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 at last combined all of those into an Ox package, called MC2Pack.
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.
State space models
With the myriads of possible models, the class of state space models provides a, in my view, useful guideline from which to start. Enlarging these models, with non-linear components, non-Gaussian error terms or combinations of the two can be rather cumbersome. Even so, I am in the process of modelling a series where these models allow for contemporaneous effects of the mean on the variance of the components of the series.