EC2203/EC4203 Quantitative Methods for Economics II

Question 1

Consider a regression of monthly expenditure (𝐸, measured in pounds) on monthly disposable income (𝐼, measured in pounds). The regression model is

𝐸 = 𝛽0 + 𝛽1𝐼 + 𝑢,

where 𝑢 is the error term.

a) List the random variables in the model and list the coefficients of the model. (3 marks)

Using a random sample of college-educated full-time workers earning £500 to £15,000 to estimate the regression model yields the following

𝐸̂ = 350.64 + 0.78 × 𝐼, 𝑅 2 = 0.030, 𝑆𝐸𝑅 = 540.30,

where the standard errors of the coefficient estimates have been omitted.

b) Explain what the coefficient estimates 350.64 and 0.78 mean. (5 marks)

c) The standard error of the regression (𝑆𝐸𝑅) is 540.30. Interpret the 𝑆𝐸𝑅. What are the units of measurement for the 𝑆𝐸𝑅? (3 marks)

d) The regression 𝑅 2 is 0.030. Interpret the 𝑅 2 . What are the units of measurement for the 𝑅 2? (3 marks)

e) What does the regression predict will be the expenditure of a person with a disposable income of £500? With a disposable income of £2,000? (3 marks)

f) Will the regression give reliable predictions for a person with a disposable income of £50,000? Why or why not? (3 marks)

g) Is it important that the error term has a normal distribution? Why or why not? (5 marks)

[Total 25 Marks]

Question 2

Data were collected from a random sample of 200 home sales from a community in 2013. Let 𝑃𝑟𝑖𝑐𝑒 denote the selling price (in $1,000s), 𝐵𝐷𝑅 denote the number of bedrooms, 𝐵𝑎𝑡ℎ denote the number of bathrooms, 𝐻𝑠𝑖𝑧𝑒 denote the size of the house (in square feet), 𝐿𝑠𝑖𝑧𝑒 denote the lot size (in square feet), 𝐴𝑔𝑒 denote the age of the house (in years), and 𝑃𝑜𝑜𝑟 denote a binary variable that is equal to 1 if the condition of the house is reported as “poor” and is equal to 0 otherwise.

An estimated regression yields

𝑃𝑟𝑖𝑐𝑒 ̂ = 109.7 (22.1) + 0.567 (1.23) 𝐵𝐷𝑅 + 26.9 (9.76) 𝐵𝑎𝑡ℎ + 0.239 (0.021) 𝐻𝑠𝑖𝑧𝑒 + 0.005 (0.00072) 𝐿𝑠𝑖𝑧𝑒 +0.10 (0.23) 𝐴𝑔𝑒 − 56.9 (12.23) 𝑃𝑜𝑜𝑟, 𝑅̅2 = 0.85, 𝑆𝐸𝑅 = 45.8

with robust standard errors in parentheses.

a) Suppose that a homeowner converts part of an existing family room in her house into a new bathroom. What is the expected increase in the value of the house? (1 mark)

b) Suppose that a homeowner adds a new bathroom to her house, which increases the size of the house by 80 square feet. What is the expected increase in the value of the house? (3 marks)

c) What is the loss of value if a homeowner lets his house run down so that its condition becomes “poor”? (1 marks)

d) Using a 5% significance level, is the coefficient on 𝐵𝐷𝑅 statistically significantly different from zero? (Hint: Test against a two-sided alternative) (5 marks)

e) Typically, four-bedroom houses sell for more than three-bedroom houses. Is this consistent with your answer to d) and with the regression more generally? (5 marks)

f) A homeowner purchases 2,500 square feet from an adjacent lot. Construct a 95% confidence interval for the change in the value of her house (hint: if 𝑍 is a standard normal random variable, Pr(𝑍 < 1.96) = 0.975.) (5 marks)

g) The 𝐹-statistic for omitting 𝐵𝐷𝑅 and 𝐴𝑔𝑒 from the regression is 𝐹 = 2.38. Are the coefficients on 𝐵𝐷𝑅 and 𝐴𝑔𝑒 statistically different from zero at the 10% significance level? (5 marks)

[Total 25 Marks]

Question 3

Consider the problem

max 4𝑥1 + 6𝑥2

s.t. 4𝑥1 + 3𝑥2 ≤ 15,

2𝑥1 + 2𝑥2 ≤ 10,

𝑥1 ≥ 0, 𝑥2 ≥ 0.

Let 𝑝1 denote the price for the constraint 4𝑥1 + 3𝑥2 ≤ 15, and let 𝑝2 denote the price for the constraint 2𝑥1 + 2𝑥2 ≤ 10.

a) What conditions must an equilibrium for this problem satisfy? (6 marks)

b) Find an equilibrium. (6 marks)

c) Are there other equilibria? If yes, identify a distinct equilibrium. If no, show that there are no other equilibria. (6 marks)

d) If the right-hand side of the second constraint goes up from 10 to 11, how much does the maximum feasible value of the objective function increase? (7 marks)

[Total 25 Marks]

Question 4

Consider the following input coefficients matrix which shows the value of inputs required to produce one unit value of output in three sectors of the economy. For example, £0.4 is the value of the third input required to produce one unit value of the second product:

𝐴 = [ 0.5 0.1 0.3

0.2 0.1 0.2

0.1 0.4 0.1 ]

a) Calculate 𝐼 − 𝐴 and using cofactor expansion across the first row, show that the determinant of 𝐼 − 𝐴 is 0.294. (7 marks)

b) Are the Hawkins-Simon conditions satisfied for this economy? Make sure you state the values of all of the parameters you are checking. (5 marks)

c) Suppose labour is the sole primary input vector. Identify the labour employment multiplier following a one unit value drop in final demand for the third product. (3 marks)

d) Suppose in fact that for every £ paid to primary inputs in sector 3, only half is paid to labour and in sectors 1 and 2 all payments to primary inputs go to labour.

i) What is the change in employment following a one unit value drop in final demand for the third product? (8 marks)

ii) What is the labour employment multiplier following a one unit value drop in final demand for the third product now? (2 marks)

[Total 25 Marks]