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88

whether we obtain similar features as in the case of foreign exchange rates. To avoid a systematic, deterministic decrease of volatility as explained by Section 5.6.4, we use forward rates for fixed time intervals. The forward rates are labeled according to the market conventions for forward rate agreements. The IxJ forward rate (e.g., the 3x6 forward rate) is the forward rate quoted at time t and applicable for the interval starting at time (t +1) and ending at time (/ + J) (/ and J are expressed in months). The corresponding time-to-start is / months and the maturity is (J - I) months.

The results of the EMA-HARCH process estimation for 3-hr ?-time intervals for the different forward rates for the LIFFE Three-Month Euromark are given in Table 8.5. The EMA-HARCH process is estimated for each forward rate by maximizing the log-likelihood and use data from April 6, 1992, to December 30, 1997, with about 16,800 observations. The distribution of the random variable s, is normal with zero mean and a unit variance. The market components (with p = 4) are similar to the ones described in the previous section. Like in the case of FX rates, the impact coefficient for the interval range from 6 hr to 1 days (second component) is very small. These results also indicate a decreasing impact of the longer-term components (corresponding to the market actors with the longest time-horizon) going from the first forward rate (i.e., whose time-to-start is closest in the future) to the last one (i.e., whose time-to-start is farthest in the future), reflecting the decrease in the volatility autocorrelation. The sum of the impacts is smaller than one in all cases, meaning that the estimated processes are stationary according to Equation 8.34.

8.4 FORECASTING SHORT-TERM VOLATILITY

The true test of the veracity of a volatility model is its ability to forecast the future behavior of volatility. This means that the data used to test the model are distinct from the data used to find the model parameters. All of the analyses described in this section are performed in an out-of-sample setting.

There is some added complexity in the case of volatility models where there is no unique definition of volatility. Andersen and Bollerslev (1998a) showed that, if the wrong estimators of volatility are taken, it is not possible to really test the forecasting quality of a model. That is why it is important to set a framework in which a forecasting performance analysis can be performed.

8.4.1 A Framework to Measure the Forecasting Performance

We choose here a path similar to that proposed in Taylor and Xu (1997). We construct a time series of realized hourly volatility, vn(t), from our time series of returns as follows:



where ah is the aggregation factor. In this case, we use data points every 10 min in #-time, so the aggregation factor is ah = 6.

Forecasts of different models are compared to the realized volatility of Equation 8.35. The one-step ahead forecasts are based on hourly returns in #-time. The advantage of using hourly returns instead of 30-min returns as in the previous section is that hourly forecasts are compatible with the historical hourly volatility defined in Equation 8.35. Four models are studied here.

The first model is used as a benchmark and is a naive historical model inspired by the effect described in Section 7.4.2 and Muller et al. (1997a) where low-frequency volatility predicts high-frequency volatility. The historical volatility is computed over a given day measured from the hourly returns. This quantity, properly normalized, is used as a predictor for the next hour volatility, v(t + 1), as defined in Equation 8.35. Formally the forecasting model Vb is

where the factor in front of the summation is here to normalize Vb to hourly

where st is i.i.d. and follows a normal distribution function with zero mean and unit variance.

The HARCH model in Equation 8.13 and the seven components of Table 8.3, introduced by Muller et al. (1997a).

The EMA-HARCH model in Equation 8.27 and 8.28 with seven components.

The three parameter models are optimized over a sample of 5 years of hourly data using the estimation procedure described in Section 9.3.5. The forecasts are then analyzed over the 5 remaining years. We term this procedure the static optimization. To account for possible changes in the model parameters, we also recompute them every year using a moving sample of 5 years. We term this procedure dynamic optimization. In this case, the performance is always tested outside of the gliding sample to ensure that the test is fully out-of-sample. In both cases, we use an out-of-sample period<of 5 years of hourly data, which represents more than 43,000 observations.

We compare the accuracy of the four forecasting models to the realized hourly volatility of Equation 8.35. The quantities of interest are the forecasting signal,

(8.36)

volatility. GARCH(l,l)is

Vgarch.i = h, = + ais2 ] + Biht-\

(8.37)

sf = v/,t - vh,t

(8.38)



where if is any of the forecasting models, and the realized signal,

sr = Vh,t+\ - vh,t (8.39)

The quantity vyif is taken in the estimation sample either directly or rescaled by the ratio of the averages and v/. This makes the forecast values on average closer to the historical volatility. In the rest of this chapter, we call the quantity, ( Vfit)/vf,the rescaled forecast.

In this formulation, performance measures proposed in Dacorogna et al. (1996) can be applied because the quantities defined in Equations 8.38 and 8.39 can take positive and negative values contrary to the volatilities, which are positive definite quantities. One of these measures is the direction quality,

& - \" slZ <8-40)

N(\vf \ sf-sr 0})

where Af is a function that gives the number of elements of a particular set of variables. It should be noted that this definition does not test the cases where either the forecast is the same as the current volatility or the volatility at time t + 1 is the same as the current one. This is, of course, unlikely to occur in our particular case. A detailed statistical discussion of this measure can be found in Pesaran and Timmerman (1992).

In addition to this measure, we use a measure that combines the size of the movements and the direction quality. It is often called the realized potential,

s\sn(sf sr) \sr\ Qr = rn (8.41)

In fact, the measures Qr and Qd are not independent and Qr is a weighted average of sign(sf sr) whereas 2Qd - 1 is the corresponding unweighted average. It is easy to show that if

Qr > 2 Qd - 1 (8.42)

the forecast of the sign of sr for large \sr \ values is better than average.

A more traditional measure such as the comparison of the absolute error of a model to a benchmark model can also be used. This benchmark model is chosen to be the historical volatility as defined in Equation 8.36, U£. The quantity

X>r -s"}odel

E \Sr - Sbl

i benchmark i (8.43)

is a quality measure, which increases with an increasing performance of the model. If Qf > 0, the model outperforms the benchmark. If Qf < 0, the benchmark outperforms the model. The second part of Equation 8.43 is similar to the known Theils [/-statistic, Makridakis et al. (1983), except that we use the absolute value instead of the squared errors.



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