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Abstract
The rate of evolution of many economic processes can depend upon the economic environment in which they are placed. For example, inventories will generally fluctuate more rapidly at higher than lower levels of sales. Such processes can be thought of as stationary on a time scale that is a nonlinear transformation of the scale on which the process is observed. This phenomenon will be referred to as "time deformation."
Three versions of the time deformation model are studied. First, the time deformation is assumed to be nonrandom. Two tests for non-random time deformation are proposed, and their asymptotic properties are derived under general conditions. Second, the time transformation is allowed to depend on an exogenous random variable. Properties of the autocovariance function and spectral density of the resultant process are presented. These properties form the basis for a consistent estimator of finitely many parameters describing the spectrum of the observable process. Finally, models in which the time deformation depends on past values of the observable process are shown to be similar to several other nonlinear time series models. In this case, the relevant parameters can be estimated using an algorithm based on the Kalman filter.
