Sample Selection and Non-response. This is based on Manski (1995), see the Appendix to this chapter. Suppose we are interested in estimating the probability than an individual who is homeless at a given date has a home six months later. Let y = 1 if the individual has a home six months later and y = 0 if the individual remains homeless. Let x be the sex of the individual and let z = 1 if the individual was located and interviewed and zero otherwise. 100 men and 31 women were initially interviewed. Six months later, only 64 men and 14 women were located and interviewed.

Of the 64 men, 21 exited homelessness. Of the 14 women only 3 exited homelessness.

(a) Compute Pr[y = 1/Male, z = 1], Pr[z = 1/Male] and the bound on Pr[y = 1/Male].

(b) Compute Pr[y = 1/Female, z = 1], Pr[z = 1/Female] and the bound on Pr[y = 1/Female].

(c) Show that the width of the bound is equal to the probability of attrition. Which bound is tighter? Why?