Estimation and specification test for diffusion models with stochastic volatility

Given the importance of continuous-time stochastic volatility models to describe the dynamics of interest rates, we propose a goodness-of-fit test for the parametric form of the drift and diffusion functions, based on a marked empirical process of the residuals built with an adapted Kalman Filter estimation. The test statistics are constructed using a continuous functional (Kolmogorov–Smirnov and Crámer–von Mises) over the empirical processes. Both, the different estimation procedures (including other alternatives as for example methods based on Markov Chain Monte Carlo or Particle filters) and the new proposed tests are compared in different simulation studies. The tests are calibrated with a specific bootstrap method using the estimation of a discrete version of the diffusion model with stochastic volatility. Finally, an application of the procedures to real data is provided.