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YEAR: 2024-25
GROUP: L41
EC2020: ELEMENTS OF ECONOMETRICS
ASSIGNMENT 1
INSTRUCTIONS
1. Answer all questions.
• All your answers must be handwritten on A4-size papers (lined or blank) and converted to PDF, to be uploaded on Canvas.
2. You are required to upload your answers latest by 4 NOVEMBER 2024 (Monday) at 11:59pm. The window period for uploading your answers will not be extended beyond this time and late submission will not be entertained. Answers sent through email will not be entertained or marked.
3. Please write your name and student ID on top of the right-hand corner of the first page of your answers.
Question 1 [8 marks]
Consider the following simple linear regression (SLR) function:
where ui is the error term and is not correlated with xi .
a) Assume that E(ui |xi ) = c , where c is a constant. Express E(yi |xi ) in terms of β0, β1, xi and c (3 marks).
b) An econometrician has rewritten model (1) as the following model
yi = α0 + α1xi + vi (2)
where vi is the error term and is not correlated with xi . Additionally, he imposes the assumption that E(vi
|xi
) = E(vi ) = 0. Using your answer obtained for part (a), show how
the parameters (intercept and slope) in model (1) and model (2) are related. (3 marks)
c) From your answers in part (a) and (b) above, what can we infer about the relationship between the zero conditional mean assumption and the error term in a regression model? (2 marks)
Question 2 [8 marks]
Consider the bivariate regression model
yi = β0 + β1xi + ui
for a given random sample of n observations {(yi , xi)}i
n
=1under the standard Assumptions SLR.1 - SLR.4 (linear in parameters, random sampling, non-zero sample variation in the regressor, and zero conditional mean).
Assuming the intercept is zero, the OLS estimator is given by:
(a) Express β 1 in terms of xi , β0 and β1. [3 marks]
(b) With your answer obtained in part (a), show that β1 in unbiased for β1 when the population intercept (β0) is zero, conditional on X = {xi , i = 1,2, … , n}. Make sure you explain your steps in the derivations by referring to Assumptions SLR.1 to SLR.5 or other necessary requirements. [5 marks]
Question 3 [8 marks]
Consider the bivariate regression model
yi = βxi + ui
for a given random sample of n observations {(yi , xi)}i
n
=1under the standard Assumptions SLR.1 - SLR.4 (linear in parameters, random sampling, non-zero sample variation in the regressor, and zero conditional mean).
Consider the following estimator:
(a) Show that β is an unbiased estimator of β [3 marks].
(b) Show that β is also a consistent estimator of β [3 marks]
(c) Since β is both unbiased and consistent, it can concluded that it is the Best Linear Unbiased Estimator (BLUE). Justify whether you agree with this claim. [2 marks]