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Applied Econometrics (Semester 2, 2023/2024) –– Assignment 2
发布时间：2024-03-27
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Applied Econometrics (Semester 2, 2023/2024) -- Assignment 2
Submit a soft copy only via iSpace before 5:00pm, Friday, 12 April 2024
This assignment paper has a total of 100 marks, and contributes 25% to the course’s overall assessment, where for Q1d, Q1e and
Q3a, do and ONLY do the REQUIRED parts based on your Student Number’s being odd or even.
CLEARLY write down your answers/solutions to each question with your Name and Student Number on some clean paper.
Necessary steps/formulas/calculations/arguments MUST be included in your answers as a good practice.
Keep FOUR (4) decimals for all calculations/results for relatively higher accuracy, unless clearly unnecessary.
For each estimated regression model, the number in parentheses below each estimated coefficient is its standard error, unless
otherwise indicated.
In testing hypotheses and/or constructing confidence intervals, using the critical values corresponding to the closest degrees of
freedom from the t-table orF-table (available to you in a separate file, together with a formula sheet, for your easy use).
Part A: Basic Concepts (30 marks)
As part of the effort to improve U.S. urban planning, the federal government conducted a study to identify key factors that determine the intensity of bus rides in cities. The following regression results are obtained based on arandom sample of 40 cities applied to the regression model:
BUSRide = β0 + β1 Income + β2 DenSity + β3ATea + U (1a)
BUSRl(̂)de = 1,013.6300 − 0.2354Income + 0.4970DenSity + 9.6778ATea (1b)
(1593.97) (0. 1023) (0.0483) (1.5076)
[n = 40, R2 = 0.7823, SSR = 50,200, 130]
where BusRide is transportation by bus measured in ‘000 passenger hours, Income is average annual income per capita in US$, Density is city density in terms of number of persons per square mile, and Area is the land area of the city in square miles.
Q1a (2 marks): If a city has average income of US$15,000, a city density of 6,000, and a landarea of 80, what is the expected or predicted amount of bus transportation?
Q1b (2 marks): If a city’s average income increases by $600 and its density and land area remain unchanged, what is the predicted change in the amount of bus transportation?
Q1c (2 marks): If City A has the same average income as City B, but A’s city density is 50 higher than that of B and at the sametime A’s land area is 10 less than B’s, what is the estimated difference in the amount of bus transportation of the two cities?
Q1d-even (3 marks) - for students with even Student Numbers only: If City C has average income lower than City D by US$850 and its land area 30 higher than that of City D, other things being the same. What is the predicted difference in the amount of bus transportation of the two cities?
Q1d-odd (3 marks) - for students with odd Student Numbers only: If City C has average income higher than City D by US$850 and its landarea 30 lower than that of City D, other things being the same. What is the predicted difference in the amount of bus transportation of the two cities?
Q1e-even (9 marks) - for students with even Student Numbers only: Construct a 95% confidence interval for the population regression coefficient β1. Do you think Income has a statistically significant effect on the amount of bus transportation at the 5% level? Briefly explain.
Q1e-odd (9 marks) - for students with odd Student Numbers only: Construct a 95% confidence interval for the population regression coefficient β2 . Do you think Density has a statistically significant effect on the amount of bus transportation at the 5% level? Briefly explain.
Q1f (4 marks): We want to test the null hypothesis that a city’s average income has no statistically significant effect on the amount of bus transportation against the two-sided alternative that it has (an effect). The computed p-value for the test can be found to be 0.0273. (i) Can we reject the null hypothesis at the 1%, 5% and 10% significance levels? Briefly explain. (ii) If the alternative hypothesis is left-sided, can we reject the null hypothesis at the 5% significance level? Also briefly explain.
Q1g (8 marks): Test whether the population model (1a) has an overall significance at the 5% level, i.e., test whether all its explanatory variables are jointly significant in affecting the explained variable BusRide. Include all information needed to draw the conclusion.
Part B: Multiple Regression Estimation (40 marks)
Suppose the model (1b) is replaced by the model below:
log(BUsRide) = a0 + a1 log(Density) + a2 log(Area) + U (2a)
which is estimated using the same sample of 40 cities in Question 1 with results as follows:
log(BUsR(̂)lde) = −7.6905 + 1.3365 log(Density) + 0.6748 log(Area) (2b)
(2.8283) (0.2620) (0. 1982)
[n = 40, R2 = 0.4246, SSR = 30.0699]
In addition, an auxiliary model relating log(Density) to log(Area) is specified,
log(Density) = Y0 + Y1 log(Area) + ε (3a)
which is also estimated using the same sample:
log(Dens(̂)lty) = 10.3150 − 0.3575 log(Area) (3b)
[n = 40, R2 = 0.2232 SSR = 11.8420]
Q2a (4 marks): Explain the meanings of the estimated regression coefficients 1.3365 and 0.6748 in model (2b).
Q2b (5 marks): Is there any multicollinearity in regression model (2b) based on the rule of thumb introduced in class?
Q2c (10 marks): A common or reasonable belief is that higher log(Area) should be associated with a higher log(BUsRide), other things held fixed. Test this belief or hypothesis at the 5% significance level (with details provided). How about at the 1% significance level?
Q2d (9 marks): Comparing the estimated sample regression models (1b) and (2b), can you name three differences based on information given?
Q2e (12 marks): If log(Density) is omitted in population model (2a) due to some reasons, then we only have the following simple regression of log(BusRide) on log(Area):
log(BUsRide) = φ0 + φ1 log(ATea) + u (4a)
which is estimated using the same sample data with results as follows:
log(BUsR(̂)lde) = φ0(̂) +φ1(̂)log(ATea) (4b)
[n = 40, SSR = 51.2234]
Q2e-1 (2 marks): Is the R2 of the estimated model (4b) higher or lower than the R2 of the estimated model (2b)? Briefly explain why this is the case.
Q2e-2 (6 marks): Find the estimated sample regression coefficients φ(̂)0 and φ(̂)1 in model (4b).
Q2e-3 (4 marks): Which of the five assumptions (e.g., MLR.1) of the Gauss-Markov theorem is violated such that the sample estimatedφ(̂)1 of model (4b) has an omitted bias from the population a1 of model (2a)?
Briefly explain.
Part C. Multiple Regression Inference (30 marks)
To better examine the determinants of bus transportation, another relevant factor, Bus fare as a percentage of Gas expenditure (FareGas), expressed in percentage points, is added to the model (1a) to give the following new model,
BUsRide = δ0 + δ1 Income + δ2 Density + δ3ATea + δ4 FaTeGas + ξ (5a)
which is estimated using the same sample data as in Questions 1-2 as follows:
BUsRl(̂)de = 1685.45 − 0.2385Income + 0.4908Density + 10.0758ATea − 6.6036FaTeGas (5b)
[n = 40, SSR = 48,741,221]
Q3a-even (13 marks) - for students with even Student Numbers only: A government consultant claims that, in terms of the effects on BusRide, a 2-percentage point decrease in FareGas is equivalent to 1 square mile increase in Area, holding other factors fixed. Describe how to test this claim in details (i.e., stating the hypotheses, performing the transformation, showing the transformed model and hypotheses, etc.), and briefly explain whether the claim is more likely to be rejected or not based on the estimated sample regression model (5b).
Q3a-odd (13 marks) - for students with odd Student Numbers only: A government consultant claims that, in terms of the impact on BusRide, FareGas is 10 times as important as Income, holding other factors fixed. Describe how to test this claim in details (i.e., stating the hypotheses, performing the transformation, showing the transformed model and hypotheses, etc.), and briefly explain whether the claim is more likely to be rejected or not based on the estimated sample regression model (5b).
Q3b (17 marks): An auxiliary model is given:
BUsRide = λ0 + λ1 Income + λ2 FaTeGas + C (6a)
and when estimated using the same sample data yields the following results:
BUsRl(̂)de = −1704.96 + 0.2558Income − 6.7051FaTeGas (6b)
[n = 40, SSE = 13,674, 174. 1341]
Test whether Density and Area are jointly significant in affecting BusRide in model (5a) at the 5% level. Show all details of your derivation. [Hint: you need to find the SST based on all information given in the question paper.]
