xxiv Editors’ Introduction
a number of other attractive specifications; for example, jump diffusions, affine
diffusions, affine jump diffusions and non-affine diffusions. In motivating alterna-
tive specifications, Dotsiset al. note some key empirical characteristics in financial
markets that underlie the rationale for stochastic volatility models, namely fat
tails, volatility clustering, leverage effects, information arrivals, volatility dynam-
ics and implied volatility. The chapter then continues by covering such issues as
specification, estimation and inference in stochastic volatility models. A compar-
ative evaluation of five models applied to the S&P 500, for daily data over the
period 1990–2007, is provided to enable the reader to see some of the models “in
action.”
One of the most significant ideas in the area of financial econometrics is that the
underlying stochastic process for an asset price is a martingale. Consider a stochas-
tic processX=(Xt,Xt− 1 ,...), which is a sequence of random variables; then the
martingale property is that the expectation (at timet−1) ofXt, conditional on the
information setIt− 1 =(Xt− 1 ,Xt− 2 ,...),isXt− 1 ; that is,E(Xt|It− 1 )=Xt− 1 (almost
surely), in which case,Xis said to be a martingale (the definition is sometimes
phrased in terms of theσ-field generated byIt− 1 , or indeed some other “filtra-
tion”). Next, define the related processY=(Xt,Xt− 1 ,...); thenYis said to be a
martingale difference sequence (MDS). The martingale property forXtranslates to
the property forYthatE(Yt|It− 1 )=0 (see, for example, Mikosch, 1998, sec. 1.5).
This martingale property is attractive from an economic perspective because of its
link to efficient markets and rational expectations; for example, in terms ofX, the
martingale property says that the best predictor, in a minimum mean squared error
(MSE) sense, ofXtisXt− 1.
In Chapter 20, J. Carlos Escanciano and Ignacio Lobato consider tests of the
martingale difference hypothesis (MDH). The MDH generalizes the MDS condition
toE(Yt|It− 1 )=μ, whereμis not necessarily zero; it implies that past and current
information (as defined inIt) are of no value, in an MSE sense, in forecasting future
values ofYt. Tests of the MDH can be seen as being translated to the equivalent
form given byE[(Yt−μ)w(It− 1 )], wherew(It− 1 )is a weighting function. A useful
means of organizing the extant tests of the MDH is in terms of the type of functions
w(.)that are used. For example, ifw(It− 1 )=Yt−j,j≥1, then the resulting MDH
test is ofE[(Yt−μ)Yt−j]=0, which is just the covariance betweenYtandYt−j.
This is just one of a number of tests, but it serves to highlight some generic issues.
In principle, the condition should hold for allj≥1 but, practically,jhas to be
truncated to some finite value. Moreover, this is just one choice ofw(It− 1 ), whereas
the MDH condition is not so restricted. Escanciano and Lobato consider issues such
as the nature of the conditioning set (finite or infinite), robustifying standard test
statistics (for example, the Ljung–Box and Box–Pierce statistics), and developing
tests in both the time and frequency domains; whilst standard tests are usually
of linear dependence, for example autocorrelation based tests, it is important to
consider tests based on nonlinear dependence. To put the various tests into context,
the chapter includes an application to four daily and weekly exchange rates against
the US dollar. The background to this is that the jury is out in terms of a judgment
on the validity of the MDH for such data; some studies have found against the