Structural Breaks and Switching Models Course (2nd Edition)

This workbook 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.

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 include Threshold Autoregressions and VAR's and 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.

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 regime, 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. These start off with simple models and work toward increasingly complex models such as MS-VAR's.

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.

The second edition adds over 100 pages, with new coverage of the ICSS test for variance breaks (Section 2.1), the “fixed regressor bootstrap” (Section 3.3), increased coverage of computation of non-linear impulse response functions in various threshold models (Sections 4.4, 5.2.3 and 6.3) and a completely rewritten section on Threshold VAR’s (5.2). We’ve reworked the various Markov switching support procedures, and the updated chapters on Markov Switching models and their examples have been revised to reflect that. In particular, Chapter 10 on Markov Switching Multivariate Regressions now has a (very) detailed description of the process of computing regime-specific impulse response functions with error bands.

Finally, the section on Markov Switching GARCH models (Section 13.2) has been completely rewritten to explain the difference between the more accurate “Dueker filter” and the more commonly used “Gray filter” with application to the same data set to demonstrate that. It’s also shown that each of the filters at times may fail (rather badly) to provide an adequate approximation to the log likelihood, producing possibly misleading results.

Workbook Contents

(342 pages, 40 examples)


1 Structural Breaks at Known Locations

1.1 Structural Breaks: Introduction
1.2 Breaks in Static Models
1.3 Breaks in Dynamic Models
1.4 RATS Tips and Tricks
Example 1.1 Models with Exogenously-determined Trend Breaks

2 Fluctuation Tests

2.1 ICSS Test for Variance Changes
2.2 Rolling Sample Estimates
Example 2.1 Simple Fluctuation Test
Example 2.2 Fluctuation Test for GARCH
Example 2.3 ICSS Analysis vs GARCH model

3 Parametric Tests

3.1 Full Coefficient Vector
3.1.1 Linear Least Squares
3.1.2 GMM
3.2 Outliers and Shifts
3.2.1 Linear Least Squares
3.2.2 ARIMA models
3.2.3 GARCH models
3.3 Fixed Regressor Bootstrap
Example 3.1 Break Analysis for GMM
Example 3.2 ARIMA Model with Outlier Handling
Example 3.3 GARCH Model with Outlier Handling
Example 3.4 Fixed Regressor Bootstrap

4 TAR Models

4.1 Estimation
4.2 Testing
4.2.1 Arranged Autoregression Test
4.2.2 Direct Threshold Tests
4.3 Forecasting
4.4 Non-linear Impulse Responses
4.5 Tips and Tricks
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 \textsc {var
5.2.1 Tsay(1998) interest rates example
5.2.2 Balke(2000) Credit Regimes
5.2.3 Impulse Response Functions
5.3 Threshold Cointegration
Example 5.1 Threshold Error Correction Model
Example 5.2 Threshold Error Correction Model: Forecasting
Example 5.3 Threshold VAR, Multiple Breaks
Example 5.4 Threshold VAR, Testing and Estimation
Example 5.5 Threshold VAR, Impulse Responses

6 STAR Models

6.1 Testing
6.2 Estimation
6.3 Forecasts and Impulse Responses
6.4 More Complicated Models
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 Markov Switching Models
8.2 Common Concepts
8.2.1 Prediction Step
8.2.2 Update Step
8.2.3 Smoothing
8.2.4 Simulation of Regimes
8.2.5 Pre-Sample Regime Probabilities
8.2.6 Pathologies
8.3 Estimation
8.3.1 Simple Example
8.3.2 Maximum Likelihood
8.3.3 EM
8.3.4 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 MSREGRESSION procedures
9.2 The example
9.2.1 Maximum Likelihood
9.2.2 EM
9.2.3 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 @MSSYSREGRESSION procedures
10.1.1 Impulse Response Functions
10.2 The example
10.2.1 Maximum Likelihood
10.2.2 EM
10.2.3 MCMC (Gibbs Sampling)
10.3 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 MSVARSETUP procedures
11.2 The example
11.2.1 Maximum Likelihood
11.2.2 EM
11.2.3 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
13.2.1 The Example
13.2.2 Dueker Filter
13.2.3 Gray Filter
Example 13.1 MS ARCH Model-Maximum Likelihood
Example 13.2 MS ARCH Model-MCMC
Example 13.3 MS GARCH Model-Approximate ML, Dueker filter
Example 13.4 MS GARCH Model-Approximate ML, Gray filter

A A General Result on Smoothing

B The EM Algorithm

C Hierarchical Priors

D Gibbs Sampling and Markov Chain Monte Carlo

E Time-Varying Transition Probabilities

E.1 EM Algorithm

F Probability Distributions

F.1 Univariate Normal
F.2 Beta distribution
F.3 Dirichlet distribution
F.4 (Scaled) Inverse Chi-Squared Distribution
F.5 Gamma Distribution
F.6 Inverse Gamma Distribution
F.7 Bernoulli Distribution
F.8 Multivariate Normal
F.9 Wishart Distribution
F.10 Inverse Wishart Distribution