You observe yt = xt + t = 12 where xt follows a stationary AR(1) process with AR parameter and innovation variance v, i.e., xt = xt 1 + t with independent innovations t N(0v) Assume all parameters (

v) are known.

(a) Identify the ACF and PACF of yt, and comment of comparisons with those of xt

(b) What is the marginal distribution of yt?

(c) What is the distribution of (ytyt 1)?

(d) What is the distribution of (yty1 yt 1)?

(e) Nowconsider as a parameter to be estimated. As a function of and conditioning on the initial value y1 what is the likelihood function p(y2 yT+1y1 )?

(f) Assume v are known. Under the reference prior p( ) constant show that the resulting posterior for based on the conditional like lihood above is normal with precision (1

)2T v and give an ex pression for the mean of this posterior.

(g) Show that, for large T the reference posterior mean above is approx imately the sample mean of the yt data.

(h) If = 0, we have the usual normal random sampling problem. For nonzero values of the above posterior for the mean of the normal data yt depends on in the posterior variance. Comment on how the posterior changes with and why this makes sense.