Predykcja szeregów czasowych w warunkach autokorelacji składników losowych

Abstract

In the article the autor discusses prediction problems in the general linear model where disturbances are autocorrelated. Suppose we have sample data for periods 1 to n from the model y=Xß+u with E(u)=0 and E(uu')=V. Suppose further that we are given the row vector xn + 1 of values of the explanatory variables in period n+1. The problem now how to estimate yn+s when E(uu')=V=σ2 uΩ, where Ω is assumed to be known, symmetric, positivedefinite matrix. If the disturbances follows a first-order scheme ut=γ1ut-1+εt where |γ1|<1, then we can see that the best linear unbiased predictor is ŷn+1=xn+sb+(γ1)s·en where b is the generalized least-squares estimator and en is the last row of the vector e=y—Xb of generalized least-squares residuals. When the elements of Ω (that is, the value of γ1) were unknown formula pictured above should still be used for prediction purposes with b and γ1 replaced by estimates.