A researcher tries two regression models to describe the relationship between a variable X > 0 and a variable Y > 0:

log Y = 1 + 2 log X + u, (1)

log(XY ) = 1 + 2 log X + v. (2)

(a) Explain whether model (2) is a restricted version of (1) or not.

(b) Using a random sample of size n, the researcher obtains (1, 2, 1 , 2 ) as the OLS estimators for (1, 2, 1 , 2 ). Let (u, v) be the residuals of the OLS estimations. Prove that u = v.

(c) Assume that the regression equation (3.1) describes the true relationship between X and Y and there exists a variable W uncorrelated with Y. What are the properties of the OLS estimator for 2 if we use the following regression specification.

log Y = 1 + 2 log X + 3W + w? (3)