Approximate maximum likelihood estimation of a threshold diffusion process

Abstract

In order to estimate the parameters of a two-regime threshold diffusion process with discretely sampled data, an approximate maximum likelihood method (AMLE) based on approximating the log-likelihood function of the observations is proposed. Both the drift and the diffusion terms are allowed to be either linear or non-linear. In order to choose the most appropriate among these four possibilities, three information criteria are employed. Further, a likelihood ratio test can help to determine whether threshold effects are present. Via simulations, the finite sample performance of the proposed AMLE is compared to an alternative quasi-likelihood estimator and the finite sample performance of the information criteria as well as the likelihood ratio test are studied. Finally, the efficacy of our approach is demonstrated with two financial time series.