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Applied Econometrics (Semester 1, 2024/2025) –– Assignment 2
Submitted to the TA or teacher in hard or printed copy before 5:00pm, Friday, 8 November 2024
Using AI tools in doing this Assignment is strongly prohibited!!!
Instructions:
1. This assignment has a total of 100 marks, contributing 25% to the course’s overall assessment.
2. CLEARLY write down your answers/solutions to each question in the space provided in THIS assignment paper, and for Q2c, Q3b, and Q3e, do and only do the REQUIRED part based on your Student Number’s being odd or even.
3. Necessary steps/formulas/calculations/arguments MUST be included in your answers/solutions as a good practice.
4. Some concepts/methods/formulas and statistical tables are provided at the end of this assignment for your easy reference.
5. For each estimated sample regression model, the number in parentheses below each estimated sample coefficient is its standard error, unless otherwise indicated.
6. Keep FOUR (4) decimals for all calculations/results for relatively higher accuracy, unless clearly unnecessary.
7. ALL logs used in this assignment/course are natural logarithms with a base of e = 2.71828…, i.e., log(x) = loge(x) = ln(x).
8. In hypothesis testing, unless otherwise instructed or clearly unnecessary, you SHOULD: (a) state the null and alternative hypotheses, (b) calculate the appropriate sample test-value (or test-statistic), (c) find the corresponding critical value, and (d) draw statistical/practical conclusions.
As an Applied Econometrics student, you were interested in and wanted to explain the fact that many people had quite different salaries. After doing some literature review, you proposed the following population regression model
(I) to examine the factors affecting people’s salaries, where salary is monthly earnings in RMB yuan (roughly around 10,000 yuan), IQ is Intelligence Quotient in points (roughly around 100), edu, Fedu, and Medu are a person’s own, his father’s and mother’s educational levels in years of formal schooling (roughly around 12 years), respectively:
log(salary) = 0 + 1 edu + 2 log(IQ) + 3Fedu + 4 Medu + u. (I)
To get representative and reliable results, you spent about one week in data collection and successfully got the relevant data from 350 randomly surveyed employees from different industries in Zhuhai. Eventually you were able to estimate the population model (I) using the ordinary least squares (OLS) method and the sample data as follows, where SSR is the commonly-used residual sum of squares:
log(salary) = 6.3815 + 0.0354 edu + 0.5356 log(IQ) – 0.0076 Fedu + 0.0058 Medu + û (II)
(1.2876) (0.0119) (0.1435) (0.0084) (0.0091)
[n = 350, R
2 = 0.1187, SSR = 48.9769]
Q1 (25 marks): Basic interpretations and calculations
Q1a (2 marks): Indicate two different factors in the error-term u of model (I) which affect (the log of) salary.
Q1b (3 marks): Explain the practical meaning of the estimated coefficient 0.0354 of edu.
Q1c (3 marks): Explain the practical meaning of the estimated coefficient 0.5356 of log(IQ).
Q1d (5 marks): Employee A’s father and mother had the same education as employee B’s father and mother, but he had 5 more years of education and 3% higher IQ than employee B, respectively. Then what is the predicted (percentage) difference in their salaries?
Q1e (6 marks): Employee C’s father and mother both had 4 more years of education than employee D’s father and mother, and he also had 4 more years of education than employee D. In order to have no expected (percentage) difference in their salaries, what should the (percentage) difference in their IQ points be?
Q1f (6 marks): Employee G had 16 years of education and 100 points of IQ, and his father and mother both had 12 years of education. What was his expected salary?
Q2 (35 marks): Testing a single restriction
Q2a (7 marks): Test whether the effect of mother’s education in (the log of) salary is significant at the 20% level.
(See Instruction 8)
Q2b (8 marks): Test the hypothesis that a 1% increase in IQ will lead to a 1% increase in salary against a left-sided alternative at the 1% significance level. (See Instruction 8)
Q2c (10 marks): First construct a 99% confidence interval for the population 1 ---- education’s (percentage) effect in salary, and then test whether 1 is equal to 0 (if your Student Number is odd) or 0.05 (if your Student Number is even) against a two-sided alternative hypothesis at the 1% significance level BASED ON this confidence interval.
Q2d (10 marks): Describe in detail how to test whether the effects of mother’s education and father’s education in (the log of) salary cancel each other (i.e., 3 + 4 = 0) by (i) introducing a new parameter, (ii) transforming or rewriting the model, and then (iii) doing an appropriate t-test.
Q3 (40 marks): Testing multiple restrictions and other relevant issues
Q3a (8 marks): Test the overall significance or all explanatory variables’ joint significance of the population model (I) at the 5% level. (See Instruction 8)
Q3b (10 marks): What is the restricted model for testing the hypothesis in model (I) that father’s and mother’s educational levels are jointly insignificant AND a 2% (if your Student Number is odd) or 3% (if your Student
Number is even) increase in IQ leads to a 1.8% increase in salary? Write down the null hypothesis (H0), alternative hypothesis (H1) and the F-value formula for testing the hypothesis. What is the critical value for this F-test at the 5% significant level?
In addition to the above estimated model (II), the following model (III) without Fedu and Medu (for father’s and mother’s education) was also estimated using the same sample data, where SSE is the explained sum of squares:
log(salary) = 6.2987 + 0.0382 edu + 0.5403 log(IQ) + (III)
(0.9062) (0.0104) (0.1375) [n = 350, SSE = 6.2942]
Also, the sample correlation coefficient between edu and log(IQ) was calculated as 0.5916.
Q3c (6 marks): Is there high multicollinearity in regression model (III) based on the rule of thumb?
Q3d (10 marks): Test whether father’s and mother’s educational levels in model (I) are jointly significant in affecting (the log of) salary at the 5% level based on the estimated models (II) and (III). (See Instruction 8)
Q3e-odd (6 marks) – for students with odd Student Numbers only: For regression model (III), if edu was omitted, discuss what bias the new coefficient of log(IQ) would have when only using log(IQ) to explain log(salary).
Q3e-even (6 marks) – for students with even Student Numbers only: For regression model (III), if log(IQ) was omitted, discuss what bias the new coefficient of edu would have when only using edu to explain log(salary).
–– End of Assignment 2 (2024-25Sem1) ––