1 Introduction
In finance the dynamic behavior of underlying economic variables and asset prices has been often
described using one-factor diffusion models, where volatility is a deterministic function of the level
of the underlying variable.1
Since determining the functional form of such diffusion processes is particularly important for
pricing contingent claims and for hedging purposes, several specification tests have been proposed,
within the class of one-factor models.
Examples include Ait-Sahalia (1996), who compares the parametric density implied by a given
null model with a nonparametric kernel density estimator. He rejects most of the commonly
employed models and argues that rejections are mainly due to nonlinearity in the drift term.2
Similar findings to those of Alt-Sahalia (1996) have been also provided by Stanton (1997) and
Jiang (1998). Durham (2003) also rejects most of the popular models; in his case rejections are
mainly due to misspecification of the volatility term. In particular, he finds implausibly high values
for the elasticity parameter in the Constant Elasticity of Variance (CEV) model, implying violation
of the stationarity assumption. Bandi (2002) applies fully nonparametric estimation of the drift
and variance diffusion terms, based on the spatial methodology of Bandi and Phillips (2003), and
finds that the drift term is very close to zero over most of the range of the short term interest rate.
Therefore, rejections of a given model seem to be due to failure of the mean reversion property
rather than to nonlinearity in the drift term. Qualitatively similar findings are obtained by Conley,
Hansen, Luttmer and Scheinkman (1997), using generalized method of moments tests based on the
properties of the infinitesimal generator of the diffusion.3
Most of the papers cited above have suggested testing and modeling procedures which are valid
under the maintained hypothesis of a one-factor diffusion data generating process. Hence, the need
of testing for the validity of the whole class of one-factor models.
This is the objective of the paper. Under minimal assumptions, the paper proposes a testing
procedure in order to distinguish between the case in which the volatility process is a deterministic
function of the level of the underlying variable and the one in which it is a function of one or more
1 Although in the financial literature there is a somewhat widespread consensus about the fact that stock prices
are better characterized by multifactor stochastic volatility models, short term interest rates are still often modeled
as a one-factor diffusion process, in which volatility is a deterministic function of the level of the variable (see e.g.
Vasicek, 1977, Brennan and Schwartz, 1979, Cox, Ingersoll and Ross, 1985, Chan, Karolyi, Longstaff and Sanders,
1992, Pearson and Sun, 1994).
2Alt-Sahalia (1996) does not reject a generalized version of the Constant Elasticity of Variance model. His results
have been revisited by Pritsker (1998), who points out the sensitivity of Alt-Sahalia’s test to the degree of dependency
in the short interest rate process.
3See also the comprehensive review on estimation of one-factor models by Fan (2003).