(possibly unobservable) factors, driven by not perfectly correlated Brownian motions. With a slight
abuse of terminology, the former class of models is referred to as one-factor models and the latter as
stochastic volatility models.4 In particular, the paper compares generic classes of one-factor versus
stochastic volatility models, without making assumptions on the functional forms of either the drift
or the variance component.
If the null hypothesis is not rejected, then one can use the different testing and modeling pro-
cedures mentioned above, based on the maintained hypothesis of a one-factor diffusion generating
process. Conversely, if the null hypothesis is rejected, then one has to perform model diagnostics
within the class of stochastic volatility models, using for example the efficient method of moments
(e.g. Chernov, Gallant, Ghysels and Tauchen, 2003), or generalized moment tests based on the
properties of the infinitesimal generator of the diffusion (see e.g. Corradi and Distaso, 2004). For
example, one can test the validity of multi factor term structure models, suggested by e.g. Duffie
and Singleton (1997), Dai and Singleton (2000, 2002).
The suggested test statistics are based on the difference between a kernel estimator of the
instantaneous variance, averaged over the sample realization on a fixed time span, and realized
volatility. The intuition behind the chosen statistic is the following: under the null hypothesis of a
one-factor model, both estimators are consistent for the underlying integrated volatility; under the
alternative hypothesis the former estimator is not consistent, while the latter is. More precisely,
building on some recent work by Bandi and Phillips (2003) and Barndorff-Nielsen and Shephard
(2004a), it is shown that the statistics weakly converge to mixed normal distributions under the
null hypothesis and diverge at an appropriate rate under the alternative. The derived asymptotic
theory is based on the time interval between successive observations approaching zero, while the
time span is kept fixed. As a consequence, the limiting behavior of the statistic is not affected by
the drift specification. Also, no stationarity or ergodicity assumption is required.
The proposed testing procedure is derived under the assumptions that the underlying variables
are observed without measurement error and that the generating processes belong to the class of
continuous semimartingales. Therefore, the provided tests are not robust to the presence of either
jumps or market microstructure effects; more precisely, when either of the two occur, the test
tends to reject the null hypothesis, even if the volatility process is a deterministic function of the
underlying variable. However, as the test is computed over a finite time span, one can first test for
the hypotheses of no jumps and no microstructure effects, and then perform the suggested testing
procedure over a time span in which neither of the hypotheses above is rejected.
4In the stochastic volatility literature, often by one-factor model one means a model in which volatility is a function
of a single stochastic factor, driven by a Brownian motion not perfectly correlated with the one driving the underlying
economic variable or the asset price.