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(a) In a survey, 10 individuals were randomly chosen from the Australian labour force
(a) In a survey, 10 individuals were randomly chosen from the Australian labour force. Among other variables, their annual income, in thousand dollars, (Income) and the number of years of education (Education) were recorded. To evaluate the relationship between Income and Education, Simon considers the following regression model: Income = Bo +B1 Education + e. The regression result is displayed below. Regression: Standard Coefficients Error Intercept 8.988 26.052 Education 7.103 1.840 ANOVA: Degrees of Freedom 1 8 Regression Residual Total Sum of Squares 6463.06 3469.04 9932.1 9 (1) What is the interpretation of the intercept and the slope estimates. [4 marks] (ii) What is the predicted annual income of an individual who finishes 12 years of education? [2 marks] (ii) Compute the regression R?. [2 marks] (iv) Simon wants to test for the presence of linear association between Income and Education. Carry out the test for Simon at the 1% significance level. [6 marks] (Hints: What are the null and altemative hypotheses? What is the test statistic? What distribution does the test statistic follow under the null hypothesis? What assumptions do you need? What is the rejection rule? What is your conclusion?) (v) Will the conclusion in part (a)(iv) change if the significance level is set to 5% instead? Explain. [3 marks]
(vi) Mimi estimated the following linear regression model: Income = 10.420 + 1.671 Education – 0.012 Sleep + v, where Sleep measures the daily amount of sleep (in hours). Explain why the slope estimate of Education obtained by Mimi is smaller than that obtained by Simon. [3 marks] (Hint: Think about the correlations between the relevant variables) (b) Since then, Simon obtained more resources for conducting the survey. He randomly selected 40 more individuals from the same population. With a larger random sample, Simon hopes that the linear regression analysis would be more informative about the relationship between Income and Education. The regression result and the scatter plot of Income and Education based on the updated sample are displayed below. Regression: 200 180 160 Intercept Education Coefficients 63.631 2.457 Standard Error 11.227 0.614 income ANOVA: Degrees of Freedom Regression 1 Residual 48 Total 49 140 120 100 80 60 40 20 0 0 Sum of Squares 13749.78 41154.40 54904.18 5 25 30 10 15 20 years of education (1) Based on the updated regression result, compute the 95% confidence interval of B1, the slope coefficient of Education. [3 marks] (ii) What is the sample correlation between Income and Education in the updated sample? [2 marks] (iii) From what you see in the scatter plot, which of the two standard assumptions of the linear regression model are likely to be violated? Explain. [4 marks]
(iv) Sally estimated the following linear regression model based on the updated sample. log(Income) = 4.106 + 0.029 Education + u. What is the economic interpretation of the slope coefficient estimate? [2 marks] (v) Sally then ran a regression of the squared residuals u obtained from part (b)(iv) on Education. The LM statistic is equal to 4.568. Suppose the significance level is set to be 5%. Perform a Breusch-Pagan test for Sally. [4 marks] (Hints: What are the null and alternative hypotheses? What distribution does the LM statistic follow under the null hypothesis? What is the rejection rule? What is your conclusion?) (vi) Baobao hypotheses that income is positively related to work hours and negatively related to sleep time. To these his hypothesis he proposes the following regression model: Income = @o + az Work + a2 Sleep + az Leisure + w, where Work measures the daily amount of work (in hours), Sleep measures the daily amount of sleep (in hours) and Leisure measures the daily amount of leisure (in hours). Baobao defines leisure as any activities other than work and sleep. Baobao expects that he would get a positive estimate of an and negative estimates of a2 and a3. (1) Is Baobao's regression model feasible? Explain. [3 marks] As a proud student of Simon, what suggestion would you give to Baobao to improve his model?
Expert Solution
Q1.
i) As the number of years of education increases by one, income increases by value equal to the coefficient of the Education variable. = 1000*7.103 = $7103
Intercept is th income when the number of years of education is zero = 1000*8.988 = USD 8988
Slope the coefficient of estimates.
ii) 12 years of education
Income = 8.988 + 12(7.103) = 94.244 = USD 94244
iii) Variance explained by the model = Sum of Squares (Regression) / Sum of Squares(Total) = 6463.06 / 9932.1
= 0.65072
65.072%
iv) Null hypothesis:
Education = 0
Alternate hypothesis:
Education > 0
t-statistic = 7.103 / 1.840 = 3.86
t(9, 0.01) = 2.821
t-statistic is greater than the critical value. Null hypothesis is rejected.
The variable education is significant at 1% level.
v) No. If a variable is significant at 1% level of significance, it is significant at 5% level of significance. It is significant at all values greater than 1.
Critical value = 1.833
vi) The sample has a larger number of individuals with high income and no education or a smaller number of years of education.
Other relevant factors could be in the error term. Sleep is controlling for the effect of education. If sleep suffers from upward bias due to omitted variable bias then that variable is in the error term, and, correlated with the Sleep variable. The sample is skewed towards those having a lower number of years of education. There are other variables that are the determinant factors of Income.
Other variables are more important in these models over Education; background, race, parental education
Q2. I) 2.457 ( + / -) (1.96 * 0.614) = [ 1.25356 , 3.66044]
1.96 is the Z-score for 95% confidence interval
2.457 is the mean; 0.614 is the standard error;
ii) Coefficient of detemination = Sum of Squares (Regression) / Sum of Squares (Total)
= 0.25043
25.043%
Correlation coefficient = 0.5004
iii) There is heteroskedasticity, variance of the error term is not constant; normality of the error term does not hold. The error terms are not independently and identically distributed.
iv) For one year increase in the number of years of education, income increases by 100* 0.029 = 2.9%
v) Chi square distribution value for 0.05; sample size = 50
LM -statistic from sample = 4.568 degree of freedom = number of samples = 1
LM - critical value (1, 0.05) = 3.841
Null hypothesis: All variances are equal
Alternate hypothesis: All variances are not equal
LM-statistic is greater than the critical value; therefore, null hypothesis is rejected;
Variances are not equal
vi) The model is feasible but the Leisure coefficient could be positive for high income individuals. This is because as the work hours increase, and, income increase; the number of leisure hours could also increase. The number of work-hours may also be less for senior management. The earning is greater for a lower number of hours of work, and, leisure on average for the same is correlated to high income.
The coefficient does not represent causality.
II. Education is an important factor; the variable effect for the Baobao model will be different if Education is considered. A higher number of years of education or entrepreneurial activity will make Leisure coefficient positive as earnings are high.
The poor cannot afford leisure, the number of hours off work, and, in sleep is low. Sleep takes the time when one is not working.
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