Unobserved components models
With the myriads of possible models, the class of unobserved component models provides a, in my view, useful guideline from which to start. Enlarging the linear and Gaussian state space models with non-linear components, non-Gaussian error terms or combinations of the two is not straightforward, though I have made inroads here to obtain results even with models at higher frequency.
Without data, an econometrician can do little. Financial econometricians tend to have loads of high quality data, allowing for a detailed analysis. New problems occurring in this field, like irregularly spaced timing of observations, differences in importance of observations on different parts of the day, those leave many unsolved riddles for future research. Now, the interest lies in following the data at close range, preferably at the tick-by-tick frequency. Present-day models however are not immediately capable of following the data so closely.
Part of the Ph.D. project was directed at distinguishing long-lasting effects in the model from sudden changes in the model parameters. ARFIMA models, fractional integration, are all topics that I looked into. This effect is often found in inflation rates, or more generally in series that are constructed/aggregated from many underlying series.
As long as the information content of the data is large enough, specifying rather precisely the location of the maximum likelihood, often a classical analysis can go quite far. On the other hand, when decisions have to be made under uncertainty, the imprecision in parameter estimates may very well influence the final outcome. In such situations, the Bayesian method of analysis may be better suited.
Bayesian statistics cannot exist without all kind of Markov Chain Monte Carlo simulation techniques, in order to find the posterior density of the parameters in the model. Over the years I used many different sampling methods, and proposed novel algorithms..