# On Global Optimization of the Likelihood Function for Linear Time Series Models

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3898-1980.

dept.math [at] unitn.it

**Luogo: **Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Sala Seminari "-1"

**Ore**: 16:30

### Relatore:

**Bernard Hanzon**(Dept Mathematics, University College Cork, Ireland)

### Abstract:

We present an approach to find the global optimum of a linear time series model in state space form. By firstly applying a partial optimization step to the likelihood function the problem can be reduced to a problem of optimization of the likelihood over a set which has compact closure. By applying a regularization the resulting problem is that of finding the optimum of a Lipschitz continuous function on a compact set which is known to have a (constructive) solution. Techniques from the theory of parametrization of linear state space systems and some basic techniques from differentiable manifold theory are used to obtain these results. Furthermore the surprising fact is shown that the "crude" maximum likelihood estimator (without regularization and compactification) does not exist in the sense that the likelihood function has an infinite, typically unattained, supremum. Comments will be made about the effect of the regularization used on the outcome.

**Contact person: **Stefano Bonaccorsi