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

reflecting strong volatility persistence, and are qualitatively similar to those
reported in BPT (2001)⁵. Furthermore, allowing for asymmetries in conditional
volatility is important, irrespective of the volatility process considered.

While not considered by BPT 2001, this study also proposes that an SV

model may be used to generate forecasts. SV models differ from GARCH models
in that conditional volatility is treated as an unobserved variable, and not as
a deterministic function of lagged returns. The simplest SV model describes
returns as

r_t = μ + σ_t u_t    u_t ~ N (0,1)                      (3)

where σ_t is the time t conditional standard deviation of r_t. SV models treats
σ_t as an unobserved (latent) variable, following its own stochastic path, the
simplest being an AR(1) process,

!°g(σ?) = a + βlog(σ^_-1)+ w_t    Wt ~ N(0,σ‰).          (4)

Similar to Koopman et al. (2004), this study extends a standard volatility

model to incorporate RV as an exogenous variable in the volatility equation.

The standard SV process in equation (4) can be extended to incorporate RV in

the following manner and is denoted by SVRV

∣°g(^σC = ^{α +} β ^log ⁽^"t—⁺ 70M≡t-J - ¾-i[^log^LJD ^{+ w}t∙    ⁽⁵⁾

Here, RV enters the volatility equation through the term log(BFt-1) — T^,_t-i

[log (σ^_-ι)]. This form is chosen due to the high degree of correlation between

⁵As the models discussed in this section will be used to generate 2,460 recursive volatility
forecasts (see Section 3) reporting parameter estimates is of little value. Here we will merely
discuss the estimated model properties qualitatively. Parameter estimates for the recursive
windows and the full sample are available on request.

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