Consider the following models:

yt = 1yt 1+ 2yt 2+ t yt = acos(2 0t)+bsin(2 0t)+ t

(1.26)

(1.27)

with t N(0v) and 0 >0 xed.

(a) Sample T = 200 observations from each model using your favorite choice of the parameters. Make sure your choice of ( 1 2) in model

(1.26) lies in the stationary region. That is, choose 1 and 2 such that 1 < 2<1, 1<1 2 and 1> 2 1.

(b) Find the MLEs of the parameters in models (1.26) and (1.27). Use the conditional likelihood for model (1.26).

(c) Find the MAPestimators of the model parameters under the reference prior. Again, use the conditional likelihood for model (1.26).

(d) Sketch p(vyF) and p( 1 2yF) for model (1.26).

(e) Sketch p(abyF) and p(vyF) for model (1.27).

(f) Perform a conjugate Bayesian analysis, i.e., repeat

(c) to

(e) assuming conjugate prior distributions in both models. Study the sensitivity of the posterior distributions to the choice of the hyperparameters in the prior.