Handbook of Economic Forecasting part 44. Research on forecasting methods has made important progress over recent years and these developments are brought together in the Handbook of Economic Forecasting. The handbook covers developments in how forecasts are constructed based on multivariate time-series models, dynamic factor models, nonlinear models and combination methods. The handbook also includes chapters on forecast evaluation, including evaluation of point forecasts and probability forecasts and contains chapters on survey forecasts and volatility forecasts. Areas of applications of forecasts covered in the handbook include economics, finance and marketing | 404 A. Harvey serial correlation in volatility is by means of the GARCH class in which it is assumed that the conditional variance of the observations is an exact function of the squares of past observations and previous variances. An alternative approach is to model volatility as an unobserved component in the variance. This leads to the class of stochastic volatility SV models. The topic is covered Chapter 15 by Andersen et al. in this Handbook so the treatment here will be brief. Earlier reviews of the literature are to be found in Taylor 1994 and Ghysels Harvey and Renault 1996 while the edited volume by Shephard 2005 contains many of the important papers. The stochastic volatility model has two attractions. The first is that it is the natural discrete time analogue though it is only an approximation of the continuous time model used in work on option pricing see Hull and White 1987 and the review by Hang 1998 . The second is that its statistical properties are relatively easy to determine and extensions such as the introduction of seasonal components are easily handled. The disadvantage with respect to the conditional variance models of the GARCH class is that whereas GARCH can be estimated by maximum likelihood the full treatment of an SV model requires the use of computer intensive methods such as MCMC and importance sampling. However these methods are now quite rapid and it would be wrong to rule out SV models on the grounds that they make unreasonably heavy computational demands. . Basic specification and properties The basic discrete time SV model for a demeaned series of returns yt may be written as yt otet a e -5h et t - IID 0 1 t 1 . T 180 where a is a scale parameter and ht is a stationary first-order autoregressive process that is ht 1 ht nt nt - IID 0 a2 181 where nt is a disturbance term which may or may not be correlated with et. If et and nt are allowed to be correlated with each other the model can pick up the kind of asymmetric behaviour .
