On the Estimation of Regression Coefficients of a Vector-Valued Time Series with a Stationary Residual

Abstract

Time series which are realizations of a vector-valued stochastic process of dimension two with a stationary disturbance are considered. Linear estimates of the regression coefficients of the time series are discussed, in particular the least-squares or classical estimate and the Markov estimate. The least-squares estimate is the estimate computed under the assumption that the components of the disturbance are orthogonal processes and orthogonal to each other. It is known that the Markov estimate is in general better than the least-squares estimate. The asymptotic behavior of the covariance matrices of the least-squares estimate and of the Markov estimate is described. Conditions under which the least-squares estimate is as good asymptotically as the Markov estimate are obtained, that is, conditions under which the least-squares estimate is efficient asymptotically in the class of linear unbiased estimates. The analogues of the results described for vector-valued time series of dimension greater than two can be seen to hold.