Asymptotic Distributions of the Least Squares Estimator for Diffusion Processes

The asymptotic distributions of the least squares estimator of the mean reversion parameter (κ) are developed in a general class of diffusion models under three sampling schemes, namely, longspan, in-fill and the combination of long-span and in-fill. The models have an affine structure in the drift function, but allow for nonlinearity in the diffusion function. The limiting distributions are quite different under the alternative sampling schemes. In particular, the in-fill limiting distribution is non-standard and depends on the initial condition and the time span whereas the other two are Gaussian. Moreover, while the other two distributions are discontinuous at κ = 0, the in-fill distribution is continuous in κ. This property provides an answer to the Bayesian criticism to the unit root asymptotics. Monte Carlo simulations suggest that the in-fill asymptotic distribution provides a more accurate approximation to the finite sample distribution than the other two distributions in empirically realistic settings. The empirical application using the U.S. Federal fund rates highlights the difference in statistical inference based on the alternative asymptotic distributions and suggests strong evidence of a unit root in the data.