Case study on married couples

6.3 Probit regression: A panel study followed 25 married couples over a period of five years. One item of interest is the relationship between divorce rates and the various characteristics of the couples. For example, the researchers would like to model the probability of divorce as a function of 238 Exercises age differential, recorded as the man’s age minus the woman’s age. The data can be found in the file divorce.dat. We will model these data with probit regression, in which a binary variable Yi is described in terms of an explanatory variable xi via the following latent variable model: Zi = ßxi + i Yi = d(c,8)(Zi), where ß and c are unknown coefficients, 1, . . . , n ~ i.i.d. normal(0, 1) and d(c,8)(z) = 1 if z > c and equals zero otherwise. a) Assuming ß ~ normal(0, t 2 ß ) obtain the full conditional distribution p(ß|y, x, z, c). b) Assuming c ~ normal(0, t 2 c ), show that p(c|y, x, z, ß) is a constrained normal density, i.e. proportional to a normal density but constrained to lie in an interval. Similarly, show that p(zi |y, x, z-i , ß, c) is proportional to a normal density but constrained to be either above c or below c, depending on yi . c) Letting t 2 ß = t 2 c = 16 , implement a Gibbs sampling scheme that approximates the joint posterior distribution of Z, ß, and c (a method for sampling from constrained normal distributions is outlined in Section 12.1.1). Run the Gibbs sampler long enough so that the effective sample sizes of all unknown parameters are greater than 1,000 (including the Zi ’s). Compute the autocorrelation function of the parameters and discuss the mixing of the Markov chain. d) Obtain a 95% posterior confidence interval for ß, as well as Pr(ß > 0|y, x).

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