De ne Ii = 1 if a family is poor and Ii = 0 if a family is not poor, and de ne a family to be poor if
its income, yi, is less than L, a government de ned level. Let
yi = Xi + i
with i N(0; 2).
a) If you have data only on Ii and Xi, which parameters of the model are estimable and how?
b) Now assume that you observe yi but your sample consists of poor families only.
i. Derive the expected value of yi given Xi in the whole population. What is the expected value
of yi given Xi in your sample (i.e. among poor families)?ii. If you estimate the model by OLS, is your estimate of a consistent estimate of the e ect of
X?
iii. Write down the log likelihood function for this model.
c) Finally, assume that you have data on all the families. However, the income yi is missing for the
poor families. How would you estimate the e ect of X on income? Write down the log likelihood
function.