Household income in marketing surveys is often reported in brackets: for example, under $15,000, $15,000$25,000, ... , over $95,000.

(a) Suppose that household income is the outcome variable in a normal linear model. Using clear and consistent notation, show that this model, in conjunction with the reporting of income in brackets, is a special case of the censored linear model.

(b) Suppose that the logarithm of household income is the covariate xtk (the last column of X) in a normal linear model. Suppose further that the normal linear model is a correct specificationin the notation of Section 3.4, p(y | X,D) = p(y | , h, X, A) for appropriate values = and h = h. Finally, suppose that household income is incorrectly assumed to be at the midpoint of the observed bracket, and $140,000 is assumed to be household income if the observed bracket is ($95,000,). Show that the pseudotrue values of and h in this situation are not and h, respectively.

(c) Let X denote columns 1 through k 1 of X, and X = [x 1,..., x T ]. Formulate a complete model incorporating the assumption that household incomes xtk are conditionally independent given the x t , with log(xtk ) N ( x t , j1). Utilize a prior distribution for and j that is independent of the prior distribution for and h.

(d) Outline a Gibbs sampling algorithm to simulate the posterior distribution of the model in (c).

(e) Given the assumptions made about p(y | X,D) in (b) and about xtk in (c), are the pseudotrue values of and h in (c) equal to and h, respectively?

(f) Suppose that in part (b) the level of household income rather than the logarithm of household income were the covariate xtk . How would this affect the Gibbs sampling algorithm in part (d)?