econometrics 12

1. Use the data in MLB1 for this exercise.

(i) Run the following model:

log(𝑠𝑎𝑙𝑎𝑟𝑦)= 𝛽0 + 𝛽1 𝑦𝑒𝑎𝑟𝑠+ 𝛽2 𝑔𝑎𝑚𝑒𝑠𝑦𝑟+ 𝛽3 𝑏𝑎𝑣𝑔+ 𝛽4 h𝑟𝑢𝑛𝑠𝑦𝑟+𝛽5 𝑟𝑏𝑖𝑠𝑦𝑟+𝑢 012345

Report your estimation results in the usual form.
(ii) Now drop the variable 𝑟𝑏𝑖𝑠𝑦𝑟. What happens to the statistical significance of h𝑟𝑢𝑛𝑠𝑦𝑟? What about

the size of the coefficient on h𝑟𝑢𝑛𝑠𝑦𝑟?
(iii) Add the variables 𝑟𝑢𝑛𝑠𝑦𝑟 (runs per year), 𝑓𝑙𝑑𝑝𝑒𝑟𝑐 (fielding percentage), and 𝑠𝑏𝑎𝑠𝑒𝑠𝑦𝑟 (stolen

bases per year) to the model from part (ii). Which of these factors are individually significant? (iv) In the model from part (iii), test the joint significance of 𝑏𝑎𝑣𝑔, 𝑓𝑙𝑑𝑝𝑒𝑟𝑐, and 𝑠𝑏𝑎𝑠𝑒𝑠𝑦𝑟.

2. Use the data in HTV to answer this question.

(i) Estimate the regression model

𝑒𝑑𝑢𝑐 = 𝛽0 + 𝛽1 𝑚𝑜𝑡h𝑒𝑑𝑢𝑐 + 𝛽2 𝑓𝑎𝑡h𝑒𝑑𝑢𝑐 + 𝛽3 𝑎𝑏𝑖𝑙 + 𝛽4 𝑎𝑏𝑖𝑙^2 + 𝑢 01234

by OLS and report the results in the usual form. Test the null hypothesis that 𝑒𝑑𝑢𝑐 is linearly related to 𝑎𝑏𝑖𝑙 against the alternative that the relationship is quadratic.

(ii) Using the equation in part (i), test 𝐻0: 𝛽1 = 𝛽2 against a two-sided alternative. What is the 𝑝-value of the test?

(iii) Add the two college tuition variables to the regression from part (i) and determine whether they are jointly statistically significant.

3. Use the data in GPA2 for this exercise.

(i) Estimate the model

𝑠𝑎𝑡 = 𝛽0 + 𝛽1 h𝑠𝑖𝑧𝑒 + 𝛽3 h𝑠𝑖𝑧𝑒^2 + 𝑢

where h𝑠𝑖𝑧𝑒 is the size of the graduating class (in hundreds), and write the results in the usual form. Is the quadratic term statistically significant?

(ii) Using the estimated equation from part (i), what is the “optimal” high school size? Justify your answer.

(iii) Is this analysis representative of the academic performance of all high school seniors? Explain.

(iv) Find the estimated optimal high school size, using 𝑙𝑜𝑔(𝑠𝑎𝑡) as the dependent variable. Is it much different from what you obtained in part (ii)?

4. Use the housing price data in HPRICE1 for this exercise.

(i) Estimate the model

log(𝑝𝑟𝑖𝑐𝑒) = 𝛽0 + 𝛽1 log (𝑙𝑜𝑡𝑠𝑖𝑧𝑒) + 𝛽2 log (𝑠𝑞𝑟𝑓𝑡) + 𝛽3𝑏𝑑𝑟𝑚𝑠 + 𝑢

and report the results in the usual OLS format.
(ii) Find the predicted value of 𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒), when 𝑙𝑜𝑡𝑠𝑖𝑧𝑒 = 20,000, 𝑠𝑞𝑟𝑓𝑡 = 2,500, and 𝑏𝑑𝑟𝑚𝑠 = 4.

(iii) For explaining variation in price, decide whether you prefer the model from part (i) or the model described below:

𝑝𝑟𝑖𝑐𝑒 = 𝛽0 + 𝛽1 𝑙𝑜𝑡𝑠𝑖𝑧𝑒 + 𝛽2 𝑠𝑞𝑟𝑓𝑡 + 𝛽3𝑏𝑑𝑟𝑚𝑠 + 𝑢