Consider a linear regression model: = 0 + 11 + 22 + 33 + 4(3)^2 + , where is the log wage, 1 is unobserved ability, 2 is the years of education, 3 is the age in years, and is uncorrelated with 1, 2 and 3.

(a) Suppose that 1 is an observed measurement of 1, for example, a test score. Assume that 1 = 1 + 1, where 1 is uncorrelated with 1, 2 and 3. Show that an OLS regression of on a constant term, 1, 2, 3 and (3)^2 gives an inconsistent estimator for 1. This problem is called an errors-in-variables problem.

(b) Could you say something about the direction of the bias?

(c) Now suppose that there is another test score available, say 1. Assume that 1 = 1 + 2, where 2 is uncorrelated with 1, 2, 3, , and 1. Show that 1 can be used as an IV for 1.