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State-space models are represented by a state equation and an observation equation (or system of equations to be more precise). These equations are parametarized by components including a transition matrix (FF in some notation) and GG respectively.

These component matrices can have large dimensions. Indeed, the log-likelihood function is a non-convex function and its maximization is difficult. Typical optimization methods used include Newton-Raphson and EM algorithm.

I am new to using state-space models and am looking for quick and robust optimization procedures used by practitioners of state-space models. Is there any literature or best practices regarding the best ways to estimate these matrices?

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If you consider gaussian state-space models, maximum likelihood is conceptually straightforward, and several packages in R (including dlm, kfas, or FKF, to name a few) include routines for doing so: you can find a short review here (disclaimer: I am the author) and a monographic issue of the Journal of Statistical Software covering state-space models and their estimation here.

Newton-Raphson methods are generally used and have been found to be generally faster (and are indeed used in the packages mentioned). The EM algorithm is quite well suited to the problem, but slower (see e.g. Shumway and Stoffer(2010), Sec. 6.3).

A particular topic which is sometimes neglected is the instability of the most naive implementations of the Kalman filter: on this it is useful (although very old) this monography by Bierman and you may find useful books like Grewal. There are also quite a lot of papers, that you will find easily googling for "square root filters" or similar expressions.

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Thanks! I look forward to working thru your paper. Is any arbitrary optimization procedure "valid" so long as it is not tapped in local minima -- for example, may I use my favorite non-convex optimizer including genetic algorithms or should I stick to the conventional methods? – Quant Guy Sep 9 '11 at 13:25
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I think genetic algorithms might do well (although they tend to be quite slow, in my limited experience). Local minima are indeed a problem: using several starting points with gradient climbing algorithms is always advisable. But the problem is not only multiple local mínima, also that quite often solutions are in the boundary of the feasible parameter space, e.g. some variances of the noises are estimated to have value zero. – F. Tusell Sep 9 '11 at 14:10

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