Problem Set

Problem 1 - You are approached by several NHL team owners to help them analyze their salary structure. The owners believe that for a certain set of players, salaries are based only on one aspect of performance, scoring goals. As such, they provide you with a sample of the salaries and goals scored by 100 different players:

_i=1∑ⁿg_i = 2400, _i=1∑ⁿm_i = 4500, _i=1∑ⁿg_im_i = 110000, _i=1∑ⁿg_i² = 60000

where g_i represents the goals scored by player i, and m_i represents the salary (in hundreds of thousands of dollars) received by player i.

1. Find the least squares regression line.

2. If a player scores eight additional goals, how much extra money does your model predict that he should be paid?

3. What is your predicted salary for a player who scores 45 goals?

Problem 2 - Suppose that you have the model

y_i = β₁ + x_iβ₂ + u_i

which you estimate by ordinary least squares.

Consider each alternative modiÖcation of the model below, and explain how the new estimates of β₁ and β₂ and the estimated standard errors di§er from those from the original model. Also discuss what will happen to R². Consider each of the alternatives separately; do not consider them cumulatively. Be as precise as possible.

1. Subtract x_i from y_i for each i.

2. Add 8 to each x_i and y_i.

3. Add 2 to each y_i.

4. Multiply each y_i by 3 and subtract 1 to each x_i.

5. Multiply each x_i by 2 and add 2 to each y_i.

Problem 3 - Show that the regression R² in the regression of Y on X equals the squared value of the correlation between X and Y, i.e., show that R² = r²_XY, where

r²XY = (1/(n-1)_i=1∑ⁿ(X_i - X^-)(Y_i - Y^-))/(√(1/(n-1)_i=1∑ⁿ(X_i-X^-)²)√(1/(n-1)i=1∑n(Y_i-Y^-)²))

Hint: To solve the problem you may want to write Y^{^}_i in the formula for R² as Y^{^}_i = β^{^}₁ + Xiβ^{^}₂ and then use the formulas for β^{^}₁ and β^{^}₂.

Problem 4 - For this problem, please use the houses_471.dta posted on Blackboard.

1. Read the data into STATA. What is the average house price in this data set?

2. Regress house prices on lot size. Based on your findings, what is your estimate of the price of a house with a lot size of ten-thousand square feet?

3. Plot the house prices against the lot size and draw the regression line found in part 2 (it is probably easiest to do this by hand, but investigate whether you can get STATA to do it)

4. Create two new variables, lp representing the logarithm of the price of a house and ll representing the logarithm of the lot size. Regress lp on ll.

Description of the data set:

Filename: houses_471.dta

This dataset contains twelve characteristics of 471 houses (thus, there are twelve variables in the data set, each of which has 471 observations). The variable names and their descriptions are listed below:

1. PRICE = the sale price of the house (in dollars)

2. LOT = the houses lot size (in square feet)

3. BDRMS = the number of bedrooms in the house

4. FB = the number of full bathrooms in the house

5. STY = the houses total number of storeys

6. DRIVE = 1 if the house has a driveway

7. REC = 1 if the house has a rec room

8. FFINBMNT = 1 if the house has a fully-furniished basement

9. GHW = 1 if the house uses gas to heat its water

10. CA = 1 if the house has air-conditioning

11. GARAGE = the number of cars that can be parked in the houses garage

12. PREF =1 if the house is located in a preferred part of the city

These defiitions can also be accessed by using the command "describe" in STATA. This command will list the variables in the data set and provide a brief description of each one.

Attachment:- Data File.rar