Semiparametric Estimation for Time-Inhomogeneous Diffusions

We develop two likelihood based approaches to semiparametrically estimate the time-inhomogeneous diffusion process: log penalized splines (P-splines) and the local log-linear method. Positive volatility is naturally embedded and this positivity is not guaranteed in most existing diffusion models. We investigate different smoothing parameter selection methods. Separate bandwidths are used for drift and volatility estimation. In the log P-splines approach, different smoothness for different time varying coefficients is feasible by assigning different penalty parameters. We also provide accompanying theorems for both approaches. Finally, we present a case study using the weekly three-month Treasury bill data from 1954 to 2004. We find that the log P-splines approach seems to capture the volatility dips in mid-1960s and mid-1990s the best.