Problem set 1:
1. Summarize the general idea of Bayesian model averaging. When would this method be preferred to choosing one of the alternative models? Provide an example related to your own research where you think model averaging could be useful.
2. Consider a linear regression model:
yt = θyt-1 + ∈t,
where θ is a scalar parameter, yt denotes the data for t = 0,1, ... , T observations and the er- ror terms have an independent normal distribution such that ∈t~N(0,5) for t = 1, ... , T and E(?t,?k) = 0 for t ≠ k.
Consider a truncated normal prior for θ:
θ~N(-1,1)(0,4),
where N(-1,1)(0,4) denotes a normal distribution with mean 0 and variance 4 on the restricted region (-1,1).
Derive the posterior density of θ for this model under the above prior distribution. Show that the posterior density of θ is zero outside the region (-1,1). Why do you think such a truncation is imposed on this model parameter?
3. Comment on the statement ‘When there is autocorrelation, you can just be a Bayesian, there is no need to worry about asymptotic tests'. Do you agree with this statement? How does this statement relate to the information matrix? Derive the Durbin Watson statistic for a model with first order autocorrelation in the disturbances.
Problem set 2:
The purpose of this exercise is to illustrate the Bayesian inference and simulation methods that have been discussed during the lectures in empirical examples. Provide a brief report on the Bayesian analysis of the extended New Keynesian Phillips Curve (NKPC) model that was presented in the re- search seminar.
1. Propose an addition to the extended NKPC model, for instance by including an equation on, say, interest rate behavior and/or an equation on possible long term expectations on infla- tion. What extra research question(s) on forecasting and policy analysis can you handle with this addition to the NKPC model? Suppose that you are interested inflation targeting, what does the literature say on this topic and how can you include that in the NKPC model? What other policy topics can be dealt with in your extended NKPC model?
2. Motivate the use of Bayesian inference for your model. In terms of results, which additional results can you obtain using a Bayesian procedure compared to using a classical/frequentist procedure for your model?
3. Summarize the prior(s) and the sampling technique(s) that you think are applicable to your model, including possible advantages and disadvantages. Speculate on the applicability of the Bayesian estimation methods on the model based on the topics covered in the lectures.