Applied Econometrics (2024/2025 Semester 1) -- Assignment 1

Instructions:

1.  This assignment paper has a total of 100 marks, and contributes 25% to the course’s overall assessment.

2. Write down your solution/answer to each question in the space provided in THIS assignment paper.

3. Necessary calculations and/or formulas MUST be included in your solutions/answers.

4. Key concepts/methods/formulas and t-table can be found from the textbook or lecture notes, or from the review document available from the course iSpace.

5. Round your calculation results to THREE (3) decimals to achieve higher accuracy, unless clearly unnecessary.

Miss Ariel, a smart girl of the beautiful Ultimate Imagination College (UIC) which currently has 9,260 students, is attending an Artistic Emotion (AE) class. Because of her learning experience, she is interested in examining whether or not harder study will lead to better learning outcome and, if yes, by how much, which can be simplified to examine the relationship between GPA (y) and average daily study-hour (x). For that purpose, yesterday she went to UIC’s University Road, to do a survey. She randomly surveyed 10 UIC students with each student’s GPA (y) and average daily study-hour (x) recorded as follows, where, taking student #5 for example, x5 = 12 andy5 = 3.3 imply that she (or he) studied 12 hours everyday in average and her (or his) latest GPA was 3.3.

Five sample sums have been calculated from this sample data set as follows:

Part A. Basic concepts (15 marks)

Q01 (2 marks): What is the appropriate population for the survey (or the random sample)?

Q02 (3 marks): How many different samples of 10 students could be drawn from the population indicated in Q01?

Q03 (10 marks): For and just for this question (Q03) only, suppose that Mr. Simon also randomly surveyed 10 students yesterday from the same population indicated in Q01.

Q03a (2 marks): Are Ariel’s and Simon’s samples the same (i.e., do they have the same 10 students?)

Q03b (2 marks): Will the two samples have the same average GPA?

Q03c (2 marks): Which of the two samples will have an average GPA closer to the population average?

Q03d (2 marks): Will the two samples produce the same sample regression model to explain GPA (y) using average daily study-hour (x)?

Q03e (2 marks): Which of the two samples will produce a sample regression model closer to the population regression model?

Part B. Calculate the sample statistics (20 marks)

Q04 (1 mark): Sample mean ofx. _____________________________________________________________

Q05 (1 mark): Sample mean of y. _____________________________________________________________

Q06 (2 marks): Sample variance of x. _____________________________________________________________

Q07 (2 marks): Sample variance of y. _____________________________________________________________

Q08 (1 mark): Sample standard deviation of x. ______________________________________________________

Q09 (1 mark): Sample standard deviation of y. ______________________________________________________

Q10 (2 marks): Sample covariance between x and y. __________________________________________________

Q11 (2 marks): Sample correlation coefficient between x and y. ________________________________________

Q12 (2 marks): Standard error of sample mean of x. __________________________________________________

Q13 (2 marks): Standard error of sample mean of y. __________________________________________________

Q14 (4 marks): Briefly explain why standard error of sample mean of y (in Q13) is much smaller than the standard deviation of y (in Q09). What is the major implication of this?

Q14a (2 marks): Reasons. _____________________________________________________________________

Q14b (2 marks): Implication. _____________________________________________________________________

Part C. Inference for population mean (30 marks)

Q15 (10 marks): Test the null hypothesis (H0) that the population’s average GPA (μy) is 3.0 against a two-sided alternative hypothesis (H1) at the 5% significance level.

Q15a (2 marks): State the two hypotheses formally in symbols.

Q15b (3 marks): Calculate the sample t-statistic. __________________________________________________

Q15c (2 marks): Find the (two-sided) critical value from the t-distribution table. __________________________

Q15d (3 marks): Draw conclusions. _____________________________________________________________

Q16 (10 marks): Test the null hypothesis (H0) that the population’s average GPA (μy) is 2.7 against a right-sided alternative hypothesis (H1) at the 5% significance level.

Q16a (2 marks): State the testing problem formally in symbols. _______________________________________

Q16b (3 marks): Calculate the sample t-statistic. __________________________________________________

Q16c (2 marks): Find the (one-sided) critical value from the t-distribution table. __________________________

Q16d (3 marks): Draw conclusions. _____________________________________________________________________

Q17 (6+4 marks): First construct a 95% confidence interval for the population mean (μx) of average daily study- hours, and then test whether μx is equal to 10 against a two-sided alternative hypothesis at the 5% significance level based on this confidence interval. How about μx = 11?

Q17a (6 marks): Confidence interval.

Q17b (4 marks): Hypothesis testing.

Part D. Simple linear regression: basic calculations and interpretations (35 marks)

This Part relates to a simple linear regression model estimated using David’s sample data (together with the results in Part B) and the ordinary least squares (OLS) method: yi =  β(ˆ)0   +  β(ˆ)1 xi + ûi 三 ŷi + ûi, where ŷi =  β(ˆ)0   +  β(ˆ)1 xi is the model-fitted or forecast yi corresponding to xi and ûi is the corresponding residual for student i (i = 1, 2, … , 10).

Q18 (3+2 marks): Find  β(ˆ)1 , and explain its meaning.

Q19 (2+2 marks): Find  β(ˆ)0  , and explain its meaning.

Q20 (2+3 marks): Find R2, which is just the squared sample correlation coefficient between y and x for simple regression, and explain its meaning.

Q21 (1 mark): Find the total sum of squares of x (SSTx). _______________________________________________

Q22 (1 mark): Find the total sum of squares of y (SST). _______________________________________________

Q23 (3 marks): Find the residual sum of squares (SSR). _______________________________________________

Q24 (2 marks): Find the standard error of regression ( ). _____________________________________________

Q25 (3 marks): Find the standard error of βˆ1. _______________________________________________________

Q26 (2 marks): Is the standard error of βˆ1 small or big? _______________________________________________

Q27 (3 marks): Find the standard error of βˆ0. ______________________________________________________

Q28 (3+3 marks): For student #5, find her model-fitted GPA (ŷ5) and comment on her actual study performance. ______________________________________________________