This workbook is based upon the content of the RATS e-course on Switching Models and Structural Breaks, offered in fall of 2010. It covers a broad range of topics for models with various types of breaks or regime shifts.

In some cases, models with breaks are used as diagnostics for models with fixed coefficients. If the fixed coefficient model is adequate, we would expect to reject a similar model that allows for breaks, either in the coefficients or in the variances. For these uses, the model with the breaks isn't being put forward as a model of reality, but simply as an alternative for testing purposes. Chapters 2 and 3 provide several examples of these, with Chapter 2 looking at "fluctuation tests" and Chapter 3 examining parametric tests.

Increasingly, however, models with breaks are being put forward as a description of the process itself. There are two broad classes of such models: those with observable regimes and those with hidden regimes. Models with observable criteria for classifying regimes are covered in Chapters 4 (Threshold Autoregressions), 5 (Threshold VAR and Cointegration) and 6 (Smooth Threshold Models). In all these models, there is a threshold trigger which causes a shift of the process from one regime to another, typically when an observable series moves across an (unknown) boundary.

There are often strong economic argument for such models (generally based upon frictions such as transactions costs), which must be overcome before an action is taken. Threshold models are generally used as an alternative to fixed coefficient autoregressions and VAR's. As such, the response of the system to shocks is one of the more useful ways to examine the behavior of the model. However, as the models are nonlinear, there is no longer a single impulse response function which adequately summarizes this. Instead, we look at ways to compute two main alternatives: the eventual forecast function, and the generalized impulse response function (GIRF).

The remaining seven chapters cover models with hidden regimes, that is models where there is no observable criterion which determines to which regime a data point belongs. Instead, we have a model which describes the behavior of the observables in each regimes, and a second model which describes the (unconditional) probabilities of the regimes, which we combine using Bayes rule to infer the posterior probability of the regimes. Chapter 7 starts off with the simple case of time independence of the regimes, while the remainder use the (more realistic) assumption of Markov switching. The sequence of chapters 8 to 11 look at increasingly complex models based upon linear regressions, from univariate, to systems, to VAR's with complicated restrictions. All of these demonstrate the three main methods for estimating these types of models: maximum likelihood, EM and Bayesian MCMC.

The final two chapters look at Markov switching in models where exact likelihoods can't be computed, requiring approximations to the likelihood. Chapter 12 examines state-space models with Markov switching, while Chapter 13 is devoted to switching ARCH and GARCH models.

(229 pages, 34 examples)

1.2 Breaks in Dynamic Models

1.3 RATS Tips and Tricks

3.1.1 Full Coefficient Vector

3.1.2 Outliers and Shifts

4.2 Testing

4.2.1 Arranged Autoregression Test

4.2.2 Fixed Regressor Bootstrap

4.3 Forecasting

4.4 Generalized Impulse Responses

5.2 Threshold VAR

5.3 Threshold Cointegration

7.2 EM Estimation

7.3 Bayesian MCMC

7.3.1 Label Switching

8.1.1 Prediction Step

8.1.2 Update Step

8.1.3 Smoothing

8.1.4 Simulation of Regimes

8.1.5 Pre-Sample Regime Probabilities

8.2 Estimation

8.2.1 Simple Example

8.2.2 Maximum Likelihood

8.2.3 EM

8.2.4 MCMC (Gibbs Sampling)

9.1.1 MSREGRESSION procedures

9.1.2 The example

9.1.3 Maximum Likelihood

9.1.4 EM

9.1.5 MCMC (Gibbs Sampling)

10.1.1 MSSYSREGRESSION procedures

10.1.2 The example

10.1.3 Maximum Likelihood

10.1.4 EM

10.1.5 MCMC (Gibbs Sampling)

10.2 Systems Regression with Fixed Coefficients

11.1.1 The example

11.1.2 MSVARSETUP procedures

11.1.3 Maximum Likelihood

11.1.4 EM

11.1.5 MCMC (Gibbs Sampling)

12.2 The Kim Filter

12.2.1 Lam Model by Kim Filter

12.2.2 Time-Varying Parameters Model by Kim Filter

12.3 Estimation with MCMC

12.3.1 Lam Model by MCMC

12.3.2 Time-varying parameters by MCMC

13.1.1 Estimation by ML

13.1.2 Estimation by MCMC

13.2 Markov Switching GARCH

F.2 Beta distribution

F.3 Gamma Distribution

F.4 Inverse Gamma Distribution

F.5 Bernoulli Distribution

F.6 Multivariate Normal

F.7 Dirichlet distribution

F.8 Wishart Distribution