Econ 551b Econometrics II
Problem Set 2

Prof. John Rust

Due: February 17, 1999

Question 1

Let where Y is random vector and is positive definite matrix. Prove . (Hint: Apply Jordan Decomposition on .)

Question 2

The Legendre Polynomials, denoted by , , ,..., can be obtained by finding the set of orthogonal vectors that span the same subspace in as 1, x, , ,... Using the Gram-Schmidt Orthogonalization Process, find the first four Legendre Polynomials. Then normalize the Legendre Polynomials so that they are orthonormal.

Question 3

Let be the OLS estimate in the regression:

And be the estimate from stepwise regression. That is, first obtain as the OLS estimate from the regression:

Then obtain as the OLS estimate from the following regression:

Where and are projections of y and on respectively, i.e., they are the predicted values in the regression of y and on respectively. Show that and are identical. Suggest when stepwise regression can be useful.

Question 4

Suppose , show that , where .

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