Intro to Econometric Theory Heinz School, Carnegie Mellon University 90-906, Spring 2004-5

Homework #2, due Friday, March 18, 2005

1. You may recall that, in the bivariate regression model,

Y = β_{1} + β_{2}X_{2} + ϵ  (1)

the OLS estimate of β_{2} is β̂_{2,OLS} = \frac{Ĉov(Y,X_{2})}{V̂ (X_{2})}. This means that if the cor-

relation (or covariance) between Y and X_{2} is positive, then β̂_{2} is positive. Consider now a slightly more complex model:

Y = β_{1}X_{1} + β_{2}X_{2} + ϵ  (2)

Let’s suppose that \overline{Y} = \overline{X_{1}} = \overline{X_{2}} = 0. Please calculate an expression for β̂_{1,OLS} and β̂_{2,OLS} in terms of the variances and covariances of Y, X_{1}, X_{2}. Using your formula, discuss whether or not it can happen that Ĉov(Y, X_{1}) > 0 while β̂_{1,OLS} < 0. If it cannot, why not? If it can, then how? Your goal is both to construct a correct argument and to explain in intuitive English why your result holds.

2. Do problem 9 on page 115 of Greene

3. Do problem 10 on page 115 of Greene