## Structural Breaks and Switching Models Course

### Workbook Preface

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.

### Workbook Contents

(229 pages, 34 examples)

#### 1 Estimation with Breaks at Known Locations

1.1 Breaks in Static Models
1.2 Breaks in Dynamic Models
1.3 RATS Tips and Tricks

#### 2 Fluctuation Tests

Example 2.1 Simple Fluctuation Test
Example 2.2 Fluctuation Test for GARCH

#### 3 Parametric Tests

3.1 LM Tests
3.1.1 Full Coefficient Vector
3.1.2 Outliers and Shifts
Example 3.1 Break Analysis for GMM
Example 3.2 ARIMA Model with Outlier Handling
Example 3.3 GARCH Model with Outlier Handling

#### 4 TAR Models

4.1 Estimation
4.2 Testing
4.2.1 Arranged Autoregression Test
4.2.2 Fixed Regressor Bootstrap
4.3 Forecasting
4.4 Generalized Impulse Responses
Example 4.1 TAR Model for Unemployment
Example 4.2 TAR Model for Interest Rate Spread

#### 5 Threshold VAR/Cointegration

5.1 Threshold Error Correction
5.2 Threshold VAR
5.3 Threshold Cointegration
Example 5.1 Threshold Error Correction Model
Example 5.2 Threshold Error Correction Model: Forecasting
Example 5.3 Threshold VAR

#### 6 STAR Models

6.1 Testing for STAR
Example 6.1 LSTAR Model: Testing and Estimation
Example 6.2 LSTAR Model: Impulse Responses

#### 7 Mixture Models

7.1 Maximum Likelihood
7.2 EM Estimation
7.3 Bayesian MCMC
7.3.1 Label Switching
Example 7.1 Mixture Model-Maximum Likelihood
Example 7.2 Mixture Model-EM
Example 7.3 Mixture Model-MCMC

#### 8 Markov Switching: Introduction

8.1 Common Concepts
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)
Example 8.1 Markov Switching Variances-ML
Example 8.2 Markov Switching Variances-EM
Example 8.3 Markov Switching Variances-MCMC

#### 9 Markov Switching Regressions

9.1 Estimation
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)
Example 9.1 MS Linear Regression: ML Estimation
Example 9.2 MS Linear Regression: EM Estimation
Example 9.3 MS Linear Regression: MCMC Estimation

#### 10 Markov Switching Multivariate Regressions

10.1 Estimation
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
Example 10.1 MS Systems Regression: ML Estimation
Example 10.2 MS Systems Regression: EM Estimation
Example 10.3 MS Systems Regression: MCMC Estimation

#### 11 Markov Switching VAR's

11.1 Estimation
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)
Example 11.1 Hamilton Model: ML Estimation
Example 11.2 Hamilton Model: EM Estimation
Example 11.3 Hamilton Model: MCMC Estimation

#### 12 Markov Switching State-Space Models

12.1 The Examples
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
Example 12.1 Lam GNP Model-Kim Filter
Example 12.2 Time-Varying Parameters-Kim Filter
Example 12.3 Lam GNP Model-MCMC
Example 12.4 Time-Varying Parameters-MCMC

#### 13 Markov Switching ARCH and GARCH

13.1 Markov Switching ARCH models
13.1.1 Estimation by ML
13.1.2 Estimation by MCMC
13.2 Markov Switching GARCH
Example 13.1 MS ARCH Model-Maximum Likelihood
Example 13.2 MS ARCH Model-MCMC
Example 13.3 MS GARCH Model-Approximate ML

E.1 EM Algorithm

#### F Probability Distributions

F.1 Univariate Normal
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