Art of Likelihood Maximization
Tohru Ozaki · 2012
We learned in Section 10.1 that with the innovation approach, we can write down the likelihood of any nonlinear state space model by transforming the time series, whether Gaussian or non-Gaussian, into white Gaussian innovations. Writing down the likelihood function of the Gaussian innovations may be easy, but it does not mean that its maximization with respect to the parameters is easy. In fact, numerical maximization of the likelihood of state space models is not so simple and easy. It needs some experience and skill even for the case of linear state space models. Numerical maximization of the likelihood is an important part of the work of the state space modeling of time series. What is difcult to nd in the most textbooks of time series analysis are “useful instructions” for obtaining proper maximum likelihood estimates after writing down the likelihood, that is, useful instructions and guidance for obviating numerical difculties during the stage of numerical optimization of the likelihood of the state space model.