Table 4.3: GARCH(l.l) parameter estimates for US stocks

Gamble

0 -I-1-1-!- --,-1-1-,-,-1

Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 - Aol --- Citibank -- General Electric --Microsoft Exxon Mobil

Figure 4.3 GARCH(l.l) volatility estimates for US stocks.

Citibank, have volatilities that are more affected by global events, such as the crises in equity markets in the autumns of 1997 and 1998.

Putting ct2 = ct2 for all / in (4.2) gives an expression for the long-term steady-state variance in a GARCH(1,1) model:

CT2 = co/(l - a - p).

(4.4)

The sum a + P must be less than 1 if the returns process is to be stationary. Only in this case will GARCH volatility term structures described in §4.4.1 converge to a long-term average level of volatility that is determined by (4.4). The sum of the a estimate and the P estimate in Table 4.2 is generally less than 1, except for Proctor and Gamble. American International Group and Walt Disney have an estimate of a + P equal to 0.993514 and 0.9954, respectively. These stocks may be better modelled by a different GARCH model, such as the integrated GARCH model described in §4.2.3, or a simple exponentially weighted moving average (EWMA) model with the smoothing constant X set equal to the estimated GARCH beta (§3.2).

The GARCH(1,1) process is the most common specification for GARCH volatility models, being relatively easy to estimate and generally having robust coefficients that are interpreted naturally in terms of long-term volatilities and short-run dynamics. However, it should be stressed that all three parameter estimates, and particularly that of , will be sensitive to the data used. Thus the choice of historic data will affect the current volatility forecasts. In particular, long-term volatility forecasts will be influenced by the inclusion of stress events in the historic data. The problems of which data to use for GARCH volatility estimation, the stability of GARCH coefficients, and the possible advantages of imposing a value for long-term volatility on the GARCH model, are all discussed in §4.3.

The sum + 3 must be less than 1 if the returns process is to be stationary. Only in this case will GARCH volatility term structures converge to a long-term average level of volatility that is determined by (4.4)

4.2.3 Integrated GARCH and the Components Model

Most financial markets have GARCH volatility forecasts that mean-revert. That is, there is a convergence in term structure forecasts to the long-term average volatility level, and by the same token the time series of any GARCH volatility forecast will be stationary. However, currencies and commodities tend to have volatilities that are not as mean-reverting as the volatility of other types of financial assets. In fact, they may not mean-revert at all. In some currency markets not only are exchange rates themselves a random walk, but the volatilities of exchange rates may also be random walks. In this case the usual stationary GARCH models will not apply.

Currencies and commodities tend to have volatilities that are not as mean-reverting as the volatility of other types of financial assets. In fact, they may not mean-revert at all

When + p = 1 we can put P = X and rewrite the vanilla GARCH (4.2) as

ct2 = + (1 - ) 2 { + Xa2 (0 X 1).

(4.5)

Market Models Table 4.4: GARCH(1, 1) parameters of the cable rate

Omega

Omega (f-stat)

Alpha

Alpha (f-stat)

Beta

Beta (f-stat)

. -06

5.31182

0.043201

10.5196

0.947753

196.236

Note that the unconditional variance (4.4) is no longer defined and term structure forecasts (§4.4.1) do not converge. Since in this case the variance process is non-stationary, (4.5) is called the integrated GARCH (I-GARCH) model.6 When = 0 the I-GARCH model (4.5) becomes an EWMA, hence EWMAs may be viewed as simple GARCH models without an and with constant term structures.

I-GARCH is often encountered in foreign exchange markets (Gallant et al., 1991). For example, using daily data on cable (sterling-US dollar) rates from 5 January 1988 to 6 October 2000, the estimated GARCH(1,1) model parameters produce an estimate of a+p = 0.991, as shown in Table 4.4. Thus the volatilities of the cable rate that are estimated by an I-GARCH model will be very similar to those estimated by an EWMA with X = 0.94, as shown in Figure 4.4a.

It is not just in foreign exchange markets that GARCH(1,1) models can become close to being integrated. For example, in Table 4.7 below (§4.3.1) the GARCH(1,1) model for the CAC equity index has an estimate of a + P = 0.993. Again, the persistence parameter is around 0.94 so the similarity of the GARCH volatility with the EWMA (X = 0.94) volatility of the CAC will be obvious, as shown in Figure 4.4b.

The currency and the equity index I-GARCH models that were mentioned above both have persistence parameters that are near 0.94, the same as the RiskMetrics daily data persistence parameter (§7.3). However, I-GARCH models can arise with other values for the persistence parameter. For example, estimating GARCH(1,1) models on the WTI spot and futures data used in Chapter 3 gives the I-GARCH(1,1) parameter estimates shown in Table 4.5. The resultant GARCH volatility estimates are shown in Figure 4.5.

The exponentially weighted moving average correlation that was shown in Figure 3.7 has a smoothing constant of 0.94, but the parameter estimates in Table 4.5 show that this value of lambda is likely to be too high if the whole data period were used. However, it is evident from Figure 4.5 that the excessively high volatility around the time of the Gulf war may be exerting a significant influence on the model parameter estimates, and if the Gulf war period were excluded from the data the beta parameter estimates are closer to

integrated time series processes are discussed in §11.1.3