2. Unit root testing with the local level model. To do this exercise, use either real data (e.g. the industrial production data from the empirical illustration in Section 8.3.2) or artificial data generated according to various data generating processes of your choice. Use the local level model with natural conjugate Normal-Gamma prior described in Section 8.2 with the second variant on the empirical Bayesian methodology described in Section 8.2.3. That is, treat

as an unknown parameter, choose a prior for
of your choice (e.g. a Gamma or Uniform prior) and obtain p.
jy/. Remember that a unit root is present in the model M1 :
> 0, but is not present in the model M2 :
D 0. You want to calculate the Bayes factor comparing M1 to M2.

(a) Derive the formula for the marginal likelihood of M2.

(b) Using your result for part (a), write a program for calculating the required Bayes factor and test the program using your data set(s).

(c) Consider an approximate strategy where you calculate the Bayes factor comparing M1 to MŁ2 :
D a where a is a very small number. Using the Savage–Dickey density ratio, derive a formula for calculating the Bayes factor comparing M1 to MŁ2 .

(d) For your data set(s) compare the approximate Bayes factor of part

(c) to that obtained in part

(b) for various values of a (e.g. a D 0:01, 0.0001, 0.0000001, etc.). How well does the approximate strategy work?