An econometrician notes that the assumed normal mixture error distri bution used in the SV model analysis is just an approximation to the sampling distribution of the data, and is worried that the speci c val ues of the (qj bj wj) might be varied to better t the data in a speci c application. To begin to explore this, assume the (bj wj) are xed and assumed appropriate, but now treat the mixing probabilities as uncer tain. That is, assume the more general model J

(ytzt 1:J)

j=1 jN(bj +ztwj)

where the J probabilities 1:J are now parameters to be estimated too.
The original q1:J are viewed as good rst guesses and used as prior means for a Dirichlet prior J p( 1:J)
j=1 qj 1 j over 0 < j <1 and subject to J j=1 j = 1 Taking a reasonably large value of the total precision such as =500 indicates that each j is likely to be close to its prior mean qj under this Dirichlet prior, but still allows for variation in the underlying sampling distribution.

(a) Conditional on 1:J the MCMC analysis of the SV model remains precisely the same, but with each qj replaced by j We now need to add one further conditional posterior distribution to generate, at each Gibbs step, a new value 1:JWhat is the appropriate conditional distribution?

(b) Implement the SV MCMC analysis now extended to incorporate this uncertainty about the normal mixture model form, resampling this distribution for 1:J at each step. Explore how the analysis results are impacted in examples.

(c) Describe how this general idea might be extended to incorporate some uncertainty about the chosen values of the (bj wj) as well as the qj