Palgrave Handbook of Econometrics: Applied Econometrics

George Dotsis, Raphael N. Markellos and Terence C. Mills 959

Whenxtis fully observed the likelihood function for estimating the parameters
is given by:

p(x;θ)=p

(

x 1 ;θ

)T∏−^1

t= 1

p

(

xt+ 1 |xt;

)

, (19.10)

wherep(x;)is the joint density function andp

(

x 1 ;

)
is the unconditional den-
sity. In many applications (for example, Aït-Sahalia and Kimmel, 2007) the density
of the initial observation is omitted since, asymptotically, it does not affect the
efficiency of the parameter estimates. Under this approach the difficulty lies in
the fact that the conditional densities,p

(

xt+ 1 |xt;

)
, of most stochastic volatility
models cannot be derived in closed form. In principle, the transitional densities
can be obtained numerically by solving the corresponding Fokker–Planck equation
(for example, Lo, 1988). The conditional densities can also be obtained by Fourier
inversion of the conditional characteristic function if the model belongs to the
affine class or by other approximating methods.
When volatility is not directly observed it has to be integrated out of the like-
lihood function. In this case, the corresponding marginal likelihood function is
given by:

p(y;θ)=

∫

RT

p(y,V;)dV=

∫

RT

p(y|V;)p(V;)dV. (19.11)

The dimension of the integral depends on the sample size, so that direct evaluation
of the likelihood function is difficult. Moreover, when latent volatility is integrated
out, the asset price alone is non-Markov due to the presence of correlation and the
conditional density of the next period’s return depends on the entire set of pre-
vious observations,Yt− 1 =

{

yt− 1 ,...,y 1

}

. The likelihood can be approximated by
simulation methods such as MCMC or estimation can be implemented by avoiding
the likelihood altogether and resorting to methods of moments estimation such as
EMM or generalized method of moments (GMM).
In the next sections we discuss estimation methods when volatility is treated as
unobserved and we then extend the analysis to the case where volatility is extracted
from option prices.

19.3.1 Simulation-based inference

19.3.1.1 Efficient method of moments

The EMM, developed by Bansalet al. (1993, 1995) and Gallant and Tauchen (1996),
is an extension of the simulated method of moments (SMM) of Duffie and Singleton
(1993). EMM avoids direct computation of the likelihood function and resorts to
efficient estimation via GMM and a cautious selection of the moment conditions.
The parameter estimates are minimum chi-squared estimators and the optimized
chi-squared criterion can be used to evaluate the statistical fitting of the various
stochastic volatility models. Hence a significant advantage of the EMM procedure
is that it allows empirical comparison between non-nested specifications, such as,
for example, SV2 and SV5. However, EMM is computationally demanding and,
as with all GMM procedures, it depends on the optimal choice of moment condi-
tions. Estimation of stochastic volatility models with EMM has been undertaken by