Stochastic Hydrol. Hydraul. 1 (1987) 141-154 StochasticHydrology and Hydraulics 9 Springer-Verlag 1987 Multivariate contemporaneous ARMA model with hydrological applications F. Camacho and A. I. McLeod Dept. of Statistics and Actuarial Sciences, The University of Western Ontario, London, Ontario N6A 5B9, Canada K. W. Hipel Dept. of Systems Design Engineering and Dept. of Statistics and Actuarial Sciences, University of Waterloo, Ontario N2L 3G1, Canada Abstract: In order to allow contemporaneous autoregressive moving average (CARMA) models to be properly applied to hydrological time series, important statistical properties of the CARMA family of models are developed. For calibrating the model parameters, efficient joint estimation procedures are investigated and compared to a set of uivariate estimation procedures. It is shown that joint estimation procedures improve the efficiency of the autoregressive and moving average parameter estimates, but no improvements are expected on the estimation of the mean vector and the variance covariance matrix of the model. The effects of the different estimation procedures on the asymp- totic prediction error are also considered. Finally, hydrological applications demonstrate the useful- ness of the CARMA models in the field of water resources. Key words: Contemporaneous ARMA models, maximum likelihood estimation, multivariate model- ling, stochastic hydrology, time series analysis I Introduction For more than two decades, hydrologists have been advocating the use of multivari- ate models for describing complex hydrological data. Recently, for example, the import of multivariate modeling in hydrology was reinforced by a number of manuscripts that appeared in a conference proceedings edited by Shen et al. (1986) and also a special monograph on time series analysis in water resources edited by Hipel (1985). When considering the general family of multivariate autoregressive moving average (ARMA) models, a particular subset of this family, called contem- poraneous ARMA or CARMA models, is well suited for modeling hydrological time series (Salas et al. 1980; Camacho et al. 1985 ). The main objective of this paper is to derive useful statistical properties of CARMA models so that they can be conveniently and properly applied to hydrological, as well as other types of time series. The contemporaneous ARMA (p,q) model, CARMA (p,q), is defined as: (ph(B)(Zh,t -- ~th) = Oh(B)ah, t h = 1 ..... k (I) where r = 1 -- ~hl B . . . . . (~hphB ph is the autoregressive (AR) operator of order Ph for series h; 0h(B) = 1 -- 0hlB ..... OhqhBqh is the moving avearge (MA) operator of order qh for series h; a t = (air , . .... akt )' is the k dimensional vector of innovations which is distributed as NID (0, A), where NID