19.16 This exercise takes up the problem of missing data discussed in Section 9.2.

Consider the regression model Yi = Xib + ui, i = 1,

c, n, where all variables are scalars and the constant term/intercept is omitted for convenience.

a. Suppose that the least squares assumptions in Key Concept 4.3 are satisfied.

Show that the least squares estimator of b is unbiased and consistent.

b. Now suppose that some of the observations are missing. Let Ii denote a binary random variable that indicates the nonmissing observations; that is, Ii = 1 if observation i is not missing, and Ii = 0 if observation i is missing.

Assume that {Ii, Xi, ui} are i.i.d.

i. Show that the OLS estimator can be written as b

n

= aa n

i = 1 IiXiXi b

-1 aa n

i = 1 IiXiYib = b + aa n

i = 1 IiXiXi b

-1 aa n

i = 1 IiXiuib.

ii. Suppose that data are missing “completely at random” in the sense that Pr(Ii = 1Xi, ui) = p, where p is a constant. Show that b n is unbiased and consistent.

iii. Suppose that the probability that the ith observation is missing depends of Xi but not on ui; that is, Pr(Ii = 1Xi, ui ) = p(Xi). Show that b n is unbiased and consistent.

iv. Suppose that the probability that the ith observation is missing depends on both Xi and ui; that is, Pr(Ii = 1Xi, ui) = p(Xi, ui). Is b n unbiased?
Is b n consistent? Explain.

c. Suppose that b = 1 and that Xi and ui are mutually independent standard normal random variables [so that both Xi and ui are distributed N(0, 1)].
Suppose that Ii = 1 when Yi Ú 0 but that Ii = 0 when Yi 6 0. Is b n unbiased? Is b n consistent? Explain.