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This chapter examines the application of the unit root hypothesis in econometric analysis, particularly in the Bayesian inference approach. It explains that testing for a unit root in a Bayesian framework is one of the most controversial topics in the economic literature. This is because testing is one of the hot topics among classical and Bayesian statisticians and because the unit root hypothesis is a point hypothesis and Bayesians do not like testing a point hypothesis because it is not natural to compare an interval which receives a positive probability with a point null hypothesis of zero mass.

Methods of testing for a unit root in an observed series are described in this chapter. Both parametric regression tests and non‐parametric adjustments to these test statistics are considered, and tables of critical values for commonly used tests are given. The chapter also uses functionals of Wiener processes to describe the asymptotic distributions of important test statistics.

Chapter 2 aims to convince the reader that economic time series show marked seasonality and an obvious trend, and, foremost, that the patterns of these trends and seasonal fluctuations do not seem to be very stable over time, nor do they seem to be similar across time series. We illustrate this for a range of quarterly US industrial production series, but other quarterly series from other countries would have yielded the same qualitative conclusion, as a glance at the relevant literature will indicate. We use graphical techniques and recently developed tests for seasonal unit roots. When economic data show evidence of unit roots, then one can conclude that the trend or the seasonal patterns are of a stochastic nature. This observation provides motivation for considering periodic models for economic data with a stochastic trend.

The authors recall the basic differences of view between classical and Bayesian analysis and note that the dispute among statisticians has not been exactly reflected in econometrics. Starting with a ‘falt prior’that is already contestable, Bayesian econometricians were happy to reproduce the results of their colleagues. This is now less the case and Bauwens and Lubrano suggest that Bayesian methods are more effective at detecting unit roots and avoiding spurious acceptance of integration of series.

In another area, that of computation, the authors suggest that there is more complementarity between classical and Bayesian econometrics and that the sort of tools used such as Gibbs sampling may be of general value. They conclude by suggesting that Bayesian methods will more than hold their own in cases where samples are small and extra information is necessary as is often the case.

Presents the important properties of integrated variables and sets out some of the preliminary asymptotic theories essential for the consideration of such processes. It explores the concepts of unit roots, non‐stationarity, orders of integration, and near integration, and demonstrates the use of the theory in understanding the behaviour of least‐squares estimators in spurious regressions and in models involving integrated data. The theoretical analysis is accompanied by evidence from Monte Carlo simulations. Several examples are also provided to illustrate the use of Wiener distribution theory in deriving asymptotic results for such models.

Long memory, unit root models and cointegration are important in linear modelling of nonstationary processes, not the least in econometrics. Recently, nonlinear generalizations of these concepts have been attempted. The framework is mathematically demanding, requiring tools that can handle both nonstationarity and nonlinearity. Two such tools are local times and null recurrent Markov chains. These are reviewed in parametric and non‐parametric cases.

This chapter reviews the theoretical literature on testing for unit roots and cointegration in panels where the time dimension (T) and the cross section dimension (N) are relatively large. The discussions cover the model and hypotheses to test; first and second generation panel unit root tests; cross-unit cointegration; finite sample properties of panel unit root tests; residual-based approaches to panel cointegration; tests for multiple cointegration; and panel cointegration in the presence of cross section dependence. Exercises are provided at the end of the chapter.

Chapter 8 tests the hypotheses made in the theoretical models of Chapter 5 and Chapter 6 on the wealth effect and the debt effect on consumption using advanced econometric methods. Since macroeconomic level time series data generally have unit roots, a cointegration analysis between these variables is performed. It is found that the wealth effect indeed was one of the main contributors to increasing consumption which more than compensated for underconsumption in the 1990s. A similar result is shown for the debt effect. Both these hypotheses are tested with respect to household data too and the results are no different from the overall macroeconomic picture.

Chapter 5 describes three sets of auxiliary methods that have emerged as add-on supplements to the traditional ARIMA model-building strategy. First, Bayesian information criteria (BIC) can be used to inform incremental modeling decisions. BICs are also the basis for the Bayesian hypothesis tests introduced in Chapter 6. Second, unit root tests can be used to inform differencing decisions. Used appropriately, unit root tests guard against over-differencing. Finally, co-integration and error correction models have become a popular way of representing the behavior of two time series that follow a shared path. We use the principle of co-integration to define the ideal control time series. Put simply, a time series and its ideal counterfactual control time series are co-integrated up the time of the intervention. At that point, if the two time series diverge, the magnitude of their divergence is taken as the causal effect of the intervention.

This chapter compares the properties of unit root processes with stationary processes, and considers alternative ways of testing for unit roots. The discussions cover difference stationary processes; unit root and other related processes; trend-stationary versus first difference stationary processes; variance ratio tests; Dickey-Fuller unit root tests; other unit root tests; and long memory processes. Exercises are provided at the end of the chapter.

This chapter outlines pitfalls of using standard inference procedures common in cross- sectional settings in time series settings and presents alternative procedures. It also addresses the issue of spurious regression and cautions the reader against the unquestioning use of cross section tools in time series settings.

The primary objective of this book has been to present a reasonably broad overview of the different classes of discrete-time dynamic models that have been proposed for empirical modeling, particularly in the process control literature. In its simplest form, the empirical modeling process consists of the following four steps: 1. Select a class C of model structures 2. Generate input/output data from the physical process P 3. Determine the model M ∊ C that best fits this dataset 4. Assess the general validity of the model M. The objective of this final chapter is to briefly examine these four modeling steps, with particular emphasis on the first since the choice of the model class C ultimately determines the utility of the empirical model, both with respect to the application (e.g., the difficulty of solving the resulting model-based control problem) and with respect to fidelity of approximation. Some of the basic issues of model structure selection are introduced in Sec. 8.1 and a more detailed treatment is given in Sec. 8.3, emphasizing connections with results presented in earlier chapters; in addition, the problem of model structure selection is an important component of the case studies presented in Secs. 8.2 and 8.5. The second step in this procedure—input sequence design—is discussed in some detail in Sec. 8.4 and is an important component of the second case study (Sec. 8.5). The literature associated with the parameter estimation problem—the third step in the empirical modeling process—is much too large to attempt to survey here, but a brief summary of some representative results is given in Sec. 8.1.1. Finally, the task of model validation often depends strongly on the details of the physical system being modelled and the ultimate application intended for the model. Consequently, detailed treatment of this topic also lies beyond the scope of this book but again, some representative results are discussed briefly in Sec. 8.1.3 and illustrated in the first case study (Sec. 8.2). Finally, Sec. 8.6 concludes both the chapter and the book with some philosophical observations on the problem of developing moderate-complexity, discrete-time dynamic models to approximate the behavior of high-complexity, continuous-time physical systems.