Consider the regression model without an intercept term, Yi = 1Xi + ui (so the true value

Consider the regression model without an intercept term, Yi = β1Xi + ui (so the true value of the intercept, β0, is zero).
a. Derive the least squares estimator of β1 for the restricted regression model Yi = β1Xi + ui. This is called the restricted least squares estimator (β1RLS) of β1 because it is estimated under a restriction, which in this case is β0 = 0.
b. Derive the asymptotic distribution of β1RLS under Assumptions #1 through #3 of Key Concept 17.1.
c. Show that β1RLS 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 β1RLS under the Gauss-Markov conditions (Assumptions #1 through #4 of Key Concept 17.1).
e. Compare the conditional variance of β1RLS in (d) to the conditional variance of the OLS estimator β1 (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 β1RLS under Assumptions #1 through #5 of Key Concept 17.1.
g. Now consider the estimator

Drive an expression for

under the Gauss-Markov conditions and use this expression to show that

Transcribed Image Text:

(Bi|X,,.. . X) var(BRX...,X) RIS A An 2 var P1