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-Aol ---Citigroup--General Electric --Microsoft ExxonMobil

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-Aol ---Citigroup -- General Electric -Microsoft ExxonMobil

Figure 4.8 (a) and (b).

One does not need to estimate the GARCH constant freely. Instead one can fix the long-term volatility level by imposing the constant before model estimation

Despite the changes in GARCH parameter estimates during the period, Figure 4.8d shows that the long-term volatility forecasts from these GARCH models do not change much. The long-term volatility of Microsoft did increase somewhat in April 2000, reflecting developments in the court case brought by the Monopolies Commission. If the GARCH parameter estimates do vary considerably when the model is rolled over time it may be that the model is not well specified. In fact there is some evidence to suggest that specification of the GARCH model will depend on the current market regime (Hamilton and Susmel, 1994).

In summary, one should use several years of daily data, enough to ensure that parameter estimates are relatively stable as the data window is rolled, but not

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-Aol ---Citigroup -- General Electric - - Microsoft Exxon Mobil

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-Aol --- Citigroup -- General Electric - - Microsoft Exxon Mobil

Figure 4.8 Parameter estimates for US stocks in 2000: (a) rolling GARCH omega; (b) rolling GARCH alpha; (c) rolling GARCH beta; (d) long-run GARCH volatilities.

so much that these estimates do not reflect changes in current market conditions. When there are outliers in the data, however far in the past, this can upset convergence of the GARCH model and give misleading results for parameter estimates. A very long data period with several outliers is unlikely to be suitable because extreme moves from very long ago can have a great influence on the long-term volatility forecasts made today.

One does not need to estimate the GARCH constant freely. Instead one can fix the long-term volatility level by imposing the constant before model estimation. To illustrate the idea, consider the GARCH(1,1) model (4.2).

Substituting for using (4.4) gives an alternative formulation of GARCH(1,1) model in terms of deviations from long-term variance a2:

a2 - a2 = a(s2 , - a2) + p(a2 , - a2).

(4.13)

Scenario analysis on long-term volatility may be quite simply extended to the whole volatility term structure using the GARCH model

Thus a GARCH(1,1) model on the deviations of squared unexpected returns from some assumed level for the long-term variance gives estimates of the GARCH reaction and persistence parameters while fixing the long-term volatility. In this way a scenario analysis on long-term volatility may be quite simply extended to the whole volatility term structure using the GARCH model.

4.3.2 Parameter Estimation Algorithms

The mathematical foundations of computer programs that estimate GARCH model parameters are now described. It is worthwhile to have some understanding of the mathematical principles used because if a statistical package does not offer in-built commands for GARCH model estimation then procedures must be written. For example, in GAUSS the GARCH procedures are explained in the manual. Several GARCH programs in GAUSS written by Ken Kroner are available free from http: weber.ucsd.edu/~mbacci/engle/ index data.html.

Following Bollerslev (1986), GARCH model parameters are normally estimated by maximum likelihood, a powerful and general parameter estimation procedure that is widely used because it almost always produces consistent, asymptotically normal and efficient estimates (§A.6). The general idea is to choose estimates of the parameters to maximize the likelihood of the data under an assumption about the shape of the distribution of the data generation process. The parameter estimation method is therefore one of optimizing a function of several variables, for which there is often no analytic solution but there are many standard routines.

Most algorithms are iterative, that is, the parameter estimates are updated using a scheme:

=eI. + A.,6„

where X, is a step length and 5, is a direction vector, chosen so that the likelihood of the data under 6,-+] is greater than the likelihood under 9,. The gradient descent methods that are used for GARCH model estimation in most packages define the direction vector in terms of the gradient of the likelihood function and the Hessian matrix of second derivatives of the likelihood function, both evaluated at 9,.

For an i.i.d. normal data generation process with mean and variance a2, the likelihood of a sample of returns r\, r2, . ., rT is