Palgrave Handbook of Econometrics: Applied Econometrics

George Dotsis, Raphael N. Markellos and Terence C. Mills 965

to derive the conditional moments of integrated volatility. For example, the first

moment has the general formEt

(

IVt,T

)
=a(τ,)Vt+b(τ,). Bollerslev and Zhou
(2002) derive the first two conditional moments of integrated volatility and apply a
GMM procedure. In order to identify the correlation they also derive in closed form

the cross-moment,Et

(

yTIVt,T

)

. Barndorff-Nielsen and Shephard (2002) develop
a QML procedure based on the time series of realized volatility. Chourdakis and
Dotsis (2008) estimate non-affine specifications using maximum likelihood and a
Markov chain approximation procedure.

19.4 Empirical comparison of volatility processes

In this section we provide an empirical comparison of the models described in
section 19.2. A comparison of the econometric methods outlined is section 19.3
is beyond the scope of this chapter. Instead, we estimate the volatility processes
autonomously using an implied volatility index.^10 This facilitates estimation but
does not allow us to make inferences on the joint dynamics of asset returns and
volatility. For reasons explained in Jones (2003) and Bakshi, Ju and Ou-Yang (2006),
we consider as a proxy for volatility the implied volatility index VXO. Hence, we

setVt≡VXO^2 tover the period 1990–2007, a total of 4,535 daily observations.
The parameters of the various processes are estimated by maximum likelihood,
which requires the conditional density functionf[V(t+τ)|V(t),](τ> 0 )of the
processVt, whereτdenotes the sampling frequency of observations (daily in our

application). For a sample{Vt}Tt= 1 , the log-likelihood function that is maximized
is given by:

&=max



T∑−τ

t= 1

log

(

f

(

Vt+τ

)∣∣

∣Vt,

)

. (19.25)

The standard errors of the estimates are retrieved from the inverse Hessian, eval-
uated at the estimates. For SV1, SV2 and SV5 the conditional density is known in
closed form (see Dotsis, Psychoyios and Skiadopoulos, 2007; Psychoyios, Dotsis
and Markellos, 2007). However, for SV3 and SV4, the transition density does not
have a closed-form solution. The density of SV4 is obtained by the approximation
method of Aït-Sahalia (1999, 2002). The transition density of the SV3 model is
obtained by Fourier inversion of the characteristic function (see Singleton, 2001;
Dotsis, Psychoyios and Skiadopoulos, 2007). The Fourier inversion of the character-
istic function provides the required conditional density functionf[V(t+τ)|V(t)]as:

f[Vt+τ|Vt,]=

1

π

∫∞

0

Re[e−isVt+τφ

(

iω,Vt,τ;

)

]dω. (19.26)

Table 19.1 shows the ML estimates for VXO. For each of the processes, the estimated
parameters (annualized), thet-statistics (within parentheses), the AIC (Akaike
information criterion) and BIC (Bayesian information criterion), and the maxi-
mized log-likelihood values (LL) are reported. Likelihood ratio tests were also used
to compare nested models: as these supported the ranking obtained from AIC, BIC
and LL, they are not reported.