1. Show that if A and B are positive semi-definite matrices of dimension Kx K, then...

 

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1. Show that if A and B are positive semi-definite matrices of dimension Kx K, then A+B will also be positive semi-definite. Under what conditions will A+B be positive definite? (10 marks) 2. Consider the following possible estimator of 3 in the linear regression model: 3= (X'X+r1) 'X'y, where r is a positive constant. (a) Find E(X). Is 3 an unbiased estimator of 3? (10 marks) (b) Find var( X). (10 marks) (1) (c) Which estimator has the largest variance - the OLS estimator b or 3? To answer this question, you need to determine whether var(b|X) - var (3X) is positive definite or negative definite. (25 marks) Hints for question 2(c): For this question, the result that any nonsingular matrix A has the property that AA-¹ = I will be very useful. You will also need the result of question 1 and the following result: If A is a symmetric matrix and B is a nonsingular matrix, both of the same order, then (i) B'AB will be positive definite if and only if A is positive definite; and (ii) B'AB will be positive semi-definite if and only if A is positive semi-definite. 1. Show that if A and B are positive semi-definite matrices of dimension Kx K, then A+B will also be positive semi-definite. Under what conditions will A+B be positive definite? (10 marks) 2. Consider the following possible estimator of 3 in the linear regression model: 3= (X'X+r1) 'X'y, where r is a positive constant. (a) Find E(X). Is 3 an unbiased estimator of 3? (10 marks) (b) Find var( X). (10 marks) (1) (c) Which estimator has the largest variance - the OLS estimator b or 3? To answer this question, you need to determine whether var(b|X) - var (3X) is positive definite or negative definite. (25 marks) Hints for question 2(c): For this question, the result that any nonsingular matrix A has the property that AA-¹ = I will be very useful. You will also need the result of question 1 and the following result: If A is a symmetric matrix and B is a nonsingular matrix, both of the same order, then (i) B'AB will be positive definite if and only if A is positive definite; and (ii) B'AB will be positive semi-definite if and only if A is positive semi-definite.

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