A Reexamination of Diffusion Estimators With Applications to Financial Model Validation
Time-homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables. Recently, Stanton proposed drift and diffusion estimators based on a higher-order approximation scheme and kernel regression method. He claimed that “higher order approximations must outperform lower order approximations” and concluded nonlinearity in the instantaneous return function of short-term interest rates. To examine the impact of higher-order approximations, we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators. We show that these estimators will reduce the numerical approximation errors in asymptotic biases, but their asymptotic variances escalate nearly exponentially with the order of approximation. Simulation studies also confirm our asymptotic results. This variance inflation problem arises not only from nonparametric fitting, but also from parametric fitting. Stanton's work also postulates the interesting question of whether the short-term rate drift is nonlinear. Based on empirical simulation studies, Chapman and Pearson suggested that the nonlinearity might be spurious, due partially to the boundary effect of kernel regression. This prompts us to use the local linear fit based on the first-order approximation, proposed by Fan and Yao, to ameliorate the boundary effect and to construct formal tests of parametric financial models against the nonparametric alternatives. Our simulation results show that the local linear method indeed outperforms the kernel approach. Furthermore, our nonparametric “generalized likelihood ratio tests” are indeed versatile and powerful in detecting nonparametric alternatives. Using this formal testing procedure, we show that the evidence against the linear drift of the short-term interest rates is weak, whereas evidence against a family of popular models for the volatility function is very strong. Application to Standard & Poor 500 data is also illustrated.