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Unformatted text preview: Economics 4261 Introduction to Econometrics Fall 2010 HOMEWORK 3 : REGRESSION THEORY AND INFERENCE Question 1 In Homework 1, we solved the model A x b . Consider arbitrary matrix A , which is conformable to vector of variables x , such that the dimensions of A x corresponds to the dimension of b . (a) Write down such requirements formally. (b) What further conditions are required to ensure b P Col r A s . (c) Suppose A is diagonalizable and b P Col r A s . Characterize the solution. (d) Now suppose that A is not diagonalizable, but is of full column rank. Characterize the solution. (e) Call the solution to (d) x . Let c A x . What is c ? More specifically, relate c to a projection matrix. .................................................................................................................. This problem is incorrect since if a matrix is diagonalizable, this does not imply a matrix is invertible. Assuming this statement is true, here are some reasonable answers: (a) Let A be n K , which means x is K 1, and finally b is n 1. (b) For b P Col r A s , we require A to have fullcolumn rank, which is K . (c) If A is diagonalizable, then A is invertible, so there exists a unique solution given by x A 1 b . (d) If A is not diagonalizable, but is of full column rank, then A 1 A is invertible, and we can project b onto the space spanned by A . This solution is in fact unique by the Projection Theorem. The solution is x p A 1 A q 1 A 1 b . (e) c A p A 1 A q 1 A 1 b , where A p A 1 A q 1 A 1 b is the unique projection matrix onto A . Call this P A . Then c P A b . kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 1 of 15 Economics 4261 Introduction to Econometrics Fall 2010 Question 2 In class, we have seen that the OLS estimator for simple linear regression can be derived from a minimization problem. Consider the linear model y X , with K explanatory variables. Define : argmin P R K SSR p q . Prove such a solution exists. Prove the solution is unique. Hint : For the second part, show that if X is fullrank, then X 1 X is positive definite. .................................................................................................................. Please stop by to see this proof. kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 2 of 15 Economics 4261 Introduction to Econometrics Fall 2010 Question 3 The sampling error is defined as . Relate this expression to . Show that E r  X s 0. .................................................................................................................. This is equivalent to proving that the OLS estimator is unbiased. See your class notes....
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This note was uploaded on 02/24/2011 for the course ECON 4261 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff
 Econometrics

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