If ????̄???? = ∑∞

????=1 ????????????????+1−????, then for models (1) and (2) of Exercise 4.1, which are of the form (1 − ????)???????? = (1 − ????????)????????, ̄???????? is an exponentially weighted moving average.

For these two models, by actual calculation, confirm that ????̄11, ̄????12, and ????̄13 satisfy the relations ???????? = ̄????????−1 + ???????? (see Exercise 4.5)
̄???????? = ̄????????−1 + (1 − ????)????????
= (1 − ????)???????? + ???? ̄????????−1