Palgrave Handbook of Econometrics: Applied Econometrics

960 Continuous-Time Stochastic Volatility Models

Gallant, Hsieh and Tauchen (1997), Andersen and Lund (1997), Andersen, Benzoni
and Lund (2002), Chernov and Ghysels (2000) and Chernovet al.(2003), while
Andersen, Chung and Sørensen (1999) explore the efficiency of EMM estimators
in a Monte Carlo study.
The starting point of the EMM procedure is to approximate the conditional den-
sity of the data as closely as possible with a discrete-time auxiliary model. This
auxiliary model is not related to any particular stochastic-volatility model, its
purpose being to capture the probabilistic structure of the data and to provide
the moment conditions that must be satisfied by a postulated stochastic-volatility
model, which in EMM applications is called the structural model. The auxiliary
model usually involves an ARMA-GARCH specification and a semi-nonparametric
density based on Hermite polynomials.
Suppose that the conditional density of the auxiliary model isfK(yt|Yt− 1 ;.a),
whereαis the parameter vector of the auxiliary model. The parameters can be
estimated by QML as follows:

a ̃=arg max

1

n

∑n

t= 0

log

[

fK(yt|Yt− 1 ;a

]

. (19.12)

The ML estimatesa ̃ensure that the quasi-score function satisfies the first-order
conditions:
1
T

∑n

t= 0

∂

∂a

lnfK

(

yt|Yt− 1 ;a ̃

)

=0. (19.13)

In the second stage, EMM uses the expectation of the score functions of the auxil-
iary model as the moment conditions that deliver the estimatesof the structural
model. The expectation is taken under the probability measure,P

(

Yt;

)
, of the
structural model:

m

(

θ,a ̃

)

=EP

[

∂lnfK(Yt;a ̃)

∂a

]

=

∫

∂lnfK(Yt;a ̃)

∂a

dP(Yt;). (19.14)

Given a set of parameters, the expectation is calculated numerically, using a long
simulated series,yˆN(θ), from the structural model, as:^7

m ̃

(

θ,a ̃

)

=

1

N

∑N

t= 1

∂lnfK(yˆt(θ)|Yˆt− 1 (θ); ̃a)

∂a

, (19.15)

and, asN→∞,m ̃

(

θ,a ̃

)

→m

(

θ,a ̃

)

. The EMM estimator is then obtained by
minimizing the quadratic function:

ˆ=arg min



mN(,a ̃)′WTmN(,a ̃), (19.16)

where the weighting matrixWTis a consistent estimate of the inverse asymptotic
covariance matrix of the auxiliary score function. Gallant and Tauchen (1996) show
how to derive the weighting matrix and prove that the parameter estimates of the
structural model are asymptotically normally distributed. Latent volatility can be
filtered out using the reprojection method of Gallant and Tauchen (1998).