1.4 Demonstrate from first principles that the least squares estimator of 𝛽1 in the primitive model where Y consists simply of a constant plus a disturbance term, 𝑌𝑖=𝛽1+𝑢𝑖 is 𝛽1̂=𝑌̅, the sample mean of Y. (First define RSS and then differentiate)
1.5 The table shows the average annual percentage rates of growth of employment, e, and real GDP, g, for 31 OECD countries for the period 2002-2007. The regression output shows the result of regressing e on g. Provide an interpretation of the coefficients.
Average annual percentage rates of growth of employment and real GDP, 2002-2007:
1.6 In Exercise 1.5, 𝑒̅=1.3603,𝑔̅=3.6508, Σ(𝑒𝑖−𝑒̅)(𝑔𝑖−𝑔̅)=21.5935, Σ(𝑔𝑖−𝑔̅)2=90.7639.
Calculate the regression coefficients and check that they are the same as in the regression output.
1.7 Does educational attainment depend on intellectual ability? In the United States, as in most countries, there is a positive correlation between educational attainment and cognitive ability. S (highest grade completed by 2011) is the number of years of schooling of the respondent. ASVABC is a composite measure of numerical and verbal ability scaled to have mean 0 and standard deviation 1 (both approximately; for further details of the measure, see Appendix B). Perform a regression of S on ASVABC and interpret the regression results (Use EAWE Data set 22).
1.9 The output shows the result of regressing the weight of the respondent in 2004, measured in pounds, on his or her height, measured in inches, using EAWE Data Set 21. Provide an interpretation of the coefficients.
1.17 Two individuals investigate the relationship between weight in 2004 and height using EAWE Data Set 21. The first individual regresses weight on height and obtains the result found in Exercise 1.9: 𝑊𝐸𝐼𝐺𝐻𝑇04̂=−177+5.07𝐻𝐸𝐼𝐺𝐻𝑇
The second individual intended to do the same but makes a mistake and regresses height on weight, obtaining the following result: 𝐻𝐸𝐼𝐺𝐻𝑇̂=59.2+0.052𝑊𝐸𝐼𝐺𝐻𝑇04
From this result the second individual derives 𝑊𝐸𝐼𝐺𝐻𝑇04̂=−1138.5+19.23𝐻𝐸𝐼𝐺𝐻𝑇
Explain why this equation is different from that fitted by the first individual.
1.20 What was the value of 𝑅2 in the educational attainment regression fitted by you in Exercise 1.7? Comment on it.
2.3 For the model 𝑌𝑖=𝛽1+𝑢𝑖, the OLS estimator of 𝛽1 is 𝛽1̂=𝑌̅.
Demonstrate that 𝛽1̂ may be decomposed into the true value plus a linear combination of the disturbance terms in the sample. Hence demonstrate that it is an unbiased estimator of 𝛽1.
2.4 An investigator correctly believes that the relationship between two variables X and Y is given by 𝑌𝑖=𝛽1+𝛽2𝑋𝑖+𝑢𝑖. Given a sample of n observations, the investigator estimates 𝛽2 by calculating it as the average value of Y divided by the average value of X. Discuss the properties of this estimator. What difference would it make if it could be assumed that 𝛽1=0?