Introduction to Econometrics 3rd Edition by James Stock, James Stock

Exercise 6

This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience.
a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent.
b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d.
i. Show that the OLS estimator can be written as
ii. Suppose that data are "missing completely at random" in the sense that where p is a constant. Show that is unbiased and consistent.
iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent.
iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain.
c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.