Modelling Nonlinear Economic Relationships.

This book appears in the Advanced Texts in Econometrics series from Oxford University Press. The authors, Clive Granger of U.C.S.D. and Timo Terasvirta of the Bank of Norway, are leading proponents of nonlinear multivariate modeling. This book provides a good introductory text on this topic, and should be accessible to graduate students and professional economists, even those with just a basic background in econometrics and time series analysis.

The authors viewpoint is stated in Chapter 1. The economy is nonlinear. At least, there are many theoretical reasons to presume that this is so. When we turn to econometrics, what does this mean for specification, estimation, and inference? In particular, what does this mean when the exact nonlinear specification is unknown? For instance, if we start with a particular nonlinear specification we usually know how to estimate the unknown parameters, and we may know quite a bit about the asymptotic properties of these estimators. The bigger difficulty is when the specification is unknown. In that case there are many possible specifications, to understate the case, and hence there is a need for exploratory statistical techniques to identify interesting properties of the data and possibly to better inform theorists of the behavior of the data. This development and refinement of what are often called stylized facts will then feed back into further developments and refinements of the economic theory, and the ongoing feedback between theory and econometric work will further our knowledge of the nonlinear multivariate economy we live in.

This book, then, is an introduction to nonlinear modeling. While linear time series models are well developed, nonlinear models are less so, and this book aims to both widen the profession's knowledge of nonlinear models and also spur further research in the development of these models.

The text contains ten chapters. Chapter 1 is an introduction, which briefly reviews some features of linear models and introduces some specific nonlinear univariate models. These include the nonlinear auto-regressive model (NLAR), the threshold autoregressive model (TAR), smooth transition threshold model (STAR), exponential autoregressive model, and time varying parameter autoregressive model. Implications for stationarity, invertibility, memory, and stability are also discussed. Chapter 2 then goes on to discuss more general classes of nonlinear models, state space models and generalizations of the linear moving average model based on the Volterra series expansion.

Chapter 3 is a very brief discussion of some nonlinear models in economic theory, and their possible implication for econometrics. Included are switching regime models, models with bifurcations and catastrophe, and models of chaos. A major theme of this chapter is that many nonlinear models used by theorists are deterministic, whereas econometricians typically view the world as stochastic. The authors do not suggest a solution to this impasse.

Chapter 4 introduces an assortment of multivariate nonlinear models that are useful for applied work. These include the nonlinear autoregressive and regression models, STAR models, bilinear models, nonlinear moving average models, random coefficient models, and heteroskedastic models such as the autoregressive conditional heteroskedasticity (ARCH) model. Chapter 5 then discusses long-memory models, models in which shocks that occur today will have a permanent effect into the distant future. The discussion begins with integrated series and unit roots, building up to a brief discussion of cointegration in linear models. There follows a nice discussion of long-memory and nonlinearity, attractors, estimation and testing for attractors, and nonlinear error correction models. Here it would have been useful to have had an application to illustrate the discussion, such as the application found in Granger and Hallman [1].

Chapter 6 discusses tests for linearity. Topics include Lagrange multiplier tests against specific alternatives, tests against unspecified alternatives, and a discussion of some properties of these tests. The chapter concludes with some advice on which test to use. Chapter 7 then begins a discussion of building nonlinear models. If tests indicate nonlinearity, this chapter provides advice on how to proceed. The discussion includes nonparametric and semiparametric models, bilinear models, neural network models, and more. There is also a section on how to evaluate these models. Three additional topics are discussed in Chapter 8. These are forecasting, aggregation, and asymmetry.

The only large-scale examples of applications occur in Chapter 9. Here there are two applications, both dealing with smooth transition regression models. The first is based on published work by Terasvirta and Anderson [3] and is an application of the smooth transition autoregression model to industrial production in thirteen of the OECD countries. The example includes linearity tests, the choice between exponential and logistic STAR models, the specification of bivariate smooth transition regression (STR) models, parameter estimation, and a discussion of how to interpret the results. The discussion is quite good and provides a nice example of how the authors think STAR modelling should proceed.

The second application is based on work by Granger, Terasvirta, and Anderson [2], and looks at non-linear modeling of the relationship between U.S. GNP and an index of leading indicators. The model is again a smooth transition regression model. Missing are any applications using other nonlinear multivariate models.

The text concludes with a brief discussion of strategies for nonlinear modeling in Chapter 10, a list of references, and an index.

References

1. Granger, Clive W. J. and Jeffrey J. Hallman, "Long Memory Processes with Attractors." Oxford Bulletin of Economics and Statistics, February 1991, 11-26.

2. Granger, C. W. J.; T. Terasvirta, and Heather Anderson. "Modelling Non-linearity over the Business Cycle," in New Research on Business Cycles, Indicators and Forecasting, edited by Jim H. Stock and Mark W. Watson. Chicago: Chicago University Press, 1993, pp. 311-25.

3. Terasvirta, T. and Heather M. Anderson, "Characterizing Nonlinearities in Business Cycles using Smooth Transition Autoregressive Models." Journal of Applied Econometrics, December 1992, S119-S139.

Dennis W. Jansen Texas A&M University