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Modelling Nonlinear Relationships between Extended-Memory Variables

C. W. J. Granger

Abstract

Many economic variables have a persistence property which may be called extended memory and the relationship between variables may well be nonlinear. This pair of properties allow for many more types of model misspecification than encountered with stationary or short-memory variables and linear relationships, and misspecifications lead to greater modelling difficulties. Examples are given using the idea of a model being balanced. Alternative definitions of extended memory are considered and a definition based on the properties of optimum forecasts is selected for later use. An important but not necessarily pervasive class of processes are those that are extended-memory but whose changes are short-memory. For this case, called I(1), standard cointegration ideas will apply. Tests of linearity are discussed in both the I(1) case, where a possible group of tests is easily found, and more generally. Similarly, methods of building nonlinear models based on familiar techniques, such as neural networks and projection pursuit, are briefly considered for I(1) and the more general case. A number of areas requiring further work in this new area are emphasized.