Modelling Australian annual mean rainfall data: a new approach based on fractional integration

The modelling of rainfall and other hydrological data has been the point of much discussion in past research. Hurst (1951) heuristically detected the presence of long range dependence (or long memory) in the well-known series of annual minima of the Nile River. These processes are so called because they display a high degree of association between observations which are distant in time. Following Hurst’s work, extensive research has been carried out to detect long memory in hydrological data. According to Montanari and Rosso (1997) there are two main reasons for the presence of the Hurst phenomenon in time series: (a) the series exhibits persistence, i.e. a strong association between observations widely separated in time, and/or (b) the mean of the series changes with time, i.e. the series is nonstationary. Nowadays, it is widely accepted that the Hurst phenomenon may be caused by both serial correlation and nonstationarity. In another study, Montanari et al. (1996) examined six rainfall time series in various sites in Italy, in order to analyse whether a linear trend and/or long memory is present in the data. Among other methods, they propose a modified variance-type estimator to detect long memory even when some types of nonstationarity are potentially present. They found a decreasing trend, though not statistically significant, in each of their records, but significant long memory in only two of them. In this article we examine the long memory property of annual Australian-averaged rainfall data using fractional integration techniques. By fractional integration we mean that the order of differencing required to get a series that is stationary may be a fractional value. A proper discussion of this idea and other relevant statistical concepts will be presented in the following section. We employ a procedure that permits us to estimate the fractional differencing parameter even in nonstationary contexts. Moreover, the presence of a changing deterministic pattern is also examined through a method that allows us to estimate the deterministic terms and the fractional orders of integration at each subsample, the break-date being endogenously determined by the model itself. There are numerous papers describing analysis of Australian rainfall data. For example, McBride and Nicholls (1983) showed that the variability of rainfall across Australia is strongly influenced by the El Niño-Southern Oscillation (ENSO) on interannual time scales. Chiew and McMahon (2003) also demonstrated a link between ENSO and Australian rainfall. More recently, Srikanthan et al. (2007) argued that long-term persistence in Australian annual rainfall data Modelling Australian annual mean rainfall data: a new approach based on fractional integration