UOB Wk 4 Applied Econometrics Sample Regression and OLS Estimator Worksheet

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Mathematics

University of Bath

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Consider the three questions in the uploaded file carefully. Answer all questions. Answers should be submitted in pdf format and the word limit for the document is 750 words. Any mathematical derivations do not count towards the word limit.

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Week 4 Assessment Brief Answer all questions. Answers should be submitted in .pdf format and the word limit for your submitted document is 750 words. Any mathematical derivations do not count towards the word limit (but note that the automatic word count in your software might include derivations). 1 Q1. Suppose that the population relationship between two variables is given by Y = βX + u. The sample regression model yi = α̂ + β̂xi + ûi is estimated, with α̂ and β̂ being the Ordinary Least Squares estimators. Assuming that E[u|X] = E[u], are α̂ and β̂ unbiased? Prove your answer. Q2. The sample regression model ri = β̂0 + β̂1 pi + ûi is estimated using OLS. ri is the annual return (expressed in percentage points) on shares of company i and pi is the earnings per share (expressed in pounds sterling) of company i within the same year. For a sample of 100 listed companies, the estimates are β̂0 = 0.2 and β̂1 = 3.1. The standard errors are 0.15 and 1.2, respectively. 1. Interpret the estimated parameters of the model. 2. Is there evidence that the return depends upon the earnings per share? 3. Might these estimates be biased? Q3. Given the estimation results in question 2: ˆ Do you think the errors would be heteroskedastic in this case? ˆ Describe how you would test for heteroskedasticity in this regression. ˆ Outline the potential consequences of heteroskedasticity in this case and how these consequences could be addressed/remedied. 2
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Running head: LINEAR REGRESSION

1

Linear Regression
Student’s Name
Institutional Affiliation
Course
Date

Linear Regression
Question 1
𝑌 = 𝑋𝛽 + ɛ
(𝛽ˆ) = (𝑋 ′ 𝑋)−1 𝑋′𝑌
𝐼𝑓 𝑂𝐿𝑆 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 𝛽ˆ 𝑖𝑠 𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑, 𝑖𝑡 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝐸(𝛽ˆ) = 𝛽
Thus, substituting 𝑋𝛽 + ɛ in the 𝛽ˆ equation, one gets,
(𝛽ˆ) = (𝑋 ′ 𝑋)−1𝑋′(𝑋𝛽 + ɛ)
Remove the brackets
(𝛽ˆ) = (𝑋 ′ 𝑋)−1 𝑋′𝑋𝛽 + (𝑋 ′ 𝑋)−1 𝑋′ɛ
Cancelling (𝑋 ′ 𝑋)−1 𝑤𝑖𝑡ℎ 𝑋′𝑋 , we remain with 𝛽
Therefore,
(𝛽ˆ) = 𝛽 + (𝑋 ′ 𝑋)−1 𝑋′ɛ
However, mean of a constant is a constant, and since X’s are fixed, and the mean of error term
equal to zero, the last part of the equation equals to 0.
𝐸(ɛ) = 0
𝜀~(0, 𝛿 2 )
Hence, 𝐸(𝛽ˆ) = 𝛽, 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑛𝑔 𝑖𝑡 𝑖𝑠 𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑
Question 2
ui is the error term that is stochastic.Pi is the independent variable, or the explanatory variable
that influence the dependent variable. In this case, earnings per share influences the annual
return. 𝛽0 is the constant, or the y-intercept, meaning, earnings at zero earnings per share. Since
𝛽0 is 0.2, it means that 0.2 of the annual return is explained by other factors besides earnings per
share. Additionally, 𝛽1 is 3.1. It indicates that for every unit change of earnings per share of
company i, the annual returns will change by 3.1 in the same year. The standard errors show
deviations from the mean. It is a measure of precision with which the regression coefficient was
measured. Since the standard errors are 0.15 and 1.2, which are small, it means that the sample
mean is close to the population mean. There is evidence that a...


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