1. Consider a linear regression model: Y = Po + BX + BX + P3X3 +...

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1. Consider a linear regression model: Y = Po + B₁X₁ + B₂X₂ + P3X3 + B4X3 + U, where Y is the log wage, X₁ is unobserved ability, X₂ is the years of education, X3 is the age in years, and U is uncorrelated with X, X₂ and X3. (a) Suppose that X₁ is an observed measurement of X₁, for example, a test score. Assume that X₁ E1, where E1 is uncorrelated with X₁, X₂ and X3. Show that an OLS regression of Y on a constant term, X₁, X₂, X3 and X3 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 Z₁. Assume that Z₁ = X₁ + &2, where E2 1 is uncorrelated with X₁, X2, X3, U, and ₁. Show that Z₁ can be used as an IV for X₁. 1 = X₁ + 1. Consider a linear regression model: Y = Po + B₁X₁ + B₂X₂ + P3X3 + B4X3 + U, where Y is the log wage, X₁ is unobserved ability, X₂ is the years of education, X3 is the age in years, and U is uncorrelated with X, X₂ and X3. (a) Suppose that X₁ is an observed measurement of X₁, for example, a test score. Assume that X₁ E1, where E1 is uncorrelated with X₁, X₂ and X3. Show that an OLS regression of Y on a constant term, X₁, X₂, X3 and X3 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 Z₁. Assume that Z₁ = X₁ + &2, where E2 1 is uncorrelated with X₁, X2, X3, U, and ₁. Show that Z₁ can be used as an IV for X₁. 1 = X₁ +