Bootstrap Specification Tests for Diffusion Processes

This paper introduces bootstrap specification tests for diffusion processes. In the one-dimensional case, the proposed test is closest to the non parametric test introduced by Ait-Sahalia (1996), in the sense that both procedures determine whether the drift and variance components of a particular continuous time model are correctly specified. However we compare cumulative distribution functions, while Ait-Sahalia compares probability densities. In the multidimensional and/or multifactor case, the proposed test is based on the comparison of empirical CDF of the actual data and the empirical CDF of the simulated data. The limiting distributions of both tests are functionals of zero mean Gaussian processes with covariance kernels that reflect data dependence and parameter estimation error (PEE). In order to obtain asymptotically valid critical values for the test, we use an empirical process version of the block bootstrap which properly accounts for the contribution of PEE. An example based on a simple version of Cox, Ingersol and Ross (1985) square root process is outlined and related Monte Carlo experiments are carried out. These experiments suggest that the test has good finite sample properties, even for samples as small as 400 observations when tests are formed using critical values constructed with as few as 100 bootstrap replications.