What are the more Advanced models for time series As far as my studies go, I did: 


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*ARIMA in all sauces

*Dynamic linear models/state space model. The basics

*VAR(IMA)

*VECM


I then tried to see if there is a model that combines some or most of the ideas underlying the previous models and I only found 


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*BVAR


So my question is, what are the models that are currently "popular" in research nowadays?
(If this question is too broad, then the models which are popular in economics.)
 A: Speaking for economics, your choice of time series model is dictated by (1) the objectives of your research and (2) the nature of the data you wish to analyze.  Within different subfields of economics these things can vary greatly (i.e. Macro vs Financial vs Industrial Organization), so you will find that the types of popular, “advanced”, time series techniques will also vary greatly even between different subfields of Economics.
One model which does tend to show up frequently across most all areas of empirical economics  and which you did not already mention is the Markov-switching model, otherwise known as the regime-switching model.  This model was made popular for economists by Hamilton(1989).  I think it may also be in Hamilton’s text book “Time Series Analysis” 
See here or here for an introduction to regime-switching models.  In essence, the regime-switching model assumes a time series switches between a finite number of states or “regimes” as time changes.  Each regime is identified by its own mean, trend and/or variance components.  I think these models are popular because they can both improve forecast accuracy and provide for very intuitive economic interpretations.  In my personal experience, I have also found regime-switching models easier to estimate than the dynamic linear models which require a Kalman filter. 
Bayesian Vector Auto-Regression, error correction, and Dynamic Stochastic Generalized Equilibrium (DSGE) models will be popular with macroeconomists and bankers for the following reasons:


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*(research question) Macro economists are interested in studying the behavior of the economy as a whole.  This includes, among other things, how variables like GDP, inflation, and unemployment react to policy decisions and other economic inputs in a dynamic setting.  Along these lines, VAR, VECM and DSGE provide insight into the correlation among such time series variables, and one can further investigate granger causality and cointegrated relationships in such frameworks.

*(data) Macro data sets have relatively few observations.  The priors
in BVAR provide a means of shrinkage which allows stable estimation
of parameter dense models on small data sets.  Other models like
DSGE use economic theory to impose enough restrictions so as to make
them estimable on smaller data sets.
Among economists doing research in the financial industry, volatility models such as GARCH, and stochastic volatility, and other models which adjust for fat tails and non-normality in time series data such as Levy process, Jump-Diffusion, etc . (there are  a lot of models under this category) are really popular.  This is because:


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*(research questions) in finance, forecasting accuracy is a huge concern so you will see many financial time series models who’s sole purpose is to better parametrize a forecast distribution or minimize forecasting error.  This is often accomplished by modeling volatility, mixture components, etc.  In addition, the fundamentals of modern portfolio theory are grounded in optimizing a volatility(risk) vs. return paradigm and the basics of mathematical finance (stochastic calculus, Ito’s lemma, etc.) lend strong intuition to many of the aforementioned models.

*(data) In general, finance data tends to be very abundant, especially with daily and intra-daily trading data.   This type of data is also characterized by volatility clustering and fat tails.  This provides the ideal setting for estimating higher moments than just the mean and for estimating more sophisticated forecast distributions.  


Of course, BVAR has been used in finance and volatility models have been utilized in Macroeconomics, I am just trying to give you an idea of how primary modeling trends can differ between sub-fields.  This is also only a gimps,  there are many more sub-fields of economics and series of literature which favor specialized modeling approaches.  Like I said at the beginning, your choice of time series model is dictated by your research goals and the nature of your data.


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*Hamilton, James D. "A new approach to the economic analysis of nonstationary time series and the business cycle." Econometrica: Journal of the Econometric Society (1989): 357-384.

