SEMIFAR models - a semiparametric framework for modelling trends, long-range dependence and nonstationarity

Time series in many areas of application often display local or global trends. Typical models that provide statistical \explanations" of such trends are, for example, polynomial regression, smooth bounded trends that are estimated nonparametrically, and di erence-stationary processes such as, for instance, integrated ARIMA processes. In addition, there is a fast growing literature on stationary processes with long memory which generate spurious local trends. Visual distinction between the large variety of possible models, and in particular between deteministic, stochastic and spurious trends, can be very di cult. Also, for some time series, several \trend generating" mechanisms may occur simulateneously. In this paper, a class of semiparametric fractional autoregressive models (SEMIFAR) is proposed that includes deterministic trends, di erence stationarity and stationarity with shortand long-range dependence. Parameters characterizing stochastic dependence and stochastic trends, including a fractional and an integer di erencing parameter, can be estimated by maximum likelihood. Deterministic trends are estimated by kernel smoothing. In combination with automatic model and bandwidth selection, the proposed method allows for exible modelling of time series and helps the data analyst to decide whether the observed process contains a stationary shortor long-memory component, a di erence stationary component, and/or a deterministic trend component. Data examples from various elds of application illustrate the method. Finite sample behaviour is studied in a small simulation study.