support the findings of Drost and Nijman: that a GARCH model that is estimated on high-frequency data does not predict lower-frequency volatility well, and that it is better to predict a high-frequency volatility with a low-frequency model.

A number of studies have shown that the aggregation properties of GARCH models are not straightforward (Guillaime et al., 1994; Ghose and Kroner, 1995; Andersen et al., 1999a; Galbraith and Zinde-Walsh, 2000). The persistence in volatility seems to be lower when measured on intra-day data than when it is measured on daily or weekly data. One would expect, for example, that fitting a GARCH(1, 1) to daily data would yield a sum of alpha and beta parameter estimates that is greater than the sum of alpha and beta parameter estimates that is obtained when fitting the same GARCH(1, 1) to 2-day returns. The square root of time rule does not hold, of course, but one should expect some similarity between the long-term volatilities that are inherent in GARCH(1, 1) implementations on different data frequencies. In fact, from the long-term variance equation (4.17), it is clear that we should observe

[1/(1 - a, - p,)] « [2/(1 - a2 - p2)] « ... « [ /(1 - a„ - p„)]

where a, and p, are the GARCH parameters based on /-day returns. This does not always turn out to be the case. Andersen and Bollerslev (1996) attribute this to a variety of volatility components in the data, possibly due to the different investment strategies of heterogeneous agents.

These observations have prompted Muller et al. (1997) and Dacorogna et al. (1996) to develop a GARCH model that is based on a number of independent volatility components in high-frequency data. In the simple ARCH model (4.1) the conditional variance is a weighted sum of squared returns. The heterogeneous interval autoregressive conditional heteroscedasticity (HARCH) model modifies this so that the squared returns are taken at different frequencies. It is extremely burdensome computationally to use all possible data frequencies, so the most recent formulation of the HARCH model in Dacorogna et al. (1998) uses exponentially smoothed squared sums of returns measured at only a few different frequencies. The partial volatilities that are used in the HARCH model are designed to capture volatility clustering at different frequencies. They are defined as

The number of returns taken in the sum, Kj, is increasing with j, so as j increases, the time span in the memory of the partial volatility also increases."

"Dacorogna et al. (1998) use Kj=\, 2, 5, 17, 65, ... for j=l, 2, 3, 4, ... . This is similar to the data preprocessing stage of a neural network, where partial sums of different frequencies are used to capture certain patterns in the data (§13.2).

The square root of time rule does not hold, of course, but one should expect some similarity between the long-term volatilities that are inherent in GARCH(1, 1) implementations on different data frequencies

The HARCH conditional variance equation is then similar to the ARCHfyj) equation (4.1) but with the partial sums ct2, . . ., ct2 in place of s2 1; . . ., s2 p. For example, one might take 1 hour as the basic interval of time, and form the partial sums of 1-hour returns by taking the sum of Kj consecutive returns for each j=l, . . ., p. Then the exponentially smoothed partial sums (4.11) are taken in the conditional variance equation

ct2 = a0 + 2 + ... + otpCT2. (4.12)

4.3 Specification and Estimation of GARCH Models

In a GARCH model there is a trade-off between having enough data for parameter estimates to be stable as the data window is rolled, and so much data that the forecasts do not properly reflect the current market conditions

How should one choose the data period for a GARCH model on daily data? Obviously the data should run up to the present date, but how far should one look back? An example of how GARCH parameter estimates depend on the choice of data period has already been given (Table 4.5). In §4.3.1 we show that in a GARCH model there is a trade-off between having enough data for parameter estimates to be stable as the data window is rolled, and so much data that the forecasts do not properly reflect the current market conditions.12 Then, in §4.3.2, a brief mathematical description of the algorithms that are used to estimate GARCH parameters is provided. However, it is unlikely that a market model developer will need to code up any programs. The most common GARCH models are already available as preprogrammed procedures in econometric packages such as the OxMetrics suite given with the CD, S-PLUS, TSP, EVIEWS, SAS, RATS and MICROFIT. A critical review of the available GARCH software is given in Brooks et al. (2001).

Even vanilla GARCH models can encounter convergence problems, although this is relatively rare; most often the cause of non-convergence is simply trying to apply the wrong model or using some rather difficult data. Section 4.3.3 gives some hints for helping GARCH models to converge; the section finishes by outlining, in §4.3.4, some simple diagnostics that will help you choose the correct specification for the GARCH model.

4.3.1 Choice of Data, Stability of GARCH Parameters and Long-Term Volatility

It is usual to take daily or intra-day returns, because GARCH effects at lower frequencies are not so apparent. In choosing the time span of historical data used for estimating a GARCH model, the first consideration is whether major market events from several years ago should be influencing forecasts today. For example, including Black Monday (October 1987) in equity GARCH models will have the effect of raising current long-term volatility forecasts by several per cent. For another example, including the Gulf war period in a crude

12In §4.2.3 it was shown that EWMA models can be considered as simple GARCH models. However, the estimation of EWMA volatility normally takes the smoothing constant parameter as given, so the data period has little influence on results. It makes no difference to the current estimate whether a time series of EWMA volatility has been taken over the past 50 days or the past 100 days. This is not the case with GARCH models.

Table 4.7: GARCH(1,1) parameter estimates and long-run volatility (daily data from 2 January 1996 to 6 October 2000)

Index Omega Alpha Beta 1-day Forecasts LR Forecasts

(xlO"5) (% Vol) (% Vol)

*These models are integrated so no long run volatility forecast exists.

oil GARCH model will give large reaction and low persistence coefficient estimates and consequently make current volatility estimates very spiky (Figure 4.5 and Table 4.5).

The long-term level of volatility to which a current volatility term structure will converge depends on the estimates of the GARCH parameters. For example, in the GARCH(1,1) model the long-term volatility is related to the GARCH constant as in (4.4). All parameter estimates, and in particular the estimate of the GARCH constant, are sensitive to the historic data used for the model. Thus even if the market has been stable for some time, the estimate of long-term volatility can be high if the data period covers several years with many extreme market movements. Therefore, in choosing how far to go back with the data, one has to take a view on whether or not current forecasts should be influenced by events that occurred many years ago.

In choosing how far to go back with the data, one has to take a view on whether or not current forecasts should be influenced by events that occurred many years ago

Table 4.7 reports the GARCH(1, 1) parameter estimates and corresponding long-run volatility forecasts for international equity indices, that were applicable on 6 October 2000, using GARCH models based on daily data since 2 January 1996.13 The corresponding GARCH volatility estimates are shown in Figure 4.7.

In the European markets, the Netherlands (AEX) and Germany (DAX) are less persistent and more reactive in volatility than France (CAC) and the UK (FTSE 100), and so their GARCH volatilities are more spiky (see Figure 4.7)

"For Taiwan the data start on 2 January 1997.