- Email: [email protected]
Modelling nonlinear economic relationships
Book reviews I International Journal of Forecasting
makes use of empirical data and subject-matter on models in the forecasting field. Although the examples are based on data-intensive explorations, the descriptions of the techniques and tools are provided only in generalized, descriptive terms, relying heavily on citations to the extensive references included with most of the papers. Just as this is not a textbook, it also is not a handbook with step-by-step instructions about how to forecast. One may learn many valuable lessons from the case studies in this specialized area of forecasting and will certainly gain an appreciation for the complexity of forecasting the health of elderly populations. This eclectic ensemble of results and methods is woven together from numerous data sources and concepts with an inherently empirical philosophy. The main contribution of this volume is to help us learn about the difficulties of forecasting health for populations by using the demographic structure of populations. As with all forecasts, we must constantly ask whether we believe the results. Do they truly guide policy or conclusions? Are the forecasting techniques merely the means to shore-up or reinforce preconceived beliefs? In any case, data lend credibility, plausibility and substance to the forecasts. But we must also be alert to the fact that forecasts can be client-driven to please the sponsors, just as the actuaries for insurance companies must conform to the business needs of insurance companies. Often we learn about forecasting in a context of external expectation or forces. In any case, the examples of this collection clearly lead us to appreciate that the art of forecasting involves information selection and judgment in development of models and screening of data.
R. Clifton Bailey Health Care Financing Administration Washington, D. C., USA The opinions and assertions are solely those of the author and do not necessarily reflect the opinions or policies of the Health Care Financing
10 (1994) 165-172
Administration or the Department Human Services.
169
of Health
and
C.W.J. Granger and T. Terasvirta, 1993, Modelling Nonlinear Economic Relationships (Oxford University Press, New York), 187 pp., hardback $30, ISBN O-19-877319-6; paperback $14.95, ISBN 0-19-877320-X. How important is nonlinear modelling for the forecasting practitioner? This is a fast developing area and the present book is a serious effort to attract applications, where until now most progress has come from theory. We have often read, especially in this journal, that models that are theoretically optimal in a linear world seem to work poorly in real forecasting. Much simpler methods, some even encompassed by the optimal model, often produce more accurate forecasts. Could nonlinearity be the explanation? Maybe, we do not know for sure yet. But at the end of this book you will find many examples of nonlinear models working better on real data than their linear counterparts - evidence that should awake the interest of any practitioner. The first five chapters provide a strict, but not too detailed introduction to nonlinear models. This is fascinating reading! Raised in a linear world the reader has to go back to square one. Not even R2 survives, shadow (auto-) correlation takes its place. Still more thrilling is that you can abandon the idea of universal stationarity. Models can be explosive, with or without attractors, but still behave reasonably (be stationary) after they hit a roof or a ceiling. The demand for nonlinear economic modelling does not emanate only from data studies; economic theory has long been formulating nonlinear relationships. Sometimes, as in the case of a Cobb-Douglas production function, the model can easily be cast into linear form. In other cases, particularly models suggested during the last decade, this is not possible. Switching regimes and saturation models are well known. The trouble is that many recently suggested
170
Book reviews I International Journal of Forecasting
nonlinear economic models are deterministic, and econometricians want to model a stochastic world. To give an idea of deterministic modelling, the authors present some important features of models that produce entirely different output for a small change in parameter (bifurcation and catastrophe) and they also demonstrate the closeness of white noise to white chaos. The book approaches its main theme in Chapter 4 where some particular nonlinear models are described. On just 10 pages one gets a good glimpse of the nonlinear descendants of the Box-Jenkins ARMA family, bilinear models where products of lagged observations and errors appear, and finally to the STARS of the show: smoorh transition regression (STR) models. Instead of abrupt switches between regimes. these models allow smooth logistic (LSTR) or exponential (ESTR) transition functions. One of the messages in this book is that although you could suggest a multitude of decent looking nonlinear models, these two, and especially their univariate versions, LSTAR and ESTAR, may be the most promising. What becomes of unit roots and cointegration in the nonlinear world? This is treated in a chapter under the Grangerian title of ‘LongMemory Models’, as opposite to short memory. These properties are connected with, but not synonymous to, nonstationarity versus stationarity. A parallel to stationarityishort memory is found in the concept ‘attractor’, which in the case of linear cointegration would be the cointegrating relationship. Here a geometrical reasoning clarifies the concepts, but strange enough there is no geometrical figure to accompany the text. In fact, the whole book contains only four figures! One reason for not getting better forecasts with a refined model could be that it has not been adequately tested. In the past 15 years or so. a large number of tests have been invented and many are now included in standard program packages. Those forecasters who routinely use these tests without detailed knowledge about how, for example, a LM test works, are advised first to consult, for example, Handbook of Econometrics, Vol. 2, before endeavouring to understand the long chapter on linearity testing.
10 (1994) 16.6172
Right at the start, the authors emphasize that one should not too lightly turn to nonlinear models because there is so much more statistical theory in the linear world. Linearity tests are divided into tests with a specific nonlinear alternative and those without it. As the authors are the first to admit, there is no lack of tests. I met tests I had never heard of before, like one against neural networks. Taking the easy way, the authors could have just described the tests and then left the reader to figure out what to do with all the instruments. But both writers know the problems of the applied researcher too well to leave it at that. There is a section on the asymptotic efficiency of tests as well as one titled: ‘Which Test to Use’. Here they pragmatically suggest that (as in the linear case) the tests be used as a model selection device, so that the alternative hypothesis, against which linearity is most strongly rejected, is regarded as a promising nonlinear explanation of the data. From the necessarily complex world of testing, in the last four chapters we are brought close to the everyday problems of applications. First, model-building strategies are discussed in more detail. The main divide is between nonparametric and parametric models. Nonlinear modelling makes use of areas of statistics and mathematics that often are not too familiar to linear modellers. Nonparametric models belong to the nonlinear world and here models use smoothing techniques developed mainly for other purposes. The reader meets kernel techniques, cubic splines and projection pursuit. the Among the parametric approaches, smooth transition models get most attention. This is easier to understand after the chapter on testing: the linear model is nested in the nonlinear one. Using the LM test, there is no need for model estimation under the nonlinear alternative. A stepwise procedure is presented that should help the practitioner in the difficult work of specifying nonlinear models. Also, good advice is given on diagnostic checking, where analysis of residuals is important, as with linear models. It is said that failure to pass a normality test usually indicates that there is an outlier. This may be true, but outliers must generally be a
Book
reviews
I International
Journal
much more serious problem in nonlinear than in linear modelling - since in a sense, an outlier is another regime, but lasting only for one period. Chapter 8 essentially contains three short essays on forecasting, aggregation and asymmetry, respectively. One-step forecasts of nonlinear models are straightforward but as soon as one hits period two, things get complicated, and exceedingly so for longer horizons. The difficulty arises as forecasts of explanatory variables (or own lags) have to be forecasted. Even if the forecast is unbiased, so that the expected value of the error is zero, this does not mean that in the nonlinear forecast function (as opposite the linear case) the error term can be ignored. Several methods for calculating forecasts for longer horizons are proposed and a combination is mentioned as a promising solution. Aggregation, both sectoral, temporal and due to systematic sampling dilutes nonlinearity, the more so the less common factors there are in the aggregated components. The ‘essay’ on asymmetry is essentially a survey of empirical business cycle studies. Different profiles for boom and slump periods cannot be described using univariate linear models. The strongest evidence of nonlinearity comes from studies of US unemployment. Nonlinearity in some manufacturing series also gets weak support here and more in the next chapter. However, there is no repetitive pattern (economic scale) of successive stages occurring with varying speed (time deformation). The final chapter contains selected applications published elsewhere, and where the authors, mainly Terbvirta, have been involved. For the practitioner this is a thrilling chapter and I will not reveal any details. To stimulate the reader’s appetite, let me just hint that the nonlinearity found in US industrial production involves explosive complex roots in a regime triggered by a large negative shock. Another important problem solved here is modelling impulses from an economically dominant country. These are all examples where rigorous statistical testing has shown nonlinear models to be
of Forecasting
10 (1994)
165-172
171
superior. The authors are more cautious when it comes to forecasting. Their results indicate that in normal situations one is not better off (but probably not much worse off either) using a nonlinear model. But after a substantial shock has hit the economy, a nonlinear model of a nonlinear relationship can be expected to be more accurate. One might add that it is precisely when something serious happens that good forecasts are in highest demand. There is not much to say against the book. It is rather short, considering the novelty of the theme. Especially the scarcity of applications, realistic examples and figures may cause some readers to find the text occasionally rather academic. There are no exercises. The references contain 233 titles for those who want to know more. However, among these there are very few referring to the literature on time dependent parameters in models for structural change, cf. the recent books: Hack1 (ed.) (1989) and Hack1 and Westlund (1991), both reviewed in IJF 1992, (no. 4). This is an important way of modelling nonlinearity. Generally speaking, the style of presentation is enjoyable. As a first edition, there are few misprints. For the next edition, which I am sure will soon be needed, I suggest a list of abbreviations be added. There are many of them and, although once explained, the reader may get confused, especially as the same letter can have two different meanings. By that time, this book will have inspired many more applications that can be added to illustrate the technique of nonlinear modelling. Lars-Erik Gller National Institute of Economic Research Stockholm. Sweden
T. Subba Rao, ed., 1993, Developments in Time Series Analysis (Chapman & Hall, London, UK), 433 pp., hardback f49.95, ISBN O-41249260- 1.
makes use of empirical data and subject-matter on models in the forecasting field. Although the examples are based on data-intensive explorations, the descriptions of the techniques and tools are provided only in generalized, descriptive terms, relying heavily on citations to the extensive references included with most of the papers. Just as this is not a textbook, it also is not a handbook with step-by-step instructions about how to forecast. One may learn many valuable lessons from the case studies in this specialized area of forecasting and will certainly gain an appreciation for the complexity of forecasting the health of elderly populations. This eclectic ensemble of results and methods is woven together from numerous data sources and concepts with an inherently empirical philosophy. The main contribution of this volume is to help us learn about the difficulties of forecasting health for populations by using the demographic structure of populations. As with all forecasts, we must constantly ask whether we believe the results. Do they truly guide policy or conclusions? Are the forecasting techniques merely the means to shore-up or reinforce preconceived beliefs? In any case, data lend credibility, plausibility and substance to the forecasts. But we must also be alert to the fact that forecasts can be client-driven to please the sponsors, just as the actuaries for insurance companies must conform to the business needs of insurance companies. Often we learn about forecasting in a context of external expectation or forces. In any case, the examples of this collection clearly lead us to appreciate that the art of forecasting involves information selection and judgment in development of models and screening of data.
R. Clifton Bailey Health Care Financing Administration Washington, D. C., USA The opinions and assertions are solely those of the author and do not necessarily reflect the opinions or policies of the Health Care Financing
10 (1994) 165-172
Administration or the Department Human Services.
169
of Health
and
C.W.J. Granger and T. Terasvirta, 1993, Modelling Nonlinear Economic Relationships (Oxford University Press, New York), 187 pp., hardback $30, ISBN O-19-877319-6; paperback $14.95, ISBN 0-19-877320-X. How important is nonlinear modelling for the forecasting practitioner? This is a fast developing area and the present book is a serious effort to attract applications, where until now most progress has come from theory. We have often read, especially in this journal, that models that are theoretically optimal in a linear world seem to work poorly in real forecasting. Much simpler methods, some even encompassed by the optimal model, often produce more accurate forecasts. Could nonlinearity be the explanation? Maybe, we do not know for sure yet. But at the end of this book you will find many examples of nonlinear models working better on real data than their linear counterparts - evidence that should awake the interest of any practitioner. The first five chapters provide a strict, but not too detailed introduction to nonlinear models. This is fascinating reading! Raised in a linear world the reader has to go back to square one. Not even R2 survives, shadow (auto-) correlation takes its place. Still more thrilling is that you can abandon the idea of universal stationarity. Models can be explosive, with or without attractors, but still behave reasonably (be stationary) after they hit a roof or a ceiling. The demand for nonlinear economic modelling does not emanate only from data studies; economic theory has long been formulating nonlinear relationships. Sometimes, as in the case of a Cobb-Douglas production function, the model can easily be cast into linear form. In other cases, particularly models suggested during the last decade, this is not possible. Switching regimes and saturation models are well known. The trouble is that many recently suggested
170
Book reviews I International Journal of Forecasting
nonlinear economic models are deterministic, and econometricians want to model a stochastic world. To give an idea of deterministic modelling, the authors present some important features of models that produce entirely different output for a small change in parameter (bifurcation and catastrophe) and they also demonstrate the closeness of white noise to white chaos. The book approaches its main theme in Chapter 4 where some particular nonlinear models are described. On just 10 pages one gets a good glimpse of the nonlinear descendants of the Box-Jenkins ARMA family, bilinear models where products of lagged observations and errors appear, and finally to the STARS of the show: smoorh transition regression (STR) models. Instead of abrupt switches between regimes. these models allow smooth logistic (LSTR) or exponential (ESTR) transition functions. One of the messages in this book is that although you could suggest a multitude of decent looking nonlinear models, these two, and especially their univariate versions, LSTAR and ESTAR, may be the most promising. What becomes of unit roots and cointegration in the nonlinear world? This is treated in a chapter under the Grangerian title of ‘LongMemory Models’, as opposite to short memory. These properties are connected with, but not synonymous to, nonstationarity versus stationarity. A parallel to stationarityishort memory is found in the concept ‘attractor’, which in the case of linear cointegration would be the cointegrating relationship. Here a geometrical reasoning clarifies the concepts, but strange enough there is no geometrical figure to accompany the text. In fact, the whole book contains only four figures! One reason for not getting better forecasts with a refined model could be that it has not been adequately tested. In the past 15 years or so. a large number of tests have been invented and many are now included in standard program packages. Those forecasters who routinely use these tests without detailed knowledge about how, for example, a LM test works, are advised first to consult, for example, Handbook of Econometrics, Vol. 2, before endeavouring to understand the long chapter on linearity testing.
10 (1994) 16.6172
Right at the start, the authors emphasize that one should not too lightly turn to nonlinear models because there is so much more statistical theory in the linear world. Linearity tests are divided into tests with a specific nonlinear alternative and those without it. As the authors are the first to admit, there is no lack of tests. I met tests I had never heard of before, like one against neural networks. Taking the easy way, the authors could have just described the tests and then left the reader to figure out what to do with all the instruments. But both writers know the problems of the applied researcher too well to leave it at that. There is a section on the asymptotic efficiency of tests as well as one titled: ‘Which Test to Use’. Here they pragmatically suggest that (as in the linear case) the tests be used as a model selection device, so that the alternative hypothesis, against which linearity is most strongly rejected, is regarded as a promising nonlinear explanation of the data. From the necessarily complex world of testing, in the last four chapters we are brought close to the everyday problems of applications. First, model-building strategies are discussed in more detail. The main divide is between nonparametric and parametric models. Nonlinear modelling makes use of areas of statistics and mathematics that often are not too familiar to linear modellers. Nonparametric models belong to the nonlinear world and here models use smoothing techniques developed mainly for other purposes. The reader meets kernel techniques, cubic splines and projection pursuit. the Among the parametric approaches, smooth transition models get most attention. This is easier to understand after the chapter on testing: the linear model is nested in the nonlinear one. Using the LM test, there is no need for model estimation under the nonlinear alternative. A stepwise procedure is presented that should help the practitioner in the difficult work of specifying nonlinear models. Also, good advice is given on diagnostic checking, where analysis of residuals is important, as with linear models. It is said that failure to pass a normality test usually indicates that there is an outlier. This may be true, but outliers must generally be a
Book
reviews
I International
Journal
much more serious problem in nonlinear than in linear modelling - since in a sense, an outlier is another regime, but lasting only for one period. Chapter 8 essentially contains three short essays on forecasting, aggregation and asymmetry, respectively. One-step forecasts of nonlinear models are straightforward but as soon as one hits period two, things get complicated, and exceedingly so for longer horizons. The difficulty arises as forecasts of explanatory variables (or own lags) have to be forecasted. Even if the forecast is unbiased, so that the expected value of the error is zero, this does not mean that in the nonlinear forecast function (as opposite the linear case) the error term can be ignored. Several methods for calculating forecasts for longer horizons are proposed and a combination is mentioned as a promising solution. Aggregation, both sectoral, temporal and due to systematic sampling dilutes nonlinearity, the more so the less common factors there are in the aggregated components. The ‘essay’ on asymmetry is essentially a survey of empirical business cycle studies. Different profiles for boom and slump periods cannot be described using univariate linear models. The strongest evidence of nonlinearity comes from studies of US unemployment. Nonlinearity in some manufacturing series also gets weak support here and more in the next chapter. However, there is no repetitive pattern (economic scale) of successive stages occurring with varying speed (time deformation). The final chapter contains selected applications published elsewhere, and where the authors, mainly Terbvirta, have been involved. For the practitioner this is a thrilling chapter and I will not reveal any details. To stimulate the reader’s appetite, let me just hint that the nonlinearity found in US industrial production involves explosive complex roots in a regime triggered by a large negative shock. Another important problem solved here is modelling impulses from an economically dominant country. These are all examples where rigorous statistical testing has shown nonlinear models to be
of Forecasting
10 (1994)
165-172
171
superior. The authors are more cautious when it comes to forecasting. Their results indicate that in normal situations one is not better off (but probably not much worse off either) using a nonlinear model. But after a substantial shock has hit the economy, a nonlinear model of a nonlinear relationship can be expected to be more accurate. One might add that it is precisely when something serious happens that good forecasts are in highest demand. There is not much to say against the book. It is rather short, considering the novelty of the theme. Especially the scarcity of applications, realistic examples and figures may cause some readers to find the text occasionally rather academic. There are no exercises. The references contain 233 titles for those who want to know more. However, among these there are very few referring to the literature on time dependent parameters in models for structural change, cf. the recent books: Hack1 (ed.) (1989) and Hack1 and Westlund (1991), both reviewed in IJF 1992, (no. 4). This is an important way of modelling nonlinearity. Generally speaking, the style of presentation is enjoyable. As a first edition, there are few misprints. For the next edition, which I am sure will soon be needed, I suggest a list of abbreviations be added. There are many of them and, although once explained, the reader may get confused, especially as the same letter can have two different meanings. By that time, this book will have inspired many more applications that can be added to illustrate the technique of nonlinear modelling. Lars-Erik Gller National Institute of Economic Research Stockholm. Sweden
T. Subba Rao, ed., 1993, Developments in Time Series Analysis (Chapman & Hall, London, UK), 433 pp., hardback f49.95, ISBN O-41249260- 1.