Outline of Chapters

When the project objectives have been defined properly, the underlying eco-nomic or other subject matter theory has been evaluated, and a suitable set of time series has been prepared, the actual econometric modeling and statistical analysis can begin. Some tools for this stage of the analysis are presented in the following chapters.

Even when the objective is a joint analysis of a set of time series, it is usually a good idea to start with exploring the special properties and characteristics of the series individually. In other words, univariate analysis of the individual series typically precede a multivariate or systems analysis. The tools available for univariate analysis are presented in Chapter 2. In that chapter, some more discussion of important characteristics is given, in particular, in anticipation of a later multivariate analysis. For example, specific attention is paid to an exploration of the trending behavior of a series. Therefore, unit root tests that can help in detecting the existence of stochastic trends form a prominent part of the chapter. With respect to the models for describing univariate DGPs, the

emphasis in Chapter 2 is on linear models for the conditional expectation or the first- and second-order moment part of a series because it is an advantage in many situations to construct simple models. Therefore, if a simple linear model is found to describe the data well, this is important information to carry on to a multivariate analysis.

At the multivariate level, linear models for the conditional mean such as vec-tor auvec-toregressions (VARs) and vecvec-tor error correction models (VECMs) are again the first choice. Given that data sets are often quite limited and that even linear models can contain substantial numbers of parameters, it is sometimes difficult to go beyond the linear model case at the multivariate level. Chapter 3 discusses VECMs and VAR models, how to specify and estimate them, how to use them for forecasting purposes, and how to perform a specific kind of causal-ity analysis. The recent empirical literature has found it useful to distinguish between the short- and long-run parts of a model. These parts are conveniently separated in a VECM by paying particular attention to a detailed modeling of the cointegration properties of the variables. Therefore, Chapter 3 emphasizes modeling of cointegrated series. In this analysis the results of preliminary unit root tests are of some importance. More generally, some univariate character-istics of the series form a basis for the choice of multivariate models and the analysis tools used at the systems level.

Once a model for the joint DGP of a set of time series of interest has been found, econometricians or economists often desire to use the model for analyz-ing the relations between the variables. The objective of such an analysis may be an investigation of the adequacy of a particular theory or theoretical argument.

Alternatively, the aim may be a check of the model specification and its ability to represent the structure of a specific market or sector of an economy properly.

Nowadays impulse responses and forecast error variance decompositions are used as tools for analyzing the relations between the variables in a dynamic econometric model. These tools are considered in Chapter 4. It turns out, how-ever, that a mechanical application of the tools may not convey the information of interest, and therefore structural information often has to be added to the analysis. Doing so results in a structural VAR (SVAR) or structural VECM (SVECM) analysis that is also covered in Chapter 4, including the resulting additional estimation and specification problems.

If sufficient information is available in the data to make an analysis of non-linearities and higher order moment properties desirable or possible, there are different ways to go beyond the linear models discussed so far. Of course, the choice depends to some extent on the data properties and also on the purpose of the analysis. An important extension that is often of interest for financial market data is to model the conditional second moments. In a univariate context, this means, of course, modeling the conditional variances. For multivariate systems, models for the conditional covariance matrices may be desired. Some models,

estimation methods, and analysis tools for conditional heteroskedasticity are presented in Chapter 5.

Nonlinear modeling of the conditional mean is considered in Chapters 6 and 7. Chapter 6 contains a description of the parametric smooth transition (STR) model, and an organized way of building STR models is discussed and illuminated by empirical examples. An STR model may be regarded as a linear model with time-varying parameters such that the parametric form of the linear model varies smoothly with two extreme “regimes” according to an observable, usually stochastic – but in some applications deterministic – variable. The smoothness of the transition from one extreme regime to the other accounts for the name of this model. The modeling strategy described in Chapter 6 is only applicable to single-equation models, and the question of how to build nonlinear systems consisting of STR equations is not addressed in this book.

The discussion in Chapter 6 also covers purely univariate smooth transition autoregressive (STAR) models that have been frequently fitted to economic and other time series.

A more general approach, as far as the form of nonlinearity is concerned, is adopted in Chapter 7, where both the conditional mean as well as the conditional variance of the DGP of a univariate series are modeled in general nonlinear form.

Estimation of the nonlinear functions is done nonparametrically using suitable local approximations that can describe general nonlinear functions in a very flexible way. The drawback of the additional flexibility is, however, that more sample information is needed to get a clear picture of the underlying structures.

Therefore, these methods can currently only be recommended for univariate time series analysis and, hence, the exposition in Chapter 7 is limited to this case.

In modern applied time series econometrics the computer is a vital tool for carrying out the analysis. In particular, the methods described in this volume rely heavily on extensive computations. Therefore, it is important to have software that does not create obstacles for the analysis by presenting only tools that are too limited. In the last chapter of this volume, software is therefore introduced that includes many of the methods and procedures considered in this book.

Clearly, the methods for econometric time series analysis are evolving rapidly;

hence, packaged, ready-to-use software can easily become obsolete. The soft-wareJMulTiintroduced in Chapter 8 is supposed to be able to decrease the time gap between the development of new methods and their availability in user-friendly form. This software provides a flexible framework for checking new methods and algorithms quickly. Readers may therefore find it useful to familiarize themselves with the software as they go through the various chapters of the book. In other words, it may be worth having a look at the final chapter at an early stage and trying out the methods by replicating the examples using theJMulTisoftware.

2 Univariate Time Series