Modified two-step method for stochastic differential equation's parameter estimation

Abstract

A previous study introduced two-step method of Stochastic Differential Equations (SDEs) for estimating the parameters of SDEs models where the selection of optimal knot is required when regression spline is used in the first step of this method. However, the choice of optimal knot is considered only for single optimal knot since it is suitable for a selected case study. Thus, modified two-step method of SDEs as an alternative to the limitation and computational difficulties in choosing optimal knot is proposed. A new non-parametric estimator which is Nadaraya- Watson (NW) kernel regression estimator is applied to replace regression splines in the first step of modified two-step method. The NW kernel regression model is later untilised in the second step to estimate the parameters of one-dimensional linear SDEs models. The outcome indicates a modification of two-step method providing better estimates for SDEs model compared to two-step method. The performance of modified two-step method is compared with the well-known established classical methods, particularly Simulated Maximum Likelihood Estimation (SMLE) and Generalised Method of Moments (GMM) by using simulated data. Results indicate GMM method is the best parameter estimation method since it outperforms other methods in terms of percentage of accuracy and computational times. Nevertheless, the differences of percentage of accuracy were not too great, and therefore, modified two-step method could be considered comparable for practical purposes. The computational time of modified two-step method is faster than SMLE method although not as good as GMM method. This however verifies that modified two-step method serves as a good alternative to the existing classical methods because it excludes the difficulty of finding transition density, moment functions and optimal knot.