Table 4.6: Asymmetric GARCH(1, 1) models for major equity indices

Parameter France Germany Japan UK US

where g(-) is an asymmetric response function defined by9

g(z,) = Xz, + ip(\z,\ - \f2jn).

The standard normal variable z, is the standardized unexpected return s,/ct,. When if > 0, and X < 0 negative shocks to returns (z, ] < 0) induce larger conditional variance responses than positive shocks.

Exponential GARCH is difficult to use for volatility forecasting because there is no analytic form for the volatility term structure

Many GARCH models have to place non-negativity constraints on the parameters to avoid generating negative variances, even though this may unduly restrain the model dynamics. The E-GARCH model eliminates the need for such constraints by formulating the conditional variance in logarithmic terms. Several studies have found that the exponential GARCH model fits financial data very well, often better than other GARCH models. Even without significant leverage effects, the logarithmic specification appears to have considerable advantages (Taylor, 1994; Heynen et al., 1994; Lumsdaine, 1995). But unfortunately, exponential GARCH is difficult to use for volatility forecasting because there is no analytic form for the volatility term structure.

The asymmetric GARCH or A-GARCH model of Engle and Ng (1993) is easier to estimate than exponential GARCH and its volatility term structure forecasts may be generated in a simple analytic way (§4.4.1). The A-GARCH model has the conditional variance equation

oj = + a(s, ! - X)2 + (3a2 ! (for > 0, a, > 0). (4.9)

Asymmetric GARCH(1,1) models for the five major equity indices using a very long period of daily data (from 17 February 1981 to 28 April 1995) have been estimated, and the results are shown in Table 4.6 and Figure 4.6. Parameter estimates and r-ratios in parentheses may be obtained using the PcGive demo on the CD, or by using the BHHH algorithm in RATS (Regression Analysis of Time Series, an econometrics package produced by Estima and available from http: www.estima.com).

9The last term (r, - J(2 )) is the mean deviation of z, since y/(2/n) = £(z,).

5 -I-1-,-,-<-!-1

Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95

- A-GARCH(1,1) - GARCH(1,1)

Figure 4.6 Asymmetric and symmetric GARCH volatility estimates for the S&P 500 index.

The estimates of the leverage coefficient X are of a similar order of magnitude as daily returns, but they are much less significant than the other GARCH coefficient estimates. This implies that the volatility estimates from these asymmetric GARCH(1,1) models will be fairly close to the volatility estimates from a symmetric GARCH model, as shown in the figure. The volatility forecasts made from an A-GARCH model differ from the forecasts made with a symmetric GARCH(1,1) model, and this is discussed in more detail in §4.4.1.

For option pricing and hedging (§4.4.2), Duan (1995) advocates the non-linear asymmetric GARCH or N-GARCH model:

r, = r- 0.5g2 + ofc,

aj = + arsl - 8 - Xf + (3a2 , (4.10)

= + X.

When volatility is stochastic the perfect markets assumption that is necessary for a risk-neutral probability measure no longer holds, but Duan shows that a form of local risk neutrality does hold if prices follow this model. Thus option prices can be calculated as discounted expected values under a unique risk neutral probability measure, in the usual way. An empirical example of the model is given in §4.4.3, where we discuss the work of Duan (1996) on using this model to fit a volatility smile surface.

Many other asymmetric GARCH models have been developed, notably the asymmetric power GARCH of Ding et al. (1993), the GJR model of Glosten et al. (1993), the threshold GARCH of Zakoian (1994), and the quadratic GARCH model of Sentana (1995). The ability of these models to capture the dynamics of conditional volatility is compared in Hagerud (1997).

4.2.5 GARCH Models for High-Frequency Data

Most of the GARCH models that are used in practice assume that the errors in the conditional mean equation, that is, the unexpected part of returns, are conditionally normally distributed. Nevertheless, the unconditional returns distributions generated by a normal GARCH model will have fat tails, because the conditional volatility is time-varying.

GARCH models for high- The normal GARCH models that have been described in this section are frequency data that have commonly used for modelling the volatility of daily returns. However, there is error distributions some evidence to suggest that normal GARCH models cannot capture the full defined by mixtures of extent of excess kurtosis in high-frequency data (Terasvirta, 1996).10 In this normal distributions are case a leptokurtic distribution, such as a Student / error distribution, could be required assume(j for the error process in the conditional mean equation (Bollerslev, 1987; Baillie and Bollerslev, 1989b; Engle and Gonzalez-Riviera, 1991). The absence of analytic derivatives makes optimization rather slow and unreliable, nevertheless t-GARCH (where the t refers to Students /-distribution) is available in many statistical packages such as MFIT (Oxford University Press) and from Ken Kroners GAUSS programs at Rob Engles home page: http: weber.ucsd.edu/~mbacci/engle/index data.html.

Other non-normal GARCH models can be programmed just by changing the likelihood function that is given in §4.3.2. In tandem with other recent developments on modelling fat tails in financial data (§10.3), GARCH models for high-frequency data that have error distributions defined by mixtures of normal distributions are required. However, these have yet to be incorporated in standard statistical packages.

Is it better to predict high-frequency volatility

with a low-frequency model, or a low-frequency volatility with a high-frequency model?

Intra-day data are useful for modelling very short-term volatility but there are some additional problems here, to do with time aggregation. In §13.1.3 it is shown that ARCH effects in very high-frequency data are far stronger than one might suppose from the analysis of daily data. The question arises whether a GARCH model estimated on intra-day returns will predict the same daily volatility as a GARCH model estimated on daily returns; and whether a GARCH model estimated on daily returns will predict the same intra-day volatility as a GARCH model estimated on intra-day returns. If not, is it better to predict high-frequency volatility with a low-frequency model, or a low-frequency volatility with a high-frequency model?

Drost and Nijman (1993) use symmetric GARCH models to imply conditional heteroscedasticity at both a lower frequency than the data used to estimate the model, and a higher frequency than the model data. Muller et al. (1997) also compare GARCH volatilities that are estimated at different frequencies. They

10High-frequency foreign exchange data are highly leptokurtic. For example, in §10.1.1 the excess kurtosis of 1-hourly returns on the DEM-LSD exchange rate is calculated as 8.34. Equity returns can be very leptokurtic even at the daily frequency (see s*4.4.2l.