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

by BPT (2001). The simplest model specification is the GJR (see Glosten,
Jagannathan and Runkle, 1993, Engle and Ng, 1991) process,

r_t = μ + ε_t    ε_t = hh_tz_t    z_t ~ N (0,1)             (1)

h_t = α₀ + ɑiɛt-i + α2St-ι≡t-ι + βht-ι

that captures the asymmetric relationship between volatility and returns, with
s_t-ι taking the value of unity when ε_t-ι < 0 and 0 otherwise. This process
nests the standard GARCH model when «₂ = 0.

Following BPT (2001), standard GARCH style models are augmented by the
inclusion of RV⁴. The most general specification of a GARCH process including
RV is given by,

rt = μ + εt    εt = y∕h^_tz_t    zt ~ N (0,1)             (2)

^ht = ^h1t ^{+ h}2t

^hιt = «o ^{+ βh}t-ι ⁺ '¹7 ι ^{+ α}2^s∕. i^ε/² i

^h2t = 7 1^h2t-1 ⁺ 7 2^½-1

and is defined as the GJRRVG model. This allows for two components to
contribute to volatility, with each component potentially exhibiting persistence.
This specification nest various other models, GJRRV if 7i = 0, GARCHRV if
7i = «2 = 0, GJR if 71 = 72 = 0 and GARCH if 71 = 72 = «2 = 0.

Parameter estimates for the GARCH and GJR models are similar to those
commonly observed for GARCH models based on various financial time series,
⁴While BPT (2001) also extend the GJR model to include the VIX index, this is not
relevant to the current study as it is the goal to seperate forecast performance of IV and
MBF.

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