Multiple streamflow time series modeling using VAR–MGARCH approach

  • Farshad FathianEmail author
  • Ahmad Fakheri-Fard
  • T. B. M. J. Ouarda
  • Yagob Dinpashoh
  • S. Saeid Mousavi Nadoushani
Original Paper


Multivariate time series modeling approaches are known as valuable methods for simulating and forecasting the temporal evolution of hydroclimatic variables. These approaches are also useful for modeling the temporal dependence and cross-dependence between variables and sites. Although multiple linear time series approaches, such as vector autoregressive (VAR) and multiple generalized autoregressive conditional heteroscedasticity (MGARCH) approaches are ordinarily applied in finance and econometrics, these methods have not been broadly applied in hydrology science. The present research employs the VAR and VAR–MGARCH methods to model the mean and conditional variance (heteroscedasticity) of daily streamflow data in the Zarrineh Rood dam watershed, in northwestern Iran. The bivariate diagonal vectorization heteroscedasticity (DVECH) model, as one of the key MGARCH models, demonstrates how the conditional variance, covariance, and correlation structures change in time between the residual time series from VAR model. In this regards, in the present study, five experiments which present different combinations of twofold streamflows (including both upstream and downstream stations) are conducted. The VAR approach is fitted to the twofold daily time series in each of the experiments with different orders. The Portmanteau test, as a formal test for demonstrating time-varying variance (or so-called ARCH effect), indicates the existence of conditional heteroscedastic behavior in the twofold residual time series obtained from the VAR models fitted to the twofold streamflows. Thus, the VAR–DVECH approach is suggested to capture the inherent heteroscedasticity in daily streamflow series. The bivariate DVECH approach indicates short-term and long-term persistency in the conditional variance–covariance structure of the twofold residuals of streamflows. Results show also that the use of the nonlinear bivariate DVECH model improves streamflow modeling efficiency by capturing the heteroscedasticity in the twofold residuals obtained from the VAR model for all experiments. The assessment criteria indicate also that the VAR–DVECH approach leads to a better performance than the VAR model.


Multiple streamflow time series VAR Conditional variance–covariance Diagonal VECH Conditional correlation Assessment criteria 



The authors wish to thank the University of Tabriz and Water Resources Management Company for data provision. The authors wish to express their appreciation to the Editor-in-Chief, Dr. George Christakos, and to two anonymous reviewers for their invaluable comments and suggestions which helped considerably improve the quality of the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Water Science and Engineering, Faculty of AgricultureVali-e-Asr University of RafsanjanRafsanjanIran
  2. 2.Department of Water Engineering, Faculty of AgricultureUniversity of TabrizTabrizIran
  3. 3.National Institute for Scientific ResearchINRS-ETEQuébecCanada
  4. 4.Department of Water Resources Engineering, Faculty of Civil, Water and Environmental Engineering, Abbaspour School of EngineeringShahid Beheshti UniversityTehranIran

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