Dynamic models involving latent 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. Latent variables can receive an economic interpretation, or they can be used as statistical tools. Particular attention will be devoted to the modeling of local explosions (bubbles) for which several approaches can be considered. To handle such models, the standard statistical methods (in particular the likelihood inference) are often in failure and alternative estimation methods (e.g. 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 and statistical inference
Chapter III: Markov-switching models
1. Hidden-Markov models
2. Markov-switching ARMA models
3. Estimation of the MS-AR(p) model and illustrations
Chapter IV: Bubble modeling
1. Rational bubbles
2. Nonlinear models (time-varying parameters, thresholds)
3. Noncausal linear models for local explosions
Chapter IV: Simulated methods
1. Simulation by acceptance-rejection
2. The Metropolis-Hastings and Gibbs algorithms
3. Examples: STAR model, stochastic volatility model.
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