ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Dynamic models with latent variables


ZAKOÏAN Jean-Michel

Department: Finance


Dynamic models involving hidden variables, or factors, constitute a rich class which is particularly important to capture the dynamic properties of economic and financial series.  Important models of this class are the state-space linear models, the hidden-Markov or Markov-switching models, the stochastic volatility models. Hidden variables can receive an economic interpretation, or they can be used as statistical tools. To handle such models, the standard statistical methods (e.g. the likelihood inference) are often in failure. Alternative estimation methods, which can be based on simulations, have to be introduced. The objective of the course is to present the main specifications, to derive their probabilistic properties and to study the appropriate inference methods for such models. Illustrations based on simulated or real economic data will also be presented.

At the end of the course, students should be able to write models in state-space form, study the probability properties of models including latent variables (existence of stationary solutions, moments, correlations, predictions) and develop appropriate statistical tools for their estimation and tests.


Chapter I: Definitions and examples

  1.  Stationary processes, ARMA and ARIMA models
  2.  Random variance models, Hidden-Markov models
  3. State-space models

Chapter II: The Kalman Filter

  1. General form of the Kalman filter
  2. Prediction and smoothing
  3. The stationary case
  4. Statistical inference
  5. Examples

Chapter III: Markov-switching models

  1. Finite-state Markov chains
  2. Hidden-Markov models
  3. Markov-switching ARMA models
  4. Estimation of the MS-AR(p) model

Chapter IV: Bayesian and simulated methods

  1. Bayesian inference and MCMC
  2. Simulation by acceptance-rejection
  3. The Metropolis-Hastings and Gibbs algorithms
  4. Examples: STAR model, stochastic volatility model


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•    Durlauf, S. and L. Blume  (Eds) (2016) Macroeconometrics and time series analysis. Springer.
•    Francq, C. and J-M. Zakoian (2019) GARCH Models. Structure, Statistical Inference and Financial Applications. 2nd Edition, Wiley. 
•    Gouriéroux, C.  and A. Monfort (1997) Time Series and Dynamic Models,  Cambridge University Press, Cambridge.
•    Frühwirth-Schnatter, S. (2006) Finite Mixture and Markov Switching Models, Springer.
•    Hamilton, J. D. (1994)  Time Series Analysis. Princeton University Press.
•    Harvey, A.C. (1989)  Forecasting, structural time series models and the Kalman filter. Cambridge University Press.