Comparison between ARMA and ARMA-PARCH Models in Modeling the Annual River Discharge (Case Study: Zarineh-Rud River in the West Azerbaijan Province)

In econometrics, the autoregressive conditional heteroscedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms; often the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average model (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroscedasticity (GARCH) model. For forecasting, combining ARIMA and ARCH models could be considered. For instance, a hybrid ARIMA-ARCH model was examined for shipping freight rate forecast. ARCH models are commonly employed in modeling financial time series that exhibit time varying volatility and volatility clustering, i.e. periods of swings interspersed with periods of relative calm. ARCH-type models are sometimes considered to be in the family of stochastic volatility models, although this is strictly incorrect since at time t the volatility is completely pre-determined (deterministic) given previous values. After the introduction of ARCH models there were enormous theoretical and practical developments in financial econometrics in the eighties. It became clear that ARCH models could efficiently and quite easily represent the typical empirical findings in financial time series, e.g. the conditional heteroscedasticity. In particular after the collapse of the Bretton Woods system and the implementation of flexible exchange rates in the seventies ARCH models are increasingly used by researchers and practitioners. However the ARCH model is only the starting point of the empirical study and relies on a wide range of specification tests. Some practically relevant disadvantages of the ARCH model have been discovered recently, for example, the definition and modelling of the persistence of shocks and the problem of modelling asymmetries. Thus a large number of extensions of the standard ARCH model have been suggested. We will discuss them in detail later. Time series models as the mathematical-physical models are able to model linear and nonlinear processes. These models contain two sectors included stochastic and deterministic components that deterministic part of model has been estimated by the observed data and stochastic part of model has been calculated by stochastic methods. So, the structure of time series models is in accordance with the structure of hydrological series if the appropriate model selected (Salas, 1993). The most applied time series models in the hydrology and water resources studies, is linear models. Nonlinear models is used and developed in the sciences related with the statistics, economics and mathematics. The application of linear time series models in hydrology has been started from the current four decades and developed by the Box-Jenkins models. Thomas and Fiering (1962) were as the first researches that had been used linear autoregressive models for the river flow analysis. The purpose of this paper is to introduce combined ARMA-PARCH model in order to modeling river discharge and increasing accuracy of hydrological modeling and comparing current ARMA models with the combined model. Time series modeling has been applied by Thomas and Fiering (1962) for the first time and has been developed by Box and Jenkins in the 1970. Among time series models AR model is the most simple model on the basis of Markov chain method. ARCH model is the first model for modeling variance series proposed by Engle (1982). Since then, it has been widely used to model volatility of financial and economic time series. The main idea of the ARCH models is in two forms: (A) The corrected average of return period is independent (B) Model is dependent and can be explained by a simple quadratic function. Using time series models is one of the applicable ways to simulate and predict hydrological series. One of the most problems in the predicting time series models is method of generating random series. In this process, with the changing generated data, predicted series will be changed. In this study, after initial investigations on the Zarineh-rud River flow series, ARMA models fitted and ARMA(1,0) model selected as the best model. Residual series of ARMA model extracted and fitted with the PARCH model. Finally two models combined and ARMA-PARCH model obtained. Results showed that by the combining ARMA and PARCH models, error (root mean square error) and accuracy of model (regression coefficient) improved about 40 and 10 percent. Also from the results it is concluded that maximum and minimum points of the average river discharge series is modeled properly with the combined model in the studied station.