On shifted multiple arma processes

Abstract

Let {X _t} be a p-dimensional ARMA (m n) process. Write where have q and r components, respecctivelyq+r=p). Put . It is proved that {Y _t} is an ARMA (m -1. n + 1) process and a procedure for evaluation of its matrices of coefficients is given. If {X _t} is an AR (m) process, then {Y _t is an AR (m -f1) process; if {X,_tis an MA(n-) process, then {Y _t} is an MA (n +1) process. Explicit formulas are derived for p = 2m = 1n = 0. The repeated use of the described procedure leads to a model for , The results can be applied in theorv of extrapolation in multiple ARMA processes when the different components of a process are known to different points of time.

Keywords:

• Multiple ARMA processes
• shift of some components
• extrapolation