Engle (1982), was applied to economic data. For financial data it is more appropriate to use a generalization of this model, the symmetric GARCH introduced by Bollerslev (1986). Following this very many different GARCH models have been developed, notably the exponential GARCH model of Nelson (1991), one of the first asymmetric GARCH models to be introduced. For excellent reviews of the enormous literature on GARCH models in finance, see Bollerslev et al, (1992, 1994) and Palm (1996).

4.2.1 ARCH

The ARCH(p) process captures the conditional heteroscedasticity of financial returns by assuming that todays conditional variance is a weighted average of past squared unexpected returns:

a2 = a0 + s2 , + ... + aps2-p

a0 > 0, a,, . . ., a, Ss 0 e,\I, ~ 7v*(0, a2).

If a major market movement occurred yesterday, the day before or up to p days ago, the effect will be to increase todays conditional variance because all parameters are constrained to be non-negative (and a0 is constrained to be strictly positive). It makes no difference whether the market movement is positive or negative, since all unexpected returns are squared on the right-hand side of (4.1).

ARCH models are not often used in financial markets because the simple GARCH models perform so much better. In fact the ARCH model with exponentially declining lag coefficients is equivalent to a GARCH(1,1) model, as is shown in (4.3) below, so the GARCH process actually models an infinite ARCH process, with sensible constraints on coefficients and using only very few parameters. The convergence of ARCH(p) models to GARCH(1,1) as p increases is illustrated in Figure 4.2. Here ARCH(5), ARCH(20) and GARCH(1,1) volatilities have been estimated on the AC. The volatility from an ARCH(5) model is too variable because the lag is too short. The ARCH(20) volatility is similar to the GARCH(1,1) volatility, except there is a certain amount of noise around the estimate that we could very well do without.

As the lag increases in an ARCH model it becomes more difficult to estimate parameters because the likelihood function becomes very flat (§4.3.3). Add to this the inadequate dynamics in an ARCH model with only a few lags, and the differences between volatility estimates in Figure 4.2 are easily accounted for. Since we need very many lags to get close to a GARCH(1,1) model, which has only three parameters, the use of standard ARCH models for financial volatility estimation is not recommended.

The use of standard ARCH models for financial volatility estimation is not recommended

20 -

40 -

50 -

30 -

10 -

0 -I-1-1-1-,-1-,-,-r-1

Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99

Figure 4.2 ARCH and GARCH volatilities of the CAC index. 4.2.2 Symmetric GARCH

The full GARCH(/>, q) model adds q autoregressive terms to the ARCH(/>) specification, and the conditional variance equation takes the form

However, it is rarely necessary to use more than a GARCH(1,1) model, which has just one lagged error square and one autoregressive term. Using the standard notation for the GARCH constant , the GARCH error coefficient a and the GARCH lag coefficient P, the symmetric GARCH(1,1) model is:

Equation (4.2) is the generic or vanilla1 GARCH model that many financial institutions use today. Note that this model may also be written

ARCH(20) - GARCH - -- ARCH(5)

07 = a0 + a, s2 , + ... + apEj-p + P,a2 , + ... + P,o7 a0 > 0, a,, . . ., a,, p„ . . ., P„ 0.

a2 = + ae2 , + Pa2 ! > 0, a, p 0

(4.2)

07 = 4- ae2 , + Pa2.,

- + as2., 4- P(co + ae2 2 4- P(co + as2 3 = (1 - P) 4- a(s2 , + pE2 2 + p2s2 3 + .

+ ...)

+ PC )))

(4.3)

so the GARCH(l.l) model is equivalent to an infinite ARCH model with exponentially declining weights, as mentioned in §4.2.1 above.

The sizes of the parameters a and P determine the short-run dynamics of the resulting volatility time series. Large GARCH lag coefficients p indicate that shocks to conditional variance take a long time to die out, so volatility is persistent. Large GARCH error coefficients a mean that volatility reacts quite intensely to market movements, and so if alpha is relatively high and beta is relatively low then volatilities tend to be more spiky. In financial markets it is common to estimate lag (or persistence) coefficients based on daily observations in excess of 0.8 and error (or reaction) coefficients no more than 0.2. Much of the discussion in this section and §4.3 concerns the values that one should expect for these reaction and persistence coefficients in different markets, and their stability over time.

Estimating vanilla GARCH models using a statistical package such as PcGive or TSP (http: www.tspintl.com) is very simple. Most market data are sufficiently well behaved for the GARCH(1,1) model to be estimated on just a few years of daily data. For example, daily returns on 20 US stocks between 1 January 1996 and 1 October 2000 were used to estimate GARCH(1,1) models in TSP, and the parameter estimates are given in Table 4.2.

Large GARCH lag coefficients P indicate that shocks to conditional variance take a long time to die out, so volatility is persistent. Large GARCH error coefficients a mean that volatility reacts quite intensely to market movements, and so if alpha is relatively high and beta is relatively low then volatilities tend to be more spiky

The GARCH models for Ford, BankOne, Rockwell and Unicom did not converge, which, following the discussion after Table 4.1, should come as no surprise.4 We already know the reasons for this: for example, in the case of Ford it is a single observation on 8 April 1998 that upsets the GARCH model, and removing it will resolve the problem. A discussion of more general convergence problems is given in §4.3.3.

Some dubious results are highlighted in italics in Table 4.3. Boeing, Hewlett Packard and AT&T all have excessive excess kurtosis (see Table 4.1) so for these stocks a non-normal symmetric GARCH model will be better than a normal one. Cisco Systems also shows a marked leverage effect, and really needs an asymmetric GARCH model to capture its volatility clustering. The result of applying symmetric normal GARCH to these four stocks is that, although the models do converge, the conditional volatilities on these stocks are very spiky (large reaction, low persistence) and not at all similar to the volatilities on the other stocks.5

Figure 4.3 shows some of the GARCH volatility estimates that are obtained from the GARCH models for America Online, Citigroup, General Electric, Microsoft and Exxon Mobil. America Online was very volatile for most of the period, and only in the last few months did its volatility come down to more normal levels. Microsoft was also very volatile around the time of the court case between it and the Department of Justice. The other stocks, notably

The result of applying symmetric GARCH to these four stocks is that, although the models do converge, the conditional volatilities on these stocks are very spiky (large reaction, low persistence) and not at all similar to the volatilities on the other stocks

4The PcGive demo version on the CD has a tutorial that will discuss this problem.

5Of course, it may be the case that some of these stocks, like Cisco, do in fact have very spiky volatilities that are not at all like the other stocks. However, it is likely that the normal GARCH model where e,\I, -~ N(0.07) should be replaced by one with a fat-tailed conditional distribution for 8,; see also the remarks at the beginning of SJ4.2.5.