You are interested on the relation between wages (y), education (z) and age (x). Your econometrician...

 

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You are interested on the relation between wages (y), education (z) and age (x). Your econometrician friend tells you that a well known estimator is defined by the solution of the following minimization problem: n minab(yi-azi - bx;)² i=1 (1) (a) [8 marks] Write down the first order condition of the minimization problem (1). (b) [8 marks] The first order conditions form a linear system of two equations in a and b. Write them in matrix format AB = B where 3 = [a, b]T. (c) [8 marks] Write down the Hessian matrix. Is the objective function globally convex? (Hint: (z) ([x²) - (Σ xizi) ² > 0) (d) [8 marks] Given that Σx² = 1 and Σ = = 2, Σΐΐΐΐ = ² Zixi 1, yizi = 4, Σ yixi = 5; what is the rank of A? Are the columns of A linearly independent? (e) [8 marks] Is there a unique solution for 3, how do you know? If there is, use Cramer's rule to find the solution for a. (f) [4 marks] Verify your answer to the previous question by finding A-¹ and solv- ing for B. (g) [6 marks] Assume now that i = ax; for all i. Is there now a unique solution to ABB? If there is not, are there infinite solutions or none? You are interested on the relation between wages (y), education (z) and age (x). Your econometrician friend tells you that a well known estimator is defined by the solution of the following minimization problem: n minab(yi-azi - bx;)² i=1 (1) (a) [8 marks] Write down the first order condition of the minimization problem (1). (b) [8 marks] The first order conditions form a linear system of two equations in a and b. Write them in matrix format AB = B where 3 = [a, b]T. (c) [8 marks] Write down the Hessian matrix. Is the objective function globally convex? (Hint: (z) ([x²) - (Σ xizi) ² > 0) (d) [8 marks] Given that Σx² = 1 and Σ = = 2, Σΐΐΐΐ = ² Zixi 1, yizi = 4, Σ yixi = 5; what is the rank of A? Are the columns of A linearly independent? (e) [8 marks] Is there a unique solution for 3, how do you know? If there is, use Cramer's rule to find the solution for a. (f) [4 marks] Verify your answer to the previous question by finding A-¹ and solv- ing for B. (g) [6 marks] Assume now that i = ax; for all i. Is there now a unique solution to ABB? If there is not, are there infinite solutions or none?