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 asymptotic prediction error are also considered. Finally, hydrological applications demonstrate the usefulness of the CARMA models in the field of water resources.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ansley, C.F. 1979: An algorithm for the exact likelihood of a mixed autoregressive moving average process. Biometrika 66, 59–65
Bloomfield, P. 1972: On the error prediction of a time series. Biometrika 59, 501–507
Box, G.E.P.; Jenkins, G.M. 1976: Time series analysis, forecasting and control. San Francisco: Holden Day
Camacho, F. 1984: Contemporaneous CARMA modelling with application. Ph.D. thesis, Dept. of Statistical and Actuarial Sciences, The University of Western Ontario, London, Canada
Camacho, F.; McLeod, A.I.; Hipel, K.W. 1985: Contemporaneous autoregressive—moving average (CARMA) modeling in hydrology. Water Res. Bulletin 21, 709–720
Camacho, F.; McLeod, A.I.; Hipel, K.W. 1987: Contemporaneous bivariate time series models. Biometrika 74, 103–113
Cipra, T. 1984: Simple correlated ARMA processes. Mathematische Operationsforschung und Statistik, Series Statistics 15, 513–525
Cox, D.R.; Hinkley, D.V. 1974: Theoretical statistics. London: Chapman and Hall
Dunsmuir, W.; Hannan, E.J. 1976: Vector linear time series models. Advances in Appl. Prob. 8, 449–464
Granger, C.W.J. 1969: Investigating casual relations by econometric models and cross spectral methods. Econometrica 37, 424–438
Granger, C.W.J.; Newbold, P. 1979: Forecasting economic time series. New York: Academic Press
Harvey, A.C. 1981: The econometric analysis of time series. Oxford: Philip Allan
Haugh, L.D. 1976: Checking the independence of two covariate stationary time series: A univariate residual cross-correlation approach. JASA 71, 378–385
Haugh, L.D.; Box, G.E.P. 1977: Identification of dynamic regression (distributed lag) models connecting two time series. JASA 72, 121–130
Hillmer, S.C.; Tiao, G.C. 1979: Likelihood function of stationary multiple autoregressive moving average models. JASA 74, 602–607
Hipel, K.W. (ed.) 1985: Time series analysis in water resources. American Water Res. Association, Bathesda, Maryland
Hipel, W.K.; McLeod, A.I.; Li, W.K. 1985: Casual and dynamic relationships between natural phenomenon. In: Anderson, O.D.; Ord, J.K.; Robinson, E.A. (eds.) Time series analysis: theory and practice 6, pp. 13–34, Amsterdam: North-Holland
Isserlis, L. 1918: On a formula for the product moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134–139
Li, W.K.; McLeod, A.I. 1981: Distribution of the residual autocorrelations in multivariate ARMA models. J. Royal Stat. Soc. 43, 231–239
McLeod, A.I. 1977: Improved Box-Jenkins estimators. Biometrika 64, 531–534
McLeod, A.L. 1978: On the distribution of residual autocorrelations in Box-Jenkins models. J. Royal Stat. Soc. 40, 296–302
McLeod, A.I. 1979: Distribution of the residual cross correlations in univariate ARMA time series models. JASA 74, 849–855
McLeod, A.I.; Holanda Sales, P.R. 1983: Algorithm AS191: An algorithm for approximate likelihood calculation of ARMA and seasonal ARMA models. Applied Statistics 32, 211–223
Moriarty, M.; Salamon, G. 1980: Estimation and forecast performance of a multivariate time series model of sales. J. Market Research 17, 558–564
Nelson, C.R. 1976: Gains in efficiency from joint estimation of systems of autoregressive—moving average processes. J. Econometrica 4, 331–348
Pierce, D.A. 1977: Relationships—and the lack thereof—between economic time series, with special reference to money and interest rates. JASA 72, 11–26
Pierce, D.A.; Haugh, L.D. 1977: Causality in temporal systems: Characterizations and survey. J. Econometrica 5, 265–293
Pierce, D.A.; Haugh, L.D. 1979: The characterization of instantaneous causality: A comment. J. Econometrica 10, 257–259
Risager, F. 1980: Simple correlated autoregressive process. Scandinarian J. Statistics 7, 49–60
Risager, F. 1981: Model checking of simple correlated autoregressive processes. Scandinarian J. Statistics 8, 137–153
Salas, J.D.; Delleur, J.W.; Yevjevich, V.; Lane, W.L. 1980: Applied modeling of hydrologic time series. Littleton, Colorado: Water Res. Publications
Shen, H.W.; Obeysekera, J.T.B.; Yevjevich, V.; Decoursey, D.G. (eds.) 1986: Multivariate analysis of hydrologic processes. Engineering Research Center, Colorado State University, Fort Collins, Colorado
Umashankar, S.; Ledolter, J. 1983: Forecasting with diagonal multiple time series models: An extension of univariate models. J. Market Research 20, 58–63
Wilson, G.T. 1973: The estimation of parameters in multivariate time series models. J. Royal Stat. Soc. 35, 76–85
Yamamoto, T. 1981: Predictions of multivariate autoregressive—moving average models. Biometrika 68, 485–492
Yevjevich, V. 1963: Fluctuation of wet and dry years. I. Paper 1, Colorado State University, Fort Collins, Colorado
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Camacho, F., McLeod, A.I. & Hipel, K.W. Multivariate contemporaneous ARMA model with hydrological applications. Stochastic Hydrol Hydraul 1, 141–154 (1987). https://doi.org/10.1007/BF01543810
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01543810