14.4 Consider the collapsed hierarchical model (14.6).

(a) Write out the matrices J, K. I,, as in Equation (14.3). Assume that i. j , and k each have two levels; make clear thc ordering of the indices

(iJ, k) labeling each row and column of these matrices.

(b) Write the marginal covariance matrix of y given p and the variance components (with entries that are combinations of the variance components).

Verify visually (without writing out the details) that the exchangeability relationships described in Exercise 14.3

(c) hold for the covariance matrix you wrote.

(d) For a more notationally challenging version, write the same matrices J, K, L for a general number of levels of each index using Kronecker product notation.