Consider an AR model of order m; yn = a m 1 yn−1 +···+a m myn−m +vn, vn ∼ N(0,σ 2 m). (1) Using the relation σ 2 m = (1 − (a m m) 2 )σ 2 m−1 , show a criterion to judge whether or not AR(m) is better...

Consider an AR model of order m; yn = a m 1 yn−1 +···+a m myn−m +vn, vn ∼ N(0,σ 2 m).

(1) Using the relation σ 2 m = (1 − (a m m) 2 )σ 2 m−1 , show a criterion to judge whether or not AR(m) is better than AR(m−1).

(2) The PARCOR coefficients (a j j of AR(j)) are estimated by using 100 observations, and are given by a 1 1 = 0.9, a 2 2 = −0.6, a 3 3 = 0.3, a 4 4 = −0.1, a 5 5 = 0.15. Assuming that C0 = 1, compute σ 2 m, for m = 1,...,5.

(3) Assuming the situation of (2), compute AICm for m = 1,···,5 and determine the best order.