JOURNAL OF APPLIED ECONOMETRICS, VOL. 9, 479-480 (1994)

Modelling Nonlinear Economic Relationships, C. W. J . GRANGER and T. University Press: 1993, ISBN 0-19-877320)3, E14.95 paperback, pp. x + 187.

TERASVIRTA. Oxford

Granger and Terasvirta (hereafter GT) aim to provide a ‘discussion of how to model nonlinear relationships between series’ (p.vii) and ultimately to provide applied workers with usable techniques. A wide variety of material is covered but this review attempts to concentrate on their message for empirical researchers.

(1) The specification of a linear model; (2) The application of one or more tests for linearity; (3) The specification and estimation of one or more nonlinear models; and (4) An evaluation of the model@) selected in (3). The main foci of the volume are (2), (3), and what nonlinear models should be considered. GT take an agnostic stance on how the initial linear model is obtained. Where d o GT differ from most existing econometric practitioners who subject their model to a battery of diagnostic tests? At the risk of oversimplification, first, they suggest the use of several tests for linearity not just the RESET test; second, they take far more seriously the nonlinear alternative treating it as a viable appropriate alternative that should be considered when the null of linearity is rejected; and, third, they have some particular nonlinear models they are willing to try. GT believe fairly simple nonlinear models are likely t o work well when non-linearities are present.

Typically, economic theory offers little or no guidance about the functional form that should be used so a crucial question is which models from the wide possible choice of nonlinear models are worth consideration by the practitioner. Although nonlinear autoregressive, bilinear, nonlinear moving average, random coefficient and neural network models are discussed, the class of nonlinear model that features most prominently and for which the most detailed modelling strategy is discussed is the smooth transition regression (STR) model. This model takes the form

The four steps of GT’s suggested strategy for modelling nonlinear relationships are:

y , = * ’ X i + F(wl)O’xt + ~t

where yt is the dependent variable, Xt and Wt are explanatory variables, a and 8 are parameters, Ur is a disturbance and F(.) is a given function that also depends on unknown parameters. Two particular functional forms for F considered are the logistic and exponential distribution functions that give rise to what are called the LSTR and ESTR models, respectively. The variables in W t , the transition variables, will typically be a linear function of the variables in Xr or lagged values of Yr - T ’ X r . An extremely brief survey of nonlinear economic models is given to suggest that smooth transition models are likely to be relevant in practice. I must admit that I am not particularly persuaded by this survey and that my general reaction to the STR class of model is that it is rather ad hoc.

However, with some particular nonlinear alternative in mind it is then possible to construct tests for linearity. Use of the Lagrange Multiplier (LM) testing principle is obviously attractive in testing a null linear model against some nonlinear alternative since it it based only on estimates of the linear model. Another attraction of the LM test compared to tests based on the unrestricted nonlinear model is the worrying finding that estimation of the alternative nonlinear model will typically indicate nonlinearity even when there is none! LM tests are derived for testing linearity against specific LSTR and ESTR

0 1994 by John Wiley & Sons, Ltd .

480 BOOK REVIEW

models. Alternative LM tests for LSTR and ESTR models based on a cubic approximation to the distribution function are also given. Not surprisingly, the LM tests lead to standard variable addition type tests where the added variables are particular powers of the variables in w, and their products with the variables in Xt. For a given transition variable, a comparison of the variables to be added for the LM tests of the LSTR and ESTR models based on cubic approximations suggests combining these variables to provide a general test against an STR model. One useful result is that the RESET test which is commonly viewed as a test against an unspecified alternative can be interpreted as an LM test of linearity against a particular LSTR model.

One concern I have is that rejection of the linear model using a linearity test will automatically lead to estimation of the nonlinear alternative in the same way that significant Durbin-Watson tests used to lead to automatic application of Cochrane-Orcutt type estimators. The reason for this concern is simple. Rejection of linearity may be caused by other problems (for example, misspecification of x,) and these possibilities need to be considered seriously.

As GT acknowledge, there are several other key problems raised by the tests for linearity that include: the sheer number of tests; several models may be locally equivalent alternatives and lead to the same testing equation; and the tests are likely to have power against other alternatives. The suggested solution to the problem of using several linearity tests leading to a rejection of linearity is to compare the strength of the rejections of linearity and to take more seriously those alternatives that are strongly rejected.

Given a rejection of the null of linearity, identification of the alternative nonlinear model is the next problem. When attention is restricted to STR alternatives, the choice of transition variables and the choice between LSTR and ESTR models proceeds in two steps. The general STR testing equation is used with several transition variables and if there is more than one rejection, the transition variable with the smallest P-value is chosen. For a given transition variable, the same general testing equation is also used to set up a sequence of nested hypotheses to choose between ESTR and LSTR models. Use of the same general testing equation for all these tests means that estimation of a nonlinear model is not required at the specification stage. The alternative STR models determined by different F(.)’s and transition variables are essentially non-nested but this literature is not appealed to. It is a moot question whether the appropriate non-nested tests would have any higher power. One clear advantage of a non-nested testing procedure is that it entertains the possibility that all models considered are rejected. Non-nested tests could also be of use in testing between models that are locally equivalent alternatives.

The first suggested step in the evaluation of a nonlinear model is the application of standard diagnostic tests for serial correlation, normality, conditional heteroscedasticity and further non-linearities. Other suggested steps include an evaluation of the long-run dynamics of the model and an evaluation of its forecast performance.

One key assumption made is that the functional form to be applied to the dependent variable is known a priori, thus ruling out the important problem of distinguishing linear and logarithmic regression models. Since there is evidence that at least one of the linearity tests discussed (RESET) works well for this problem, this also raises the question of how well the other linearity tests work for this problem. Obviously, the practitioner needs to keep this alternative in mind.

GT have performed an extremely useful service by focusing our attention on nonlinear modelling and some particular types of nonlinear models, and by laying out a modelling strategy for these models. As they freely acknowledge, there are still many unanswered questions awaiting researchers in this field.

COLIN McKENZIE Osaka School of International Public Policy, Osaka University, Toyonaka 560, Japan and Economic Growth Center, Department of Economics, Yale University, PO Box 208269, New Haven, Connecticut 06520-8269, USA