SEMIFAR models—a semiparametric approach to modelling trends, long-range dependence and nonstationarity

Abstract

Time series in many areas of application often display local or global trends. Statistical “explanations” of such trends are, for example, polynomial regression, smooth bounded trends that are estimated nonparametrically, and difference-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 deterministic, stochastic and spurious trends can be very difficult. For some time series, several “trend generating” mechanisms may occur simultaneously. Here, a class of semiparametric fractional autoregressive models ( SEMIFAR) is proposed that includes deterministic trends, difference stationarity and stationarity with short- and long-range dependence. The components of the model can be estimated by combining maximum likelihood estimation with kernel smoothing in an iterative plug-in algorithm. The method helps the data analyst to decide whether the observed process contains a stationary short- or long-memory component, a difference stationary component, and/or a deterministic trend component. Data examples from climatology, economics and dendrochronology illustrate the method. Finite sample behaviour is studied in a small simulation study.