Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions

The aim of this paper is to find approximate log-transition density functions for multivariate time-inhomogeneous diffusions in closed form. There are many empirical evidences supporting that the data generating process governing dynamics of many economics variables might vary over time because of economic climate changes or time effects. One possible way to explain the time-dependent dynamics of state variables is to model the drift or volatility terms as functions of time t as well as state variables. A way to find closed-form likelihood expansion for a multivariate time-homogeneous diffusion has been developed by Ait-Sahalia (2008). This research is built on his work and extends his results to time-inhomogeneous cases. We conduct Monte Carlo simulation studies to examine performance of the approximate transition density function when it is used to obtain ML estimates. The results reveal that our method yields a very accurate approximate likelihood function, which can be a good candidate when the true likelihood function is unavailable as is often the case.