Improvement of Extrapolation in Multiple Time Series

Abstract

Let x_t be a p-dimensional random process with zero expectation and with finite second moments. We shall investigate the discrete case, when t=…,−1,0,1,‖. Assume that the vectors x_t-1,X_t-2,… are known and that we wish to get the best linear extrapolation ${{\rm{\hat X}}_{\rm{t}}}$ of the vector X_t. Our extrapolation ${{\rm{\hat X}}_{\rm{t}}}$. can be calculated either by a method given in Rozanov [3] or by a well known iterative procedure, which is briefly described in our Section 2. The accuracy of ${{\rm{\hat X}}_{\rm{t}}}$ is usually measured by the residual variance matrix

$${\Delta _{\rm{X}}}{\rm{ = E(}}{{\rm{X}}_{\rm{t}}}{\rm{ - }}{{\rm{\hat X}}_{\rm{t}}})({{\rm{X}}_{\rm{t}}}{\rm{ - }}{{\rm{\hat X}}_{\rm{t}}})',$$

where the prime denotes the transposition. If the diagonal elements of the matrix Δ_X are too large, the extrapolation ${{\rm{\hat X}}_{\rm{t}}}$ is not satisfactory and we should like to improve it. It may happen that we can observe another discrete q-dimensional process {Y_t}, also with zero expectation and with finite second moments. If {Y_t} is correlated with {X_t}, we can use the information contained in {X_t} and improve the extrapolation of the process {X_t} Denote ${{\rm{W}}_{\rm{t}}}{\rm{ = }}({{\rm{X'}}_{\rm{t}}}{\rm{,}}{{\rm{Y'}}_{\rm{t}}})'$. The simplest case is that we know vectors w_t-1,w_t-2’… so “tha”t it is possible to calculate the best linear extrapolation ${{\rm{\hat W}}_{\rm{t}}}$ of the vector W_t. in an usual way. The first p components of ${{\rm{\hat W}}_{\rm{t}}}$ will be denoted by ${{\rm{\hat X}}_{\rm{t}}}(1,1)$. We can also say that ${{\rm{\hat X}}_{\rm{t}}}(1,1)$ is the best linear extrapolation of X_t in the case that the vectors x_t-l,Y_t-l.X_t-2,Y_t-2,… are known Surprisingly, it may happen even in “regular” models that ${{\rm{\hat X}}_{\rm{t}}}{\rm{ = }}{{\rm{\hat X}}_{\rm{t}}}(1,1)$ see Andċl [1]