Assume that a time series follows an AR model of order 1, yn = ayn−1 +vn, vn ∼ N(0,1). (1) When the noise term vn is not a white noise but follows an autoregressive process of order 1, vn = bvn−1 +wn,...

Assume that a time series follows an AR model of order 1, yn = ayn−1 +vn, vn ∼ N(0,1).

(1) When the noise term vn is not a white noise but follows an autoregressive process of order 1, vn = bvn−1 +wn, show that yn follows an AR model of order 2.

(2) Obtain the autocovariance function Ck , k = 0,1,2,3 of the contaminated series xn defined by xn = yn +wn,wn ∼ N(0,0.1).