Testing for One-Factor Models versus Stochastic Volatility Models

4 Concluding remarks

This paper provides a testing procedure which allows to discriminate between one-factor and
stochastic volatility models. Hence, it allows to distinguish between the case in which the volatil-
ity of an asset is a function of the asset itself (and therefore the volatility process is Markov and
predictable in terms of its own past), and the case in which it is a diffusion process driven by a
Brownian motion, which is not perfectly correlated with the Brownian motion driving the asset.
The suggested test statistics are based on the difference between a kernel estimator of the instan-
taneous variance, averaged over the sample realization on a fixed time span, and realized volatility.
The intuition behind is the following: under the null hypothesis of a one-factor model, both es-
timators are consistent for the true underlying integrated (daily) volatility; under the alternative
hypothesis the former estimator is not consistent, while the latter is. More precisely, we show that
the proposed statistics weakly converge to well defined 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 specifi-
cation. Also, no stationarity or ergodicity assumption is required. The finite sample properties of
the suggested statistic are analyzed via a small Monte Carlo study. Under the null hypothesis, the
asset process is modelled as a version of the Cox, Ingersoll and Ross (1985) model with a mean
reverting component in the drift. Thus, volatility is a square root function of the asset itself. Under
the alternative, the asset and volatility processes are generated according to a stochastic volatility
model, where volatility is modelled as a square root diffusion. The empirical sizes and powers of
the proposed tests are reasonably good across various m/n ratios.

15

<<   14   15   16   17   18   19   20   >>

More intriguing information