A seasonal periodic long memory model for monthly river flows
Introduction
It is well known since the early work by Hurst on Nile data that river flows show long, persistent changes in mean which may be characterized by long memory. Additional to long memory, most river flow data display pronounced seasonality, both in mean and in variance.
Lawrance and Kottegada (1977) presented an overview of early results on the statistical modeling of river flows. One of the main objectives of these modeling efforts is to develop simulation models which can be used for the design and operation of reservoirs. Brochu (1978) noticed that even modest improvements in the operation of large reservoir systems can result in multi-million dollar savings per year. The time series analysis of river flow data has remained an innovating research area. Novel statistical models for simulation, forecasting and diagnostic analysis have been introduced for river flow data and other new methods have been tried on river flow data soon after their introduction, see e.g. McLeod and Hipel (1978) and Hipel and McLeod, 1978a, Hipel and McLeod, 1978b.
Noakes et al. (1988) compared the one-year-ahead forecasting ability of ARMA models, fractional Gaussian noise models, Fractional ARMA models, and Markov and nonparametric regression models for yearly river flow series, analysed earlier by Hipel and McLeod, 1978a, Hipel and McLeod, 1978b. They showed that it is hard to find significant differences between the models, but the simple fractional models seemed too restrictive for the four series they studied.
The evidence on the adequacy of statistical models for seasonality is much clearer. Not only the mean and variance of monthly river flows depend on the season. Other characteristics like skewness and autocorrelation do as well, as shown by Moss and Bryson (1974).
The skewness is usually taken care of by an a priori log transformation of the series and the seasonally dependent autocorrelations are successfully modeled using periodic autoregressive moving average (PARMA) models, see e.g. Vecchia and Ballerini (1991). Periodic autoregressive (PAR) models have definite computational advantages over PARMA models. PAR models are easy to identify using periodic partial autocorrelation functions, they are easy to estimate using least squares, from the Yule–Walker equations or by Maximum Likelihood, see e.g. McLeod (1994). PAR models for monthly riverflow modeling and simulation were originally introduced by Thomas and Fiering (1962).
Noakes et al. (1985) compared the short-term forecasting ability of seasonal ARIMA, deseasonalized ARMA and periodic autoregressive (PAR) models on 30 monthly river flow series. The results clearly suggest that periodic autoregressive models identified by the partial autocorrelation function, provided the most accurate forecasts. They also established the superiority of the natural log transformation over other Box–Cox transformations, Box and Cox (1964), in a classical likelihood framework.
Although the explicit modeling of long range dependence may not be too useful for point-forecasting, especially if the process is stationary, it can still be important for confidence interval forecasting, see e.g. Ray (1993). It also plays a deciding role in hypothesis testing and in the development of simulation models. It is, for example, important to take account of long range dependence if one wants to do inference about the (seasonal) long run mean of a process, as emphasized by Beran (1994) in the first chapter of his monograph on “Statistics for Long-Memory Processes”. Neglecting long-range dependence may result in gross downward biases in estimates of the uncertainty about the mean. This is especially relevant if one wants to test for structural stability of the correlation structure and the mean, where proper estimation of the variance of the (sub)sample means is crucial.
Beran and Terrin (1996) reanalyzed yearly minimum water levels for the river Nile (622-1281) using an ARFIMA(0,d,0) model and found significant changes in the correlation structure over time. The analysis of structural change in long geophysical time series is particularly interesting for climate research. Atkinson et al. (1997) used recent structural time series for the annual flow of the Nile (1871–1970) to illustrate new tests for structural breaks and found that the process could well be described by white noise allowing for a couple of additive outliers and a structural break due to the building of the Aswan dam in 1899. MacNeill et al. (1991) surveyed earlier analyses of those Nile data.
A class of models, which seems to have been overlooked in the literature on river flow modeling is the class of seasonal long memory models, introduced by Carlin et al. (1985) for economic time series. These models describe long range dependence in the seasonal pattern time series, and focus on the correlation structure at yearly intervals. Our model focuses on this aspect as well. We discuss the relationship between our model and other seasonal long memory models in more detail in Section 3.1 below. In our model we combine seasonal long memory allowing for the now well established periodic variation in the autocovariance function. The combination of these two features may explain both the long memory apparent in yearly series of minima of river flows, the so-called Joseph effect, and the absence of long memory in aggregate yearly river flow data. In this paper we specify and estimate such a seasonal periodic model for monthly river flows.
Droughts and floods are phenomena that are typical for special seasons of the year. If we look specifically at autocorrelations for data of a specific month we may notice the long nonperiodic cycles, whereas we overlook them if we aggregate flow data over the year. This is of course more likely to happen if the seasonal long range dependence occurs for months with relatively small flows.
There seems to be a misunderstanding among practitioners that seasonal (fractional) differencing as applied in seasonal AR(F)IMA modeling, and periodic modeling, as in PAR models, are substitutes for describing seasonal phenomena. Our application shows they are complements: seasonal parameters can be periodic as well!
The specification of our model does not involve new statistical problems. The models can easily be estimated using existing software for ARFIMA analysis. We basically extend the periodic AR(3) model of McLeod (1994), introducing error terms which display seasonal fractional integration which varies from month to month. For application we consider the monthly Fraser river flow data at Hope, B.C., made available by Ian McLeod at Statlib. The current URL is http://lib.stat.cmu.edu/datasets/.
We show how we can capture the interesting long memory characteristic, which appears evident from the periodic autocorrelation functions at long yearly lags. Statistical analysis shows seasonal long memory to be significant, especially for the spring and the autumn months. Our statistical analysis provides an additional test on model adequacy and can be used as a parametric complement to the residual serial correlation tests for periodic models developed by Vecchia and Ballerini (1991), McLeod (1994) and Franses (1996) and residual correlation tests for long memory models by Beran (1992) and Robinson (1994).
The outline of our paper is as follows. In Section 2 we present some characteristics of the monthly Fraser river flow data. In Section 3 we propose the novel seasonal periodic long memory model, compare it with related models and discuss estimation issues and available software. Section 4 provides the empirical analysis and Section 5 concludes.
Section snippets
Data and memory characteristics
Let yt denote a monthly time series, t=1,2,⋯, n, where yt concerns log-transformed data of the monthly mean river flows in cubic feet per second, following the analyses in Vecchia and Ballerini (1991) and McLeod (1994). Let Ym,T denote these observations, m=1,2,3,⋯,12 and T=1,2,⋯, N, so that m denotes the number of the month and T denotes the number of the year. We have N years of subsequent observations with monthly data. To simplify notation we only use complete years with observations
A seasonal periodic long memory model
The basic idea of our model is very simple. It is a standard Seasonal Autoregressive Integrated Moving Average (p,0,0)×(P,D,Q)12 model (SARIMA), where the integration parameter d and the MA order for monthly lags, q are zero and where the other parameters of the model, i.e. the seasonal AR and MA parameters, the seasonal integration parameter D and the AR parameters for the monthly lags are allowed to vary from month to month. Furthermore, we allow for fractional D, so that the seasonal pattern
Empirical results
We first check McLeod's periodic AR(3) model using the familiar diagnostic regression tests implemented in PcGive 9.0, Hendry and Doornik (1996). We simply regressed the April series on a constant and the series for March, February and January. The May series were also regressed on the three previous months and so forth. These regression showed no substantial problems with the model specification at yearly lags, with regard to nonnormality and short run serial correlation. McLeod (1994) already
Conclusion
We extended the periodic AR model for the monthly data for the logs of the Fraser river flow, developed by McLeod (1994) with a periodic seasonal long memory innovation proces. The statistical model detected the presence of long non-periodic cycles which are especially clear in sample correlations for the data for the month of March. We estimated the model using Gaussian Maximum Likelihood. If the long memory is not too pronounced and the MA-part of the model is clearly invertible one can use
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