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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...