Assessing Persistence in Discrete Nonstationary Time-Series Models

The aim of this paper is to examine the application of measures of persistence in a range of time-series models nested in the framework of Cramer (1961). This framework is a generalization of the Wold (1938) decomposition for stationary time-series which, in addition to accommodating the standard I(0) and I(1) models, caters for a broad range of alternative processes. Two measures of persistence are considered in some detail, namely the long-run impulse-response and variance-ratio functions. Particular emphasis is given to the behaviour of these measures in a range of non-stationary models specified in discrete time. We document the conflict that arises between different measures, applied to the same model, as well as conflict arising from the use of a given measure in different models. Precisely which persistence measures are time dependent and which are not, is highlighted. The nature of the general representation used also helps to clarify which shock the impulse-response function refers to in the case of models where more than one random disturbance impinges on the time series