View - Statistics - University of Washington

61E[(Y i − µ)(Y (i−j) − µ)] = E[β W +1 (Y i−(W +1) − µ)(Y (i−j) − µ)+β W (Y i−W − µ)(Y (i−j) − µ)+β W −1 (Y i−(W −1) − µ)(Y (i−j) − µ)+β 1 (Y i−1 − µ)(Y (i−j) − µ) + ɛ i (Y (i−j) − µ)](4.66)Let γ j denote the covariance of Y i and Y i−j . Note that γ j = γ −j . Since E[Y i −µ] = 0, equation 4.66 is actually equal to γ j . We can now rewrite equation 4.66 interms of γ.γ j = β W +1 γ W +1−j + β W γ W −j + β W −1 γ W −1−j + β 1 γ 1−j + I(j = 0)σ 2 ɛ (4.67)where I(j = 0) is an indicator function equal to 1 when j = 0 and 0 otherwise.We now have an equation for γ j for arbitrary j in terms of β values, σɛ 2 , andother γ j values. The equations for γ 0 to γ W +1 form a system of W + 2 equationswith W + 2 unknowns (treating the β and σɛ2 values as constant). This can besolved for γ 0 ; this will yield an equation of the form given in equation 4.68.γ 0 = h(β W +1 , β W , β W −1 , β 1 )σɛ 2 (4.68)Here, h(β W +1 , β W , β W −1 , β 1 ) is a function which can be found algebraically forany given value of W . For the more general AR(P) model, equation 4.68 willcontinue to hold, except that h will be a function of β 1 to β P . Since γ 0 is thevariance of Y , we can rewrite equation 4.68 as equation 4.69.σ 2 ɛ =σ 2 Yh(β W +1 , β W , β W −1 , β 1 )(4.69)