Persistence and Nonstationary Models

The aim of this paper is to examine the measurement 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 alternative nonstationary processes. Three measures of persistence are considered, namely the long-run impulse response, variance ratio and autocorrelation functions. Particular emphasis is given to the behaviour of these measures in a range of nonstationary models. We document 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 clarify what shock the impulse response function refers to in the case of models where more than one random disturbance impinges on the time series.