Let xt be a stationary AR(1) process given by xt = xt 1 + x t with N(0vx) Let yt = xt + y t with y t uncorrelated with the process xt and with y t N(0vy)

(a) Find the spectrum of yt .

(b) Simulate 500 observations from the process above, yt t = 1 : 500 using parameters = 09 vx = vy = 1

(c) Draw the periodogram of the 500 time series data points you simu lated in part (b).

(d) Fit a Bayesian autoregressive model to the simulated data y1:500 us ing a reference analysis as explained in Chapter 2 (you can pick the optimal model order using BIC). Obtain samples from the posterior distribution of the AR model parameters and the corresponding ob servational variance in this AR setting. Finally, draw samples of the resulting posterior spectral densities computed using posterior sam ples of the AR model parameters. Compare these to the true spectral density.