  We propose a test for model specification of a parametric diffusion process
based on a kernel estimation of the transitional density of the process. The
empirical likelihood is used to formulate a statistic, for each kernel
smoothing bandwidth, which is effectively a Studentized $L_2$-distance between
the kernel transitional density estimator and the parametric transitional
density implied by the parametric process. To reduce the sensitivity of the
test on smoothing bandwidth choice, the final test statistic is constructed by
combining the empirical likelihood statistics over a set of smoothing
bandwidths. To better capture the finite sample distribution of the test
statistic and data dependence, the critical value of the test is obtained by a
parametric bootstrap procedure. Properties of the test are evaluated
asymptotically and numerically by simulation and by a real data example.
