Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 1

Consider the regression model without an intercept term,
(so the true value of the intercept, 0 , is zero).
a. Derive the least squares estimator of 1 for the restricted regression model
This is called the restricted least squares estimator
of 1 because it is estimated under a restriction, which in this case is 0 = 0.
b. Derive the asymptotic distribution of
under Assumptions #1 through #3 of Key Concept 17.1.
c. Show that
is linear [Equation (5.24)] and, under Assumptions #1 and #2 of Key Concept 17.1, conditionally unbiased [Equation (5.25)].
d. Derive the conditional variance of
under the Gauss Maikov conditions (Assumptions #1 through #4 of Key Concept 17.1).
e. Compare the conditional variance of
in (d) to the conditional variance of the OLS estimator
(from the regression including an intercept) under the Gauss-Markov conditions. Which estimator is more efficient Use the formulas for the variances to explain why.
f. Derive the exact sampling distribution of
under Assumptions #1 through #5 of Key Concept 17.1.
g. Now consider the estimator
Derive an expression for
under the Gauss-Markov conditions and use this expression to show that