Testing for One-Factor Models versus Stochastic Volatility Models

The rest of this paper is organized as follows. In Section 2, the testing procedure is outlined
and the relevant limit theory is derived. Section 3 reports the findings from a Monte Carlo exercise,
in order to assess the finite sample behavior of the proposed tests. Concluding remarks are given
in Section 4. All the proof are gathered in the Appendix.

In this paper, -^p→, -^d→ and -^a.→^s. denote respectively convergence in probability, in distribution
and almost sure convergence. We write 11.∣ for the indicator function, |_wJ for the integer part of
w, I j for the identity matrix of dimension J and Z ~ MN (∙, ∙) to denote that the random variable
Z is distributed as a mixed normal.

2 Testing for One-Factor vs Stochastic Volatility Models

2.1 Set-Up

As discussed above, our objective is to device a data driven procedure for deciding between one-
factor diffusion models and stochastic volatility models, under minimal assumptions.

We consider the following class of one-factor diffusion models

d Xt = μ ( Xt )d t + σt d W1 ,t

σt = σ (Xt)                                              (1)

and the following class of stochastic volatility models

d Xt = μ ( Xt )d t + σt d W1 ,t
σ_t² = g(ft)

dft = b(ft)dt+σ1(ft)dW2,t,                                 (2)

where ft is typically an unobservable state variable driven by a Brownian motion, W2,t, possibly but
not perfectly correlated with the Brownian motion driving X_t , thus allowing for possible leverage
effects.

The models in (1) encompass the class of parametric specifications analyzed by Alt-Sahalia
(1996), and they also allows for generic nonlinearities. The models in (2) include the square root
stochastic volatility of Heston (1993), the Garch diffusion model (Nelson, 1990), the lognormal
stochastic volatility model of Hull and White (1987) and Wiggins (1987), and are also related to
the class of eigenfuction stochastic volatility models of Meddahi (2001). Note that ft may be a
multidimensional process, thus allowing for multifactor stochastic volatility processes. Also, the
one-factor model may be possibly nested within the stochastic volatility model, in the sense that we
can allow for the specification σ_t² = σ² (Xt) g(ft). Andersen and Lund (1997) and Durham (2003)

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