Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 12 1 2005 10.5194/npg-12-55-2005 http://www.nonlin-processes-geophys.net/12/55/2005/ http://www.nonlin-processes-geophys.net/12/55/2005/npg-12-55-2005.html http://www.nonlin-processes-geophys.net/12/55/2005/npg-12-55-2005.pdf 55 66 2005-01-21 Testing and modelling autoregressive conditional heteroskedasticity of streamflow processes W. Wang P. H. A. J. M Van Gelder J. K. Vrijling J. Ma Faculty of Water Resources and Environment, Hohai University, Nanjing, 210098, China Faculty of Civil Engineering & Geosciences, Section of Hydraulic Engineering, Delft University of Technology, P.O.Box 5048, 2600 GA Delft, The Netherlands Yellow River Conservancy Commission, Hydrology Bureau, Zhengzhou, 450004, China Conventional streamflow models operate under the assumption of constant variance or season-dependent variances (e.g. ARMA (AutoRegressive Moving Average) models for deseasonalized streamflow series and PARMA (Periodic AutoRegressive Moving Average) models for seasonal streamflow series). However, with McLeod-Li test and Engle's Lagrange Multiplier test, clear evidences are found for the existence of autoregressive conditional heteroskedasticity (i.e. the ARCH (AutoRegressive Conditional Heteroskedasticity) effect), a nonlinear phenomenon of the variance behaviour, in the residual series from linear models fitted to daily and monthly streamflow processes of the upper Yellow River, China. It is shown that the major cause of the ARCH effect is the seasonal variation in variance of the residual series. However, while the seasonal variation in variance can fully explain the ARCH effect for monthly streamflow, it is only a partial explanation for daily flow. It is also shown that while the periodic autoregressive moving average model is adequate in modelling monthly flows, no model is adequate in modelling daily streamflow processes because none of the conventional time series models takes the seasonal variation in variance, as well as the ARCH effect in the residuals, into account. Therefore, an ARMA-GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) error model is proposed to capture the ARCH effect present in daily streamflow series, as well as to preserve seasonal variation in variance in the residuals. The ARMA-GARCH error model combines an ARMA model for modelling the mean behaviour and a GARCH model for modelling the variance behaviour of the residuals from the ARMA model. Since the GARCH model is not followed widely in statistical hydrology, the work can be a useful addition in terms of statistical modelling of daily streamflow processes for the hydrological community.