# Hw1_answer_econ120c_ - Economics 120C Professor Yongil Jeon Summer 2006 Name Student ID Answer to Homework#1 Summer 2006(Midterm Exam Summer 2005

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Economics 120C Name: _________________________ Professor Yongil Jeon Summer 2006 Student ID#: _________________________ Answer to Homework #1 – Summer 2006 (Midterm Exam, Summer 2005) 1. (15 points) Consider the simple model i i i u X Y + = β for which the variance of the error term is known to be 2 2 2 i i X σ = . (a) (5 points) Is the OLS estimate of unbiased, consistent, and efficient (carefully justify the answers)? Answer: Because the errors still satisfy the MLR assumptions 1 through 4, including the zero conditional mean assumption 0 ) | ( = i i X u E . The OLS estimates are unbiased and consistent. However, since the errors are heteroscedastic, estimates are inefficient, according to the Gauss-Markov Theorem. (b) (5 points) Describe step-by-step how you will go about obtaining the weighted least squares estimate of . Answer: Divide both sides of the equation i i i u X Y + = by i X so that the transformed equation is i i i i X u X Y + = The variance of i i X u is constant since () . 1 1 2 2 2 2 2 = = = i i i i i i X X u Var X X u Var Hence, we can apply OLS to the above equation and obtain estimates that are BLUE. (c) (5 points) Explain derive the WLS estimate of and give a geometric interpretation of the estimate. Answer: Because the model has just a constant term, the OLS estimate is simply the mean of the transformed dependent variable. Thus = = n i i i X Y n 1 1 ˆ . Geometrically, connect each of the observation points ( ) , i i X Y to the origin and compute its slope. The average of these slopes is the WLS estimate of .

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2 Answer to HW #1, ECON 120C, Summer 2006 2. (25 points) Consider the relationship between the expenditure on travel ( i E ) and the total income ( i Y ) given by i i i u Y E + + = 1 0 β You are certain that the error term is heteroscedastic with 2 2 1 0 2 ) ( i i i i P a P a a u Var + + = = σ , where i P is the population. You have data on E, Y, and P. (a) (5 points) State the null and alternative hypotheses for no heteroscedasticity. Answer: , 0 : 2 1 0 = = a a H vs : 1 H at least one of them is not zero. (b) (5 points) Describe the regressions to be run for carrying out the test. Answer : (1) Regress i E against a constant and i Y , (2) compute the residuals i i i Y E u 1 0 ˆ ˆ ˆ = (3) regress 2 ˆ i u against a constant, i P and 2 i P . (c)
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## This note was uploaded on 07/17/2008 for the course ECON 120C taught by Professor Stohs during the Spring '08 term at UCSD.

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Hw1_answer_econ120c_ - Economics 120C Professor Yongil Jeon Summer 2006 Name Student ID Answer to Homework#1 Summer 2006(Midterm Exam Summer 2005

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