third and most important topic for future research is the evaluation of different volatility measures. The results of this paper show that high frequency data may be used to obtain more frequent evaluation of volatility but not for better long-run volatility measurement. This however, may be due to using them in the context of standard GARCH models which are then extrapolated forward. It would ( ) be interesting to think of other uses of high frequency data, for example to retrieve a noisy estimate of daily volatility which can then be directly used in models based on daily data.