Consider the random coefficient model (14.7).

(a) Write the Level 1 conditional likelihood (given the variance components)

for fit, i.e., the density of the part of the data that depends on /Il.

(b) What is the posterior distribution of 8, I y,. A, 02,T ? (Hint:c ombine a normal prior and a normal likelihood as in Section 14.2.1, but with a multivariate distribution.)

(c) Write the posterior mean conditional on 1, n2, T as

d, =

fi: + (I - B)CB, fl), where fi; = x t i is the Level 2 regression prediction and 8, is the least-squares estimate of p, based on y,.

Note the similarity to the shrinkage factor in Exercise 14.8.

14.8*