We propose an optimal test procedure for testing the marginal density
functions of a class of nonlinear diffusion processes. The proposed
test is not only an optimal one but also avoids undersmoothing. An
adaptive test is constructed, and its asymptotic properties are
investigated. To show the asymptotic properties, we establish some
general results for moment inequalities and asymptotic distributions
for strictly stationary processes under the α-mixing condition.
These results are applicable to some other estimation and testing of
strictly stationary processes with the α-mixing condition. An
example of implementation is given to demonstrate that the proposed
model specification procedure is applicable to economic and financial
model specification and can be implemented in practice. To ensure the
applicability and implementation, we propose a computer-intensive
simulation scheme for the choice of a suitable bandwidth involved in
the kernel estimation and also a simulated critical value for the
proposed adaptive test. Our finite sample studies support both the
proposed theory and the simulation procedure.The authors thank the co-editor and three anonymous referees
for their constructive comments and suggestions. The first author also
thanks Song Xi Chen for some constructive suggestions, in particular the
suggestion on using the local linear form instead of the
Nadaraya–Watson kernel form in equation (2.6), and Yongmiao Hong for
sending a working paper. The authors acknowledge comments from seminar
participants at the International Chinese Statistical Association Meeting
in Hong Kong in July 2001, the Western Australian Branch Meeting of the
Statistical Society of Australia in September 2001, the University of
Western Australia, and Monash University. Thanks also go to the Australian
Research Council for its financial support.