Are combination forecasts of S&P 500 volatility statistically superior?

RV and the latent volatility process and represents the incremental information
contained in the RV series. It is noted that equation (5) nests the standard SV
model as a special case by imposing the restriction 7 = O.⁶ The SV models
appear to capture the same properties of the volatility process as the GARCH-
type models. In both instances, volatility is found to be a persistent process,
and the inclusion of RV as an exogenous variable is important.

In addition to GARCH and SV approaches, it is possible to utilise estimates

of RV to generate forecasts of future volatility. These forecasts can be generated
by directly applying time series models, both short and long memory, to daily
measures of RV, RV_t. In following ADBL (2003) and Koopman et al. (2004)
ARMA(2,l) and ARFIMA(l,d,0) processes are utilised. In its most general
form an ARFIMA(p,d,q) process may be represented as

^A(L) (1 — L)^d (x_t — μ_xt) = B(L) ε_t,                   ⁽⁶⁾

where A(L) and B(L) are coefficient polynomials of order p and q. The degree
of fractional integration is d. A general ARMA(p,q) process applied to x_t is
defined under the restriction of d =O. In the context of this work ARMA(2,l)
and ARFIMA(l,d,0) were estimated with x_t = √R½ and x_t = In (√RVt) [RB:
Which one doe we use??]. These variable transformations are applied to
reduce the skewness and kurtosis of the observed volatility data (Andersen
et al., 2003). In the ARMA (2,l) case, parameter estimates reflect the com-
mon feature of volatility persistence. Allowing for fractional integration in the
⁶Numerous estimation techniques may be applied to the model in equations 3 and 4 or
5. In this instance the nonlinear filtering approach proposed by Clements, Hurn and White
(2003) is employed. This approach is adopted as it easily accommodates exogenous variables
in the state equation.

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