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)
Preface 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 2.1 Simple Fluctuation Test 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 3.1 Break Analysis for GMM 3.2 ARIMA Model with Outlier Handling 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 4.1 TAR Model for Unemployment 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 5.1 Threshold Error Correction Model 5.2 Threshold Error Correction Model: Forecasting 5.3 Threshold VAR 6 STAR Models 6.1 Testing for STAR 6.1 LSTAR Model: Testing and Estimation 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 7.1 Mixture Model-Maximum Likelihood 7.2 Mixture Model-EM 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) 8.1 Markov Switching Variances-ML 8.2 Markov Switching Variances-EM 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) 9.1 MS Linear Regression: ML Estimation 9.2 MS Linear Regression: EM Estimation 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 10.1 MS Systems Regression: ML Estimation 10.2 MS Systems Regression: EM Estimation 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) 11.1 Hamilton Model: ML Estimation 11.2 Hamilton Model: EM Estimation 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 12.1 Lam GNP Model-Kim Filter 12.2 Time-Varying Parameters-Kim Filter 12.3 Lam GNP Model-MCMC 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 13.1 MS ARCH Model-Maximum Likelihood 13.2 MS ARCH Model-MCMC 13.3 MS GARCH Model-Approximate ML A A General Result on Smoothing B The EM Algorithm C Hierarchical Priors D Gibbs Sampling and Markov Chain Monte Carlo E Probability Distributions E.1 Univariate Normal E.2 Beta distribution E.3 Gamma Distribution E.4 Inverse Gamma Distribution E.5 Bernoulli Distribution E.6 Multivariate Normal E.7 Dirichlet distribution E.8 Wishart Distribution Bibliography Index