Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 7

Let Y denote a Bernoulli( ) random variable with 0 1. Suppose we are interested in estimating the odds ratio, = /(1- ), which is the probability of success over the the probability of failure. Given a random sample {Y 1 , …, Y n }, we know that an unbiased and consistent estimator of is , the proportion of successes in n trials. A natural estimator of is G = /(1 - ), the proportion of successes over the proportion of failures in the sample.
(i) Why is G not an unbiased estimator of
(ii) Use PLIM.2(iii)
PLIM.2 If plim ( T n ) = and plim ( U n ) = , then
(i) plim( T n _+ U n ) = + ;
(ii) plim( T n U n ) = ;
(iii) plim( T n / U n ) = / , provided 0.
to show that G is a consistent estimator of .