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Structural Break Or Long Memory: an Empirical Survey on Daily Rainfall

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Structural Break Or Long Memory: an Empirical Survey on Daily Rainfall EGU Journal Logos (RGB) Open Access Open Access Open Access Advances in Annales Nonlinear Processes Geosciences Geophysicae in Geophysics Open Access Open Access Natural Hazards Natural Hazards and Earth System and Earth System Sciences Sciences Discussions Open Access Open Access Atmospheric Atmospheric Chemistry Chemistry and Physics and Physics Discussions Open Access Open Access Atmospheric Atmospheric Measurement Measurement Techniques Techniques Discussions Open Access Open Access Biogeosciences Biogeosciences Discussions Open Access Open Access Climate Climate of the Past of the Past Discussions Open Access Open Access Earth System Earth System Dynamics Dynamics Discussions Open Access Geoscientific Geoscientific Open Access Instrumentation Instrumentation Methods and Methods and Data Systems Data Systems Discussions Open Access Open Access Geoscientific Geoscientific Model Development Model Development Discussions Open Access Open Access Hydrol. Earth Syst. Sci., 17, 1311–1318, 2013 Hydrology and Hydrology and www.hydrol-earth-syst-sci.net/17/1311/2013/ doi:10.5194/hess-17-1311-2013 Earth System Earth System © Author(s) 2013. CC Attribution 3.0 License. Sciences Sciences Discussions Open Access Open Access Ocean Science Ocean Science Discussions Structural break or long memory: an empirical survey on daily Open Access Open Access rainfall data sets across Malaysia Solid Earth Solid Earth Discussions F. Yusof1, I. L. Kane1, and Z. Yusop2 1Department of Mathematical Sciences, Faculty of Science University of Technology Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia Open Access Open Access 2Institute of Environmental and Water Resources Management, University of Technology Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia The Cryosphere The Cryosphere Discussions Correspondence to: I. L. Kane ([email protected]) Received: 8 October 2012 – Published in Hydrol. Earth Syst. Sci. Discuss.: 31 October 2012 Revised: 25 January 2013 – Accepted: 17 March 2013 – Published: 8 April 2013 Abstract. A short memory process that encounters occa- in distinguishing between non-stationarity, sample variability sional structural breaks in mean can show a slower rate of and long-term climatic fluctuations (Brath et al., 1999). decay in the autocorrelation function and other properties of Studies by Cheung (1993) and Diebold and Inoue (2001) fractional integrated I (d) processes. In this paper we em- have shown that there is a bias in favour of finding long- ployed a procedure for estimating the fractional differencing memory processes when structural breaks are not accounted parameter in semiparametric contexts proposed by Geweke for in a time series. Observed long memory behaviour can be and Porter-Hudak (1983) to analyse nine daily rainfall data due to neglected structural breaks. The presence of breaks sets across Malaysia. The results indicate that all the data sets in a time series and/or of long memory behaviour in the exhibit long memory. Furthermore, an empirical fluctuation break-free series could indicate whether the series shows real process using the ordinary least square (OLS)-based cumula- breaks or long memory. A further insight can be obtained by tive sum (CUSUM) test for the break date was applied. Break estimating the difference parameter on the subseries identi- dates were detected in all data sets. The data sets were parti- fied by splitting the original series according to the estimated tioned according to their respective break date, and a further break dates: in the case of erroneous long memory identifi- test for long memory was applied for all subseries. Results cation, the subseries are expected to show short memory. show that all subseries follows the same pattern as the origi- If a time series has one or more structural breaks, then the nal series. The estimate of the fractional parameters d1 and d2 series has one or more discontinuities in the data-generating on the subseries obtained by splitting the original series at the process (DGP). In this case a structural break method will break date confirms that there is a long memory in the data report a number of breaks which will divide the series into generating process (DGP). Therefore this evidence shows a regimes which are of different subpopulations. The statistical true long memory not due to structural break. properties of these subpopulations within the regimes will need to be estimated. The estimated differences will be the result of actual differences between the samples. The task of estimating the break dates can be accomplished within the 1 Introduction framework of least squares regression (Rea et al., 2009). In the last decade, a lot of interest has been paid to the Recent analyses have claimed the possible presence of non- issue of confusing long memory and occasional structural stationarity is produced by the presence of either trend or breaks in mean (see among others Diebold and Inoue, 2001; long-term cyclic fluctuations. However, it is well known that Granger and Hyung, 2004; and Smith, 2005). Indeed, there is a reliable assessment of the presence of non-stationarity in evidence that a stationary short memory process that encoun- hydrological records is not an easy task because of the lim- ters occasional structural breaks in mean can show a slower ited extension of the available data sets. This causes difficulty Published by Copernicus Publications on behalf of the European Geosciences Union. 1312 F. Yusof et al.: Structural break or long memory rate of decay in the autocorrelation function and other prop- in Europe, Canada and USA to detect the existence of long erties of fractionally integrated (I (d)) processes (Cappelli memory. The results show that 29 out of the 31 daily series and Angela, 2006). Therefore, a time series with structural exhibit long memory as confirmed by at least three meth- breaks can generate a strong persistence in the autocorrela- ods, whereas the other two series are indicated to have long tion function, which is an observed behaviour of a long mem- memory with two methods. Gil-Alana (2012) analysed the ory process. On the contrary, long memory processes may UK monthly rainfall data from a long-term persistence view- cause breaks to be detected spuriously. In this paper, we used point using different modelling approaches, taking into ac- the term true long memory to refer to fractionally integrated count the strong dependence and the seasonality in the data. series. The results indicate that the most appropriate model is the The literature on the tests to distinguish between true long one that presents cyclical long-run dependence with the or- memory and various spurious long memory models has been der of integration being positive though small, and the cy- steadily growing. For example, Berkes et al. (2006) and cles having a periodicity of about a year. Other applications Shao (2011) proposed a testing procedure to discriminate a of long memory models to hydrological time series can be stationary long memory time series from a short-range de- found in Hosking (1984), Koutsoyiannis (2003, 2005a, b, pendent time series with change points in the mean. Their 2006), Koscielny-Bunde et al. (2006), Rybski et al. (2006), null hypothesis corresponds to changes in the mean, and their Langousis and Koutsoyiannis (2006), and Mudelsee (2007). alternative is that the series is stationary with long mem- Several authors did a reasonably good job in testing for ory. The test statistic is a modification of the cumulative long memory in hydro-meteorological time series; however, sum (CUSUM-type) test, which is quite popular in the lit- is it really a long memory process or a short memory with erature of change point detection (e.g. Galeano, 2007; Shao structural break? This is the main concern of this paper. In and Zhang, 2010). order to have an insight on “which is which”, we propose fit- Since the work of Hurst (1951) that detected the presence ting long memory and structural break separately following of long-term persistence in minimum annual flow series of Cappelli and Angela (2006). In case both provide a plausible the Nile River, a lot of studies have been carried out for test- explanation of the DGP of the data at hand, the long mem- ing and modelling long memory in hydrological processes. ory and structural break analysis are repeated on the break- For example, Montanari et al. (1997) studied the monthly free series and on the filtered series, respectively. However and daily inflows of Lake Maggiore, Italy, using fractional an in Malaysia and other tropical regions, little or not much has autoregressive integrated moving average (ARIMA) model. been done in this area. Rao and Bhattacharya (1999) studied the memory of four Heavy rainfall could bring disaster such as floods and land- hydrologic time series in Midwestern United States by test- slides. Of course, the shortage of rainfall could also affect the ing the null hypothesis that there is only short-term mem- water management system in such a way it could cause prob- ory in the series using a modified version of the rescaled lems to the economic activities. Therefore, there is a need range method. The main conclusion from the study is that to investigate the characteristic of rainfall of a country inten- there is little evidence of long-term memory in monthly hy- sively and comprehensively. Modelling of daily rainfall using drologic series. However, for annual series, the evidence for various mathematical models has been done throughout the lack of long-term memory is inconclusive, mainly because world to give a better understanding about the rainfall pattern the number of observations is small and the power of the test and its characteristics (e.g. Suhaila and Abdul Aziz, 2008). based on modified rescaled range is low with small samples. The purpose of this paper is to apply a simple strategy to Koutsoyiannis (2002) proposed a simple explanation of the study whether the daily rainfall data sets of nine weather sta- Hurst phenomenon based on the fluctuation of a hydrologi- tions across Malaysia exhibit true long memory behaviour, cal process at different temporal scales.
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