Modeling rainfall–runoff relationship using multivariate GARCH model

The traditional hydrologic time series approaches are used for modeling, simulating and forecasting conditional mean of hydrologic variables but neglect their time varying variance or the second order moment. This paper introduces the multivariate Generalized Autoregressive Conditional Heteroscedasticity (MGARCH) modeling approach to show how the variance–covariance relationship between hydrologic variables varies in time. These approaches are also useful to estimate the dynamic conditional correlation between hydrologic variables. To illustrate the novelty and usefulness of MGARCH models in hydrology, two major types of MGARCH models, the bivariate diagonal VECH and constant conditional correlation (CCC) models are applied to show the variance–covariance structure and cdynamic correlation in a rainfall–runoff process. The bivariate diagonal VECH-GARCH(1,1) and CCC-GARCH(1,1) models indicated both short-run and long-run persistency in the conditional variance–covariance matrix of the rainfall–runoff process. The conditional variance of rainfall appears to have a stronger persistency, especially long-run persistency, than the conditional variance of streamflow which shows a short-lived drastic increasing pattern and a stronger short-run persistency. The conditional covariance and conditional correlation coefficients have different features for each bivariate rainfall–runoff process with different degrees of stationarity and dynamic nonlinearity. The spatial and temporal pattern of variance–covariance features may reflect the signature of different physical and hydrological variables such as drainage area, topography, soil moisture and ground water fluctuations on the strength, stationarity and nonlinearity of the conditional variance–covariance for a rainfall–runoff process.