1. Consider a linear regression model: Y = Po + BX + BX + P3X3 + B4X3 + U, where Y is the log wage, X is unobserved ability, X is the years of education, X3 is the age in years, and U is uncorrelated with X, X and X3. (a) Suppose that X is an observed measurement of X, for example, a test score. Assume that X E1, where E1 is uncorrelated with X, X and X3. Show that an OLS regression of Y on a constant term, X, X, X3 and X3 gives an inconsistent estimator for . 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 Z. Assume that Z = X + &2, where E2 1 is uncorrelated with X, X2, X3, U, and . Show that Z can be used as an IV for X. 1 = X +