Suppose that your linear regression model includes a constant term, so that in the linear regression model y = X + " ; (1) the matrix of explanatory...

1. Suppose that your linear regression model includes a constant term, so that in the

linear regression model

y = X + " ; (1)

the matrix of explanatory variables X can be partitioned as follows: X = [i X1]. The

OLS estimator of  can thus be partitioned accordingly into b0 = [b0 b0

1], where b0

is the OLS estimator of the constant term and b1 is the OLS estimator of the slope

coecients.

(a) Find the inverse of the matrix X0X. (hint: Apply result (A-74).)

(b) Use partitioned regression to derive formulas for b1 and b0. (Note: Question 5 of

Problem Set 1 asks you to do this without using partitioned regression.)

(c) Derive var(b1 j X) using (B-87). How is your answer related to your answer to

part (a)?

(d) What is var(b0 j x)? (You should be able to answer this question without doing

any further derivations, using your answers to parts (a) - (c).)

2. Suppose that instead of estimating the full regression model including the constant

term, you have estimated instead a model in deviations from means; i.e., you have

regressed M0y on M0X1. We can write the estimating equation in this case as

M0y = M0X11 +M0" ; (2)

Call the OLS estimator of 1 in this equation eb

1.

(a) Derive eb

1. How does it compare to b1 in question 1?

(b) Let the residuals vector for equation (2) be ee

. Show that ee

is identical to e, the

vector of OLS residuals for equation (1).