Adaptive Testing In Continuous-Time Diffusion Models

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.